Totally geodesic subvarieties in the moduli space of curves

TL;DR

This paper investigates the dimensions of totally geodesic subvarieties within the moduli space of abelian varieties, establishing bounds when these subvarieties are contained in the Torelli locus for genus g ≥ 4.

Contribution

It provides a new upper bound on the dimension of such subvarieties, advancing understanding of the geometry of the moduli space and Torelli locus.

Findings

Dimension bound:  Y ext{ is in Torelli locus, then } ext{dim } Y \u2264 (7g - 2)/3.

Applicable for genus g  4.

Enhances knowledge of the structure of totally geodesic subvarieties.

Abstract

In this paper we study totally geodesic subvarieties $Y \subset \mathsf{A}_g$ of the moduli space of principally polarized abelian varieties with respect to the Siegel metric, for $g\geq 4$. We prove that if $Y$ is generically contained in the Torelli locus, then $\dim Y \leq (7g -2)/3$.

Equations

$$X⟼X^{\ast }{|}_{{\mathcal{H}}_J^{1,0}}\in \mathrm{Hom}({\mathcal{H}}^{1,0},{\mathcal{H}}^{0,1})$$

$$T_J{S}_g\simeq W_J:=\{L\in \mathrm{Hom}({\mathcal{H}}^{1,0},{\mathcal{H}}^{0,1}):\mathcal{Q}(L\alpha ,\bar{\beta })+\mathcal{Q}(\alpha ,\overline{L\beta })=0\}.$$

$$0$$

$${\Omega }_{\mathcal{C},d}^1=\ker \{d:{\Omega }_{\mathcal{C}}^1\to {\Omega }_{\mathcal{C}}^2\}\subseteq {\Omega }_{\mathcal{C}}^1$$

$$0$$

$${\partial }_b:H^0(C_b,{\omega }_{C_b})⟶H^1(C_b,{\mathcal{O}}_{C_b})\otimes T_b^{\ast }B,{\partial }_b(\omega ):={\xi }_b\cup \omega ,$$

$$\mathcal{K}=\ker \partial .$$

$${\mathcal{K}}_b=\ker {\sigma }_b(v)=\ker {\xi }_b(v).$$

$$0$$

$$E_b:={\Omega }_{\mathcal{C}}^1{|}_{C_b}.$$

$$\Phi_{b}\colon\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 23.1822pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-23.1822pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\bigwedge^{2}H^{0}(E_{b})}}$}}}}}}}{\hbox{\kern 47.1822pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{H^{0}(\bigwedge^{2}E_{b})\simeq H^{0}(\omega_{C_{b}})}.\ignorespaces\ignorespaces\ignorespaces\ignorespaces\hbox{\kern 6.0pt\vrule height=3.0pt,depth=3.0pt,width=0.0pt}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 47.1822pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces$$

$$0$$

$$m_b(v_1,v_2)=[{\Phi }_b(\widetilde{v_1},\widetilde{v_2})]\in H^0({\omega }_{C_b})/⟨v_1,v_2⟩$$

$$0$$

$${\tau }_x:T_{x}B\otimes {\mathcal{K}}_x⟶{\mathcal{H}}_x/{\mathcal{K}}_x,{\tau }_x(v\otimes e):=\pi (({\nabla }_v\tilde{e})(x)),$$

$${\nabla }_{\dot{\gamma }}^{\ast }(\xi (s))=({\nabla }_{\dot{\gamma }}^{\mathrm{Hom}}\xi )(s)+\xi ({\nabla }_{\dot{\gamma }}^{hdg}s).$$

$${\nabla }_{\dot{\gamma }}^{\mathrm{Hom}}\dot{\gamma }={\nabla }_{\dot{\gamma }}^{Levi-Civita}\dot{\gamma }=0,{\nabla }_{\dot{\gamma }}^{hdg}\tilde{e}=0.$$

$${\nabla }_{\dot{\gamma }(t)}\tilde{e}={\nabla }_{\dot{\gamma }(t)}^{hdg}\tilde{e}\equiv 0.$$

$${\tau }_x(v,e)=\pi ({\nabla }_{\dot{\gamma }(t)}\tilde{e})(t_0)=0.$$

$$\dim (\ker \cup \xi )\ge g-k_0-1,$$

$$\displaystyle k_{0}:=\Bigl{\lceil}\frac{c-1}{2}\Bigr{\rceil}.$$

$$\xi \in \ker ({\rho }_D:H^1(C,T_C)⟶H^1(C,T_C(D))$$

$$\dim \ker (\cup \xi :H^0(C,{\omega }_C)\to H^1(C,{\mathcal{O}}_C))\ge g-\mathrm{deg}D=g-k_0-1.$$

$$c<\frac{2g-7}{3}.$$

$$2k_0\le g-4.$$

Full text

The authors were supported by PRIN 2015 Moduli spaces and Lie Theory, INdAM - GNSAGA, FAR 2016 (Pavia) “Varietà algebriche, calcolo algebrico, grafi orientati e topologici” and MIUR, Programma Dipartimenti di Eccellenza (2018-2022) - Dipartimento di Matematica “F. Casorati”, Università degli Studi di Pavia.

1. 1 Introduction
2. 2 Preliminaries on weight $1$ variations of the Hodge structure.
3. 3 A lemma on geodesics
4. 4 Proof of the Theorem

1. Introduction

Denote by $\mathsf{M}_{g}$ the moduli space of smooth complex projective curves of genus $g$, by $\mathsf{A}_{g}$ the moduli space of principally polarized abelian varieties and by $j:\mathsf{M}_{g}{\rightarrow}\mathsf{A}_{g}$ the period map. The Torelli locus $\mathsf{T}_{g}$ is the closure of $j(\mathsf{M}_{g})$ in $\mathsf{A}_{g}$. It is interesting to relate $\mathsf{T}_{g}$ to the geometry of $\mathsf{A}_{g}$ as a locally symmetric variety. We refer to [19, 4, 5, 10] for more information and motivation. In particular we are interested in totally geodesic subvarieties $Y$ of $\mathsf{A}_{g}$, i.e. algebraic subvarieties that are images of totally geodesic submanifolds of Siegel space $\operatorname{\mathfrak{S}}_{g}$. Shimura subvarieties are an important subclass of totally geodesic subvarieties, related to Hodge theory and arithmetics [20]. One expects that there are very few totally geodesic subvarieties of $\mathsf{A}_{g}$ that are generically contained in $\mathsf{T}_{g}$ i.e. such that $Y\subset\mathsf{T}_{g}$ and $Y\cap j(\mathsf{M}_{g})\neq\emptyset$. As for Shimura varieties, following Coleman and Oort, one expects that for large $g$ there are no such varieties generically contained in $\mathsf{T}_{g}$, see [19].

An important step in the study of the extrinsic geometry of $\mathsf{T}_{g}$ inside $\mathsf{A}_{g}$ was the computation of the second fundamental form of the period map (which is an embedding outside the hyperelliptic locus [21]). This was accomplished in [6] and refined in [5]. Unfortunately this leads only rarely to explicit formulae. But it is enough to get an upper bound for the dimension of a totally geodesic subvariety $Y$ generically contained in $\mathsf{T}_{g}$ in terms of the gonality of a point of $Y\cap j(\mathsf{M}_{g})$, see [5]. From this one gets a bound without gonality assumptions: $\dim Y\leq 5(g-1)/2$, as soon as $g\geq 4$ and $Y$ is not contained in the hyperellitptic locus.

In this paper we prove the following.

Theorem 1.1**.**

Let $Y$ be a totally geodesic subvariety of $\mathsf{A}_{g}$ that is generically contained in the Torelli locus. If $g\geq 4$, then $\dim Y\leq(7g-2)/3$.

Our proof contains two new ideas. First of all, we use directly the geodesic curves in $\operatorname{\mathfrak{S}}_{g}$. We are able to relate them to Hodge theory (see Lemma 3.3). The Hodge bundle of the (real one-dimensional) family of abelian varieties represented by the geodesic has nice properties with respect to the Fujita decomposition [9, 2]). Such properties do not hold for general families [12].

Secondly, our proof depends heavily on some recent results obtained in [22, 11]. In fact starting from the geodesics, we build a complex one-dimensional family of curves (contained in the $Y$) and we apply the above-mentioned results to this family. This allows to split the proof in two cases: in the first case one of the main theorems in [22, 11] yields a map of the whole family onto a fixed curve. From this one easily gets a better bound on $\dim Y$. In the second case one is able to control the Clifford index and the gonality of a point of $Y$. Then an application of the bound in [5] yields the result.

The plan of the paper is the following: in §2 we recall some definitions and some results from [22, 11]. In §3 we study the relation of geodesics to Hodge theory. In §4 we prove Theorem 1.1.

We refer to [15, 7, 14, 16, 17, 23] for related results obtained by different methods.

Acknowledgements. The authors would like to thank Paola Frediani for interesting discussions.

2. Preliminaries on weight $1$ variations of the Hodge

structure.

The sheaf of $k$-forms (resp. $(p,q)$-forms) on a complex manifold $M$ will be denoted by ${\mathcal{A}}^{k}_{M}$ (resp. ${\mathcal{A}}^{p,q}_{M}$). The sheaf holomorphic $k$-forms is denoted by $\Omega^{k}_{M}$.

2.1.

Given an exact sequence $0{\longrightarrow}E{\longrightarrow}F\stackrel{{\scriptstyle\pi}}{{{\longrightarrow}}}G{\longrightarrow}0$ of holomorphic vector bundles over a complex manifold $M$ and a (complex) connection $\nabla$ on $F$, the second fundamental form $\sigma\in{\mathcal{A}}^{1,0}_{M}(E^{*}\otimes G)$ is defined in the following way: given $u\in E_{x}$, extend $u$ to a local section $\tilde{u}$ of $E$ and set $\sigma(u):=\pi((\nabla\tilde{u})(x))$. When $\sigma\equiv 0$, we have $\nabla s\in{\mathcal{A}}^{1}_{M}(F)$ for any section $s\in\Gamma(F)$. This means that the connection $\nabla$ restricts to the bundle $F$ and defines a connection there.

If $\nabla$ is flat, $\sigma$ is in fact holomorphic, i.e. $\sigma\in H^{0}(B,\Omega^{1}_{B}\otimes E^{*}\otimes G)$.

2.2.

Let $({\mathbb{H}}_{\mathbb{Z}},{\mathcal{H}}^{1,0},{\mathcal{Q}})$ be a polarized variation of the Hodge structure (shortly, PVHS) of weight $1$ over a complex manifold $B$. Here ${\mathbb{H}}_{\mathbb{Z}}$ denotes the local system of lattices, ${\mathcal{H}}^{1,0}$ the Hodge bundle (that in weight $1$ determines the Hodge filtration) and ${\mathcal{Q}}$ the polarization. We also let ${\mathbb{H}}_{\mathbb{C}}={\mathbb{H}}_{\mathbb{Z}}\otimes_{\mathbb{Z}}{\mathbb{C}}$ denote the local system of complex vector spaces and ${\mathcal{H}}={\mathbb{H}}_{\mathbb{C}}\otimes_{\mathbb{C}}{\mathcal{O}}_{B}$ the associated holomorphic flat bundle with flat connection $\nabla$. This flat holomorphic connection is in fact defined by setting the kernel equal to ${\mathbb{H}}_{\mathbb{C}}$. The Hodge metric is defined by $h(v,w):=i{\mathcal{Q}}(v,\bar{w})$. It is positive definite on ${\mathcal{H}}^{1,0}$. Hence the orthogonal projection $p:{\mathcal{H}}\to{\mathcal{H}}^{1,0}$ is well-defined and $\nabla^{hdg}=p\nabla_{|{\mathcal{H}}^{1,0}}:{\mathcal{H}}\to{\mathcal{H}}^{1,0}\otimes{\mathcal{A}}^{1}_{B}$ is the Chern connection of the Hermitian bundle $({\mathcal{H}}^{1,0},h)$.

2.3.

We recall the definition of Siegel upper half-space. Let $\omega=\sum_{i}dx_{i}\wedge dy_{i}$ be the standard symplectic form on $V:={\mathbb{R}}^{2g}$. If $J\in\operatorname{End}V$, $J^{2}=-\operatorname{id}_{V}$ and $J^{*}\omega=\omega$, then $g_{J}(x,y):=\omega(x,Jy)$ is a nondegenerate symmetric bilinear form on ${\mathbb{R}}^{2g}$. The Siegel upper half-space is defined as $\operatorname{\mathfrak{S}}_{g}:=\{J\in\operatorname{End}V:J^{2}=-\operatorname{id}_{V},J^{*}\omega=\omega,g_{J}$ is positive definite$\}$. It is a symmetric space of the non-compact type.

Set $V_{\mathbb{C}}:=V\otimes{\mathbb{C}}$. For any $J\in\operatorname{\mathfrak{S}}_{g}$ the space $V$ can be endowed with the complex structure $J$. We denote by $V_{J}^{1,0}\subset V_{\mathbb{C}}$ the space of its vectors of type $(1,0)$. We also set ${\mathcal{H}}^{1,0}_{J}:=\operatorname{Ann}(V_{J}^{0,1})\subset V_{\mathbb{C}}^{*}$. Let $\Lambda\subset V_{\mathbb{C}}^{*}$ denote the set of forms that are integer-valued on ${\mathbb{Z}}^{2g}\subset V$. $\Lambda$ is a lattice in $V^{*}$. The symplectic form $\omega$ induces an isomorphism $\phi:V\cong V^{*}$ in the usual way. We denote by ${\mathcal{Q}}$ the symplectic form on $V^{*}$ obtained by transporting $\omega$ to $V^{*}$ via $\phi$. With these data we get a polarized variation of the Hodge structure on $\operatorname{\mathfrak{S}}_{g}$: the local system is ${\mathbb{H}}_{\mathbb{Z}}:=\operatorname{\mathfrak{S}}_{g}\times\Lambda$, Hodge bundle is ${\mathcal{H}}^{1,0}$ and the polarization is ${\mathcal{Q}}$.

2.4.

If $J\in\operatorname{\mathfrak{S}}_{g}$, then $T_{J}\operatorname{\mathfrak{S}}_{g}=\{X\in\operatorname{End}V:XJ+JX=0$, $\omega(Xx,y)+\omega(x,Xy)=0,\forall x,y\in V\}$. The Siegel upper half-space has an integrable complex structure that on $T_{J}\operatorname{\mathfrak{S}}_{g}$ acts by the rule $X\mapsto JX$. If $X\in T_{J}\operatorname{\mathfrak{S}}_{g}$, we can complexify $X$ and its complexification (still denoted by $X$) maps $V^{1,0}_{J}$ to $V_{J}^{0,1}$ and vice versa. The transpose of $X$, denoted by $X^{*}$ maps therefore ${\mathcal{H}}^{1,0}_{J}$ to ${\mathcal{H}}^{0,1}_{J}:=\overline{{\mathcal{H}}^{1,0}_{J}}$. The map

[TABLE]

yields an isomorphism

[TABLE]

Therefore we can identify the tangent bundle $T\operatorname{\mathfrak{S}}_{g}$ with the subbundle $W\subset\operatorname{Hom}({\mathcal{H}}^{1,0},{\mathcal{H}}^{0,1})$ defined by (2.1).

The Hodge bundle ${\mathcal{H}}^{1,0}$ is provided with the connection $\nabla^{hdg}$, as happens for every VHS. Since ${\mathcal{H}}^{0,1}\cong({\mathcal{H}}^{1,0})^{*}$, the connection $\nabla^{hdg}$ induces a connection on ${\mathcal{H}}^{0,1}$ that we denote by $\nabla^{*}$. So we get an induced connection $\nabla^{\operatorname{Hom}}$ on the bundle $\operatorname{Hom}({\mathcal{H}}^{1,0},{\mathcal{H}}^{0,1})$. The Levi-Civita connection of the symmetric metric coincides with the restriction of $\nabla^{\operatorname{Hom}}$ to $W\cong T\operatorname{\mathfrak{S}}_{g}$.

2.5.

Assume now that $B$ is a Riemann surface and that $({\mathbb{H}}_{\mathbb{Z}},{\mathcal{H}}^{1,0},{\mathcal{Q}})$ is a PVHS of weight 1 on $B$. Consider the exact sequence

[TABLE]

Consider on ${\mathcal{H}}$ the flat connection and let $\sigma:{\mathcal{H}}^{1,0}\to{\mathcal{H}}/{\mathcal{H}}^{1,0}\otimes\Omega_{B}^{1}$ be the corresponding second fundamental form, as defined in 2.1. Using $\nabla$ and $\sigma$ we define two vector subbundles of ${\mathcal{H}}^{1,0}$.

Definition 2.6**.**

Let ${\mathbb{U}}$ denote the subsheaf of ${\mathcal{H}}^{1,0}$ spanned by $\nabla^{hdg}$-flat sections (equivalently by $\nabla$-flat sections). Set

• $(i)$

${\mathcal{U}}:={\mathbb{U}}\otimes{\mathcal{O}}_{B},$**

• $(ii)$

${\mathcal{K}}:=\ker(\sigma:{\mathcal{H}}^{1,0}{\longrightarrow}{\mathcal{H}}/{\mathcal{H}}^{1,0}\otimes\Omega_{B}^{1})$.

We call ${\mathcal{U}}$ the unitary flat bundle and ${\mathcal{K}}$ the kernel bundle of the variation, respectively.

By definition ${\mathcal{U}}$ is a holomorphic flat bundle. Since $\nabla^{hdg}_{|{\mathcal{U}}}$ is the metric connection, it is unitary. Since $\nabla$ is flat, $\sigma$ is holomorphic, so ${\mathcal{K}}$ is a coherent subsheaf. If $\sigma$ is a morphism of constant rank, then ${\mathcal{K}}$ is also a vector subbundle of ${\mathcal{H}}^{1,0}$.

Proposition 2.7**.**

We have ${\mathcal{U}}\subset{\mathcal{K}}$.

Proof.

If $u\in{\mathcal{U}}_{x}$, there is a section $s\in\Gamma(A,{\mathbb{U}})$ defined on a neighbourhood $A$ of $x$ such that $s(x)=u$. Since $\nabla^{hdg}s=\nabla s\equiv 0$, $\sigma(u)=\pi^{0,1}((\nabla s)(x))=0$. ∎

2.8.

The VHS we are interested in come from families of curves. Let $f:{\mathcal{C}}\to B$ be a smooth family of genus $g$ curves. This is a proper and submersive morphism $f$ from a smooth complex surface ${\mathcal{C}}$ to a smooth complex curve $B$ whose fibres are curves of genus $g$. The map $f$ defines a PVHS $({\mathbb{H}}_{\mathbb{Z}},{\mathcal{H}}^{1,0},{\mathcal{Q}})$ (called geometric) by taking the local system ${\mathbb{H}}_{\mathbb{Z}}:=R^{1}f_{\ast}{\mathbb{Z}}$ and the Hodge bundle ${\mathcal{H}}^{1,0}:=f_{\ast}\Omega^{1}_{{\mathcal{C}}/B}=f_{\ast}\omega_{{\mathcal{C}}/B}$. The polarization ${\mathcal{Q}}$ is given by the intersection form. As usual we also have the flat bundle ${\mathcal{H}}=R^{1}f_{\ast}{\mathbb{C}}\otimes{\mathcal{O}}_{B}$ with the Gauss Manin connection $\nabla$, whose flat sections define the local system $R^{1}f_{\ast}{\mathbb{C}}$.

We consider the bundles ${\mathcal{U}}$ and ${\mathcal{K}}$ of this VHS. In this case both ${\mathcal{U}}$ and ${\mathcal{K}}$ can be described in terms of the submersion $f:{\mathcal{C}}\to B$ using holomorphic $1$-forms on ${\mathcal{C}}$. We outline shortly this description, referring to [22] and [11] for details. Let

[TABLE]

be the subsheaf of closed holomorphic $1$-forms on ${\mathcal{C}}$.

To describe ${\mathcal{K}}$ consider the exact sequence

[TABLE]

defined by duality using the morphism $df:T_{\mathcal{C}}\to f^{*}T_{B}$. Here the cokernel $\Omega^{1}_{{\mathcal{C}}/B}$ is the sheaf of relative differentials and $\Omega^{1}_{{\mathcal{C}}/B}\simeq\omega_{{\mathcal{C}}/B}$ since $f$ is smooth. Pushing forward (2.3) to $B$, we get the exact sequence

[TABLE]

By a fundamental result of Griffiths (see [13] and [24, Ch. 10]) $\partial$ is the vector bundle morphism that acts on the fibre over $b\in B$ as follows:

[TABLE]

where $\xi_{b}:T_{b}B{\rightarrow}H^{1}(C_{b},T_{C_{b}})$ is the Kodaira-Spencer map. Using the isomorphism ${\mathcal{H}}_{b}/{\mathcal{H}}^{1,0}_{b}\cong H^{0,1}(C_{b})\cong H^{1}(C_{b},{\mathcal{O}}_{C_{b}})$ one can identify $\partial_{b}$ with the second fundamental form $\sigma_{b}$ of (2.2). Consequently, we have that

[TABLE]

Since $\dim B=1$, if $v\in T_{b}B$ is a non-zero vector, then we have

[TABLE]

In particular $\sigma$ has constant rank and ${\mathcal{K}}$ is a vector subbundle of $f_{*}\omega_{{\mathcal{C}}/B}$. Moreover the sequence

[TABLE]

is exact.

We recall shortly the definition of Massey products. Fix $b\in B$ and a generator $v\in T_{b}B$. Using $v$ we get an isomorphism $T_{b}B\cong{\mathbb{C}}$. So the exact sequence (2.3) restricted to the fibre $C_{b}$ reads

[TABLE]

We set for simplicity

[TABLE]

The extension class of (2.6) is $\xi_{b}(v)\in H^{1}(C_{b},T_{C_{b}})$. It follows that $\det E_{b}\cong\omega_{C_{b}}$. So we get the adjoint map, first defined in [3]:

[TABLE]

The long cohomology exact sequence associated to (2.6) starts as follows:

[TABLE]

Given $v_{1},v_{2}$ in ${\mathcal{K}}_{b}=\ker\cup\xi_{b}$, we lift them, i.e. we take vectors $\tilde{v_{1}},\tilde{v_{2}}\in H^{0}(C_{b},E_{b})$ such that $p(\tilde{v}_{i})=v_{i}$. Let $\langle v_{1},v_{2}\rangle\subset H^{0}(C_{b},\omega_{C_{b}})$ denote the span of $v_{1},v_{2}$.

Definition 2.9**.**

The element

[TABLE]

is independent of the choice of the liftings and it is called Massey product of $v_{1},v_{2}\in\ker\cup\xi_{b}$.

We notice that $m_{b}(v_{1},v_{2})=[\Phi_{b}(\tilde{v_{1}},\tilde{v_{2}})]=0$ if and only if $\Phi_{b}(\tilde{v_{1}},\tilde{v_{2}})\in\langle v_{1},v_{2}\rangle$. Since we are assuming that $f$ is submersive, we have ${\mathcal{K}}_{b}=\ker\cup_{\xi_{b}}$ by (2.5), so the Massey product is defined for any $v_{1},v_{2}\in{\mathcal{K}}_{b}$ and for any $b\in B$.

We now consider the bundle ${\mathcal{U}}$ for the VHS coming from the family $f:{\mathcal{C}}{\rightarrow}B$. On ${\mathcal{C}}$ there is an exact sequence of sheaves

[TABLE]

We push it forward to $B$. Since $f$ is a submersion with compact connected fibres, $f_{\ast}{\mathbb{C}}\simeq{\mathbb{C}}_{B}$, $f_{\ast}{\mathcal{O}}_{{\mathcal{C}}}\simeq{\mathcal{O}}_{B}$ and $\omega_{B}=\operatorname{coker}(d:{\mathbb{C}}\to{\mathcal{O}}_{b})=\operatorname{coker}(f_{\ast}d:f_{\ast}{\mathbb{C}}\to f_{\ast}{\mathcal{O}}_{{\mathcal{C}}})$. So we get the exact sequence

[TABLE]

By the Splitting Lemma [22, Lemma 3.2] the local system ${\mathbb{U}}$ underlying ${\mathcal{U}}$ fits into the above exact sequence as

[TABLE]

Since ${\mathcal{U}}\subset{\mathcal{K}}$, we can restrict our attention to Massey products on ${\mathcal{U}}$.

Proposition 2.10**.**

Assume that the Massey products of ${\mathcal{U}}$ are all zero, i.e. $m(v_{1},m_{2})=0$ for any $v_{1},v_{2}\in{\mathcal{U}}_{b}$ and for any $b\in B$. Then, for any local flat frame $s_{1},\dots,s_{k}$ of ${\mathcal{U}}$, there are $\omega_{1},\dots,\omega_{m}\in H^{0}(B,f_{\ast}\Omega^{1}_{{\mathcal{C}},d})$ such that $\pi(\omega_{j})=s_{j}$ and $\omega_{i}\wedge\omega_{j}=0$ for any $i$ and $j$.

See [22, Prop. 4.3] for the proof. Massey products on ${\mathcal{U}}$ contain deep geometric information as is shown by the following tubular version of the classical Castelnuovo de-Franchis theorem.

Theorem 2.11** ([11]).**

Let $f:S\to\Delta$ be a submersive family of smooth projective curves over a disk $\Delta$. Let $\omega_{1},\ldots,\omega_{k}\in H^{0}\left(S,\Omega_{S}^{1}\right)$ ($k\geq 2$) be closed holomorphic 1-forms such that $\omega_{i}\wedge\omega_{j}=0$ for every $i,j$, and whose restrictions to a general fibre $F$ are linearly independent. Then (possibly after shrinking $\Delta$) there exist a projective curve $C$ and a morphism $\phi:S\to C$ such that $\omega_{i}\in\phi^{*}H^{0}\left(C,\omega_{C}\right)$ for every $i$.

3. A lemma on geodesics

3.1.

Let $B\subset\operatorname{\mathfrak{S}}_{g}$ be a complex submanifold with $\dim_{\mathbb{C}}B=1$ and consider the restriction to $B$ of the tautological PVHS on $\operatorname{\mathfrak{S}}_{g}$ introduced in 2.3. Consider the exact sequence

[TABLE]

Using on ${\mathcal{H}}$ the flat connection $\nabla$ we get a second fundamental form as described in 2.1:

[TABLE]

where $\tilde{e}$ is a local section of ${\mathcal{K}}$ such that $\tilde{e}(x)=e$. Moreover $\tau$ is a holomorphic section of $\Omega^{1}_{B}\otimes{\mathcal{K}}^{*}\otimes{\mathcal{H}}/{\mathcal{K}}$.

Lemma 3.2**.**

If $\tau\equiv 0$ on $B$, then ${\mathcal{K}}={\mathcal{U}}$.

Proof.

If $\tau\equiv 0$, the connection $\nabla$ preserves the subbundle ${\mathcal{K}}$. If $u\in{\mathcal{K}}_{x}$, let $s$ be a local section of ${\mathcal{H}}$ such that $s(x)=u$ and $\nabla s=0$. Then $s$ lies in ${\mathcal{K}}$ since ${\mathcal{K}}$ is preserved by the parallel displacement of $\nabla$. Thus $s$ in fact lies in ${\mathbb{U}}$ and $u\in{\mathcal{U}}_{x}$. Hence ${\mathcal{K}}\subset{\mathcal{U}}$. The opposite inclusion is always true. ∎

If $\gamma:{\mathbb{R}}{\rightarrow}\operatorname{\mathfrak{S}}_{g}$ is a non-constant geodesic and $-\infty<a<b<+\infty$, we call $\Gamma:=\gamma([a,b])$ a geodesic segment.

Lemma 3.3**.**

Let $B\subset\operatorname{\mathfrak{S}}_{g}$ be a complex submanifold with $\dim_{\mathbb{C}}B=1$. If $B$ contains a geodesic segment, then ${\mathcal{K}}={\mathcal{U}}$.

Proof.

By the previous lemma it is enough to show that $\tau\equiv 0$ on $B$. Since $\tau$ is holomorphic, if we prove that $\tau$ vanishes on $\Gamma$, then $\tau\equiv 0$ on all $B$ by the identity principle.

To show that $\tau=0$ on $\Gamma$ fix $x=\gamma(t_{0})\in\Gamma$, $v\in T_{x}B$ and $e\in{\mathcal{K}}_{x}$. Since $T_{x}B$ has complex dimension 1, $v=\lambda\dot{\gamma}(t_{0})$ for some $\lambda\in{\mathbb{C}}$, so it is enough to consider $v=\dot{\gamma}(t_{0})$. Let $\tilde{e}=\tilde{e}(t)$ be the section of ${\mathcal{K}}$ over $\Gamma$ obtained by parallel translation of the vector $e$ with respect to the connection $\nabla^{hdg}$. A priori $\tilde{e}$ is only a section of ${\mathcal{H}}^{1,0}$. We claim that in fact $\tilde{e}(t)\in{\mathcal{K}}_{\gamma(t)}$.

Set ${\mathcal{H}}^{0,1}:={\mathcal{H}}/{\mathcal{H}}^{1,0}$. Since ${\mathcal{H}}^{0,1}\cong({\mathcal{H}}^{1,0})^{*}$, the connection $\nabla^{hdg}$ induces a connection on ${\mathcal{H}}^{0,1}$ that we denote by $\nabla^{*}$. We get an induced connection on $\operatorname{Hom}({\mathcal{H}}^{1,0},{\mathcal{H}}^{0,1})$ that we denote by $\nabla^{\operatorname{Hom}}$. The tangent bundle $T\operatorname{\mathfrak{S}}_{g}$ is a subbundle of $\operatorname{Hom}({\mathcal{H}}^{1,0},{\mathcal{H}}^{0,1})$ and the Levi-Civita connection for the symmetric metric agrees with the restriction of the connection on $\operatorname{Hom}({\mathcal{H}}^{1,0},{\mathcal{H}}^{0,1})$. So if $\xi=\xi(t)$ is a section of $T\operatorname{\mathfrak{S}}_{g}$ and $s=s(t)$ is a section of ${\mathcal{H}}^{1,0}$ we have

[TABLE]

Take $\xi=\dot{\gamma}$ and $s=\tilde{e}$. We have

[TABLE]

So $\nabla^{*}_{\dot{\gamma}}(\dot{\gamma}(\tilde{e}))\equiv 0$. Recall now that $e=\tilde{e}(t_{0})\in{\mathcal{K}}_{x}$. Moreover ${\mathcal{K}}_{\gamma(t)}=\{u\in{\mathcal{H}}^{1,0}_{x}:\dot{\gamma}(t)(u)=0\}$. So $\dot{\gamma}(\tilde{e})(t_{0})=0$. Since we have checked that $\dot{\gamma}(\tilde{e})$ is a parallel section over $\Gamma$, we conclude that $\dot{\gamma}(\tilde{e})\equiv 0$ on $\Gamma$. This means that $\tilde{e}(t)\in{\mathcal{K}}_{\gamma(t)}$ for any $t$, as claimed.

To conclude the proof, notice that on ${\mathcal{K}}$ the connections $\nabla^{hdg}$ e $\nabla$ coincide. So

[TABLE]

Since $\tilde{e}$ is a section of ${\mathcal{K}}$ we can use it to compute the second fundamental form and we get

[TABLE]

∎

Lemma 3.4**.**

Let $f:I=[a,b]{\rightarrow}M$ be a real analytic map in a complex manifold $M$. Then there is an open subset $A\subset{\mathbb{C}}$ containing $I$ and a holomorphic extension $h:A{\rightarrow}M$ of $f$.

Proof.

We can easily find a finite family of disks $\{D_{i}\}_{i=1}^{m}$ centred at points $t_{i}\in I$ such that (a) $f(D_{i}\cap I)$ is contained in the domain of a chart $U_{i}$ $(U_{i},\phi_{i}=(z^{1}_{i},\ldots,z_{i}^{n}))$ of $M$, (b) on $D_{i}$ there are $n$ holomorphic functions $h_{i}^{j}$, $j=1,\ldots,n$, (c) $z^{j}_{i}\circ f=h_{i}^{j}$ on $D_{i}\cap I$. The function $f_{i}:=\phi_{i}{-1}\circ(h_{i}^{1},\ldots,h_{i}^{n})$ is a holomorphic extension of $f|_{D_{i}\cap I}$ to $D_{i}$. We claim that these functions glue together. Indeed we start by setting $h_{1}:=f_{1}$ on $D_{1}$ and we proceed inductively. Assume that a holomorphic extension $h_{k-1}$ is given on $D_{1}\cup\cdots\cup D_{k-1}$. We claim that $h_{k-1}=f_{k}$ on $D_{k}\cap(D_{1}\cup\cdots\cup D_{k-1})$. Indeed this set is connected and contains a subinterval of $I$. $h_{k-1}=f=f_{k}$ on this subinterval. Therefore by the identity principle $h_{k-1}=f_{k}$ on $D_{k}\cap(D_{1}\cup\cdots\cup D_{k-1})$. Thus using $h_{k-1}$ and $f_{k}$ we get a well-defined holomorphic extension $h_{k}$ on $D_{1}\cup\cdots\cup D_{k}$. At the end it is enough to set $h:=h_{n}$. ∎

4. Proof of the Theorem

Proposition 4.1**.**

Assume that $g\geq 3$ and let $Y\subset\mathsf{M}_{g}$ be a subvariety of of codimension $c$. For a smooth point $y\in Y$, there is $\xi\in T_{y}Y$ such that

[TABLE]

where

[TABLE]

Proof.

Restricting $Y$ we can assume that it embeds in the Kuranishi family. So $T_{y}Y\hookrightarrow H^{1}(C,T_{C})$ where $[C]=y$. Consider the bicanonical image $X:=\operatorname{Bic}(C):=\phi_{|2K_{C}|}(C)\subset{\mathbb{P}}(H^{1}(C,T_{C}))={\mathbb{P}}H^{0}(C,\omega_{C}^{\otimes 2})^{\vee}$. Denote by $S^{k}X$ the variety of $k$-secants of $X$. Since $\dim S^{k}X=2k+1$, we have $\dim S^{k_{0}}X\geq c=\operatorname{codim}({\mathbb{P}}(T_{y}Y)\subset{\mathbb{P}}(H^{1}(C,T_{C})))$. Hence $S^{k_{0}}\cap{\mathbb{P}}(T_{y}Y)$ contains at least some point $[\xi]$. By construction there is an effective divisor $D$ of degree $k_{0}+1$ such that

[TABLE]

where $\rho_{D}$ is the map induced by the inclusion $T_{C}\hookrightarrow T_{C}(D)$. Indeed, given an effective divisor $D$, denote by $\langle D\rangle$ the intersection of all hyperplanes $H\subset{\mathbb{P}}H^{0}(C,\omega^{\otimes 2})^{\vee}$ such that $D\leq\phi_{|2K_{C}|}^{*}H$. Set $X_{k_{0}+1}:=\{(D,p)\in C^{(k_{0}+1)}\times{\mathbb{P}}H^{0}(C,\omega^{\otimes 2})^{\vee}:p\in\langle D\rangle\}$ and denote by $p_{2}$ the second projection. Then $S^{k_{0}}X=p_{2}(X_{k_{0}+1})$. So $[\xi]\in\langle D\rangle$ for some $D\in C^{(k_{0}+1)}$, which yields (4.3). But

[TABLE]

(See e.g. [1, Lemma 2.3].) ∎

Proof of Theorem 1.1.

We argue by contradiction, assuming the existence of a totally geodesic subvariety $Y\subset\mathsf{A}_{g}$ that is generically contained in $\mathsf{M}_{g}$ and with $\dim Y>(7g-2)/3$. If $c$ denotes the codimension of $Y\cap\mathsf{M}_{g}$ in $\mathsf{M}_{g}$, this is equivalent to

[TABLE]

Observe that for $k_{0}$ defined in (4.2) this implies

[TABLE]

Observe also that by dimension $Y$ is not contained in the hyperelliptic locus. Fix a smooth point $y\in Y$ that represents a non-hyperelliptic curve. By Proposition 4.1 there is $\xi\in T_{y}Y$ such that $\xi\neq 0$ and

[TABLE]

where $k_{0}$ is defined as in (4.2). Let $\Delta$ be a polydisk and let $F:{\mathcal{X}}{\rightarrow}\Delta$ be a Kuranishi family with $[X_{0}]=y$. The moduli map $\pi:\Delta{\rightarrow}\mathsf{M}_{g}$, $\pi(t):=[X_{t}]$ is finite, satisfies $\pi(0)=y$ and its image is a neighbourhood of $y$. The period mapping $j$ can be lifted to a map $\tilde{j}:\Delta{\rightarrow}\operatorname{\mathfrak{S}}_{g}$.

[TABLE]

Thus $Y^{\prime}:=\tilde{j}(\pi^{-1}(Y))$ is a germ of totally geodesic submanifold of $\operatorname{\mathfrak{S}}_{g}$, that contains the point $y^{\prime}:=\tilde{j}(0)$ and is contained in $\tilde{j}(\Delta)$. Let $\gamma:{\mathbb{R}}{\rightarrow}\operatorname{\mathfrak{S}}_{g}$ be the geodesic in $\operatorname{\mathfrak{S}}_{g}$ such that $\gamma(0)=y^{\prime}$ and $\dot{\gamma}(0)=d\tilde{j}(\xi)$. Since $\xi\neq 0$ the curve $\gamma$ is non-constant. Moreover it is real analytic since the Siegel metric is real analytic. Fix $\varepsilon>0$ such that $\gamma([-\varepsilon,\varepsilon])\subset Y^{\prime}$. By Lemma 3.4 there is an open subset $A\subset{\mathbb{C}}$ containing $[-\varepsilon,\varepsilon]$ and a holomorphic extension $h:A{\rightarrow}Y^{\prime}$ of $\gamma$. Restricting $A$ we can assume that $h$ is an embedding and that it avoids the hyperelliptic locus. Set $B:=h(A)$. Since $B\subset Y^{\prime}\subset\tilde{j}(\Delta)$ we can restrict the Kuranishi family to $\tilde{j}^{-1}(B)\simeq B$ and we get a universal family of curves $f:{\mathcal{C}}{\rightarrow}B$ such that $b_{0}:=h(0)=\tilde{j}(y)$. The VHS of this family is simply the restriction to $B$ of the tautological VHS on $\operatorname{\mathfrak{S}}_{g}$ described in 2.3. We consider the bundles ${\mathcal{U}}$ and ${\mathcal{K}}$ for this VHS on $B$ (see Definition 2.6). By Lemma 3.3 we have ${\mathcal{U}}={\mathcal{K}}$. But using (2.4) we see that ${\mathcal{K}}_{b_{0}}=\ker(\cup\dot{\gamma}(0))=\ker(\cup\xi)$. At this point we use the bound (4.6). Summing up ${\mathcal{K}}={\mathcal{U}}$ has rank at least $g-k_{0}-1$.

Now we consider the Massey products of ${\mathcal{K}}={\mathcal{U}}$ on $B$ (see definition 2.9).

Assume first that these are identically 0. Take a basis $u_{1},\dots,u_{m}$ of ${\mathcal{K}}_{b}$. Up to shrinking $B$ we can extend these vectors to flat sections $u_{1},\dots,u_{m}$ of ${\mathcal{U}}$ on $B$. By Proposition 2.10, there exist unique liftings to sections $\omega_{1},\ldots,\omega_{m}$ of $f_{\ast}\Omega^{1}_{{\mathcal{C}},d}$ (i.e. of closed holomorphic $1$-forms on ${\mathcal{C}}$) such that $\omega_{i}\wedge\omega_{j}=0$, for any $i,j$. By Theorem 2.11 (i.e. [11, Theorem 1.5]), we get a a morphism $\phi:{\mathcal{C}}{\rightarrow}C^{\prime}$ onto a genus $g^{\prime}\geq 2$ smooth compact curve $C^{\prime}$, whose restriction to every fibre of $f$ gives a non-constant degree $n$ morphism $\phi:C_{b}{\rightarrow}C^{\prime}$ such that $\omega_{1},\ldots,\omega_{m}\in\phi^{\ast}H^{0}(\omega_{C^{\prime}})$. It follows immediately that $\operatorname{rk}{\mathcal{U}}\leq g^{\prime}$. But also $g^{\prime}\leq\operatorname{rk}{\mathcal{U}}$, since any section given by pull back from $C^{\prime}$ is flat and has wedge zero with the others. So we conclude that $\operatorname{rk}{\mathcal{U}}=g^{\prime}$. Recalling the bound on $\operatorname{rk}{\mathcal{K}}$ established above, we get

[TABLE]

Since $f$ is non isotrivial by construction, $n\geq 2$, so by the Riemann-Hurwitz formula

[TABLE]

But we were assuming (4.4) and hence (4.5). So we would get $g-3\leq 2k_{0}\leq g-4$, which is clearly absurd. This shows that the Massey products cannot vanish identically.

So there is $b\in B$ and $u_{1},u_{2}\in{\mathcal{K}}_{b}$, with $m(u_{1},u_{2})\neq 0$. In particular there are $\tilde{u}_{i}\in H^{0}(E_{b})$ with $\tilde{u}_{1}\wedge\tilde{u}_{2}\neq 0$. Before using this information, we need to recall a construction already used e.g. in [18, p. 428-429]. Consider the sequence (2.6) with extension class $\xi:=\xi_{b}(v)$. From the associated cohomology sequence

[TABLE]

we get $h^{0}(E_{b})=\dim\ker(\cup\xi)+1$. Recall that $\det E_{b}\cong\omega_{C_{b}}$ and the definition of the adjunction map (2.8). We claim that $\ker\Phi_{b}$ contains a non-zero decomposable element. Indeed we have $h^{0}(E_{b})=\dim\ker(\cup\xi)+1\geq g-k_{0}$, hence $2h^{0}(E_{b})-4\geq 2g-2k_{0}-4$. But recall that we are assuming (4.4). Hence from (4.5) we get

[TABLE]

Since $g\geq\dim\Lambda^{2}H^{0}(E_{b})-\dim\ker\phi=\dim{\mathbb{P}}(\Lambda^{2}H^{0}(E_{b}))-\dim{\mathbb{P}}(\ker\phi)$ and $\dim{\mathbb{G}}(2,\Lambda^{2}H^{0}(E_{b}))=2h^{0}(E_{b})-4$, it follows from (4.7) that ${\mathbb{G}}(2,\Lambda^{2}H^{0}(E_{b}))\cap{\mathbb{P}}(\ker\phi)\neq\oslash$, so there is a decomposable element $s_{1}\wedge s_{2}\in\ker\phi-\{0\}$, as claimed. In other words there are linearly independent sections $s_{1},s_{2}\in H^{0}(E)$ such that $s_{1}(x)$ and $s_{2}(x)$ are always proportional. Let $\mathfrak{F}$ be the subsheaf of ${\mathcal{O}}_{C_{b}}(E_{b})$ generated by $s_{1}$ and $s_{2}$. The saturation $\mathfrak{L}$ of $\mathfrak{F}$ is the sheaf of sections of a line bundle $L$. By construction $s_{1},s_{2}\in H^{0}(L)$ are linearly independent, so $h^{0}(L)\geq 2$. The quotient sheaf ${\mathcal{O}}_{C_{b}}(E_{b})/\mathfrak{L}$ is also the sheaf of sections of a line bundle $M$. Thus

[TABLE]

$s_{3}:=\beta\circ i$ is a section of $M$. We claim that $s_{3}\not\equiv 0$. Indeed if $s_{3}\equiv 0$, we would have $i(1)=\alpha(\sigma)$ for a nonvanishing section $\sigma\in H^{0}(L)$. But then $L$ would be trivial and $s_{1}$ and $s_{2}$ would be linearly dependent. Thus $s_{3}\not\equiv 0$. Let $D$ be the divisor of zeros of $s_{3}$. Then $M={\mathcal{O}}_{C_{b}}(D)$. Since $\det E_{b}=L\otimes M\cong\omega_{C_{b}}$, $L=\omega_{C_{b}}(-D)$.

Now we are finally able to use $\tilde{u_{1}}$ and $\tilde{u_{2}}$. Since $\tilde{u}_{1}\wedge\tilde{u}_{2}\neq 0$, the sections $\tilde{u}_{i}$ do not lie both in $L$. Hence at least one of them has a non-trivial image $s_{4}$ in $M$. The section $s_{3}$ and $s_{4}$ are independent, so $h^{0}(M)=h^{0}({\mathcal{O}}(D))\geq 2$. From the diagram one gets

[TABLE]

(see e.g. [1, Lemma 2.3]). Since $h^{0}(D)\geq 2$ and $h^{0}(\omega_{C}(-D))\geq 2$, the divisor $D$ contributes to the Clifford index. Therefore $\operatorname{Cliff}(C_{b})\leq\operatorname{Cliff}(D)$. It is known that $\operatorname{gon}(C_{b})\leq\operatorname{Cliff}(C_{b})+3$, see [8, Thm .2.3]. By the Lemma and 4.1 $\operatorname{Cliff}(D)\leq g-\dim\ker(\cup\xi)\leq k_{0}+1$. Thus

[TABLE]

Now we can apply [5, Theorem 4.2]: since $Y\subset\mathsf{M}_{g}$ is totally geodesic and $[C_{b}]\in Y$ is not hyperelliptic, we have

[TABLE]

For $g\geq 4$ we have $g-2\geq(2/3)g-1$. Hence

[TABLE]

But this gives $(2g-7)/3\leq c$, which yields the desired contradiction with (4.4). ∎