$p$-curvature operators and Satake-type phenomenon for $\frak{sl}_2$ KZ equations with $\kappa=\pm 2$

TL;DR

This paper investigates the $rak{sl}_2$ KZ equations at specific parameters, establishing a Satake-type correspondence to analyze $p$-curvature operators and solution spaces in characteristic $p$, revealing new phenomena about solution generation.

Contribution

It introduces a Satake-type correspondence for $rak{sl}_2$ KZ equations at $ abla= ext{pm} 2$, enabling analysis of $p$-curvature and solution spaces in characteristic $p$ with novel examples.

Findings

Not all solutions are generated by $p$-hypergeometric solutions in characteristic $p$ for $ abla=2$.

Established a Satake-type correspondence for $rak{sl}_2$ KZ equations.

Analyzed the dimension of solution spaces in characteristic $p$.

Abstract

The $\frak{sl}_2$ KZ differential equations with values in the tensor power of the fundamental representation with parameter $\kappa=\pm 2$ are considered. A Satake-type correspondence is established over complex numbers and subsequently reduced to finite characteristic. This correspondence enables the study of the KZ equations on the lower weight subspaces of the tensor power in terms of the wedge powers of the weight subspace of the weight just below the highest weight. We apply this approach to analyze the $p$-curvature operators associated with our KZ equations, evaluate the dimension of the solution space in characteristic $p$, and determine whether all solutions are generated by the so-called $p$-hypergeometric solutions. In particular, we show that not all solutions of the KZ equations with $\kappa=2$ in characteristic $p$ are generated by $p$-hypergeometric solutions. Previously, no such examples were known.

Equations

$$\left(\frac{\partial }{\partial z_i}-\frac{1}{\kappa }\sum _{j\ne i}\frac{P^{(i,j)}-1}{z_i-z_j}\right)N(z)=0,i=1,\dots ,2g+1,$$

$$N^{\gamma }(z)={\int }_{\gamma (z)}\prod _{i=a}^{2g+1}(t_1-z_a{)}^{-1/\kappa }\left(\frac{1}{t_1-z_1},\dots ,\frac{1}{t_1-z_{2g+1}}\right)d{t}_1$$

$$\prod _{i=a}^{2g+1}(t_1-z_a{)}^{(p-1)/2}\left(\frac{1}{t_1-z_1},\dots ,\frac{1}{t_1-z_{2g+1}}\right)$$

$$\prod _{i=a}^{2g+1}(t_1-z_a{)}^{(p+1)/2}\left(\frac{1}{t_1-z_1},\dots ,\frac{1}{t_1-z_{2g+1}}\right)$$

$$e=\left(\begin{array}{ccc}0& 1& \\ 0& 0& \end{array}\right),f=\left(\begin{array}{ccc}0& 0& \\ 1& 0& \end{array}\right),h=\left(\begin{array}{ccc}1& 0& \\ 0& -1& \end{array}\right).$$

$$w_J=w_{j_1}\otimes \cdots \otimes w_{j_n},$$

$$\mathrm{Sing}{L}^{\otimes n}[n-2r]=\{w\in L^{\otimes n}[n-2r]\mid ew=0\}.$$

$$\dim \mathrm{Sing}{L}^{\otimes n}[n-2r]=\dim L^{\otimes n}[n-2r]-\dim L^{\otimes n}[n-2r+2]=\left(\genfrac{}{}{0ex}{}{n}{r}\right)-\left(\genfrac{}{}{0ex}{}{n}{r-1}\right).$$

$$\dim \mathrm{Sing}{L}^{\otimes n}[n-2r]=\left(\genfrac{}{}{0ex}{}{n-1}{r}\right)-\left(\genfrac{}{}{0ex}{}{n-1}{r-2}\right).$$

$${\tilde{H}}_i(z)=\sum _{j\ne i}\frac{{\Omega }^{(i,j)}}{z_i-z_j},i=1,\dots ,n,$$

$${\partial }_i=\frac{\partial }{\partial z_i},{\tilde{\nabla }}_i^{\mathrm{KZ},\kappa }={\partial }_i-\frac{1}{\kappa }{\tilde{H}}_i(z),i=1,\dots ,n,\kappa \in {\mathbb{C}}^{\times }.$$

$$[{\tilde{H}}_i(z_1,\dots ,z_n),x\otimes 1\otimes \cdots \otimes 1+\cdots +1\otimes \cdots \otimes 1\otimes x]=0.$$

$$\left({\partial }_i-\frac{1}{\kappa }{\tilde{H}}_i(z)\right)\tilde{N}(z_1,\dots ,z_n)=0,i=1,\dots ,n,$$

$$\tilde{N}(z)=N(z)\prod _{1⩽i<j⩽n}(z_i-z_j{)}^{1/2\kappa },$$

$$\left({\partial }_i-\frac{1}{\kappa }\sum _{j\ne i}\frac{P^{(i,j)}-1}{z_i-z_j}\right)N(z_1,\dots ,z_n)=0,i=1,\dots ,n,$$

$${\nabla }_i^{\mathrm{KZ},\kappa }={\partial }_i-\frac{1}{\kappa }\sum _{j\ne i}\frac{P^{(i,j)}-1}{z_i-z_j}.$$

$$S(w_I,w_J)={\delta }_{I,J}.$$

$${\partial }_{i}S(x(z),y(z))=S({\nabla }_i^{\mathrm{KZ},\kappa }x(z),y(z))+S(x(z),{\nabla }_i^{\mathrm{KZ},-\kappa }y(z))$$

$$\Phi (t,z)=\prod _{1⩽i<j⩽r}(t_i-t_j{)}^2\prod _{a=1}^n\prod _{i=1}^r(t_i-z_a{)}^{-1}.$$

$${\mathrm{Sym}}_{t_1,\dots ,t_k}f=\sum _{\sigma \in S_k}f(t_{\sigma (1)},\dots ,t_{\sigma (k)}),{\mathrm{Ant}}_{t_1,\dots ,t_k}f=\sum _{\sigma \in S_k}(-1{)}^{|\sigma |}f(t_{\sigma (1)},\dots ,t_{\sigma (k)}),$$

$${\nu }_J(t,z)$$

$$N_J^{\gamma }(z)={\int }_{\gamma (z)}{\nu }_J.$$

$$N^{\gamma }(z)=\sum _{J\in {\mathcal{I}}_r}{N}_J^{\gamma }(z)w_J.$$

$$\left({\partial }_i-\frac{1}{2}\sum _{j\ne i}\frac{P^{(i,j)}-1}{z_i-z_j}\right)I(z_1,\dots ,z_{2g+1})=0,i=1,\dots ,2g+1.$$

$$\Psi (x,z)=\prod _{a=1}^{2g+1}(x-z_a),{\eta }_i^{(k)}(x,z)=\Psi (x,z{)}^{-1/2}\frac{x^{k}dx}{x-z_i}.$$

$$\Phi (t,z{)}^{1/2}$$

$${\nu }_{i_1,\dots ,i_r}$$

$$N_{i_1,\dots ,i_r}^{\gamma }(z)={\int }_{\gamma (z)}{\nu }_{i_1,\dots ,i_r}.$$

$$N^{\gamma }(z)=\sum _{1⩽i_1<\cdots <i_r⩽2g+1}{N}_{i_1,\dots ,i_r}^{\gamma }(z)w_{\{i_1,\dots ,i_r\}}$$

$${\nu }_{i_1}(t_1)=\frac{\Psi (t_1,z{)}^{-1/2}d{t}_1}{t_1-z_{i_1}}={\eta }_{i_1}^{(0)}(t_1).$$

Full text

We apply this approach to analyze the $p$-curvature operators associated with our KZ equations, evaluate the dimension of the solution space in characteristic $p$, and determine whether all solutions are generated by the so-called $p$-hypergeometric solutions. In particular, we show that not all solutions of the KZ equations with $\kappa=2$ in characteristic $p$ are generated by $p$-hypergeometric solutions. Previously, no such examples were known.

*∗⋆**Department of Mathematics, University of North Carolina at Chapel Hill

Chapel Hill, NC 27599-3250, USA*

$\kern-3.99994pt{}^{\circ}\mskip-0.99998mu$*Department of Mathematical Sciences, Indiana University Indianapolis

402 North Blackford St, Indianapolis, IN 46202-3216, USA*

Key words: KZ equations, $p$-hypergeometric solutions, $p$-curvature

2020 Mathematics Subject Classification: 11D79, 12H25 (32G34, 33C05, 33E30, 33E50)

††footnotetext: ${}^{*}\mskip-0.99998mu$E -mail: belkale@email.unc.edu , supported in part by NSF grant DMS - 2302288

${}^{\circ}\mskip-0.99998mu$E -mail: emukhin@iupui.edu , supported in part by Simons Foundation grant 709444

${}^{\star}\mskip-0.99998mu$E -mail: anv@email.unc.edu 

1. 1 Introduction

2. 2 ${\mathfrak{sl}}_{2}$ KZ equations

3. 2.1 Definition of KZ equations

4. 2.2 Gauge transformation

5. 2.3 Solutions of KZ equations over ${\mathbb{C}}$

6. 2.4 KZ equations for $\kappa=2$

7. 2.5 KZ equations for $\kappa=2$ and $r=1$

8. 2.6 KZ equations for $\kappa=-2$

9. 2.7 KZ equations for $\kappa=-2$ and $r=1$

10. 3 Wedge-power construction

11. 3.1 Wedge-power of KZ equations on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ with $\kappa=2$

12. 3.2 Curves and their primitive cohomology

13. 3.3 Primitive part and KZ equations

14. 3.4 Wedge-power of KZ equations on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ with $\kappa=-2$

15. 3.5 Shapovalov form

16. 4 Proof of Theorem 3.1

17. 4.1 Hyperelliptic curves

18. 4.2 KZ equations

19. 4.3 Proof of Proposition 4.2

20. 4.4 Proof of Claim Claim

21. 4.5 Proof of Lemma 4.5

22. 4.6 Proof of Theorem 3.1

23. 5 Maps $T(z)$ and $\bar{T}(z)$

24. 5.1 Cohomological relations

25. 5.2 Map $T(z)$

26. 5.3 Poincaré element

27. 5.4 Map $\bar{T}(z)$

28. 5.5 Formula for $\bar{T}(z)$ for $r=2$

29. 5.6 Formula for $\bar{T}(z)$ for arbitrary $r$

30. 6 $p$-hypergeometric solutions for $\kappa=2$

31. 6.1 $p$-integrals

32. 6.2 $p$-hypergeometric solutions in ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$ for $\kappa=2$

33. 6.3 $p$-hypergeometric solutions for $\kappa=2$ and $r=1$

34. 6.4 $p$-hypergeometric solutions for wedge-power and $\kappa=2$

35. 6.5 Comparison of solutions in characteristic $p$ for $\kappa=2$

36. 6.6 Reduction of $T(z)$ modulo $p$

37. 7 $p$-hypergeometric solutions for $\kappa=-2$

38. 7.1 $p$-hypergeometric solutions in ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$ for $\kappa=-2$

39. 7.2 $p$-hypergeometric solutions for wedge-power and $\kappa=-2$

40. 7.3 Renumbering $p$-hypergeometric solutions $\bar{M}^{\ell_{1},\ell_{2}}(z)$

41. 7.4 Renumbering $p$-hypergeometric solutions $\bar{M}^{\ell_{1},\dots,\ell_{r}}(z)$

42. 7.5 Reduction of $\bar{T}(z)$ modulo $p$

43. 7.6 Comparison of solutions in characteristic $p$ for $\kappa=-2$

44. 7.7 Orthogonal relations on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$

45. 8 $p$-curvature for $\kappa=\pm 2$

46. 8.1 Definition of $p$-curvature

47. 8.2 $p$-curvature on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$

48. 8.3 $p$-curvature operators on $\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$

49. 8.4 Kernel of $p$-curvature on $\wedge^{2}V$

50. 8.5 Poincare pairing and good bases

51. 8.6 Solutions in ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-3]$ in characteristic $p$ for $\kappa=-2$

52. 9 $p$-curvature on $\wedge^{3}V$

53. 9.1 Zeros of $p$-curvature on $V_{r,0}$ and $V_{0,r}$

54. 9.2 Intersection of kernels in $V_{2,1}$

55. 9.3 Intersection of kernels in $V_{1,2}$

56. 9.4 Intersection of kernels in $V_{1,r-1}$

57. A Kodaira-Spencer maps and $p$-curvature

58. A.1 Cohomology classes on $C(z)$

59. A.2 Kodaira-Spencer maps

1. Introduction

The KZ equations form a system of linear partial differential equations that exhibit profound connections with theoretical physics, representation theory, algebraic geometry, and integrable systems. In 1984, Vladimir Knizhnik and Alexander Zamolodchikov first introduced these equations within the context of conformal field theory to depict the behavior of correlation functions of vertex operators in two-dimensional quantum field theories, [KZ]. Since then, the equations have emerged as a central object in both mathematical physics and pure mathematics, bridging seemingly disparate fields such as Lie algebras, braid groups, hypergeometric integrals, congruences in number theory, and geometric topology.

In this paper we consider the example of the KZ differential equations associated with the tensor power $L^{\otimes(2g+1)}$ of the two-dimensional irreducible representation $L$ of the Lie algebra $\mathfrak{sl}_{2}$ with standard generators $e,f,h$. The KZ equations are the system of differential equations on an $L^{\otimes(2g+1)}$-valued function $N(z)$, $z=(z_{1},\dots,z_{2g+1})$,

[TABLE]

where $P^{(i,j)}$ is the permutation of the $i$-th and $j$-th factors of $L^{\otimes(2g+1)}$ and $\kappa$ is a scalar parameter of the equations.

The differential operators $\nabla^{\operatorname{KZ},\kappa}_{i}=\partial_{i}-\frac{1}{\kappa}\sum_{j\neq i}\frac{P^{(i,j)}-1}{z_{i}-z_{j}}$ define the KZ connection on the trivial bundle $L^{\otimes(2g+1)}\times\mathbb{A}^{2g+1}\to\mathbb{A}^{2g+1}$, where $\mathbb{A}^{2g+1}$ is the affine space with coordinates $z_{1},\dots,z_{2g+1}$. The connection is flat for any value of $\kappa$ and has singularities over the union of all diagonal hyperplanes in $\mathbb{A}^{2g+1}$.

As a rule one considers the KZ equations over the field of complex numbers. Then the KZ differential equations have multidimensional hypergeometric solutions, [SV1].

More precisely, the differential operators $\nabla^{\operatorname{KZ},\kappa}_{i}$ commute with the $\mathfrak{sl}_{2}$-action on $L^{\otimes(2g+1)}$. Hence the KZ connection has invariant subbundles ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]\times\mathbb{A}^{2g+1}\to\mathbb{A}^{2g+1}$ where ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]\subset L^{\otimes(2g+1)}$ is the subspace of vectors $v$ such that $ev=0$ and $hv=(2g+1-2r)v$. Here $r=0,\dots,g$. All $L^{\otimes(2g+1)}$-valued solutions of the KZ equations are generated by solutions with values in these subbundles.

We have $\dim{\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]=\binom{2g}{r}-\binom{2g}{r-2}$. For $r=0$, $\dim{\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1]=1$, and the KZ equations on a ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1]$-valued (scalar) function $N(z)$ take the form $\frac{\partial}{\partial z_{i}}N(z)=0$, $i=1,\dots,2g+1$, with obvious solutions. (In this case $P^{(i,j)}-1=0$.)

We have $\dim{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]=2g$ if $r=1$. In a natural basis of $L^{\otimes(2g+1)}$, the vectors of ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ are identified with $2g+1$-vectors $(N_{1},\dots,N_{2g+1})$ with $\sum_{i}N_{i}$=0, and the ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$-valued hypergeometric solutions are given by the vectors of one-dimensional hyperelliptic integrals,

[TABLE]

where $\gamma(z)$ is a flat family of Pochhammer cycles. The solutions depend on the choice of Pochhammer cycles. All ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$-valued solutions are linear combinations of such vectors of integrals.

For $r>1$, the ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$-valued hypergeometric solutions solutions are given by analogous $r$-dimensional hypergeometric integrals, [CF, DJMM, SV1, V2, V3], but considerably more complicated.

Notice that solutions of KZ equations over complex numbers have remarkable properties. For example, it was shown in [L, BBM, BF, BFM] recently, that when $\kappa$ is an arbitrary nonzero rational number, the spaces of solutions of KZ equations carry the structure of variations of mixed Hodge structures with coefficients in a cyclotomic field.

In this paper we study the KZ differential equations with parameters $\kappa=2$ and $-2$.

Our first result, Theorem 5.2 and formula (LABEL:T(z)), describes the ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$-valued solutions of the KZ equations with $\kappa=2$ in terms of the $r$-fold wedge products of the ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$-valued solutions with $\kappa=2$. The transformation of solutions is given by a linear map $T(z):\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]\to{\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$ which is a homogeneous polynomial in $z_{1},\dots,z_{2g+1}$ of degree $\binom{r}{2}$.

We construct a natural direct decomposition of the space $\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ with the first direct summand denoted by $\mathcal{P}_{r}^{\operatorname{KZ}}(z)$ and called the primitive part. We show that $T(z)|_{\mathcal{P}_{r}^{\operatorname{KZ}}(z)}:\mathcal{P}_{r}^{\operatorname{KZ}}(z)\to{\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$ is an isomorphism of the corresponding KZ connections with $\kappa=2$ and show that $T(z)$ restricted to other direct summands equals zero.

Theorem 5.6 describes an analogous statement for $\kappa=-2$. In that case, we construct a map $\bar{T}(z):{\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]\to\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ which defines an embedding of the corresponding KZ connections with $\kappa=-2$. The map $\bar{T}(z)$ is a homogeneous polynomial in $z_{1},\dots,z_{2g+1}$ of degree $\binom{r}{2}$.

Formula (LABEL:bar_TTT) presents a useful map $\tilde{T}(z):{\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]\to\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$. We show that $\tilde{T}(z)$ equals $\bar{T}(z)$ up to a nonzero multiplicative constant, Theorem 5.14.

Remark**.**

The quantum Satake principle in [GM, GGI] says that the quantum connection on the Grassmannian $\operatorname{Gr}(r,n)$ is the $r$-th wedge of the quantum connection on the projective space $\operatorname{P}^{n-1}$. The quantum connection on $\operatorname{Gr}(r,n)$ is a version of the KZ connection on the weight space $L^{\otimes n}[n-2r]$, while the quantum connection on $\operatorname{P}^{n-1}$ is a version of the KZ connection on the weight space $L^{\otimes n}[n-2]$, see, for example, [MO, GRTV, CDG, CV]. Therefore our Theorems 5.2 and 5.6 exhibit a flavor of the quantum Satake phenomenon.

Remark**.**

The development of the Satake-type operators $T(z)$ and $\bar{T}(z)$ was inspired by [BFM, Proposition 12.1], which demonstrated that the KZ connection on ${\operatorname{Sing}}L^{\otimes(2g+2)}[0]$ with $\kappa=2$ is isomorphic to the local system of the $g$-th primitive cohomology groups (tensored with $\mathbb{C}$) of the hyperelliptic genus $g$ curve obtained as a double cover of $\operatorname{P}^{1}$ branched over the $2g+2$ points (not over infinity).

Our construction of the Satake-type operators $T(z)$ and $\bar{T}(z)$ was motivated by applications in characteristic $p$.

In [SV2], the differential KZ equations were considered modulo a prime $p$. It turned out that modulo $p$ the KZ equations have a family of polynomial solutions. The construction of these solutions is analogous to the construction of the hypergeometric solutions over complex numbers, and these polynomial solutions are called the $p$-hypergeometric solutions.

The ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$-valued $p$-hypergeometric solutions of the KZ equations with $\kappa=2$ are defined as follows. For any positive integer $\ell_{1}$, denote by $N^{\ell_{1}}(z)$ the coefficient of the monomial $t_{1}^{\ell_{1}p-1}$ in the vector of polynomials

[TABLE]

in the variables $t_{1},z_{1},\dots,z_{2g+1}$. For any positive integer $\ell_{1}$, the vector of polynomials $N^{\ell_{1}}(z)$ is a ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$-valued solution modulo $p$ of the KZ equations with $\kappa=2$, [SV2]. The solutions $N^{\ell_{1}}(z)$, $1\leqslant\ell_{1}\leqslant g$, are nonzero, linearly independent, and called the ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$-valued $p$-hypergeometric solutions with $\kappa=2$. For other positive integers $\ell_{1}$ the solutions $N^{\ell_{1}}(z)$ equal zero, [V4].

The ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$-valued $p$-hypergeometric solutions of the KZ equations with $\kappa=-2$ are defined similarly. For any positive integer $\ell_{1}$, denote by $\bar{N}^{\ell_{1}}(z)$ the coefficient of the monomial $t_{1}^{\ell_{1}p-1}$ in the vector of polynomials

[TABLE]

in the variables $t_{1},z_{1},\dots,z_{2g+1}$. For any positive integer $\ell_{1}$, the vector of polynomials $\bar{N}^{\ell_{1}}(z)$ is a ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$-valued solution modulo $p$ of the KZ equations with $\kappa=-2$, [SV2]. The solutions $\bar{N}^{\ell_{1}}(z)$, $1\leqslant\ell_{1}\leqslant g$, are nonzero, linearly independent, and called the ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$-valued $p$-hypergeometric solutions with $\kappa=-2$. For other positive integers $\ell_{1}$ the solutions $\bar{N}^{\ell_{1}}(z)$ equal zero.

The ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$-valued solutions modulo $p$ of the KZ equations with $\kappa=2$ are defined analogously. For any positive integers $\ell_{1},\dots,\ell_{r}$, one considers the coefficient $N^{\ell_{1},\dots,\ell_{r}}(z)$ of the monomial $t_{1}^{\ell_{1}p-1}\dots t_{r}^{\ell_{r}p-1}$ in a suitably defined vector of polynomials $\Psi_{2}(t_{1},\dots,t_{r},z_{1},\dots$, $z_{2g+1})$. Then $N^{\ell_{1},\dots,\ell_{r}}(z)$ is a ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$-valued solution modulo $p$ of the KZ equations with $\kappa=2$.

The ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$-valued solutions modulo $p$ of the KZ equations with $\kappa=-2$ are defined as coefficients $\bar{N}^{\ell_{1},\dots,\ell_{r}}(z)$ of the monomials $t_{1}^{\ell_{1}p-1}\dots t_{r}^{\ell_{r}p-1}$ in a suitably defined another vector of polynomials $\Psi_{-2}(t_{1},\dots,t_{r},z_{1},\dots$, $z_{2g+1})$.

The solutions $N^{\ell_{1},\dots,\ell_{r}}(z)$ and $\bar{N}^{\ell_{1},\dots,\ell_{r}}(z)$ are quite intricate. Determining which of these solutions are nonzero, understanding the linear relations among them, and establishing the dimensions of the solution spaces they generate pose significant challenges. Additionally, an important question is whether the $\mathrm{KZ}$ equations admit more solutions in characteristic $p$ than the $p$-hypergeometric solutions. We address these issues by applying our Satake-type operators $T(z)$ and $\bar{T}(z)$ in characteristic $p$.

We show that for all prime $p$ with finitely many exceptions, the maps $T(z)|_{\mathcal{P}_{r}^{\operatorname{KZ}}(z)}:\mathcal{P}_{r}^{\operatorname{KZ}}(z)\to{\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$ and $\bar{T}(z):{\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]\to\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ can be reduced modulo $p$ and the reduced maps preserve the properties of these maps over complex numbers, see Sections 6.6, 7.5 and Corollaries 6.5, 7.8. These statements in particular show that all isomorphism-type characteristics of the KZ connections with $\kappa=2$ on $\mathcal{P}_{r}^{\operatorname{KZ}}(z)$ and on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$ are identified by $T(z)$. See for example, Corollary 6.5, Theorem 8.10, and Section 8.6.

In Theorem 6.4, we show that $T(z)$ transforms wedges of the ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$-valued $p$-hypergeometric solutions to the ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$-valued $p$-hypergeometric solutions, $T(z)(N^{\ell_{1}}(z)\wedge\dots\wedge N^{\ell_{r}}(z))=N^{\ell_{1},\dots,\ell_{r}}(z)$. Similarly, in Theorem 7.9, for $\kappa=-2$, the map $\bar{T}(z)$ expresses the $r$-fold wedges $\bar{N}^{\ell_{1}}(z)\wedge\dots\wedge\bar{N}^{\ell_{r}}(z)$ in terms of the $p$-hypergeometric solutions $\bar{N}^{m_{1},\dots,m_{r}}(z)$.

The statements of Theorems 6.4 and 7.9 were unexpected. Initially, it was not clear why an isomorphism between two systems of differential equations would preserve the property of solutions being $p$-hypergeometric.

Using Theorem 7.9, we show that set of all $p$-hypergeometric solutions $\bar{N}^{\ell_{1},\dots,\ell_{r}}(z)$ spans a space of dimension $\binom{g}{r}$, see Corollary 7.10.

In Theorem 7.11, we prove orthogonality relations S\big{(}N^{\ell_{1},\dots,\ell_{r}}(z),\bar{N}^{m_{1},\dots,m_{r}}(z)\big{)}=0 where $S$ is the Shapovalov form on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$ and $\ell_{1},\dots,\ell_{r}$ and $m_{1},\dots,m_{r}$ are arbitrary positive integers. These relations are generalizations of the orthogonal relations in [VV1]. Cf. the orthogonal relations in [MV] for simplest qKZ difference equations.

The $p$-curvature operators $C_{i}:=\left(\nabla^{\mathrm{KZ},\kappa}_{i}\right)^{p}$, for $i=1,\dots,2g+1$, serve as a tool to analyze the solution space of the $\mathrm{KZ}$ equations in characteristic $p$. These operators act on sections of the $\mathrm{KZ}$ connection and commute with multiplication of sections by functions on the base. In other words, the $C_{i}$ are linear operators on the fibers of the $\mathrm{KZ}$ connection. It is known that the intersection of their kernels, $\bigcap_{i=1}^{2g+1}\ker C_{i}$, coincides with the space generated by flat local sections of the $\mathrm{KZ}$-connection $\nabla^{\mathrm{KZ},\kappa}$, see the Cartier descent theorem [K1, Theorem 5.1].

Determining the $p$-curvature operators is quite nontrivial, as it requires computing the $p$-th powers of differential operators. Cf. [EV], where the spectrum of the $p$-curvature operators was explicitly determined.

An explicit formula for the $p$-curvature operators of the $\mathrm{KZ}$ connection with fiber

${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ was obtained in [VV1]. This formula naturally extends to a formula for the $p$-curvature operators of the $\mathrm{KZ}$ connection with fiber $\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$.

In this paper, we analyze the case $r=2$ and $\kappa=\pm 2$, and then using the operators $T(z)$ and $\bar{T}(z)$, demonstrate that the intersection of the kernels of the $p$-curvature operators on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-3]$ coincides with the space spanned by the $p$-hypergeometric solutions. As a consequence, we conclude that for $\kappa=\pm 2$, all ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-3]$-valued solutions in characteristic $p$ are $p$-hypergeometric, and the dimension of the solution space in these cases equals $\binom{g}{2}$, see Theorems 8.9 and 8.13.

We also analyze the $p$-curvature operators on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-5]$ for $\kappa=2$. We observe that the intersection of kernels of the $p$-curvature operators become larger than the space generated by the $p$-hypergeometric solutions, see Corollary 9.10. Hence the corresponding KZ equations have solutions which are not $p$-hypergeometric. These are the first examples where not all solutions in characteristic $p$ are $p$-hypergeometric. It would be interesting to find an effective construction of these new solutions.

Plan of the paper

In Section 2, we define the KZ equations and the ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$-valued hypergeometric solutions over complex numbers. In Section 3, we define the KZ equations with values in $\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ and their hypergeometric solutions. We also introduce the primitive part of $\wedge^{r}H_{1}(C)$ and formulate Theorem 3.1. Theorem 3.1 is proved in Section 4. Explicit formulas for $T(z)$ and $\bar{T}(z)$ are developed in Section 5. We describe the $p$-hypergeometric solutions for $\kappa=2$ in Section 6 and the $p$-hypergeometric solutions for $\kappa=-2$ in Section 7. The orthogonal relations are presented in Section 7.7. The $p$-curvature operators for $r=2$ are studied in Section 8. The $p$-curvature operators for $r=3$ are analyzed in Section 9.

In Appendix A we identify the action of the $p$-curvature operators with the action of the corresponding Kodaira-Spencer maps and apply our results on the $p$-curvature maps to the Kodaira-Spencer maps.

Acknowledgments

The authors thank G. Cotti, R. Rimányi, and V. Vologodsky for useful discussions. The third author thanks IHES for hospitality in July-August 2025.

2. ${\mathfrak{sl}}_{2}$ KZ equations

2.1. Definition of KZ equations

Consider the complex Lie algebra ${\mathfrak{sl}_{2}}$ with generators $e,f,h$ and relations $[e,f]=h,\,[h,e]=2e,\,[h,f]=-2f$. Consider the complex vector space $L$ with basis $w_{1},w_{2}$ and the ${\mathfrak{sl}_{2}}$-action,

[TABLE]

The ${\mathfrak{sl}_{2}}$-module $L^{\otimes n}$ has a basis labeled by subsets $J\subset\{1,\dots,n\}$,

[TABLE]

where $j_{i}=1$ if $j_{i}\notin J$ and $j_{i}=2$ if $j_{i}\in J$.

Consider the weight decomposition of $L^{\otimes n}$ into eigenspaces of $h$, $L^{\otimes n}=\sum_{r=0}^{n}L^{\otimes n}[n-2r]$. The vectors $w_{J}$ with $|J|=r$ form a basis of $L^{\otimes n}[n-2r]$. Denote by ${\mathcal{I}_{r}}$ the set of all $r$-element subsets of $\{1,\dots,n\}$.

Define the space of singular vectors of weight $n-2r$,

[TABLE]

This space is nonempty if and only if $n\geqslant 2r$. We have

[TABLE]

We shall also use the formula

[TABLE]

Define the Casimir element $\Omega=\frac{1}{2}h\otimes h+e\otimes f+f\otimes e\,\in\,{\mathfrak{sl}_{2}}\otimes{\mathfrak{sl}_{2}}$. It acts on $L\otimes L$ as $P-\frac{1}{2}$ where $P$ is the permutation of factors.

Define the linear operators on $L^{\otimes n}$ depending on parameters $z=(z_{1},\dots,z_{n})$,

[TABLE]

where $\Omega^{(i,j)}:L^{\otimes n}\to L^{\otimes n}$ is the Casimir operator acting in the $i$-th and $j$-th tensor factors. Denote

[TABLE]

The operators $\partial_{i}-\frac{1}{\kappa}\tilde{H}_{i}(z)$ define the KZ connection on the trivial bundle over ${\mathbb{C}}^{n}$ with fiber $L^{\otimes n}$. The connection has singularities over the diagonal hyperplanes in ${\mathbb{C}}^{n}$. The connection is flat. For any $x\in{\mathfrak{sl}_{2}}$ and $i=1,\dots,n,$ we have

[TABLE]

The system of differential equations

[TABLE]

on an $L^{\otimes n}$-valued function $\tilde{N}(z_{1},\dots,z_{n})$ is called the system of KZ equations with parameter $\kappa$, see [KZ, EFK]. By property (2.2), the system of KZ equations can be considered with values in any particular space ${\operatorname{Sing}}L^{\otimes n}[n-2r]$.

2.2. Gauge transformation

If $\tilde{N}(z)$ satisfies the KZ equations (2.3), then the function $N(z)$, defined by

[TABLE]

satisfies the equations

[TABLE]

which we also call the system of KZ equations. Denote

[TABLE]

We have $[\nabla^{\operatorname{KZ},\kappa}_{i},\nabla^{\operatorname{KZ},\kappa}_{j}]=0$ for all $i,j$.

Define the nondegenerate symmetric bilinear form $S$ on $L^{\otimes n}$ by the formula

[TABLE]

The form is called the (tensor) Shapovalov form. The KZ connections $\nabla^{\operatorname{KZ},\kappa}$ and $\nabla^{\operatorname{KZ},-\kappa}$ are dual with respect to the Shapovalov form,

[TABLE]

for any $x(z),y(z)$.

The Shapovalov form restricted to every subspace ${\operatorname{Sing}}L^{\otimes n}[n-2r]$ is nondegenerate.

2.3. Solutions of KZ equations over ${\mathbb{C}}$

Denote $t=(t_{1},\dots,t_{r})$. Define the master function

[TABLE]

Define the symmetrization and anti-symmetrization of a given differential form $f(t_{1},\dots,t_{k})$ by the formulas:

[TABLE]

For $J\in{\mathcal{I}_{r}}$ define the differential form

[TABLE]

Given a flat family $\gamma(z)$ in $\{z\}\times{\mathbb{C}}^{r}_{t}$ of $r$-dimensional cycles of the twisted homology defined by the multivalued function $\Phi(t,z;\kappa)^{1/\kappa}$, denote

[TABLE]

Consider the $L^{\otimes n}[n-2r]$-valued function

[TABLE]

Theorem 2.1**.**

The function $N^{\gamma}(z)$ takes values in ${\operatorname{Sing}}L^{\otimes n}[n-2r]$ and satisfies the KZ equations (2.5).

This theorem and its generalizations can be found, for example, in [CF, DJMM, SV1, V2, V3]. The solutions in Theorem 2.1 are called the hypergeometric solutions of the KZ equations.

Theorem 2.2** ([V2, Theorem 12.5.5]).**

If $\kappa\in{\mathbb{C}}^{\times}$ is generic, then all solutions of the KZ equations (2.5) have this form.

For special values of the parameter $\kappa$, the hypergeometric solutions of the KZ equations may span only a proper subspace of the space of all solutions. For a discussion of the relations of this subspace of solutions and conformal blocks in conformal field theory see [FSV1, FSV2, BF].

2.4. KZ equations for $\kappa=2$

In this paper, the integer $n$ is odd, $n=2g+1,$ and $\kappa=\pm 2$. For $\kappa=2$, the KZ equations are

[TABLE]

Denote

[TABLE]

Then

[TABLE]

For $1\leqslant i_{1},\dots,i_{r}\leqslant 2g+1$, we have

[TABLE]

Notice that the differential form $\nu_{i_{1},\dots,i_{r}}$ is skew-symmetric with respect to permutation of indices $i_{1},\dots,i_{r}$.

Given a flat family $\gamma(z)$ in $\{z\}\times{\mathbb{C}}^{r}_{t}$ of $r$-dimensional cycles of the twisted homology defined by the multivalued function $\Phi(t,z;\kappa)^{1/2}$, define

[TABLE]

By Theorem 2.1, the $L^{\otimes(2g+1)}[2g+1-2r]$-valued function

[TABLE]

takes values in ${\operatorname{Sing}}L^{\otimes n}[2g+1-2r]$ and satisfies the KZ equations (2.5) over ${\mathbb{C}}$ with $\kappa=2$.

2.5. KZ equations for $\kappa=2$ and $r=1$

In that case, previous formulas take the following form. For $i_{1}=1,\dots,2g+1$, we have

[TABLE]

Consider the family of projective hyperelliptic curves $C(z)$ depending on $z$ and defined by the affine equation $y^{2}=\prod_{a=1}^{2g+1}(t_{1}-z_{a}).$ For distinct $z_{1},\dots,z_{2g+1}$, the curve is a nonsingular curve of genus $g$. Given a flat family $\gamma(z)\in H_{1}(C(z))$, we define

[TABLE]

By Theorem 2.1, the $L^{\otimes n}[2g-1]$-valued function

[TABLE]

takes values in ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ and satisfies the KZ equations (2.5) with $\kappa=2$.

Recall that ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ consists of the vectors $\sum_{i_{1}=1}^{2g+1}c_{i_{1}}w_{\{i_{1}\}}$ with $\sum_{i_{1}=1}^{2g+1}c_{i_{1}}=0$.

Theorem 2.3** ([V1, Formula (1.3)]).**

All solutions of the KZ equations with $\kappa=2$ and values in ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ have this form. Namely, the complex vector space of solutions of the form $N^{\gamma}(z)$ is $2g$-dimensional.

This theorem follows from the determinant formula for multidimensional hypergeometric integrals in [V1], in particular, from [V1, Formula (1.3)].

The ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$-valued KZ equations with $\kappa=2$ were analyzed in [VV3] over $p$-adic fields.

2.6. KZ equations for $\kappa=-2$

For $\kappa=-2$, the KZ equations are

[TABLE]

Then

[TABLE]

For $1\leqslant i_{1},\dots,i_{r}\leqslant 2g+1$, we have

[TABLE]

Given a flat family $\gamma(z)$ in $\{z\}\times{\mathbb{C}}^{r}_{t}$ of $r$-dimensional cycles of the twisted homology defined by the multivalued function $\Phi(t,z;\kappa)^{-1/2}$, define

[TABLE]

By Theorem 2.1, the $L^{\otimes n}[2g+1-2r]$-valued function

[TABLE]

takes values in ${\operatorname{Sing}}L^{\otimes n}[2g+1-2r]$ and satisfies the KZ equations (2.5) over ${\mathbb{C}}$ with $\kappa=-2$.

2.7. KZ equations for $\kappa=-2$ and $r=1$

In that case, for $i_{1}=1,\dots,2g+1$, we have

[TABLE]

Recall our family of projective hyperelliptic curves $C(z)$ depending on $z$ and defined by the affine equation $y^{2}=\prod_{a=1}^{2g+1}(t_{1}-z_{a})$. Given a flat family $\gamma(z)\in H_{1}(C(z))$, we define

[TABLE]

By Theorem 2.1, the $L^{\otimes n}[2g-1]$-valued function

[TABLE]

takes values in ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ and satisfies the KZ equations (2.5) with $\kappa=-2$.

Theorem 2.4** ([V1, Formula (1.3)]).**

All solutions of the KZ equations with $\kappa=-2$ and values in ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ have this form. Namely, the complex vector space of solutions of the form $\bar{N}^{\gamma}(z)$ is $2g$-dimensional.

3. Wedge-power construction

3.1. Wedge-power of KZ equations on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ with $\kappa=2$

The KZ equations on $L^{\otimes(2g+1)}[2g-1]$ with $\kappa=2$ induce a system of differential equations with values in $\wedge^{r}L^{\otimes(2g+1)}[2g-1]$ with the invariant subspace

[TABLE]

The integral representations for solutions look as follows.

For $1\leqslant i_{1},\dots,i_{r}\leqslant 2g+1$, define

[TABLE]

For any flat families $\gamma_{1}(z),\dots,\gamma_{r}(z)\in H_{1}(C(z))$, denote

[TABLE]

Notice that $\mu_{i_{1},\dots,i_{r}}$ is skew-symmetric with respect to permutations of the indices $i_{1},\dots,i_{r}$. Hence $M_{i_{1},\dots,i_{r}}^{\gamma_{1}\times\dots\times\gamma_{r}}$ is skew-symmetric with respect to permutations of the indices $i_{1},\dots,i_{r}$.

Define a basis in $\wedge^{r}L^{\otimes(2g+1)}[2g-1]$ by the formulas

[TABLE]

For any flat families $\gamma_{1}(z),\dots,\gamma_{r}(z)\in H_{1}(C(z))$, the vector

[TABLE]

is a solution of the KZ equations over the field ${\mathbb{C}}$ with $\kappa=2$ and values in

$\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$. By Theorem 2.3, all solutions over the field ${\mathbb{C}}$ with values in

$\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ have this form. Thus we obtain an isomorphism

[TABLE]

of the Gauss-Manin connection on $\wedge^{r}H_{1}(C)$ and the KZ connection with $\kappa=2$ on

$\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$.

3.2. Curves and their primitive cohomology

For a smooth connected genus $g$ curve $C$ over the field of complex numbers, we may identify the homology and cohomology of the curve. This identification arises by the Poincare pairing and is given by an element $\delta\in\wedge^{2}H^{1}(C)$ and also by an element $\Delta\in\wedge^{2}H_{1}(C)$ which we can think of as symplectic forms.

For any integer $r$ with $0\leqslant r\leqslant g$, we have the primitive cohomology ${\mathcal{P}}^{r}(C)$ of $\wedge^{r}{H}^{1}(C)$ given as the kernel of the map $\wedge^{r}{H}^{1}(C)\to\wedge^{2g-r+2}{H}^{1}(C)$ induced by “wedging with $\delta$” $g-r+1$ times. We may also view ${\mathcal{P}}^{r}(C)$ as the quotient of $\wedge^{r}{H}^{1}(C)$ by \operatorname{Image}\big{(}\delta\wedge\,:\,\wedge^{r-2}{H}^{1}(C) \to\wedge^{r}{H}^{1}(C)\big{)}. The rank of ${\mathcal{P}}^{r}(C)$ is $\binom{2g}{r}-\binom{2g}{r-2}$.

These constructions can also be carried out in homology groups and we get primitive parts ${\mathcal{P}}_{r}(C)$ in $\wedge^{r}H_{1}(C)$. When the curve $C$ varies in a family, the homology and cohomology groups carry the Gauss-Manin connection, and the Poincare elements are flat. Therefore we get local systems of primitive homology and cohomology groups $\mathcal{P}_{r}(C)$ and $\mathcal{P}^{r}(C)$ for $0\leqslant r\leqslant g$.

For the family of curves $C(z)$ considered before, the local systems of primitive cohomology and homology groups are irreducible, and pairwise distinct. We will review this setting in detail in Section 4.

It is known that we get a direct decomposition,

[TABLE]

3.3. Primitive part and KZ equations

Notice that $\dim\mathcal{P}^{r}=\dim{\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$. This observation extends to the following theorem.

Theorem 3.1**.**

For $0\leqslant r\leqslant g$, there is a nonzero homomorphism

[TABLE]

of the Gauss-Manin connection on $\wedge^{r}H_{1}(C)$ to the KZ connection on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$ with $\kappa=2$. The nonzero homomorphism is unique up to a multiplicative constant.

The map $\mathcal{T}$ restricted to $\mathcal{P}_{r}(C(z))$ is an isomorphism, and restricted to other summands in (3.2) equals zero.

The map $\mathcal{T}$ is given by the formula,

[TABLE]

where $\gamma_{1}(z),\dots,\gamma_{r}(z)\in H_{1}(C(z))$ are flat families, and $N^{\gamma_{1}\times\dots\times\gamma_{r}}_{i_{1},\dots,i_{r}}(z)=\int_{\gamma_{1}(z)\times\dots\times\gamma_{r}(z)}\nu_{i_{1},\dots,i_{r}}$.

Theorem 3.1 is proved in Section 4.6.

A polarized variation of Hodge structures is an integrally defined local system with the extra data of a Hodge filtration (tensored with ${\mathbb{C}}$) and has a non-degenerate bilinear pairing called the polarization (plus Griffiths transversality and other data). These local systems tensored with ${\mathbb{C}}$ have a Hodge decomposition. The indices $p,q$ in $H^{p,q}$ that appear here add up to the weight of the Hodge structure.

The local system of primitive cohomology groups $\mathcal{P}_{r}(C(z))$ with ${\mathbb{C}}$ coefficients is the local system of primitive cohomology groups with integer coefficients (which is known to be a polarized variation of the Hodge structures) tensored with ${\mathbb{C}}$. Therefore, Theorem 3.1 implies that the KZ local system, which is over the complex numbers, is isomorphic to a polarized variation of pure Hodge structures of weight $r$ tensored with complex numbers:

Corollary 3.2**.**

The monodromy of the KZ equations on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$ with $\kappa=2$ is irreducible and isomorphic to a polarized variation of pure Hodge structures of weight $r$ tensored with ${\mathbb{C}}$.

Remark**.**

It was shown in [BFM, Proposition 12.1] that the KZ connection on ${\operatorname{Sing}}L^{\otimes(2g+2)}[0]$ with $\kappa=2$ is isomorphic to the local system of the $g$-th primitive cohomology groups of the hyperelliptic genus $g$ curve obtained as a double cover of $\mathbb{P}^{1}$ branched over the $2g+2$ points (not over infinity). Note that there is a Tate twist in [BFM] but that can be ignored for the statement on local systems. Some farther expectations were formulated in [BFM, Section 12.2.2] when not all tensor factors are $L$, but still only singular vectors of weight zero were considered, see also [BFM, Remark 12.14 ].

3.4. Wedge-power of KZ equations on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ with $\kappa=-2$

The KZ equations on $L^{\otimes(2g+1)}[2g-1]$ with $\kappa=-2$ induce a system of differential equations with values in $\wedge^{r}L^{\otimes(2g+1)}[2g-1]$ with the invariant subspace

[TABLE]

The integral representations for solutions look as follows. For $1\leqslant i_{1},\dots,i_{r}\leqslant 2g+1$, define

[TABLE]

For any flat families $\gamma_{1}(z),\dots,\gamma_{r}(z)\in H_{1}(C(z))$, denote

[TABLE]

Notice that $\bar{\mu}_{i_{1},\dots,i_{r}}$ is skew-symmetric with respect to permutations of the indices $i_{1},\dots,i_{r}$. Hence $\bar{M}_{i_{1},\dots,i_{r}}^{\gamma_{1}\times\dots\times\gamma_{r}}$ is skew-symmetric with respect to permutations of the indices $i_{1},\dots,i_{r}$.

For any flat families $\gamma_{1}(z),\dots,\gamma_{r}(z)\in H_{1}(C(z))$, the vector

[TABLE]

is a solution of the KZ equations over the field ${\mathbb{C}}$ with values in $\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ and $\kappa=-2$. By Theorem 2.4, all solutions over the field ${\mathbb{C}}$ with values in $\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ have this form. Thus we obtain an isomorphism

[TABLE]

of the Gauss-Manin connection on $\wedge^{r}H_{1}(C)$ and the KZ connection with $\kappa=-2$ on

$\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$.

3.5. Shapovalov form

For any $2\leqslant r\leqslant g$, the Shapovalov form on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ induces a nondegenerate symmetric bilinear form $S^{\wedge^{r}}$ on $\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$, also called the Shapovalov form. The KZ connections on $\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ with $\kappa=2$ and $-2$ are dual with respect to $S^{\wedge^{r}}$.

4. Proof of Theorem 3.1

4.1. Hyperelliptic curves

Let $\mathcal{C}_{2g+1}$ be the configuration space of $2g+1$ distinct points on $\mathbb{A}^{1}_{\mathbb{C}}$. Recall that $n=2g+1$ and we have a family of hyperelliptic curves $C(z)$ over $\mathcal{C}_{2g+1}$. There are corresponding local systems of homology and cohomology groups over $\mathcal{C}_{2g+1}$.

We introduce some formal notation to facilitate some of the arguments in this section. Let $\pi:C\to\mathcal{C}_{2g+1}$ be the family of smooth hyperelliptic curves given by the normalisation of the closure of $y^{2}=\prod_{i=1}^{2g+1}(z_{i}-t)\subset\mathcal{C}_{2g+1}\times\mathbb{A}^{2}$ in $\mathcal{C}_{2g+1}\times\mathbb{P}^{2}$. The fiber of $\pi$ over $(z_{1},\dots,z_{2g+1})$ is the curve $C(z)$ defined earlier.

We denote by $\mathcal{H}^{1}(C)$ the local system of rank $2g$ on $S$ given by $H^{1}(\pi_{*}{\mathbb{Q}}_{C})$. We have a canonical injection $\delta:{\mathbb{Q}}_{S}\to\wedge^{2}{\mathcal{H}}^{1}(C)$ given by the Poincaré pairing. For any integer $r$ with $0\leqslant r\leqslant g$, we have a canonical sub-local system $\mathcal{P}^{r}(C)$ of $\wedge^{r}(\mathcal{H}^{1}(C))$ given as the kernel of the map $\wedge^{r}\mathcal{H}^{1}(C)\to\wedge^{2g-r+2}\mathcal{H}^{1}(C)$ induced by “wedging with $\delta$” $g-r+1$ times. Here $\delta$ is the Poincaré form. Note that $\mathcal{P}^{r}(C)$ is a variation of pure Hodge structures and its $H^{r,0}$ equals $\wedge^{r}\mathcal{H}^{1,0}(C)$ and is hence nonzero. Fiber by fiber this is local system of primitive cohomology groups considered in Section 3.2.

Remark 4.1**.**

For a smooth projective connected curve $X$ over the complex numbers, let $J(X)=H^{0}(X,\Omega^{1})^{*}/H_{1}(X,\mathbb{Z})$ denote the Jacobian variety of $X$. Recall also that $J(X)$ is the $\operatorname{Pic}^{0}(X)$ of the curve $X$ (group of divisors of degree [math]) by the Abel-Jacobi theorem, and thus can be defined in any characteristic. Topologically $J(X)$ is a torus $(S^{1})^{2g}$, and therefore $H^{r}(J(X))=\wedge^{r}H^{1}(J(X))$, further the Abel-Jacobi map $X\to J(X)=\operatorname{Pic}^{0}(X)$ given by $p\mapsto[p]-[p_{0}]$ gives an isomorphism $H^{1}(J(X))=H^{1}(X)$. Therefore the $r$-th cohomology of $J(X)$ equals $\wedge^{r}H^{1}(X)$.

Let $\pi^{\prime}:J(C)\to\mathcal{C}_{2g+1}$ be family of Jacobian varieties of $C$ corresponding to the family of curves $\pi:C\to\mathcal{C}_{2g+1}$. The cohomology groups $H^{r}(\pi^{\prime}_{*}\mathbb{Q}_{J(C)})$ are identified with $\wedge^{r}\mathcal{H}^{1}(C)$. It is known that $J(C)$ is a family of principally polarized abelian varieties, with polarization given by $\delta\in\wedge^{2}\mathcal{H}^{1}(C)=H^{2}(\pi^{\prime}_{*}\mathbb{Q}_{J(C)})$. The local system $\mathcal{P}^{r}(C)$ is identified with the local system of $r$-th primitive cohomology groups of the family of polarized varieties $J(C)$.

The monodromy group of the topological local system $\mathcal{H}^{1}(C)$ on $\mathcal{C}_{2g+1}({\mathbb{C}})$ is Zariski dense in the group $\mathrm{Sp}_{2g}$ (see, e.g., [ACN, Theorem 1 (2)]). Since $\mathcal{P}^{r}(C)$ corresponds to the $r$-th fundamental representation of $\mathrm{Sp}_{2g}$ (see, e.g., [FH, Theorem 17.5]) the monodromy representation of $\mathcal{P}^{r}(C)$ is absolutely irreducible. We also obtain that the local systems $\mathcal{P}^{r}(C)$ for $r=0,\dots,g$ are pairwise distinct irreducible monodromy representations.

Let $X_{r}=C\times_{S}C\times_{S}\dots\times_{S}C$, where there are $r$ factors and let $\pi_{r}:X_{r}\to S$ be the structure map. By the Kunneth formula there is a natural injection $\otimes^{r}\mathcal{H}^{1}(C)\to H^{r}(\pi_{r,*}{\mathbb{Q}}_{X_{r}})$ and the image of $\wedge^{r}\mathcal{H}^{1}(C)$ lies in the space of $\Sigma_{r}$-invariants $H^{r}(\pi_{r,*}{\mathbb{Q}}_{X_{r}})^{\Sigma_{r}}$. Consequently, we have an injection $\mathcal{P}^{r}(C)\to H^{r}(\pi_{r,*}{\mathbb{Q}}_{X_{r}})^{\Sigma_{r}}$.

4.2. KZ equations

Let $\mathfrak{g}=\mathfrak{sl}_{2}$ and $\kappa=2$. Let $\vec{\varpi}^{2g+1}:=(\varpi,\dots,\varpi)$ with $\varpi$ appearing $2g+1$ times, where $\varpi:=\varpi_{1}$ is the fundamental dominant weight of $\mathfrak{sl}_{2}$. Let $r\in\mathbb{Z}$ with $1\leqslant r\leqslant g$.

Let $\mathcal{KZ}_{2}$ be the local system on $\mathcal{C}_{2g+1}$ corresponding to the dual of the KZ connection on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$ with $\kappa=2$. It has the same rank as that of $\mathcal{P}^{r}(C)$, see (2.1).

Proposition 4.2**.**

The local system $\mathcal{KZ}_{2}$ on $\mathcal{C}_{2g+1}$ is isomorphic to the local system $\mathcal{P}^{r}(C)$, up to tensoring with a rank one local system on $\mathcal{C}_{2g+1}$.

4.3. Proof of Proposition 4.2

The proof is an adaptation of the methods and proof in Section 12 of [BFM] where a family of hyperelliptic curves which are unramified over $\infty\in\mathbb{P}^{1}$ (and only a weight space of [math] is considered at infinity) is considered in the basic geometric setting. There are some additional ingredients which we will point out at appropriate places.

Let $\widehat{U}\to U\subseteq\mathcal{C}_{2g+1}\times\mathbb{A}_{\mathsf{k}}^{r}$ be the covering given by the tuples $({z},t_{1},\dots,t_{r},u)$ such that $u^{2}=\Phi(t,z)$ (see (2.9)). This covering has Galois group $\Sigma_{r}\times\mu_{2}$.

Let $C^{0}\subset C$ be the open subset of points on the affine part (i.e., $t\neq\infty$) such that $t\neq z_{i}$ for all $i$ and, for any $r>0$, let $X^{0}_{r}=C^{0}\times_{S}C^{0}\times_{S}\dots\times_{S}C^{0}$ with $r$ factors and let $\Delta_{r}$ be the union of the diagonals $t_{i}=t_{j}$, $1\leqslant i<j\leqslant r$ in $X^{0}_{r}$.

There is an unramified Galois covering $X^{0}_{r}\setminus\Delta_{r}\to\widehat{U}$ given by

[TABLE]

The composition $X_{0}^{r}\setminus\Delta_{r}\to{U}$ is an unramified Galois covering with Galois group $\Sigma_{r}\times\mu_{2}^{r}$. For any $r>0$, let $\pi_{r}^{0}:X^{0}_{r}\setminus\Delta_{r}\to\mathcal{C}_{2g+1}$ be the structure map. Keeping track of isotypical components, we get a map (not a priori an injection)

[TABLE]

where $\tau:\Sigma_{r}\times\mu_{2}^{r}\to\mu_{2}$ is trivial on $\Sigma_{r}$ and the product on the second (the triviality on the $\Sigma_{r}$ factor is because of the factor $\prod(t_{i}-t_{j})$ in $\Phi(t,z)$). Let $I$ be the image of this map.

We have also a map of local systems $\mathcal{P}^{r}(C)\to H^{r}(\pi^{0}_{r,*}{\mathbb{Q}}_{X^{0}_{r}-\Delta})^{\tau^{\prime}}$ and we let its image be $J$; the invariance under $\tau^{\prime}$ uses that the hyperelliptic involution acts by multiplication by $-1$ on $H^{1}(\pi_{*}{\mathbb{Q}}_{C})$.

Claim**.**

The local system $I\cap J$ is nonzero.

We first show that the claim implies the proposition and then prove the claim. Since $J$ is nonzero by the claim and the monodromy representation of $\mathcal{P}^{r}(C)$ is irreducible, we deduce that $J\simeq\mathcal{P}^{r}(C)$ and then the claim further implies that $I\cap J=J$. The rank equality of $\mathcal{KZ}_{2}$ and $\mathcal{P}^{r}(C)$ implies that $I=J$ and the map (4.3) is injective. The proposition follows.

4.4. Proof of Claim Claim

So far modulo the rank verification the argument was entirely following the lines of Section 12 of [BFM]. At this point instead of a residue computation, we follow a different method still involving global holomorphic forms of the top dimension.

Consider the space of top degree holomorphic differentials

[TABLE]

where $X_{r}$ is as before. By results of Deligne [D], $H^{r,0}(\pi_{r,*}{\mathbb{Q}}_{X_{r}})^{\tau^{\prime}}$ maps injectively into $H^{g}(\pi^{0}_{r,*}{\mathbb{Q}}_{X^{0}_{g}-\Delta})^{\tau^{\prime}}$. Since $r\leqslant g$, $H^{r,0}(\mathcal{P}^{r}(C))$ is nonzero.

To prove the claim it suffices to show that $H^{r,0}(\pi_{r,*}{\mathbb{Q}}_{X_{r}})^{\tau^{\prime}}\subseteq H^{g}(\pi^{0}_{r,*}{\mathbb{Q}}_{X^{0}_{g}-\Delta})^{\tau^{\prime}}$ is in the image $I$ of the map (4.3). We can prove this statement fiberwise on $\mathcal{C}_{2g+1}$.

This follows from [BF] diagram (4.5): To make the transition to the notation in that paper, we first note that there is a $\Sigma_{r}$ equivariant rank one local system $\mathcal{L}$ on $U$ such that there is a map [SV1] (see Remark 4.7 for further explanation of the notation):

[TABLE]

where $\chi$ is the sign character. Here we note that $H^{r}(U,\mathcal{L})^{\chi}$ is the same as the target $H^{r}(\pi^{0}_{r,*}{\mathbb{Q}}_{X^{0}_{r}\setminus\Delta_{r}})^{\tau}$ of the map 4.3, i.e., $H^{r}(\pi^{0}_{r,*}{\mathbb{Q}}_{X^{0}_{r}\setminus\Delta_{r}})^{\tau}$. We want the image of (4.4) to contain $H^{r,0}(\pi_{r,*}{\mathbb{Q}}_{X_{r}})^{\tau^{\prime}}$. We need the following consequence of [L] (as noted implicitly in [BF]) which is valid in a more general context:

Lemma 4.5**.**

The image of (4.4) contains the image of the compactly supported cohomology $H^{r}_{c}(U,\mathcal{L})^{\chi}\to H^{r}(U,\mathcal{L})^{\chi}$.

Given this, the desired equality now follows from the fact that global top degree holomorphic forms on a smooth proper variety lift to compactly supported cohomology classes on Zariski open subsets. Hence the image of $H^{r,0}(\pi_{r,*}{\mathbb{Q}}_{X_{r}})^{\tau^{\prime}}$ in $H^{r}(\pi^{0}_{r,*}{\mathbb{Q}}_{X^{0}_{r}\setminus\Delta_{r}})^{\tau}$ lifts to compactly supported cohomology of the fibers which equals $H^{r}_{c}(U,\mathcal{L})^{\chi}$. This tells us that $H^{r,0}(\pi_{r,*}{\mathbb{Q}}_{X_{r}})^{\tau^{\prime}}\subseteq J\subseteq H^{r}(U,\mathcal{L})^{\chi}$ is in the image of $H^{r}_{c}(U,\mathcal{L})^{\chi}\to H^{r}(U,\mathcal{L})^{\chi}$ (identified with $I$) as desired.

4.5. Proof of Lemma 4.5

By Looijenga’s refinement [L] of the results of Schechtman-Varchenko [SV2] (see Proposition 4.7 in [BF]), the KZ space $\mathcal{KZ}_{2}$ is identified with the image of a map

[TABLE]

Remark 4.7**.**

We explain some of the notation used above: Let $\mathcal{L}$ be the rank one $\Sigma_{r}$ equivariant local system local system on $U$ given by the connection $d+d\log\Phi(t,z)$ on the trivial bundle, and $\chi:\Sigma_{r}\to\mu_{2}$ the sign character.

Let $P$ be a smooth projective compactification of $U$ such that the complement $P-U=\cup_{\alpha}E_{\alpha}$ is a divisor with simple normal crossings. Let $a_{\alpha}\in\mathbb{Q}$ be the residue of $d\log\Phi(t,z)$ over $E_{\alpha}$.

Let $V^{\prime}=P-\cup^{\prime}E_{\alpha}$ where the union is over $\alpha$ such that $a_{\alpha}$ is not in $\mathbb{Z}$ or is an integer $>0$, and let $q:U\to V^{\prime}$ be the inclusion. Similarly $V\subseteq V^{\prime}$ is the open subset of $P$ given by $V=P-\cup^{\prime}E_{\alpha}$ where the union is over $\alpha$ such that $a_{\alpha}$ is not in $\mathbb{Z}$ or is an integer $\geqslant 0$. Let $j:U\to V$ be the inclusion.

This image has a subquotient given by the image of $H^{r}_{c}(U,\mathcal{L})^{\chi}\to H^{r}(U,\mathcal{L})^{\chi}$ (see the bottom square in equation (4.7) of [BF]),

[TABLE]

The KZ space is identified with the image of the right vertical arrow. The image of this space in $H^{r}(U,\mathcal{L})^{\chi}$ is the image of (4.4). It contains the image of the left vertical arrow because of the above diagram.

Remark 4.9**.**

Proposition 4.2 is valid also for $\kappa=-2$. One can see this by the same argument as above with $\Phi(t,z)$ replaced by its inverse. We could also appeal to duality because the KZ system for $\kappa=-2$ is dual to the system for $\kappa=2$, and the primitive cohomology groups are self dual since $\mathcal{H}^{1}(C)$ is self dual.

4.6. Proof of Theorem 3.1

The local system ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$ with $\kappa=2$ is dual to $\mathcal{KZ}_{2}$ in Proposition 4.2. Hence Proposition 4.2 implies that the local system ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$ with $\kappa=2$ is isomorphic to the dual of $\mathcal{P}^{r}(C)$ which is $\mathcal{P}^{r}(C)$ by the self-duality of representations of $\mathrm{Sp}_{2g}$. The local system of homology groups of $C$ is isomorphic to the local system of the first cohomology groups by Poincaré duality.

Therefore to prove the first part of Theorem 3.1, we are reduced to showing that there is a unique up to scalars map

[TABLE]

The desired statement now follows by Schur’s lemma since $\wedge^{r}\mathcal{H}^{1}(C)$ is a direct sum corresponding to the decomposition of the $r$-th wedge product of the vector representation of the symplectic group:

[TABLE]

where $r_{0}=0$ if $r$ is even and $1$ if $r$ is odd, and the $\mathcal{P}^{i}(C),i=0,\dots,g,$ are pairwise distinct irreducible representations of the monodromy group.

We finally prove the explicit formula for $\mathcal{T}$. When $\kappa=2$, the image of (4.6) maps to $H^{M}(V,\mathcal{L})^{\chi}$ which is equal to $H^{r}(\pi^{0}_{r,*}{\mathbb{Q}}_{X^{0}_{r}})^{\tau}$ (compare with (4.3)) so that we may ignore the singularities on the diagonals. These maps are also consistent with connections. Now $X^{0}_{r}$ is a product of affine curves, so that their second cohomology groups are zero. Therefore $H^{r}(\pi^{0}_{r,*}{\mathbb{Q}}_{X^{0}_{r}})^{\tau}$ is fiber by fiber the tensor product of the first cohomologies of the curves.

This implies that flat families of elements in $\wedge^{r}H_{1}(C^{0})$ where $C^{0}=C-\{z_{1},\dots,z_{n},\infty\}$ give rise to solutions (possibly zero) to the KZ system on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$ for $\kappa=2$ by well-defined integrals over cycles. Because of Proposition 4.2 (using that $I$ is contained in the image of $\wedge^{r}\mathcal{H}^{1}(C)$, these solutions only depend upon the image of $\wedge^{r}H_{1}(C^{0})$ in $\wedge^{r}H_{1}(C)$, and generate all solutions to the KZ system on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$. This gives the surjective map in Theorem 3.1.

Since the other summands $\mathcal{P}_{i}(C(z))$ for $i\neq r$ in (3.2) are irreducible representations of the monodromy group different from $\mathcal{P}_{r}(C(z))\cong{\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$, the map $\mathcal{T}$ restricted to these other summands in (3.2) is zero.

This concludes the proof of Theorem 3.1.

Remark 4.10**.**

We may think of $H_{1}(C)$ as the space parametrizing solutions to the KZ equations on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ with $\kappa=2$. Therefore we may want to ask for an explicit surjection of local systems of the form

[TABLE]

with $\kappa=2$. This is carried out in Theorem 5.2.

5. Maps $T(z)$ and $\bar{T}(z)$

5.1. Cohomological relations

Denote by $K$ the $(2g+1)\times(2g+1)$-matrix with all entries equal to 1. Let $Z=\operatorname{diag}(z_{1},\dots,z_{2g+1})$. For $g=1$,

[TABLE]

Denote

[TABLE]

Recall the notation (2.12),

[TABLE]

The function $\Psi(t_{1},z)^{-1/2}$ and differential form $\eta_{i}^{(k)}(t_{1},z)$ are well-defined rational function and rational differential form on the curve $C(z)$.

The next theorem describes the cohomological relations in $H^{1}(C(z))$ between the cohomology classes of the differential forms $\eta_{i}^{(k)}(t_{1},z)$. Denote by $\eta^{(k)}(t_{1},z)$ the column $2g+1$-vector $(\eta_{1}^{(k)}(t_{1},z),\dots,\eta_{2g+1}^{(k)}(t_{1},z))^{\intercal}$.

Theorem 5.1**.**

The cohomology classes $\left[\eta^{(k)}\right]$ of the vectors $\eta^{(k)}$ of differential forms satisfy the relations,

[TABLE]

Proof.

The theorem follows from the relation

[TABLE]

To prove (5.4), notice that

[TABLE]

Hence

[TABLE]

Combining (5.5) and (5.6) we obtain

[TABLE]

which is (5.4). ∎

Denote

[TABLE]

This is a $(2g+1)\times(2g+1)$-matrix, $T^{(k)}(z)=(T^{(k)}_{ij}(z))$. The entries $T^{(k)}_{ij}(z)$ are homogeneous polynomials in $z$ of degree $k$. By Theorem 5.1, we have

[TABLE]

Notice that for $k=0$, we have $T^{(0)}_{ia}(z)=\delta_{ia}$. Applying these formulas, we obtain a formula,

[TABLE]

5.2. Map $T(z)$

For any flat families $\gamma_{1}(z),\dots,\gamma_{r}(z)\in H_{1}(C(z))$ and $1\leqslant i_{1}<$ …$<i_{r}\leqslant 2g+1$, denote

[TABLE]

cf. formula (2.22). These are coordinates of the hypergeometric solution

[TABLE]

of the KZ equations with values in ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$ and $\kappa=2$, see formula (2.16). By formula (2.4), we have

[TABLE]

Applying formula (5.10) to the right-hand side we obtain

[TABLE]

Notice that $M^{\gamma_{1}\times\dots\times\gamma_{r}}_{a_{1},\dots,a_{r}}(z)$ are skew-symmetric functions of the indices $a_{1},\dots,a_{r}$. Notice also that the products $\prod_{j=1}^{r}T^{(\sigma(j)-1)}_{i_{j},a_{j}}(z)$ are homogeneous polynomials in $z_{1},\dots,z_{2g+1}$ of degree $\binom{r}{2}$.

For example, for $r=2$, we have

[TABLE]

Formula (5.13) gives a linear expression for the solution $N^{\gamma_{1}\times\dots\times\gamma_{r}}(z)$ in terms of the solution

[TABLE]

Define a map

[TABLE]

The map $T(z)$ is a homomorphism of the KZ connection on $\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ with $\kappa=2$ to the KZ connection on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$ with $\kappa=2$, see (5.13).

Recall the isomorphism

[TABLE]

in (3.4). Recall the homomorphism,

[TABLE]

of Theorem 3.1. Clearly, we have

[TABLE]

This formula and the uniqueness statement in Theorem 3.1 give us the following theorem.

Theorem 5.2**.**

*The map $T(z)$ defines a nonzero epimorphism of the KZ connection on

$\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ with $\kappa=2$ to the KZ connection on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$ with $\kappa=2$. A nonzero homomorphism of the KZ connection on $\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ with $\kappa=2$ to the KZ connection on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$ with $\kappa=2$ is unique up to a multiplicative constant. ∎*

Corollary 5.3**.**

All solutions over ${\mathbb{C}}$ of the KZequations with values in ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$ and $\kappa=2$ are hypergeometric, that is, given by formula (5.11). ∎

Remark**.**

Notice that the homomorphisms $\mathcal{T}$ and $\mathcal{I}$ in formula (5.18) are given by hypergeometric integrals, which are quite nontrivial multi-valued complex analytic functions, while the map $T(z)$ is given by a matrix whose entries are homogeneous polynomials in $z$ of degree $\binom{r}{2}$ with coefficients in the field of rational numbers.

5.3. Poincaré element

Recall the Poincare element $\Delta(z)\in\wedge^{2}H_{1}(C(z))$. Denote $\mathcal{D}(z)=\mathcal{I}(\Delta(z))$ where $\mathcal{I}:\wedge^{2}H_{1}(C(z))\to\wedge^{2}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ is the isomorphism in (5.16) for $r=2$.

Lemma 5.4** ([VV2, Theorem 4.11]).**

We have

[TABLE]

where $D_{a}(z)=\prod_{j\neq a}(z_{a}-z_{j})$ and $\frac{2}{z_{i_{1}}-z_{i_{2}}}\left(\frac{1}{D_{i_{1}}(z)}+\frac{1}{D_{i_{2}}(z)}\right)$ is the Poincaré pairing of the cohomological classes of the differential forms $\nu_{i_{1}}$ and $\nu_{i_{2}}$ on the curve $C(z)$.

The image under $\mathcal{I}$ of the direct decomposition in (3.2) becomes

[TABLE]

where $\mathcal{P}^{\operatorname{KZ}}_{a}(z):=\mathcal{I}(\mathcal{P}_{a}(C(z))$. We have the following corollary of Theorem 3.1 and formula (5.18).

Corollary 5.5**.**

The map

[TABLE]

is an isomorphism of the KZ connections with $\kappa=2$, and $T(z)$ restricted to other direct summands in (5.3) equals zero. ∎

5.4. Map $\bar{T}(z)$

Recall the map

[TABLE]

in formula (LABEL:T(z)). Consider the dual map

[TABLE]

Recall that the KZ connection on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$ with $\kappa=-2$ is dual to the KZ connection on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$ with $\kappa=2$ under the Shapovalov form $S$, while the KZ connection on $\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ with $\kappa=-2$ is dual to the KZ connection on $\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ with $\kappa=2$ under the Shapovalov form $S^{\wedge^{r}}$. Using these two dualities in (5.23) we obtain a map

[TABLE]

defined by the formula

[TABLE]

for all $v\in\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ and $w\in{\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$. Clearly, the map $\bar{T}(z)$ is given by a matrix whose entries are homogeneous polynomials in $z$ of degree $\binom{r}{2}$ with coefficients in the field of rational numbers.

Theorem 5.2 and formula (5.25) imply the following theorem.

Theorem 5.6**.**

*The map $\bar{T}(z)$ in (5.24) defines an embedding of the KZ connection on

${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$ with $\kappa=-2$ to the KZ connection on

$\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ with $\kappa=-2$. A nonzero homomorphism*

[TABLE]

of these connections is unique up to a multiplicative constant. ∎

5.5. Formula for $\bar{T}(z)$ for $r=2$

For $r=2$ and $\kappa=-2$, we have

[TABLE]

The problem is to express the cohomological classes of the differential forms $\bar{\mu}_{i_{1},i_{2}}$ in terms of the cohomological classes of the differential forms $\bar{\nu}_{i_{1},i_{2}}$.

Proposition 5.7**.**

The cohomology class of the form $\bar{\mu}_{i_{1},i_{2}}$, $1\leqslant i_{1}<i_{2}\leqslant 2g+1$, equals the cohomology class of the differential form

[TABLE]

Proof.

We have

[TABLE]

The proposition follows from the following lemma. Denote

[TABLE]

Lemma 5.8**.**

*The cohomology class of the form $\bar{\omega}_{b}$ equals the cohomology class of the form

$-\frac{1}{2g+1}\sum_{a\neq b}z_{a}\bar{\nu}_{a,b}\,.$*

Proof.

Denote

[TABLE]

The differential of $\bar{\eta}_{b}$ is given by the formula

[TABLE]

Notice that

[TABLE]

These identities imply the lemma. ∎

∎

Fix $z^{0}=(z_{1}^{0},\dots,z_{2g+1}^{0})$ with distinct coordinates. Fix non-intersecting oriented loops $c_{1}$, $c_{2}$ on the curve $C(z^{0})$ which define distinct nonzero homology classes $\gamma_{1}(z^{0}),\gamma_{2}(z^{0})\in H_{1}(C(z^{0}))$. Extend the classes $\gamma_{1}(z^{0}),\gamma_{2}(z^{0})$ to families $\gamma_{1}(z),\gamma_{2}(z)\in H_{1}(C(z))$ flat under the Gauss-Manin connection. Then the vector

[TABLE]

is a nonzero multi-valued solution of the KZ equations with values in $\wedge^{2}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ and $\kappa=-2$.

The product $c_{1}\times c_{2}$ of loops defines a two-dimensional homology class $\gamma(z^{0})$ on the graph of the two-valued function $\Psi(t_{1},t_{2},z)^{-1/2}$. Extend $\gamma(z^{0})$ to a family $\gamma(z)$ of two-dimensional homology classes flat with respect to the Gauss-Manin connection. Then the vector

[TABLE]

is a nonzero multi-valued solution of the KZ equations with values in ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-3]$ and $\kappa=-2$.

By Proposition 5.7, we have

[TABLE]

This is a linear expression of the solution $\bar{M}^{\gamma_{1}\times\gamma_{2}}(z)$ in terms of the solution $\bar{N}^{\gamma}(z)$.

The monodromy of the KZ equations on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-3]$ with $\kappa=-2$ is irreducible by Theorem 3.1. Thus, analytic continuation of the solution $\bar{N}^{\gamma}(z)$ generates the whole space of all multi-valued solutions of the KZ equations on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-3]$ with $\kappa=-2$. Therefore, formula (5.30) well-defines a nonzero morphism

[TABLE]

of the KZ connection on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-3]$ with $\kappa=-2$ to the KZ connection on

$\wedge^{2}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ with $\kappa=-2$. Here $(\bar{N}_{i_{1},i_{2}})$ are coordinates of vectors in

${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-3]$ and $(\bar{M}_{i_{1},i_{2}})$ are coordinates of vectors in $\wedge^{2}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$.

Theorem 5.9**.**

For $r=2$, the maps $\bar{T}(z)$ and $\tilde{T}(z)$ are equal up to a nonzero multiplicative constant.

Proof.

This statement is a corollary of Theorem 5.6. ∎

5.6. Formula for $\bar{T}(z)$ for arbitrary $r$

In this section, we define a map $\tilde{T}(z)$ for any $r$. Recall

[TABLE]

Let $q$ be a nonnegative integer such that $0\leqslant q\leqslant r$. Let $a=(a_{1}\geqslant a_{2}\geqslant\dots\geqslant a_{q}\geqslant 0)$ be a non-increasing sequence of nonnegative integers. Let $J=\{1\leqslant j_{q+1}<i_{q+2}<\dots<j_{r}\leqslant 2g+1\}$.

Denote

[TABLE]

Define the degree of $\beta_{q,a,J}$ by the formula

[TABLE]

The forms of minimal degree 0 are exactly the forms $\bar{\nu}_{i_{1},\dots,i_{r}}$ defined in (2.21). If $\deg\beta_{q,a,J}>0$, then $q>0$ and $a_{1}\geqslant 0$.

Consider two differential forms: $\beta_{q,a,J}$ and $\beta_{q^{\prime},a^{\prime},J^{\prime}}$. We say that the form $\beta_{q^{\prime},a^{\prime},J^{\prime}}$ is lexicographically smaller than the form $\beta_{q,a,J}$ if $\deg\beta_{q^{\prime},a^{\prime},J^{\prime}}<\deg\beta_{q,a,J}$ or $\deg\beta_{q^{\prime},a^{\prime},J^{\prime}}=\deg\beta_{q,a,J}$ and the sequence $a^{\prime}$ is lexicographically smaller than the sequence $a$.

If $f(z_{1},\dots,z_{2g+1})$ is a homogeneous polynomial in $z_{1},\dots,z_{2g+1}$ of degree $m$, then we say that the product $f(z_{1},\dots,z_{2g+1})\beta_{q,a,J}$ is a homogeneous differential $r$-form of degree $m+\deg\beta_{q,a,J}$.

For a differential form $\beta_{q,a,J}$ with $\deg\beta_{q,a,J}>0$, we define

[TABLE]

Proposition 5.10**.**

Let $\deg\beta_{q,a,J}>0$. Then

[TABLE]

where the summation is over $(q^{\prime},a^{\prime},J^{\prime})$ such that $\beta_{q^{\prime},a^{\prime},J^{\prime}}(z)$ is lexicographically smaller than $\beta_{q,a,J}(z)$. Here $f_{q^{\prime},a^{\prime},J^{\prime}}(z)$ are homogeneous polynomials in $z$ with half-integer coefficients, and for all nonzero summands $f_{q^{\prime},a^{\prime},J^{\prime}}(z)\beta_{q^{\prime},a^{\prime},J^{\prime}}$ we have $\deg(f_{q^{\prime},a^{\prime},J^{\prime}}(z)\beta_{q^{\prime},a^{\prime},J^{\prime}})=\deg\beta_{q,a,J}$.

Proof.

We have

[TABLE]

where

[TABLE]

Calculating the product $\frac{d\Phi}{\Phi}\wedge\alpha_{q,a,J}$ we obtain the proposition. Here are the main elements of the calculation:

[TABLE]

[TABLE]

∎

Corollary 5.11**.**

The cohomological class $\left[\beta_{q,a,J}\right]$ of the form $\beta_{q,a,J}$ equals

[TABLE]

Repeatedly applying Corollary 5.11 we obtain the following statement.

Corollary 5.12**.**

Let $\deg\,\beta_{q,a,J}>0$. Then the cohomological class $\left[\beta_{q,a,J}\right]$ of the form $\beta_{q,a,J}$ can be presented as a linear combination

[TABLE]

of the cohomological classes $[\bar{\nu}_{i_{1},\dots,i_{r}}]$ of the forms $\bar{\nu}_{i_{1},\dots,i_{r}}$ defined in (2.21). Here every $h_{i_{1},\dots,i_{r}}(z)$ is a homogeneous polynomial in $z$ of degree $\deg\beta_{q,a,J}$ or equals zero. If $h_{i_{1},\dots,i_{r}}(z)$ is nonzero, then its coefficients are rational numbers. In reduced form, the denominators of the coefficients are not divisible by any prime integer $p>2g+5+a_{1}-2r$.

For $1\leqslant i_{1},\dots,i_{r}\leqslant 2g+1$, recall

[TABLE]

Similarly to (5.5), we may write

[TABLE]

Applying Corollary 5.12 to this differential form we obtain the following statement.

Proposition 5.13**.**

The cohomological class of the form $\bar{\mu}_{i_{1},\dots,i_{r}}$ can be written as a sum

[TABLE]

of the cohomological classes of the forms $\bar{\nu}_{a_{1},\dots,a_{r}}$. Here every $h_{a_{1},\dots,a_{r}}^{i_{1},\dots,i_{r}}(z)$ is a homogeneous polynomial in $z$ of degree $\binom{r}{2}$ or equals zero. If $h_{a_{1},\dots,a_{r}}^{i_{1},\dots,i_{r}}(z)$ is nonzero, then its coefficients are rational numbers. In reduced form the denominators of the coefficients are not divisible by any prime integer $p>2g+4-r$. The differential forms $\bar{\nu}_{i_{1},\dots,i_{r}}$ are defined in (2.21). ∎

Similarly to (5.31), we obtain a well-defined nonzero morphism

[TABLE]

of the KZ connection on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$ with $\kappa=-2$ to the KZ connection on

$\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ with $\kappa=-2$. Here $(\bar{N}_{i_{1},\dots,i_{r}})$ are coordinates of vectors in

${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$ and $(\bar{M}_{i_{1},\dots,i_{r}})$ are coordinates of vectors in $\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$.

Theorem 5.14**.**

For $r\geqslant 2$, the maps $\bar{T}(z)$ and $\tilde{T}(z)$ are equal up to a nonzero multiplicative constant.

Proof.

This statement is a corollary of Theorem 5.6. ∎

Corollary 5.15**.**

All solutions over ${\mathbb{C}}$ of the KZequations with values in ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$ and $\kappa=-2$ are hypergeometric, that is, given by formula (5.11). ∎

6. $p$-hypergeometric solutions for $\kappa=2$

6.1. $p$-integrals

Let $P(t_{1},\dots,t_{r})$ be a polynomial with coefficients in a field ${\mathbb{K}}$ of characteristic $p$. Given positive integers $\ell_{1},\dots,\ell_{r}$ denote by $\int_{\ell_{1},\dots,\ell_{r}}P(t_{1},\dots,t_{r})$ the coefficient of $t_{1}^{p\ell_{1}-1}\dots t_{r}^{p\ell_{r}-1}$ in $P(t_{1},\dots,t_{r})$, see [V5]. For $i=1,\dots,r$ we have

[TABLE]

In this paper, we consider polynomials $P(t_{1},\dots,t_{r},z_{1},\dots,z_{2g+1})\in{\mathbb{K}}[t_{1},\dots,t_{r},z_{1},$ …, $z_{2g+1}]$. In that case $\int_{\ell_{1},\dots,\ell_{r}}P(t_{1},\dots,t_{r},z_{1},\dots,z_{2g+1})$ denotes the coefficient of $t_{1}^{p\ell_{1}-1}\dots t_{r}^{p\ell_{r}-1}$ in $P(t_{1},\dots,t_{r},z_{1},\dots,z_{2g+1})$.

6.2. $p$-hypergeometric solutions in ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$ for $\kappa=2$

Let $p$ be an odd prime and ${\mathbb{K}}$ a field of characteristic $p$. In this section we remind the construction of $p$-hypergeometric solutions of the KZ equations over the field ${\mathbb{K}}$.

Recall $\Psi(x,z)=\prod_{a=1}^{2g+1}(x-z_{a})$. For $1\leqslant i_{1},\dots,i_{r}\leqslant 2g+1$, define

[TABLE]

cf. (2.4). These are polynomials in $t,z$ with coefficients in ${\mathbb{K}}$. For positive integers $\ell_{1},\dots,\ell_{r}$ denote

[TABLE]

cf. (2.22). These are polynomials in $z$ with coefficients in ${\mathbb{F}}_{p}\subset{\mathbb{K}}$.

Notice that $N_{i_{1},\dots,i_{r}}^{\ell_{1},\dots,\ell_{r}}$ are skew-symmetric with respect to permutations of the indices $\ell_{1},\dots,\ell_{r}$.

Lemma 6.1**.**

We have $N_{i_{1},\dots,i_{r}}^{\ell_{1},\dots,\ell_{r}}=0$ if at least one of $\ell_{1},\dots,\ell_{r}$ is greater than $g$. ∎

Recall the standard basis $w_{\{i_{1},\dots,i_{r}\}}$, $1\leqslant i_{1}<\dots<i_{r}\leqslant 2g+1$, of $L^{\otimes(2g+1)}[2g+1-2r]$.

Theorem 6.2** ([SV2, Theorem 2.4]).**

For any positive integers $\ell_{1},\dots,\ell_{r}$, the vector

[TABLE]

*is a solution in characteristic $p$ of the KZ equations (2.11) with $\kappa=2$ and values in

${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$.*

Theorem 6.3** ([V6, Theorem 3.5]).**

The solutions $N^{\ell_{1},\dots,\ell_{r}}(z)$, $1\leqslant\ell_{1}<\dots<\ell_{r}\leqslant g$, are linearly independent over the field ${\mathbb{K}}(z)$ if $p>2g-2r+3$.

For $r=1$, this theorem is proved in [V4, Theorem 3.1].

The solutions $N^{\ell_{1},\dots,\ell_{r}}(z)$, $1\leqslant\ell_{1}<\dots<\ell_{r}\leqslant g$, are called the $p$-hypergeometric solutions.

6.3. $p$-hypergeometric solutions for $\kappa=2$ and $r=1$

In that case, previous formulas take the following form. For $i_{1}=1,\dots,2g+1$ and $\ell_{1}=1,\dots,g$, we have

[TABLE]

By Theorem 6.2, the $L^{\otimes n}[2g-1]$-valued function

[TABLE]

takes values in ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ and satisfies the KZ equations (2.11) with $\kappa=2$.

6.4. $p$-hypergeometric solutions for wedge-power and $\kappa=2$

The KZ equations on $L^{\otimes(2g+1)}[2g-1]$ with $\kappa=2$ induce a system of differential equations with values in $\wedge^{r}L^{\otimes(2g+1)}[2g-1]$ with the invariant subspace

[TABLE]

The $p$-hypergeometric solutions look as follows. For $1\leqslant i_{1},\dots,i_{r}\leqslant 2g+1$, define the polynomials

[TABLE]

cf. (3.1). For positive integers $\ell_{1},\dots,\ell_{r}$ denote

[TABLE]

cf. (2.22). These are polynomials in $z$ with coefficients in ${\mathbb{K}}$.

Notice that $\mu_{i_{1},\dots,i_{r}}^{(p)}$ is skew-symmetric with respect to permutations of the indices $i_{1},\dots,i_{r}$. Hence $M_{i_{1},\dots,i_{r}}^{\ell_{1},\dots,\ell_{r}}$ is skew-symmetric with respect to permutations of $i_{1},\dots,i_{r}$. Notice also that $M_{i_{1},\dots,i_{r}}^{\ell_{1},\dots,\ell_{r}}$ is skew-symmetric with respect to permutations of the indices $\ell_{1},\dots,\ell_{r}$.

Recall the basis $w^{\{i_{1},\dots,i_{r}\}}$, $1\leqslant i_{1}<\dots<i_{r}\leqslant 2g+1$, in $\wedge^{r}L^{\otimes(2g+1)}[2g-1]$. By Theorem 6.2, for any positive integers $\ell_{1},\dots,\ell_{r}$, the vector

[TABLE]

is a solution of the KZ equations in characteristic $p$ with $\kappa=2$ and values in

$\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$.

6.5. Comparison of solutions in characteristic $p$ for $\kappa=2$

The product $\prod_{j=1}^{r}T^{(\sigma(j)-1)}_{i_{j},a_{j}}(z)$ in formula (5.11) can be reduced modulo $p$ if $p$ does not divide $\prod_{i=1}^{r-1}(2g+1-2i)$, see the denominator in formula (5.1).

Theorem 6.4**.**

Assume that $p$ does not divide $\prod_{i=1}^{r-1}(2g+1-2i)$. Then for any positive integers $\ell_{1},\dots,\ell_{r}$, we have

[TABLE]

Proof.

The proof follows from formula (5.10). ∎

Theorem 6.4 expresses the $r$-dimensional $p$-hypergeometric solutions $N^{\ell_{1},\dots,\ell_{r}}(z)$ in terms of $r$-fold wedge-products $M^{\ell_{1},\dots,\ell_{r}}_{a_{1},\dots,a_{r}}(z)$ of one-dimensional $p$-hypergeometric solutions.

For example, for $r=2$, we have

[TABLE]

6.6. Reduction of $T(z)$ modulo $p$

The map

[TABLE]

defines an isomorphism over the field ${\mathbb{C}}$ between the KZ equations with $\kappa=2$ and values in $\mathcal{P}_{r}^{\operatorname{KZ}}(z)$ and the KZ equations with $\kappa=2$ and values in ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$. This map can be reduced modulo $p$, if $p$ does not divide $\prod_{i=1}^{r-1}(2g+1-2i)$. It is clear that the reduction modulo $p$ remains an isomorphism for all prime $p$ with finitely many exceptions.

Corollary 6.5**.**

For all prime $p$ with finitely many exceptions, all isomorphism-type characteristics of the KZ differential equations on $\mathcal{P}_{r}^{\operatorname{KZ}}(z)$ and ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$ are identified by $T(z)|_{\mathcal{P}_{r}^{\operatorname{KZ}}(z)}$.

For example, the solution space of these two systems of differential equations have the same dimension, also, the collections of the $p$-curvature operators of these systems of differential equations are isomorphic under $T(z)|_{\mathcal{P}_{r}^{\operatorname{KZ}}(z)}$, see Section 8 below. Notice that in general, it is easier to study the KZ equations on $\mathcal{P}_{r}^{\operatorname{KZ}}(z)$ due to available explicit formulas.

7. $p$-hypergeometric solutions for $\kappa=-2$

7.1. $p$-hypergeometric solutions in ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$ for $\kappa=-2$

For $n=2g+1$ and $\kappa=-2$, the KZ equations are

[TABLE]

Let $p$ be an odd prime and ${\mathbb{K}}$ a field of characteristic $p$. In this section we remind the construction of $p$-hypergeometric solutions of the KZ equations over the field ${\mathbb{K}}$.

Recall $\Psi(x,z)=\prod_{a=1}^{2g+1}(x-z_{a})$. For $1\leqslant i_{1},\dots,i_{r}\leqslant 2g+1$, define

[TABLE]

cf. (6.1). These are polynomials in $t,z$ with coefficients in ${\mathbb{K}}$. For positive integers $\ell_{1},\dots,\ell_{r}$ denote

[TABLE]

cf. (6.2). These are polynomials in $z$ with coefficients in ${\mathbb{F}}_{p}\subset{\mathbb{K}}$.

Theorem 7.1** ([SV2, Theorem 2.4]).**

Let $1\leqslant r\leqslant g$. Then for any positive integers $\ell_{1},\dots,\ell_{r}$, the vector

[TABLE]

*is a solution in characteristic $p$ of the KZ equations (7.1) with $\kappa=-2$ and values in

${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$.*

Notice that these solutions are symmetric with respect to permutations of the indices $\ell_{1},\dots,\ell_{r}$.

For $r=1$, these objects take the following form. For $i_{1}=1,\dots,2g+1$ and any positive integer $\ell_{1}$, we have

[TABLE]

By Theorem 7.1, the $L^{\otimes(2g+1)}[2g-1]$-valued function

[TABLE]

takes values in ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ and satisfies the KZ equations (2.11) with $\kappa=-2$ in characteristic $p$.

For degree reason, it is easy to see that $\bar{N}^{\ell_{1}}(z)$ is the zero solution if $\ell_{1}>g$.

Theorem 7.2**.**

[SlV*, Theorem 3.2]**

If $p>2g+1$, then the solutions $\bar{N}^{\ell_{1}}(z)$, $\ell_{1}=1,\dots,g$, are linearly independent over the field ${\mathbb{K}}(z)$.*

The solutions $\bar{N}^{\ell_{1}}(z)$, $\ell_{1}=1,\dots,g$, are called the $p$-hypergeometric solutions.

Theorem 7.3** (Orthogonality, [VV1, Corollary 1.10]).**

If $p>2g+1$, then the $p$-hypergeometric solutions $N^{\ell}(z)$, $\ell=1,\dots,g$, of the KZ equations on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ with $\kappa=2$ and the $p$-hypergeometric solutions $\bar{N}^{m}(z)$, $m=1,\dots,g$, of the KZ equations on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ with $\kappa=-2$ are orthogonal under the Shapovalov form,

[TABLE]

for all positive integers $\ell,m$.

7.2. $p$-hypergeometric solutions for wedge-power and $\kappa=-2$

The KZ equations on $L^{\otimes(2g+1)}[2g-1]$ with $\kappa=-2$ induce a system of differential equations with values in $\wedge^{r}L^{\otimes(2g+1)}[2g-1]$ with the invariant subspace

[TABLE]

We consider the case $1\leqslant r\leqslant g$. The $p$-hypergeometric solutions look as follows. For $1\leqslant i_{1},\dots,i_{r}\leqslant 2g+1$, define the polynomials

[TABLE]

cf. (3.1). For positive integers $\ell_{1},\dots,\ell_{r}$ denote

[TABLE]

cf. (2.22). These are polynomials in $z$ with coefficients in ${\mathbb{K}}$.

Notice that $\bar{\mu}_{i_{1},\dots,i_{r}}^{(p)}$ is skew-symmetric with respect to permutations of the indices $i_{1},\dots,i_{r}$. Hence $\bar{M}_{i_{1},\dots,i_{r}}^{\ell_{1},\dots,\ell_{r}}$ is skew-symmetric with respect to permutations of $i_{1},\dots,i_{r}$. Notice also that $\bar{M}_{i_{1},\dots,i_{r}}^{\ell_{1},\dots,\ell_{r}}$ is skew-symmetric with respect to permutations of the indices $\ell_{1},\dots,\ell_{r}$.

Recall the basis $w^{\{i_{1},\dots,i_{r}\}}$, $1\leqslant i_{1}<\dots<i_{r}\leqslant 2g+1$, in $\wedge^{r}L^{\otimes(2g+1)}[2g-1]$. By Theorem 7.1, for any positive integers $\ell_{1},\dots,\ell_{r}$, the vector

[TABLE]

is a solution of the KZ equations in characteristic $p$ with $\kappa=-22$ and values in

$\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$. By Theorem 7.2, the solutions $\bar{M}^{\ell_{1},\dots,\ell_{r}}(z)$, $1\leqslant\ell_{1}<\dots<\ell_{r}\leqslant g$, are linearly independent over the field ${\mathbb{K}}(z)$.

7.3. Renumbering $p$-hypergeometric solutions $\bar{M}^{\ell_{1},\ell_{2}}(z)$

Recall that

[TABLE]

We have $\bar{M}^{\ell_{1},\ell_{2}}(z)=-\bar{M}^{\ell_{2},\ell_{1}}(z)$. If $1\leqslant\ell_{1}<\ell_{2}$, then $\bar{M}^{\ell_{1},\ell_{2}}(z)\neq 0$ if an only if $\ell_{2}\leqslant g$. The solutions $\bar{M}^{\ell_{1},\ell_{2}}(z)$, $1\leqslant\ell_{1}<\ell_{2}\leqslant g$ are linearly independent over ${\mathbb{K}}(z)$.

Denote

[TABLE]

Clearly, we have $\tilde{M}^{\ell_{1},\ell_{2}}(z)=\tilde{M}^{\ell_{2},\ell_{1}}(z)$ and

[TABLE]

Lemma 7.4**.**

For any positive integers $\ell_{1},\ell_{2}$, the vector $\tilde{M}^{\ell_{1},\ell_{2}}(z)$ is a solution in characteristic $p$ of the KZ equations with values in $\wedge^{2}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ and $\kappa=-2$. These solutions have the following properties:

1. $\operatorname{(i)}$

$\tilde{M}^{\ell_{1},\ell_{2}+1}(z)+\tilde{M}^{\ell_{1}-1,\ell_{2}+2}(z)+\dots+\tilde{M}^{1,\ell_{1}+\ell_{2}}(z)=-\bar{M}^{\ell_{1},\ell_{2}}(z).$** 2. $\operatorname{(ii)}$

For $1\leqslant\ell$, we have

[TABLE] 3. $\operatorname{(iii)}$

We have $\tilde{M}^{\ell_{1},\ell_{2}}(z)=0$ if $\ell_{1}>g+1$ or $\ell_{1}>g+1$, also,

[TABLE]

∎

For example, $\tilde{M}^{1,3}(z)=-\bar{M}^{1,2}(z)$, $\tilde{M}^{2,2}(z)=2\bar{M}^{1,2}(z)$, and $\tilde{M}^{2,2}(z)+2\tilde{M}^{1,3}(z)=0$.

Corollary 7.5**.**

The vectors $\tilde{M}^{\ell_{1},\ell_{2}}(z)$ with positive integers $\ell_{1}$ and $\ell_{2}$ generate a ${\mathbb{K}}(z)$-vector space of dimension $\binom{g}{2}$. The vectors $\tilde{M}^{\ell_{1},\ell_{2}+1}(z)$, $1\leqslant\ell_{1}<\ell_{2}\leqslant g$, form a basis of that space. ∎

7.4. Renumbering $p$-hypergeometric solutions $\bar{M}^{\ell_{1},\dots,\ell_{r}}(z)$

Denote

[TABLE]

Clearly, the vectors $\tilde{M}^{\ell_{1},\dots,\ell_{r}}(z)$ are symmetric with respect to the permulations of $\ell_{1},\dots,\ell_{r}$.

Let $\sigma\in S_{r}$ and $\sigma=(\sigma_{1},\dots,\sigma_{r})$. Then

[TABLE]

For example, for $r=2$ we have formula (7.12).

Lemma 7.6**.**

For any positive integers $\ell_{1},\dots,\ell_{r}$, the vector $\tilde{M}^{\ell_{1},\dots,\ell_{r}}(z)$ is a solution in characteristic $p$ of the KZ equations with values in $\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ and $\kappa=-2$. ∎

Lemma 7.7**.**

The vectors $\tilde{M}^{\ell_{1},\dots,\ell_{r}}(z)$ with positive integers $\ell_{1},\dots,\ell_{r}$ generate a ${\mathbb{K}}(z)$-vector space of dimension $\binom{g}{r}$.

Proof.

For $r=2$, the lemma follows from Lemma 7.4. We give a proof for $r=3$. For $r>3$ the proof is similar.

The goal is to express an arbitrary $\bar{M}^{\ell_{1},\ell_{2},\ell_{3}}(z)$, $1\leqslant\ell_{1}<\ell_{2}<\ell_{3}\leqslant g$ as a linear combination of $\tilde{M}^{a_{1},a_{2},a_{3}}(z)$. We use the lexicographical order on the set of triples $(\ell_{1},\ell_{2},\ell_{3})$. The lexicographically minimal triple is $(1,2,3)$ and $\bar{M}^{1,2,3}(z)=\tilde{M}^{1,3,5}(z)$.

Assume that the set of all triples is partitioned into two subsets $X$ and $Y$ where for every $(\ell_{1},\ell_{2},\ell_{3})\in X$ the polynomial $\bar{M}^{\ell_{1},\ell_{2},\ell_{3}}(z)$ is already expressed as a linear combination of the polynomials $\tilde{M}^{a_{1},a_{2},a_{3}}(z)$ and for every $(\ell_{1},\ell_{2},\ell_{3})\in Y$ the polynomial $\bar{M}^{\ell_{1},\ell_{2},\ell_{3}}(z)$ is not yet expressed as a linear combination of the polynomials $\tilde{M}^{a_{1},a_{2},a_{3}}(z)$.

Choose the lexicographically minimal triple $(\ell_{1},\ell_{2},\ell_{3})\in Y$ and consider the polynomial $\bar{M}^{\ell_{1},\ell_{2},\ell_{3}}(z)$. Then

[TABLE]

The last five polynomials $\bar{M}^{\ell_{1},\ell_{2}-1,\ell_{3}+1}(z)$, $\bar{M}^{\ell_{1}-1,\ell_{2}+1,\ell_{3}}(z)$, $\bar{M}^{\ell_{1}-1,\ell_{2}-1,\ell_{3}}(z)$, $\bar{M}^{\ell_{1}-2,\ell_{2}+1,\ell_{3}+1}(z)$, $\bar{M}^{\ell_{1}-2,\ell_{2},\ell_{3}+2}(z)$ are either zero or correspond to solutions with triples in $X$. Hence formula (7.4) provides an expression for $\bar{M}^{\ell_{1},\ell_{2},\ell_{3}}(z)$ in terms of $\tilde{M}^{a_{1},a_{2},a_{3}}(z)$. This proves the lemma for $r=3$. ∎

7.5. Reduction of $\bar{T}(z)$ modulo $p$

Recall that the map $\bar{T}(z)$ in (5.24),

[TABLE]

defines an embedding over the field ${\mathbb{C}}$ of the KZ connection with $\kappa=-2$ and values in ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$ to the KZ connection with $\kappa=-2$ and values in $\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$.

The map $\bar{T}(z)$ is defined by formula (5.25) in terms of $T(z)$. Since $T(z)$ can be reduced modulo $p$ for all $p$ with finitely many exceptions, and the reduction for all $p$ with finitely many exceptions keeps the properties of $T(z)$ over complex numbers, we obtain the following corollary.

Corollary 7.8**.**

The reduction modulo $p$ of the map $\bar{T}(z)$ is well-defined for all $p$ with finitely many exceptions, and the reduction is an embedding in characteristic $p$ of the KZ connection with $\kappa=-2$ and values in ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$ to the KZ connection with $\kappa=-2$ and values in $\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$.

Recall also that the map (LABEL:bar_TTT),

[TABLE]

is proportional to $T(z)$ and defines an embedding over the field ${\mathbb{C}}$ of the KZ connection with $\kappa=-2$ and values in ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$ to the KZ connection with $\kappa=-2$ and values in $\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$.

It is clear that this map can be reduced modulo $p$ for all $p$ with finitely many exceptions. It is also clear that the reduction modulo $p$ remains an embedding for all prime $p$ with finitely many exceptions.

7.6. Comparison of solutions in characteristic $p$ for $\kappa=-2$

Theorem 7.9**.**

Assume that $p$ is such that $\tilde{T}(z)$ can be reduced modulo $p$. Then for any positive integers $\ell_{1},\dots,\ell_{r}$, we have

[TABLE]

Proof.

In characteristic $p$, we have

[TABLE]

Then continuing like in the proof of Proposition 5.13 we conclude that the $p$-integral of $\tilde{\mu}_{i_{1},\dots,i_{r}}^{(p)}$ associated with positive integers $\ell_{1},\dots,\ell_{r}$ is given by the right-hand side in formula (7.20). ∎

Corollary 7.10**.**

Assume that $p$ is such that $\bar{T}(z)$ is an embedding. Then the $p$-hypergeometric solutions $\bar{N}^{\ell_{1},\dots,\ell_{r}}(z)$ generate a ${\mathbb{K}}(z)$-vector space of dimension $\binom{g}{r}$.

For example, for $g=2$, the vectors $\bar{N}^{\ell_{1},\ell_{2}}(z)$ generate a one-dimensional space ${\mathbb{K}}(z)$-vector space with a basis vector $\bar{N}^{1,3}(z)$. We also have

[TABLE]

7.7. Orthogonal relations on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$

Recall the orthogonal relations of Theorem 7.3,

[TABLE]

where $S$ is the Shapovalov form on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ and $N^{\ell_{1}}(z),\bar{N}^{\ell_{2}}(z)$ are

${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$-valued $p$-hypergeometric solutions of the KZequations with $\kappa=2$ and $-2$, respectively. This implies

[TABLE]

for all positive integers $\ell_{1},\dots,\ell_{r}$ and $m_{1},\dots,m_{r}$.

Theorem 7.11**.**

For all prime $p$ with finitely many exceptions, we have

[TABLE]

*for all positive integers $\ell_{1},\dots,\ell_{r}$ and $m_{1},\dots,m_{r}$. Here $S$ is the Shapovalov form on

${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$ and $N^{\ell_{1},\dots,\ell_{r}}(z),\bar{N}^{m_{1},\dots,m_{r}}(z)$ are ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$-valued $p$-hypergeometric solutions of the KZequations with $\kappa=2$ and $-2$, respectively.*

Proof.

Recall that $\bar{T}(z)=\operatorname{const}\tilde{T}(z)$ by Theorem 5.14. Hence

[TABLE]

∎

8. $p$-curvature for $\kappa=\pm 2$

8.1. Definition of $p$-curvature

([K1, Section 5], also see, for example, [EV, Section 2.4].) Let ${\mathbb{K}}$ be a field of characteristic $p$. Let $\mathcal{V}$ be a ${\mathbb{K}}$-vector bundle over base $X$. Let $\nabla$ be a flat connection on $\mathcal{V}$. If $x_{1},...,x_{n}$ are local coordinates on $X$, then we have commuting operators $\nabla_{1},...,\nabla_{n}$ of covariant partial derivatives with respect to these coordinates, which act on the space $\Gamma_{\rm rat}(\mathcal{V})$ of rational sections of $\mathcal{V}$ and determine $\nabla$. The operators

[TABLE]

are ${\mathbb{K}}(X)$-linear and called the $p$-curvature operators of $\nabla$. We have $[\nabla_{i},C_{j}]=[C_{i},C_{j}]=0$ for all $i,j$.

An application of $p$-curvature is the computation of the space of local flat sections of $\nabla$. It is known that the intersection of kernels $\cap_{i=1}^{n}\operatorname{ker}C_{i}$ of the $p$-curvature operators coincides with the space generated by flat sections of $\nabla$, see the Cartier descent theorem [K1, Theorem 5.1].

In this section we shall discuss the $p$-curvature of the KZ equations with $\kappa=2$ on the spaces ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$, $\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$, and ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$. The $p$-curvature operators on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ are described in [VV1]. Using this information we describe the $p$-curvature on $\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$. Finally, using the isomorphism $T(z)$, we obtain information on the $p$-curvature on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$.

8.2. $p$-curvature on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$

Assume that $p>2g+1$. Let ${\mathbb{K}}$ be a field of characteristic $p$. Denote $D_{a}(z)=\prod_{j\neq a}(z_{a}-z_{j}).$ Recall the $p$-hypergeometric solutions $N^{\ell}(z)$, $\ell=1,\dots,g$, of the KZ equations on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ with $\kappa=2$ and the $p$-hypergeometric solutions $\bar{N}^{\ell}(z)$, $\ell=1,\dots,g$, of the KZ equations on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ with $\kappa=-2$.

We have coordinates $z=(z_{1},\dots,z_{2g+1})$ on the base of the KZ connection and hence we have the corresponding $p$-curvature operators $C_{a}(z)$, $a=1,\dots,2g+1$.

Theorem 8.1** ([VV1, Formula (5.14)]).**

For $a=1,\dots,2g+1$, the $p$-curvature operators of the KZ equations on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ with $\kappa=2$ are given by the formula

[TABLE]

where $v\in{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ and $S$ is the Shapovalov form.

This formula shows that $C_{a}(z)$ is an operator of rank 1 whose image is generated by the vector $\sum_{\ell=1}^{g}z_{a}^{p(\ell-1)}N^{\ell}(z)$ and whose kernel is the orthogonal hyperplane in ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ to the vector $\sum_{m=1}^{g}z_{a}^{p(m-1)}\bar{N}^{m}(z)$ under the Shapovalov form.

To simplify calculations we define the normalized $p$-curvature operators by the formula

[TABLE]

This operator has the same image and kernel as $C_{a}(z)$. The operator $\tilde{C}_{a}(z)$ is a homogeneous polynomial in $z$ of degree $(2g-1)p$.

Denote

[TABLE]

This is a $2g$-dimensional vector space over ${\mathbb{K}}$. To study the $p$-curvature operators we fix $z=(z_{1},\dots,z_{n})$ with distinct coordinates and introduce on $V$ a basis depending on $z$, the basis consisting of vectors $v_{1},\dots,v_{g}$, $w_{1},\dots,w_{g}$ and associated coordinates $x_{1},\dots,x_{g}$, $y_{1},\dots,y_{g}$ such that

[TABLE]

Namely, we define

[TABLE]

Then we choose vectors $v_{1},\dots,v_{g}$ with the property

[TABLE]

The vectors $v_{1},\dots,v_{g}$ are defined uniquely up to addition of linear combinations of $w_{1}\dots,w_{g}.$ The vectors $v_{1},\dots,v_{g},$ $w_{1},\dots,w_{g}$ form a basis of $V$. Finally, using conditions (8.4) we uniquely determine the associated coordinates $y_{1},\dots,y_{g}$. We call this basis preliminary good.

Denote $V_{0,1}=\langle w_{i}\mid i=1,\dots,g\rangle\subset V$.

Lemma 8.2**.**

The subspace $V_{0,1}$ is KZ-invariant. The projections of the vectors $v_{1},\dots,v_{g}$ to $V/V_{0,1}$ do not depend on the choice of a preliminary good basis of $V$. The vectors $v_{1},\dots,v_{g}$ project to a KZ-flat basis of $V/V_{0,1}$.

Proof.

The first two statements are trivial since $V_{0,1}$ is generated by solutions $w_{1},\dots,w_{g}$. To prove the second statement we differentiate the identity $\delta_{ij}=x_{j}(v_{i})=S(v_{i},\bar{N}^{j})$, where $i,j=1,\dots,g$. We obtain

[TABLE]

By the orthogonality relations (7.6), the subspace $V_{0,1}$ is the annihilator of the subspace $\langle\bar{N}^{j}(z)\mid j=1,\dots,g\rangle$ under the Shapovalov form. Hence $\nabla_{a}^{\operatorname{KZ},\kappa=2}v_{i}\in V^{0,1}$ and the lemma follows. ∎

For $a=1,\dots,2g+1$, define $2g+1$ vectors of $V$ and $2g+1$ linear functions on $V$:

[TABLE]

We have

[TABLE]

Formula (8.2) takes the following form,

[TABLE]

It follows from (8.6), that $\cap_{a=1}^{2g+1}\operatorname{ker}\tilde{C}_{a}(z)=\langle w_{1},\dots,w_{g}\rangle$ has dimension $g$. Hence we obtain the following corollary.

Corollary 8.3** ([VV1, Theorem 1.8]).**

All solutions in characteristic $p$ of the KZ equations on ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$ with $\kappa=2$ are generated by the $p$-hypergeometric solutions $N^{\ell}(z)$, $\ell=1,\dots,g$.

8.3. $p$-curvature operators on $\wedge^{r}{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-1]$

Fix $z=(z_{1},\dots,z_{2g+1})$ such that $z_{1}^{p},\dots,z_{2g+1}^{p}$ are distinct. Let $v_{1},\dots,v_{g}$, $w_{1},\dots,w_{g}$ be a preliminary good basis of $V$. For any $r$, the normalized $p$-curvature operator $\tilde{C}_{a}(z)$ is defined on $\wedge^{r}V$ by the formula

[TABLE]

Clearly we have

[TABLE]

for any $r$.

Let $r\leqslant g$. It is convenient to introduce the decomposition

[TABLE]

where $V_{r,k}$ is generated by vectors of the form $v_{i_{1}}\wedge\dots\wedge v_{i_{k}}\wedge w_{j_{1}}\wedge\dots\wedge w_{j_{r-k}}$. We have $\tilde{C}_{a}(z)(V_{k,r-k})\subset V_{k-1,r-k+1}$. In particular this means that if $v=\sum_{k=0}^{r}v_{k,r-k}$ is such that $v_{k,r-k}\in V_{k,r-k}$ for all $k$ and $\tilde{C}_{a}(z)(v)=0$, then $\tilde{C}_{a}(z)(v^{k,r-k})=0$ for every $k$.

We also introduce a filtration on $\wedge^{r}V$,

[TABLE]

where $V_{k}=V_{k,r-k}\oplus V_{k-1,r-k+1}\oplus\dots\oplus V_{0,k}$. By Lemma 8.2, this filtration does not depend on the choice of a preliminary good basis of $V$. The projections to $V_{k}/V_{k-1}$ of the vectors $v_{i_{1}}\wedge\dots\wedge v_{i_{k}}\wedge w_{j_{1}}\wedge\dots\wedge w_{j_{r-k}}$ also do not depend on the choice of a preliminary good basis. The filtration is invariant with respect to the KZ connection on $\wedge^{r}V$ with $\kappa=2$. The projections to $V_{k}/V_{k-1}$ of the vectors $v_{i_{1}}\wedge\dots\wedge v_{i_{k}}\wedge w_{j_{1}}\wedge\dots\wedge w_{j_{r-k}}$ are flat sections of the bundle with fiber $V_{k}/V_{k-1}$.

8.4. Kernel of $p$-curvature on $\wedge^{2}V$

Define

[TABLE]

Let $v_{1},\dots,v_{g},$ $w_{1},\dots,w_{g}$ be a preliminary good basis on $V$ with associated coordinates $x_{1},\dots,x_{g},$ $y_{1},\dots,y_{g}$.

Theorem 8.4**.**

We have

[TABLE]

where $\delta=v_{1}\wedge w_{1}+\dots+v_{g}\wedge w_{g}$ .

Proof.

Recall $\wedge^{2}V=V_{2,0}\oplus V_{1,1}\oplus V_{0,2}$. It is clear that $V_{0,2}\subset\mathcal{K}_{2}$.

Lemma 8.5**.**

We have $\mathcal{K}_{2}\cap V_{1,1}=\langle\delta\rangle$ and $\mathcal{K}_{2}\cap V_{2,0}=0$.

The lemma implies the theorem.

Proof of lemma. Let $x\in V_{1,1}$, $x=\sum_{i=1}^{g}v_{i}\wedge\sum_{j=1}^{g}c_{ij}w_{j}$ . Then $\tilde{C}_{a}x=K_{a}\wedge\sum_{i,j=1}^{g}z_{a}^{(i-1)p}c_{ij}w_{j}$ . We have $x\in\mathcal{K}_{2}$ if and only if the vector $x_{a}:=\sum_{i,j=1}^{g}z_{a}^{(i-1)p}c_{ij}w_{j}$ is proportional to the vector $K_{a}=w_{1}+z_{a}^{p}w_{2}+\dots+z_{a}^{p(g-1)}w_{g}$ for $a=1,\dots,2g+1$. That is, for $i=1,\dots,g-1$, we have the equations

[TABLE]

This equation can be written as

[TABLE]

Given $i$, the system of $g+1$ equations,

[TABLE]

implies that

[TABLE]

It is easy to see that the union of these systems of equations for $i=1,\dots,g$ implies that

[TABLE]

where $\epsilon$ is a scalar parameter. This proves the first equality in Lemma 8.5.

Let $y=\sum_{1\leqslant i<j\leqslant g}c_{ij}v_{i}\wedge v_{j}\in V_{2,0}$. Then $\tilde{C}_{a}(y)=K_{a}\wedge\sum_{1\leqslant i<j\leqslant g}c_{ij}(z_{a}^{p(i-1)}v_{j}-z_{a}^{p(j-1)}v_{i})$. If $\tilde{C}_{a}(y)=0$, then

[TABLE]

The system consisting of the first equation for $a=1,\dots,g-1$ gives $c_{1,2}=c_{1,3}=\dots=c_{1,g}=0$. Then the system consisting of the second equation for $a=1,\dots,g-1$ gives $c_{2,3}=\dots=c_{2,g}=0$, and so on. This proves the second equality in Lemma 8.5. ∎

8.5. Poincare pairing and good bases

Fix $z=(z_{1},\dots,z_{2g+1})$ such that $z_{1}^{p},\dots,z_{2g+1}^{p}$ are distinct. Let $v_{1},\dots,v_{g},$ $w_{1},\dots,w_{g}$ be a preliminary good basis of $V$. Let $\mathcal{D}(z)\in\wedge^{2}V$ be the element corresponding to Poincare pairing. The element $\mathcal{D}(z)$ has the following properties.

1. (i)

The element $\mathcal{D}(z)$ defines a non-degenerate skew-symmetric form on the space $V^{*}$ dual to $V$. 2. (ii)

We have $\tilde{C}_{a}(\mathcal{D}(z))=0$, $a=1,\dots,2g+1$, since $\mathcal{D}(z)$ is invariant with respect to the Gauss-Manin connection. 3. (iii)

The kernel $\mathcal{P}_{2}^{\operatorname{KZ}}(z)$ of the map

[TABLE]

of multiplication by $(\mathcal{D}(z)\wedge)^{g-1}$ is invariant under each $p$-curvature operator $\tilde{C}_{a}$, since the kernel of this map is preserved by the Gauss-Manin connection.

Lemma 8.6**.**

Up to proportionality, the element $\mathcal{D}(z)$ equals $\delta+\sum_{1\leqslant i<j\leqslant g}c_{ij}w_{i}\wedge w_{j}$ with suitable coefficients $c_{ij}$.

Proof.

Property (ii) implies that $\mathcal{D}(z)\in\langle\delta,w_{i}\wedge w_{j}\mid 1\leqslant i<j\leqslant g\rangle$. Property (i) implies that $\mathcal{D}(z)$ is not proportional to a linear combination $\sum_{1\leqslant i<j\leqslant g}c_{ij}w_{i}\wedge w_{j}$ . ∎

Let $\mathcal{D}(z)$ be proportional to $\delta+\sum_{1\leqslant i<j\leqslant g}c_{ij}w_{i}\wedge w_{j}$ for some coefficients $c_{ij}$. Define a new preliminary good basis $\bar{v}_{1},\dots,\bar{v}_{g},$ $\bar{w}_{1},\dots,\bar{w}_{g}$ of $V$ by the formulas $v_{i}=\bar{v}_{i}+\sum_{j=i+1}^{g}c_{ij}w_{j}$, $w_{i}=\bar{w}_{i}$ , $i=1,\dots,g$. Then

[TABLE]

We say that a basis $v_{1},\dots,v_{g},$ $w_{1},\dots,w_{g}$ of $V$ is good if it is preliminary good and $\mathcal{D}(z)$ is proportional to $\sum_{i=1}^{g}v_{i}\wedge w_{i}$ . Good bases exist since the above basis $\bar{v}_{1},\dots,\bar{v}_{g},$ $\bar{w}_{1},\dots,\bar{w}_{g}$ is good. A good basis is not unique.

Let $v_{1},\dots,v_{g},$ $w_{1},\dots,w_{g}$ be a good basis of $V$. Define the decomposition $\mathcal{P}_{2}^{\operatorname{KZ}}=\mathcal{P}_{2,0}\oplus\mathcal{P}_{1,1}\oplus\mathcal{P}_{0,2}$, $\mathcal{P}_{i,j}=\mathcal{P}_{2}^{\operatorname{KZ}}(z)\cap V_{i,j}$.

Lemma 8.7**.**

We have $\mathcal{P}_{2,0}=V_{2,0}$, $\mathcal{P}_{0,2}=V_{0,2}$. The space $\mathcal{P}_{1,1}$ is generated by the elements $v_{i}\wedge w_{j}$ with $i\neq j$, and by the elements $\sum_{i=1}^{g}c_{i}v_{i}\wedge w_{i}$ such that $\sum_{i=1}^{g}c_{i}=0$. ∎

Theorem 8.8**.**

We have

[TABLE]

Proof.

We have

[TABLE]

Clearly, these vectors for $a=1,\dots,g+1$ span the subspace

[TABLE]

Similarly, we show that the vectors $\tilde{C}_{a}(v_{i}\wedge v_{i+1})$ with $i=2,\dots,g-1$ span the subspace

[TABLE]

Clearly these subspaces span $\mathcal{P}_{1,1}$.

To show the second equality in (8.10) we prove a stronger statement,

[TABLE]

Indeed, for $1\leqslant i<j\leqslant g$, we have

[TABLE]

It is easy to see that the elements $K_{b}\wedge K_{a}$ span $\mathcal{V}_{0,2}$. ∎

Theorem 8.9**.**

Let $p>2g+1$. Then all solutions of the KZ equations in characteristic $p$ with values in $\mathcal{P}_{2}^{\operatorname{KZ}}(z)\subset\wedge^{2}V$ and $\kappa=2$ are generated by the $p$-hypergeometric solutions $M^{\ell_{1},\ell_{2}}(z)$, $1\leqslant\ell_{1}<\ell_{2}\leqslant g$.

Proof.

On the one hand, these $p$-hypergeometric solutions are linearly independent over the field ${\mathbb{K}}(z)$ by Theorem 6.3 for $r=2$, hence they span the space $V_{0,2}$ of dimension $\binom{g}{2}$. On the other hand, $\mathcal{K}_{2}\cap\mathcal{P}_{2}^{\operatorname{KZ}}(z)=V_{0,2}$ is the same space by Theorem 8.4. This proves the theorem. ∎

Theorem 8.10**.**

For all prime $p$ with finitely many exceptions, all solutions of the KZ equations in characteristic $p$ with values in ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-3]$ and $\kappa=2$ are generated by the $p$-hypergeometric solutions $N^{\ell_{1},\ell_{2}}(z)$, $1\leqslant\ell_{1}<\ell_{2}\leqslant g$.

Proof.

On the one hand, under these assumptions on $p$, the map $T(z)|_{\mathcal{P}_{2}^{\operatorname{KZ}}(z)}:\mathcal{P}_{2}^{\operatorname{KZ}}(z)\to{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-3]$ is an isomorphism in characteristic $p$ between the KZ equations with $\kappa=2$ and values in $\mathcal{P}_{2}^{\operatorname{KZ}}(z)$ and the KZ equations with $\kappa=2$ and values in ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-3]$. Hence these two systems of equations have the same dimension $\binom{g}{2}$ of the space of solutions. On the other hand, the $p$-hypergeometric solutions $N^{\ell_{1},\ell_{2}}(z)$, $1\leqslant\ell_{1}<\ell_{2}\leqslant g$, span the space of dimension $\binom{g}{2}$. This proves the theorem. ∎

8.6. Solutions in ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-3]$ in characteristic $p$ for $\kappa=-2$

Let $C_{a}^{-}(z)$, $a=1,\dots,2g+1,$ denote the $p$-curvature operators of the KZ equations with values in

${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-3]$ and $\kappa=-2$. Fix $z$ with distinct coordinates. Recall the $p$-curvature operators $C_{a}(z)|_{\mathcal{P}_{2}^{\operatorname{KZ}}(z)}:\mathcal{P}_{2}^{\operatorname{KZ}}(z)\to\mathcal{P}_{2}^{\operatorname{KZ}}(z)$ for $\kappa=2$ of Section 8.2.

For all prime $p$ with finitely many exceptions, the space ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-3]$ with operators $C_{a}^{-}(z)$ is isomorphic to the space $(\mathcal{P}_{2}^{\operatorname{KZ}}(z))^{*}$ (dual to the space $\mathcal{P}_{2}^{\operatorname{KZ}}(z)$) with operators $-\left(C_{a}(z)|_{\mathcal{P}_{2}^{\operatorname{KZ}}(z)}\right)^{*}$, dual to the operators $C_{a}(z)|_{\mathcal{P}_{2}^{\operatorname{KZ}}(z)}$ with coefficient $-1$. This follows from formula (2.8) and the isomorphism $T(z)|_{\mathcal{P}_{2}^{\operatorname{KZ}}(z)}:\mathcal{P}_{2}^{\operatorname{KZ}}(z)\to{\operatorname{Sing}}L^{\otimes(2g+1)}[2g-3]$ for $\kappa=2$ in characteristic $p$.

Lemma 8.11**.**

The space

[TABLE]

has dimension $\binom{g}{2}$.

Proof.

The space $\mathcal{K}^{-}$ is the annihilator of the subspace $\sum_{a=1}^{2g+1}\operatorname{Im}(C_{a}(z):\mathcal{P}_{2}^{\operatorname{KZ}}(z)\to\mathcal{P}_{2}^{\operatorname{KZ}}(z))$. By Theorem 8.8, the annihilator has dimension of the space $\mathcal{P}_{2,0}$ which is $\binom{g}{2}$. ∎

Corollary 8.12**.**

For all prime $p$ with finitely many exceptions, the space of solutions in characteristic $p$ of the KZ equations with values in ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-3]$ and $\kappa=-2$ is of dimension $\binom{g}{2}$.

Theorem 8.13**.**

*In characteristic $p$, all solutions of the KZ equations with values in

${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-3]$ and $\kappa=-2$ are generated by the $p$-hypergeometric solutions $\bar{N}^{\ell_{1},\ell_{2}}(z)$, see formula (7.3).*

Proof.

By Corollary 7.10, the $p$-hypergeometric solutions generate a space of dimension $\binom{g}{2}$. Now the theorem follows from Corollary 8.12. ∎

9. $p$-curvature on $\wedge^{3}V$

Choose a good basis of $V$.

9.1. Zeros of $p$-curvature on $V_{r,0}$ and $V_{0,r}$

Lemma 9.1**.**

Let $\mathcal{P}_{r}^{\operatorname{KZ}}(z)\subset\wedge^{r}V$ be the primitive subspace. Then $\tilde{C}_{a}(\mathcal{P}_{r}^{\operatorname{KZ}}(z))\subset\mathcal{P}_{r}^{\operatorname{KZ}}(z)$ for every $a$. ∎

We have a decomposition $\mathcal{P}_{r}^{\operatorname{KZ}}(z)=\oplus_{k=0}^{r}\mathcal{P}_{k,r-k}$, $\mathcal{P}_{k,r-k}=\mathcal{P}_{r}^{\operatorname{KZ}}(z)\cap V_{k,r-k}$. Denote

[TABLE]

Clearly $\mathcal{K}_{0,r}=\mathcal{P}_{0,r}=V_{0,r}$.

Proposition 9.2**.**

For any $r$, $1\leqslant r\leqslant g$, we have $\mathcal{K}_{r,0}=0$.

Proof.

We need the following linear algebra lemma. Let $y_{1},\dots,y_{n}$ be a basis of a vector space $Y$ and $y_{1}^{*},\dots,y_{n}^{*}$ the dual basis of the dual space $Y^{*}$. For $\omega\in\wedge^{k}Y$ and $y^{*}\in Y^{*}$, let $i_{y^{*}}\omega$ denote the interior product of $y^{*}$ and $\omega$.

Lemma 9.3**.**

Let $\omega\in\wedge^{k}Y$ and $i_{y_{a}^{*}}\omega=0$ for $a=1,\dots,n$. Then $\omega=0$. ∎

Let $\omega=\sum_{1\leqslant i_{1}<\dots<i_{r}\leqslant g}c_{i_{1},\dots,i_{g}}v_{i_{1}}\wedge\dots\wedge v_{i_{r}}$ be an element of $V_{r,0}$. Then

[TABLE]

We have $\tilde{C}_{a}(\omega)=0$ for all $a$ if and only if $i_{L_{a}}\omega=0$ for all $a$. By Lemma 9.3. this implies that $\omega=0$ since the vectors $L_{a}$, $a=1,\dots,g$, are linearly independent. ∎

9.2. Intersection of kernels in $V_{2,1}$

Lemma 9.4**.**

For $q\geqslant 3$, the intersection of kernels of the reduced $p$-curvature operators $\tilde{C}_{m}:V_{2,1}\to V_{1,2}$, $m=1,\dots,2g+1$, equals zero.

Proof.

A vector $\alpha\in V_{2,1}$ has the form

[TABLE]

We set $d_{a,b;c}=0$ if $b\leqslant a$ or if any of the indices $a,b,c$ is not in the set $\{1,\dots,g\}$. Similarly to previous reasonings, we observe that $\alpha$ lies in the intersection of kernels if its coefficients $d_{a,b:c}$ satisfy the system of equations

[TABLE]

where the equations are labeled by the parameters $s,k,i,j$ with the only condition $j<k$. Equation (9.1) comes as the coefficient of $z_{m}^{p(s-2)}v_{i}\wedge w_{j}\wedge w_{k}$ in the vector $\tilde{C}_{m}\alpha$.

This system of equations implies that all coefficients $d_{a,b;c}$ are equal to zero. Indeed, if $c>a$ and $b>a$, we choose $i=b,\,k=c,\,j=c-a,\,s=c$, and equation (9.1) says that $d_{a,b;c}=0$. If $c<b$ and $b>a$, we choose $i=a,\,j=c,\,s=c+g+1,\,k=c+g-b+1$, and equation (9.1) says that $d_{a,b;c}=0$. ∎

9.3. Intersection of kernels in $V_{1,2}$

Introduce the following $2g-2$ elements of $V_{1,2}$:

[TABLE]

where $k=1,\dots,g-1$. Denote by $X_{1,2}$ the span of these elements.

Let $Y_{1,2}$ be the intersection of kernels of the reduced $p$-curvature operators $\tilde{C}_{m}:V_{1,2}\to V_{0,3}$, $m=1,\dots,2g+1$.

Lemma 9.5**.**

The elements $\alpha_{k},\beta_{k}$,   $k=1,\dots,g-1$, are linearly independent and $X_{1,2}\subset Y_{1,2}$.

Proof.

Consider the lexicographical ordering on the elements $v_{a}\wedge w_{b}\wedge w_{c}$. Then the lexicographically minimal summand of $\alpha_{k}$ is $v_{1}\wedge w_{1}\wedge w_{k}$ and the lexicographically minimal summand of $\beta_{k}$ is $v_{2}\wedge w_{1}\wedge w_{k}$. These minimal summands are distinct. Hence $\alpha_{k},\beta_{k}$,   $k=1,\dots,g-1$, are linearly independent.

We have

[TABLE]

Similarly we show that $\tilde{C}_{m}\beta_{k}=0$.

∎

Lemma 9.6**.**

The intersection $X_{1,2}\cap\mathcal{P}_{1,2}$ has dimension $g-2$.

Proof.

Recall that $\mathcal{P}_{1,2}$ is the kernel of the map $f:V_{1,2}\to V_{g-1,g}$ of multiplication by the $(g-2)$-th wedge-power of the element $\sum_{i=1}^{g}v_{i}\wedge w_{i}$. The elements

[TABLE]

form a basis of $V_{g-1,g}$.

We have $f(X_{1,2})=V_{g-1,g}$ because of $f(\beta_{i})=-(g-i)!\gamma_{i}$, $i=1,\dots,g-1$, and $f(\beta_{g-1})=(g-1)!\gamma_{g}$. Since $f(X_{1,2})=V_{g-1,g}$, we obtain that $\dim X_{1,2}\cap\mathcal{P}_{1,2}=2g-2-g=g-2$. ∎

Theorem 9.7**.**

We have $X_{1,2}=Y_{1,2}$.

Proof.

Define the set

[TABLE]

The elements of $S$ are the lexicographically minimal summands of $\alpha_{k}$ and $\beta_{k}$ with $k=1,\dots,g-1$. The theorem clearly follows from the following lemma.

Lemma 9.8**.**

If $\alpha$ is a nonzero element of $Y_{1,2}$, then the lexicographically minimal summand of $\alpha$ belongs to the set $S$.

Proof of lemma. Let

[TABLE]

We set $d_{a;b,c}=0$ if $c\leqslant b$ or if any of the indices $a,b,c$ is not in the set $\{1,\dots,g\}$. Similarly to previous reasonings, we observe that $\alpha$ lies in the intersection of kernels if its coefficients $d_{a;b,c}$ satisfy the system of equations

[TABLE]

where the equations are labeled by the parameters $s,i,j,k$ with the only condition $i<j<k$. Equation (9.3) comes as the coefficient of $z_{m}^{p(s-2)}w_{i}\wedge w_{j}\wedge w_{k}$ in $\tilde{C}_{m}\alpha$.

Claim 1. If $a>c>b$, then $d_{a;b,c}=0$.

Indeed, choose $i=b,\,j=c,\,s=c+g+1,\,k=c+g-a+1$. Then (9.3) implies that $d_{a;b,c}=0$.

Claim 2. If $1<b<c$, then $v_{a}\wedge w_{b}\wedge w_{c}$ cannot be the lexicographically minimal summand of $\alpha$.

Indeed, choose $j=b,\,i=1,\,s=a+1,\,k=c$. Then equation (9.3) takes the form

[TABLE]

Claim 2 follows since the first term is the lexicographically largest of these three terms.

Claim 3. If $b=1$ and $2<a<c+1$, then $v_{a}\wedge w_{1}\wedge w_{c}$ cannot be the lexicographically minimal summand of $\alpha$.

Indeed, choose $s=c+2,\,j=c+2-a,\,i=1,\,k=c$. Then (9.3) takes the form

[TABLE]

In this equation, the first coefficient is zero by Claim 1, and the element $v_{2}\wedge w_{1}\wedge w_{c+2-a}$ is lexicographically smaller that $v_{a}\wedge w_{1}\wedge w_{c}$. Claim 3 follows.

Claims 1-3 imply the lemma. ∎

∎

Corollary 9.9**.**

The space of solutions modulo $p$ with values in $\mathcal{P}_{3}^{\operatorname{KZ}}(z)$ and $\kappa=2$ is of dimension $\binom{g}{3}+g-2$. The space of solutions has a $\binom{g}{3}$-dimensional subspace generated the $p$-hypergeometric solutions $w_{i_{1}}\wedge w_{i_{2}}\wedge w_{i_{3}}$ with $1\leqslant i_{1}<i_{2}<i_{3}\leqslant g$. The projection of any solution to $\mathcal{P}_{3}^{\operatorname{KZ}}(z)/\mathcal{P}_{0,3}$ lies in $X_{1,2}\cap\mathcal{P}_{1,2}$.

Corollary 9.10**.**

For all prime $p$ with finitely many exceptions, the space of solutions of the KZ equations with values in ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g-5]$ and $\kappa=2$ in characteristic $p$ has dimension $\binom{g}{3}+g-2$. That space has a $\binom{g}{3}$-dimensional subspace generated the $p$-hypergeometric solutions $N^{\ell_{1},\ell_{2},\ell_{3}}$ with $1\leqslant\ell_{1}<\ell_{2}<\ell_{3}\leqslant g$.

9.4. Intersection of kernels in $V_{1,r-1}$

We expect that the intersection in $V_{1,r-1}$ of the kernels of the $p$-curvature operators is of dimension $(r-1)\binom{g-1}{r-2}$ and has a basis consisting of elements analogous to the elements $\alpha_{k},\beta_{k}$, $k=1,\dots,g-1$, in (9.2).

For simplicity we define these elements in $V_{1,3}$ for $g>3$. For $i=0,1,2$ and $1\leqslant k_{1}<k_{2}\leqslant g-1$, let

[TABLE]

cf. (9.2). It is easy to see that these elements lie in the intersection of the kernels.

A general problem is to describe the dimension of the intersection of kernels in $\mathcal{P}_{r}^{\operatorname{KZ}}(z)$ for arbitrary $r$ and determine the dimension of the space of solutions of the KZ equations in ${\operatorname{Sing}}L^{\otimes(2g+1)}[2g+1-2r]$ for $\kappa=\pm 2$.

Appendix A Kodaira-Spencer maps and $p$-curvature

A.1. Cohomology classes on $C(z)$

Fix $z=(z_{1},\dots,z_{2g+1})$ with distinct coordinates. Recall $\Psi(t,z)=\prod_{a=1}^{2g+1}(t-z_{a})$. Consider the following differential 1-forms on the hyperelliptic curve $C(z)$:

[TABLE]

The forms $\omega_{i}$ have zero residues at all points of $C(z)$ and define elements $\left[\omega_{i}\right]\in H^{1}(C(z))$. The forms $\left[\omega_{i}\right]$, $i=1,\dots,2g+1$, span $H^{1}(C(z))$ and satisfy the relation $\sum_{i=1}^{2g+1}\left[\omega_{i}\right]=0$, see [VV1, Theorem 4.2]. The forms $\mu_{k}$, $k=1,\dots,g,$ are regular on $C(z)$ and give a basis of the Hodge subspace $F^{1}\subset H^{1}(C(z))$, [VV1, Lemma 4.4].

The derivatives of the elements $\left[\mu_{k}\right]$ with respect to the Gauss-Manin connection are given by the formula

[TABLE]

[VV1, Lemma 4.5].

Consider the linear functions $(\cdot,\mu_{k})$, $k=1,\dots,g$, on $H^{1}(C(z))$ where $(\cdot,\cdot)$ is the Poincaré pairing. These linear functions are zero on $F^{1}$ and define a coordinate system on $H^{1}(C(z))/F^{1}$. Let $\nu_{k}$, $k=1,\dots,g$, be the dual basis of $H^{1}(C(z))/F^{1}$, $(\nu_{i},\omega_{j})=\delta_{ij}$. For any element $\gamma\in H^{1}(C(z))$, its projection $\bar{\gamma}$ to $H^{1}(C(z))/F^{1}$ equals the linear combination

[TABLE]

Consider the direct sum $W=F^{1}\oplus H^{1}(C(z))/F^{1}$ with basis $\mu_{k},\nu_{k}$, $k=1,\dots,g$. The element $\sum_{k=1}^{g}\nu_{k}\wedge\mu_{k}$ in $\wedge^{2}W$ corresponds to the Poincare pairing element.

The Poincaré pairing of $[\omega_{i}]$ and $[\mu_{k}]$ is given by the formula

[TABLE]

where $D_{i}(z)=(z_{i}-z_{1})\dots(z_{i}-z_{i-1})(z_{i}-z_{i+1})\dots(z_{i}-z_{n})$, [VV1, Theorem 4.7].

A.2. Kodaira-Spencer maps

Recall the Kodaira-Spencer maps

[TABLE]

By definition, the map $\operatorname{KS}_{a}(z)$ sends the class $[\mu_{k}]$ of a regular 1-form to the projection $\overline{\nabla_{\!\!\frac{\partial}{\partial z_{a}}}[\mu_{k}]}$ in $H^{1}(C(z))/F^{1}$ of the class $\nabla_{\!\!\frac{\partial}{\partial z_{a}}}[\mu_{k}]$.

We extend the Kodaira-Spencer map to a map $W\to W$ by setting $\operatorname{KS}_{a}(z)|_{H^{1}(C(z))/F^{1}}=0$.

Lemma A.1**.**

For $a=1,\dots,2g+1$, the Kodaira-Spencer map $\operatorname{KS}_{a}(z):W\to W$ is defined by the formula:

[TABLE]

Proof.

This formula follows from (A.1), (A.2), and (A.3). ∎

Define the reduced Kodaira-Spencer map by the formula

[TABLE]

The reduced map has the same kernel and image as the map $\operatorname{KS}_{a}(z)$.

Corollary A.2**.**

Formula (A.6) for the reduced Kodaira-Spencer maps $\widetilde{\operatorname{KS}}_{a}:W\to W$, $a=1,\dots,2g+1$, can be identified with the formula (8.6) for the reduced $p$-curvature operators $\tilde{C}_{a}(z):V\to V$, $a=1,\dots,g$, if $\mu_{k},\nu_{k},z_{a}$ in formula (A.6) are identified with $v_{k},w_{k},z_{a}^{p}$ in Section 8.2.

Notice that the $p$-curvature operators are defined in characteristic $p$, while the Kodaira–Spencer maps are objects of deformation theory and do not require characteristic restrictions to be defined.

Remark**.**

The Katz $p$-curvature formula, [K2, Theorem 3.2] presents a $p$-curvature operator as a composition of three maps where the middle map is the Kodaira-Spencer map and the first and third maps are suitable Cartier operators, see also [VV1, Fromula (4.36)]. The identification in Corollary A.2 hides the first and third operators of that composition by choosing a good basis in Section 8.

The identification in Corollary A.2 enables us to extend the results from Sections 8 and 9 to describe the intersection of the kernels of Kodaira–Spencer maps acting on $\wedge^{r}W$. For instance, the results in Section 9 imply the following corollary.

Corollary A.3**.**

The intersection of kernels of the Kodaira-Spencer maps $\operatorname{KS}_{a}(z):\wedge^{3}W\to\wedge^{3}W$ is of dimension $\binom{g}{3}+2g-2$, and the intersection of kernels with the primitive part $\mathcal{P}^{3}\subset\wedge^{3}W$ is of dimension $\binom{g}{3}+g-2$.