Stabilization of the Homotopy Groups of the Moduli Spaces of $k$-Higgs Bundles

TL;DR

This paper proves the stabilization of homotopy groups for moduli spaces of rank 2 and 3 k-Higgs bundles, extending previous results and providing new proofs that do not rely on stratification coincidences.

Contribution

It offers a new proof of homotopy group stabilization for moduli spaces of k-Higgs bundles and extends the results from rank two to rank three.

Findings

Homotopy groups of $\

$ ightarrow$ stabilization established for ranks 2 and 3.

New proof methods avoiding stratification coincidence assumptions.

Abstract

The work of Hausel proves that the Bia\l{}ynicki-Birula stratification of the moduli space of rank two Higgs bundles coincides with its Shatz stratification. He uses that to estimate some homotopy groups of the moduli space of $k$-Higgs bundles of rank two. Unfortunately, those two stratifications do not coincide in general. Here, the objective is to present a different proof of the stabilization of the homotopy groups of $\mathcal{M}^k(2,d)$, and generalize it to $\mathcal{M}^k(3,d)$, the moduli spaces of $k$-Higgs bundles of degree $d$, and ranks two and three respectively, using the results from the works of Hausel and Thaddeus, among other tools.

Equations

$${\mathcal{M}}^k(r,d)\hookrightarrow {\mathcal{M}}^{k+1}(r,d)$$

$${\pi }_j\left({\mathcal{M}}_{\Lambda }^k(r,d)\right)\stackrel{\cong }{\to }{\pi }_j\left({\mathcal{M}}_{\Lambda }^{\infty }(r,d)\right)$$

$$\mu (E):=\frac{\mathrm{deg}(E)}{\mathrm{rk}(E)}=\frac{d}{r}.$$

$$f(E,\Phi )=\frac{1}{2\pi }\Vert \Phi {\Vert }_{L^2}^2=\frac{i}{2\pi }{\int }_X\mathrm{tr}(\Phi {\Phi }^{\ast }),$$

$$E=\underset{j=1}{\overset{n}{⨁}}{E}_j\text{such that}\Phi :E_j\to E_{j+1}\otimes K\text{for}j⩽n-1\text{and}\Phi (E_n)=0.$$

$${\mathrm{deg}}_{\sigma }(T):=\mathrm{deg}(E_1)+\mathrm{deg}(E_2)+\sigma \cdot \mathrm{rk}(E_2),$$

$${\mu }_{\sigma }(T):=\frac{{\mathrm{deg}}_{\sigma }(T)}{\mathrm{rk}(E_1)+\mathrm{rk}(E_2)}$$

$${\mathcal{N}}_{\sigma }={\mathcal{N}}_{\sigma }(\mathbf{r},\mathbf{d})={\mathcal{N}}_{\sigma }(r_1,r_2,d_1,d_2)$$

$${\mathcal{N}}_{\sigma }^s={\mathcal{N}}_{\sigma }^s(\mathbf{r},\mathbf{d})$$

$$E\stackrel{{\Phi }^k}{\to }E\otimes K\otimes {\mathcal{O}}_X(kp)=E\otimes K(kp)$$

$$i_k:{\mathcal{M}}^k(r,d)\to {\mathcal{M}}^{k+1}(r,d)$$

$$\big{[}(E,\Phi^{k})\big{]}\longmapsto\big{[}(E,\Phi^{k}\otimes s_{p})\big{]}$$

$$\left\{\begin{array}[]{r c l}\mathbb{E}^{k}&\to&X\times\mathcal{M}^{k}(r,d)\\ \boldsymbol{\Phi}^{k}&\in&H^{0}\big{(}\operatorname{End}(\mathbb{E}^{k})\otimes\pi_{2}^{*}(K(kp))\big{)}\end{array}\right.$$

$$({\mathbb{E}}^k,{\mathbf{\Phi }}^k{)}_P\cong ({\mathbb{F}}^k,{\mathbf{\Psi }}^k{)}_P\forall P=(E,{\Phi }^k)\in {\mathcal{M}}^k(r,d),$$

$$({\mathbb{E}}^k,{\mathbf{\Phi }}^k)\cong ({\mathbb{F}}^k\otimes {\pi }_2^{\ast }(\mathcal{L}),{\mathbf{\Psi }}^k\otimes {\pi }_2),$$

$$({\mathrm{Id}}_X\times i_k{)}^{\ast }({\mathbb{E}}^{k+1})\cong {\mathbb{E}}^k.$$

$$\big{(}\mathbb{E}^{k},\boldsymbol{\Phi}^{k}\otimes\pi_{1}^{*}(s_{p})\big{)}\to X\times\mathcal{M}^{k}$$

$$\big{(}\mathbb{E}^{k},\boldsymbol{\Phi}^{k}\otimes\pi_{1}^{*}(s_{p})\big{)}=(\mathrm{Id}_{X}\times i_{k})^{*}\big{(}\mathbb{E}^{k+1},\boldsymbol{\Phi}^{k+1}\big{)}.$$

$$\mathrm{Vect}(X):=\Big{\{}V\to X\colon V\ \textmd{is a top. vector bundle}\Big{\}}\Big{/}\cong$$

$$[V]\oplus [W]:=[V\oplus W]$$

$$K^{0}(X)=K\big{(}\mathrm{Vect}(X)\big{)}:=\Big{\{}[V]-[W]\Big{\}}\Big{/}\sim$$

$$[V]-[W]\sim [V\oplus U]-[W\oplus U]$$

$$K^{\ast }(X)=K^0(X)\oplus K^1(X)$$

$$\{x_1,...,x_{2g},x_{2g+1},x_{2g+2}\}\subset K^{\ast }(X)=K^0(X)\oplus K^1(X),$$

$$[{\mathbb{E}}^k]=\sum _{j=0}^{2g}{x}_j\otimes e_j^k$$

$$\pi_{j}(i_{k}):\pi_{j}\big{(}\mathcal{M}^{k}(r,d)\big{)}\to\pi_{j}\big{(}\mathcal{M}^{k+1}(r,d)\big{)}$$

$$\sum _{j=0}^{2g}{x}_j\otimes e_j^k=[{\mathbb{E}}^k]=[({\mathrm{Id}}_X\times i_k{)}^{\ast }({\mathbb{E}}^{k+1})]=\sum _{j=0}^{2g}{x}_j\otimes i_k^{\ast }(e_j^{k+1})$$

$$BU(1)\to B\mathcal{G}\to B\bar{\mathcal{G}}$$

$$B\mathcal{G}\cong BU(1)\times B\bar{\mathcal{G}}.$$

$$B\bar{\mathcal{G}}\simeq {\mathcal{M}}^{\infty }=\underset{k\to \infty }{\lim }{\mathcal{M}}^k.$$

Full text

Stabilization of the Homotopy Groups of

the Moduli Spaces of $k$-Higgs Bundles

October 30th, 2017

Ronald A. Zúñiga-Rojas111Supported by Universidad de Costa Rica through CIMM (Centro de Investigaciones Matemáticas y Metamatemáticas), Project 820-B5-202. This work is based on the Ph.D. Project [31] called “Homotopy Groups of the Moduli Space of Higgs Bundles”, supported by FEDER through Programa Operacional Factores de Competitividade-COMPETE, and also supported by FCT (Fundação para a Ciência e a Tecnologia) through the projects PTDC/MAT-GEO/0675/2012 and PEst-C/MAT/UI0144/2013 with grant reference SFRH/BD/51174/2010.

Centro de Investigaciones Matemáticas y Metamatemáticas CIMM

Universidad de Costa Rica UCR

San José 11501, Costa Rica

e-mail: ronald.zunigarojas@ucr.ac.cr

Abstract. The work of Hausel proves that the Białynicki-Birula stratification of the moduli space of rank two Higgs bundles coincides with its Shatz stratification. He uses that to estimate some homotopy groups of the moduli spaces of $k$-Higgs bundles of rank two. Unfortunately, those two stratifications do not coincide in general. Here, the objective is to present a different proof of the stabilization of the homotopy groups of $\mathcal{M}^{k}(2,d)$, and generalize it to $\mathcal{M}^{k}(3,d)$, the moduli spaces of $k$-Higgs bundles of degree $d$, and ranks two and three respectively, over a compact Riemann surface $X$, using the results from the works of Hausel and Thaddeus, among other tools.

Keywords: Moduli of Higgs Bundles, Variations of Hodge Structures, Vector Bundles.

AMS 2010 MSC classes: Primary 55Q52; Secondaries 14H60, 14D07.

Introduction

In this work, we estimate some homotopy groups of the moduli spaces of $k$-Higgs bundles $\mathcal{M}^{k}(r,d)$ over a compact Riemann surface $X$ of genus $g>2$. This space was first introduced by Hitchin [20]; and then, it was worked by Hausel [16], where he estimated some of the homotopy groups working the particular case of rank two, and denoting $\displaystyle\mathcal{M}^{\infty}=\lim_{k\to\infty}\mathcal{M}^{k}$ as the direct limit of the sequence.

The co-prime condition $\operatorname{GCD}(r,d)=1$ implies that $\mathcal{M}^{k}(r,d)$ is smooth. We shall do the estimate with Higgs bundles of fixed determinant $\det(E)=\Lambda\in\mathcal{J}^{d}$, where $\mathcal{J}^{d}$ is the Jacobian of degree $d$ line bundles on $X$, to ensure that $\mathcal{N}(r,d)$ and $\mathcal{M}(r,d)$ are simply connected. Denote $\mathcal{M}^{k}_{\Lambda}$ as the moduli space of $k$-Higgs bundles with determinant $\Lambda$, and $\displaystyle\mathcal{M}^{\infty}_{\Lambda}=\lim_{k\to\infty}\mathcal{M}^{k}_{\Lambda}$ as the direct limit of these moduli spaces, as before. Hence, the group action \pi_{1}(\mathcal{M}^{k}_{\Lambda})\rotatebox[origin={c}]{180.0}{\circlearrowright}\pi_{n}(\mathcal{M}^{\infty}_{\Lambda},\mathcal{M}^{k}_{\Lambda}) will be trivial.

Hausel [16] estimates the homotopy groups $\pi_{n}(\mathcal{M}^{k}(2,1))$ using two main tools: first the coincidence mentioned before between the Białynicki-Birula stratification and the Shatz stratification; and second, the well-behaved embeddings $\mathcal{M}^{k}(2,1)\hookrightarrow\mathcal{M}^{k+1}(2,1)$. These inclusions are also well-behaved in general for $\operatorname{GCD}(r,d)=1$; nevertheless, those two stratifications above mentioned do not coincide in general (see for instance [11]).

In this paper, our estimate is based on the embeddings

[TABLE]

and their good behavior, notwithstanding the non-coincidence between stratifications when the rank is $r=3$. The paper is organized as follows: in section 1 we recall some facts about vector bundles and Higgs bundles; in section 2, we present the cohomology ring $H^{n}(\mathcal{M}^{k})$; in section 3, we discuss the most relevant results about the cohomology and the homotopy of the moduli spaces $\mathcal{M}^{k}$; finally, in section 4, subsection 4.1 we estimate the homotopy groups of $\mathcal{M}^{k}$ under the assumption that $\pi_{1}(\mathcal{M}^{k})$ acts trivially on $\pi_{n}(\mathcal{M}^{\infty},\mathcal{M}^{k})$, and hence, in subsection 4.2 we present and prove the main result:

Theorem**.**

(Corollary 4.14) Suppose the rank is either $r=2$ or $r=3$, and $\operatorname{GCD}(r,d)=1$. Then, for all $n$ exists $k_{0}$, depending on $n$, such that

[TABLE]

for all $k\geqslant k_{0}$ and for all $j\leqslant n-1$.

1 Preliminary definitions

Let $X$ be a compact Riemann surface of genus $g>2$ and let $K=T^{*}X$ be the canonical line bundle of $X$. Note that, algebraically, $X$ is also a nonsingular complex projective algebraic curve.

Definition 1.1**.**

A Higgs bundle over $X$ is a pair $(E,\Phi)$ where $E\to X$ is a holomorphic vector bundle and $\Phi\colon E\to E\otimes K$ is an endomorphism of $E$ twisted by $K$, which is called a Higgs field. Note that $\Phi\in H^{0}(X;\operatorname{End}(E)\otimes K)$.

Definition 1.2**.**

For a vector bundle $E\to X$, we denote the rank of $E$ by $\operatorname{rk}(E)=r$ and the degree of $E$ by $\deg(E)=d$. Then, for any smooth bundle $E\to X$ the slope is defined to be

[TABLE]

A vector bundle $E\to X$ is called semistable if $\mu(F)\leqslant\mu(E)$ for any $F$ such that $0\subsetneq F\subseteq E$. Similarly, a vector bundle $E\to X$ is called stable if $\mu(F)<\mu(E)$ for any nonzero proper subbundle $0\subsetneq F\subsetneq E$. Finally, $E$ is called polystable if it is the direct sum of stable subbundles, all of the same slope.

Definition 1.3**.**

A subbundle $F\subset E$ is said to be $\Phi$-invariant if $\Phi(F)\subset F\otimes K$. A Higgs bundle is said to be semistable [respectively, stable] if $\mu(F)\leqslant\mu(E)$ [resp., $\mu(F)<\mu(E)$] for any nonzero $\Phi$-invariant subbundle $F\subseteq E$ [resp., $F\subsetneq E$]. Finally, $(E,\Phi)$ is called polystable if it is the direct sum of stable $\Phi$-invariant subbundles, all of the same slope.

Fixing the rank $\operatorname{rk}(E)=r$ and the degree $\deg(E)=d$ of a Higgs bundle $(E,\Phi)$, the isomorphism classes of polystable bundles are parametrized by a quasi-projective variety: the moduli space $\mathcal{M}(r,d)$. Constructions of this space can be found in the work of Hitchin [20], using gauge theory, or in the work of Nitsure [27], using algebraic geometry methods.

An important feature of $\mathcal{M}(r,d)$ is that it carries an action of $\mathbb{C}^{*}$: $z\cdot(E,\Phi)=(E,z\cdot\Phi)$. According to Hitchin [20], $(\mathcal{M},I,\varOmega)$ is a Kähler manifold, where $I$ is its complex structure and $\varOmega$ its corresponding Kähler form. Furthermore, $\mathbb{C}^{*}$ acts on $\mathcal{M}$ biholomorphically with respect to the complex structure $I$ by the action mentioned above, where the Kähler form $\varOmega$ is invariant under the induced action $e^{i\theta}\cdot(E,\Phi)=(E,e^{i\theta}\cdot\Phi)$ of the circle $\mathbb{S}^{1}\subset\mathbb{C}^{*}$. Besides, this circle action is Hamiltonian, with proper momentum map $f\colon\mathcal{M}\to\mathbb{R}$ defined by:

[TABLE]

where $\Phi^{*}$ is the adjoint of $\Phi$ with respect to the hermitian metric on $E$ which provides the Hitchin-Kobayashi correspondence (see Hitchin [20]), and $f$ has finitely many critical values.

There is another important fact mentioned by Hitchin (see the original version in Frankel [8], and its application to Higgs bundles in Hitchin [20]): the critical points of $f$ are exactly the fixed points of the circle action on $\mathcal{M}$.

If $(E,\Phi)=(E,e^{i\theta}\Phi)$ then $\Phi=0$ with critical value $c_{0}=0$. The corresponding critical submanifold is $F_{0}=f^{-1}(c_{0})=f^{-1}(0)=\mathcal{N}$, the moduli space of semistable bundles. On the other hand, when $\Phi\neq 0$, there is a type of algebraic structure for Higgs bundles introduced by Simpson [29]: a variation of Hodge structure, or simply a VHS, for a Higgs bundle $(E,\Phi)$ is a decomposition:

[TABLE]

It has been proved by Simpson [30] that the fixed points of the circle action on $\mathcal{M}(r,d)$, and so, the critical points of $f$, are these VHS, where the critical values $c_{\lambda}=f(E,\Phi)$ will depend on the degrees $d_{j}$ of the components $E_{j}\subset E$. By Morse theory, we can stratify $\mathcal{M}$ in such a way that there is a nonzero critical submanifold $F_{\lambda}:=f^{-1}(c_{\lambda})$ for each nonzero critical value $0\neq c_{\lambda}=f(E,\Phi)$ where $(E,\Phi)$ represents a fixed point of the circle action, or equivalently, a VHS. We then say that $(E,\Phi)$ is an $(r_{1},\dots,r_{n})$-VHS, where $r_{j}=\operatorname{rk}(E_{j})\ \forall j$.

Definition 1.4**.**

A holomorphic triple on $X$ is a triple $T=(E_{1},E_{2},\phi)$ consisting of two holomorphic vector bundles $E_{1}\to X$ and $E_{2}\to X$ and a homomorphism $\phi\colon E_{2}\to E_{1}$, i.e., an element $\phi\in H^{0}(\operatorname{Hom}(E_{2},E_{1}))$.

There are certain notions of $\sigma$-degree:

[TABLE]

and $\sigma$-slope:

[TABLE]

which give rise to notions of $\sigma$-stability of triples. The reader may consult the works of Bradlow and García-Prada [5]; Bradlow, García-Prada and Gothen [6]; and Muñoz, Ortega and Vázquez-Gallo [26] for the details.

With this notions, one can construct:

[TABLE]

the moduli space of $\sigma$-polystable triples $T=(E_{1},E_{2},\phi)$ such that $\operatorname{rk}(E_{j})=r_{j}$ and $\deg(E_{j})=d_{j}$, and

[TABLE]

the moduli space of $\sigma$-stable triples, where $(\mathbf{r},\mathbf{d})=(r_{1},r_{2},d_{1},d_{2})$ is the type of the triple $T=(E_{1},E_{2},\phi)$.

We mention the moduli space $\mathcal{N}_{\sigma}(r_{1},r_{2},d_{1},d_{2})$ of $\sigma$-stable triples because they are closely related to some of the critical submanifolds $F_{\lambda}$.

Definition 1.5**.**

Fix a point $p\in X$, and let $\mathcal{O}_{X}(p)$ be the associated line bundle to the divisor $p\in\operatorname{Sym}^{1}(X)=X$. A $k$-Higgs bundle (or Higgs bundle with poles of order $k$) is a pair $(E,\Phi^{k})$ where:

[TABLE]

and where the morphism \Phi^{k}\in H^{0}\big{(}X,\operatorname{End}(E)\otimes K(kp)\big{)} is what we call a Higgs field with poles of order $k$. The moduli space of $k$-Higgs bundles of rank $r$ and degree $d$ is denoted by $\mathcal{M}^{k}(r,d)$. For simplicity, we will suppose that $\operatorname{GCD}(r,d)=1$, and so, $\mathcal{M}^{k}(r,d)$ will be smooth.

There is an embedding

[TABLE]

[TABLE]

where $0\neq s_{p}\in H^{0}(X,\mathcal{O}_{X}(p))$ is a nonzero fixed section of $\mathcal{O}_{X}(p)$.

All the results mentioned for $\mathcal{M}(r,d)$, hold also for $\mathcal{M}^{k}(r,d)$.

2 Generators for the Cohomology Ring

According to Hausel and Thaddeus [17, (4.4)], there is a universal family $(\mathbb{E}^{k},\boldsymbol{\Phi}^{k})$ over $X\times\mathcal{M}^{k}$ where

[TABLE]

and from now on, we will refer $(\mathbb{E}^{k},\boldsymbol{\Phi}^{k})$ as a universal $k$-Higgs bundle. Note that $(\mathbb{E}^{k},\boldsymbol{\Phi}^{k})$ satisfies the Universal Property: in general, for any family $(\mathbb{F}^{k},\boldsymbol{\Psi}^{k})$ over $X\times M$, there is a morphism $\eta\colon M\to\mathcal{M}^{k}$ such that $(\mathrm{Id}_{X}\times\eta)^{*}(\mathbb{E}^{k},\boldsymbol{\Phi}^{k})=(\mathbb{F}^{k},\boldsymbol{\Psi}^{k})$. It means that, for $M=\mathcal{M}^{k}$ whenever exists $(\mathbb{F}^{k},\boldsymbol{\Psi}^{k})$ such that

[TABLE]

then, there exists a unique bundle morphism $\xi\colon\mathbb{F}^{k}\to\mathbb{E}^{k}$ such that

[TABLE]

commutes: $p_{2}=p_{1}\circ\xi$.

The universal bundle extends then to the following: if $(\mathbb{E}^{k},\boldsymbol{\Phi}^{k})$ and $(\mathbb{F}^{k},\boldsymbol{\Psi}^{k})$ are families of stable $k$-Higgs bundles parametrized by $\mathcal{M}^{k}(r,d)$, such that $(\mathbb{E}^{k},\boldsymbol{\Phi}^{k})_{P}\cong(\mathbb{F}^{k},\boldsymbol{\Psi}^{k})_{P}$ for all $P=(E,\Phi^{k})\in\mathcal{M}^{k}(r,d)$, then there is a line bundle $\mathcal{L}\to\mathcal{M}^{k}(r,d)$ such that

[TABLE]

where $\pi_{2}\colon X\times\mathcal{M}^{k}(r,d)\to\mathcal{M}^{k}(r,d)$ is the natural projection and the endomorphisms satisfy $\boldsymbol{\Phi}^{k}\cong\boldsymbol{\Psi}^{k}\otimes\pi_{2}(\sigma_{P})\cong\boldsymbol{\Psi}^{k}$, where $\sigma_{P}$ is a section of $X\times\mathcal{M}^{k}\to\mathcal{M}^{k}$. For more details, see Hausel and Thaddeus [17, (4.2)].

Remark 2.1*.*

Do not confuse $\pi_{2}$ with $p_{2}$ (neither $\pi_{1}$ with $p_{1}$); $\pi_{j}$ are the natural projections of the cartesian product, while $p_{j}$ are the bundle surjective maps:

[TABLE]

If we consider the embedding $i_{k}\colon\mathcal{M}^{k}(r,d)\to\mathcal{M}^{k+1}(r,d)$ for general rank, we get that:

Proposition 2.2**.**

Let $(\mathbb{E}^{k},\boldsymbol{\Phi}^{k})$ be a universal Higgs bundle. Then:

[TABLE]

Proof.

Note that

[TABLE]

is a family of $(k+1)$-Higgs bundles on $X$, where $\pi_{1}\colon X\times\mathcal{M}^{k}\to X$ is the natural projection. So, by the universal property:

[TABLE]

∎

Consider

[TABLE]

the set of equivalence classes of topological vector bundles taken by isomorphism between them. Define the operation

[TABLE]

and consider the abelian semi-group \big{(}\mathrm{Vect}(X),\oplus\big{)}. Denote by

[TABLE]

the abelian $K$-group of topological vector bundles on $X$, where

[TABLE]

for every topological vector bundle $U\to X$.

Let $K^{1}(X)$ be the odd $K$-group of $X$ and let

[TABLE]

be the $K$-ring described by Atiyah [1, Chapter II].

In this case, $K^{*}(X)$ is torsion free since the Riemann surface $X$ is also a projective algebraic variety. Then, as a consequence of the Künneth Theorem (see for instance Atiyah [1, Corollary 2.7.15.] or [2, Main Theorem]), there is an isomorphism:

[TABLE]

The reader may see Markman [25] for the details. Furthermore, Markman [25] chooses bases $\{x_{1},\ ...,\ x_{2g}\}\subset K^{1}(X),$ and $\{x_{2g+1},\ x_{2g+2}\}\subset K^{0}(X)$ to get a total basis

[TABLE]

and, since there is a universal bundle $\mathbb{E}^{k}\to X\times\mathcal{M}^{k}$, we get the Künneth decomposition:

[TABLE]

where $x_{0}\in K^{0}(X)=\mathrm{span}\{x_{2g+1},\ x_{2g+2}\}$, $e_{0}^{k}\in K^{0}(\mathcal{M}^{k})$, $x_{j}\in K^{1}(X)$, and $e_{j}^{k}\in K^{1}(\mathcal{M}^{k})$ for $j=1,\ \dots,\ 2g$. Finally, Markman [25] considers the Chern classes $c_{j}(e_{i}^{k})\in H^{2j}(\mathcal{M}^{k},\mathbb{Z})$ for $e_{i}^{k}\in K^{*}(\mathcal{M}^{k})$ and proves that:

Theorem 2.3** (Markman [25, Th. 3]).**

The cohomology ring H^{*}\big{(}\mathcal{M}^{k}(r,d),\mathbb{Z}\big{)} is generated by the Chern classes of the Künneth factors of the universal vector bundle.  $\boxminus$

3 Preliminary Results

Let $i_{k}\colon\mathcal{M}^{k}\hookrightarrow\mathcal{M}^{k+1}$ be the embedding given by the tensorization map of the $k$-Higgs field $(E,\Phi^{k})\longmapsto(E,\Phi^{k}\otimes s_{p})$, where $s_{p}$ is a fixed nonzero section of $L_{p}$. We want to prove that the map

[TABLE]

stabilizes as $k\to\infty$. But first, we need to present some preliminary results to conclude that.

Proposition 3.1**.**

Let $i_{k}\colon\mathcal{M}^{k}\hookrightarrow\mathcal{M}^{k+1}$ be the embedding above mentioned. Consider the $K$-classes $e_{i}^{k}\in K(\mathcal{M}^{k})$. Then i_{k}^{*}\big{(}c_{j}(e_{i}^{k+1})\big{)}=c_{j}(e_{i}^{k}).

Proof.

By Proposition 2.2, and by the naturality of the Chern classes:

[TABLE]

we have that i_{k}^{*}\big{(}e_{i}^{k+1}\big{)}=e_{i}^{k} and hence i_{k}^{*}\big{(}c_{j}(e_{i}^{k+1})\big{)}=c_{j}(e_{i}^{k}). ∎

Corollary 3.2**.**

Let $i_{k}\colon\mathcal{M}^{k}\hookrightarrow\mathcal{M}^{k+1}$ be the embedding above mentioned. Then, the induced cohomology homomorphism $i_{k}^{*}\colon H^{*}(\mathcal{M}^{k+1},\mathbb{Z})\twoheadrightarrow H^{*}(\mathcal{M}^{k},\mathbb{Z})$ is surjective.

Proof.

The result is an immediate consequence of Theorem 2.3 and Proposition 3.1. ∎

Definition 3.3**.**

A gauge transformation is an automorphism of $E$. Locally, a gauge transformation $g\in Aut(E)$ is a $C^{\infty}(E)$-function with values in $GL_{r}(\mathbb{C})$. A gauge transformation $g$ is called unitary if $g$ preserves a hermitian inner product on $E$. We will denote $\mathcal{G}$ as the group of unitary gauge transformations. Atiyah and Bott [3] denote $\bar{\mathcal{G}}$ as the quotient of $\mathcal{G}$ by its constant central $U(1)$-subgroup. We will follow this notation too. Moreover, denote $B\mathcal{G}$ and $B\bar{\mathcal{G}}$ as the classifying spaces of $\mathcal{G}$ and $\bar{\mathcal{G}}$, respectively.

We get the fibration

[TABLE]

of classifying spaces, which splits actually as the product

[TABLE]

Then, the generators of $H^{*}(B\mathcal{G})$ give generators for $H^{*}(B\bar{\mathcal{G}})$ and so, $B\bar{\mathcal{G}}$ is a free graded commutative algebra on those generators, since $B\mathcal{G}$ is, and consequently, $B\bar{\mathcal{G}}$ is free of torsion. The reader may see Atiyah and Bott [3, Sec.  9.] and Hausel [16, Chap. 3] for the details.

Let $\displaystyle\mathcal{M}^{\infty}:=\lim_{k\to\infty}\mathcal{M}^{k}=\bigcup_{k=0}^{\infty}\mathcal{M}^{k}$ be the direct limit of the spaces \big{\{}\mathcal{M}^{k}(r,d)\big{\}}_{k=0}^{\infty}. Hausel and Thaddeus [17] prove that:

Theorem 3.4** (Hausel and Thaddeus [17, (9.7)]).**

The classifying space of $\bar{\mathcal{G}}$ is homotopically equivalent to the direct limit of the spaces $\mathcal{M}^{k}(r,d)$:

[TABLE]

•  $\boxminus$*

Assumption 3.5**.**

Unless otherwise stated, from now on, we will assume that the rank is either $r=2$ or $r=3$.

Theorem 3.6**.**

H^{*}\big{(}\mathcal{M}^{k}(r,d)\big{)}* is torsion free for all $k$.*

Proof.

The proof uses the following result of Frankel [8, Corollary 1]:

[TABLE]

In fact, the result of Frankel is more general. The specific case of moduli spaces of Higgs bundles holds because the proper momentum Hitchin map $f(E,\Phi)$ described in (1.2) is a perfect Morse-Bott function, since we are taking $\mathrm{GCD}(r,d)=1$.

In both cases, $r=2$ and $r=3$, the moduli space of stable vector bundles corresponds to the first critical submanifold: $F_{0}=f^{-1}(c_{0})=f^{-1}(0)=\mathcal{N}$, which is indeed torsion free (see Atiyah and Bott [3, Theorem  9.9.]).

When $\operatorname{rk}(E)=2$, Hitchin notes that the nontrivial critical submanifolds, or $(1,1)$-VHS, are of the form

[TABLE]

and $F_{d_{1}}^{k}$ is isomorphic to the moduli space of $\sigma_{H}$-stable triples $\mathcal{N}_{\sigma_{H}}(1,1,\bar{d},d_{1})$, where $\sigma_{H}$ is giving by \sigma_{H}=\deg\big{(}K(kp)\big{)}=2g-2+k and $\bar{d}=d_{2}+2g-2+k$, by the map:

[TABLE]

Furthermore, by Hitchin [20], $\mathcal{N}_{\sigma_{H}}(1,1,\bar{d},d_{1})$ is isomorphic to the cartesian product $\mathcal{J}^{d_{1}}(X)\times\operatorname{Sym}^{\bar{d}-d_{1}}(X)$. Hence:

[TABLE]

which, by Macdonald [23, (12.3)], is indeed torsion free. 2. 2.

When $\operatorname{rk}(E)=3$, there are three kinds of nontrivial critical submanifolds:

2.1. $(1,2)$-VHS of the form

[TABLE]

In this case, there are isomorphisms between the $(1,2)$-VHS and the moduli spaces of triples $F^{k}_{d_{1}}\cong\mathcal{N}_{\sigma_{H}(k)}(2,1,\tilde{d}_{1},\tilde{d}_{2})$, where $\tilde{d}_{1}=d_{2}+2(2g-2+k)$ and $\tilde{d}_{2}=d_{1}$, and where the isomorphism is giving by a map similar to the above mentioned.

By Muñoz, Ortega, Vázquez-Gallo [26, Theorem 4.8. and Lemma 6.1.], when working with $\mathcal{N}_{\sigma}(2,1,\tilde{d}_{1},\tilde{d}_{2})$, they find that either the flip loci $S_{\sigma_{c}}^{+}$ is the projectivization of a bundle of rank $r^{+}=\tilde{d}_{1}-d_{M}-\tilde{d}_{2}$ over

[TABLE]

where $\displaystyle d_{M}=\frac{\sigma_{c}+\tilde{d}_{1}+\tilde{d}_{2}}{3}\in\mathbb{Z}$, or the flip loci $S_{\sigma_{c}}^{-}$ is the projectivization of a bundle of rank $r^{-}=2d_{M}-\tilde{d}_{1}+g-1$ over

[TABLE]

with $\displaystyle d_{M}\in\mathbb{Z}$ as above. Hence, by Macdonald [23, (12.3)], the flip loci $S_{\sigma_{c}}^{+}$ and $S_{\sigma_{c}}^{-}$ are free of torsion for $\sigma_{c}\in I$. Therefore, $\mathcal{N}_{\sigma_{H}(k)}(2,1,\tilde{d}_{1},\tilde{d}_{2})$ is also torsion free, and so is $F^{k}_{d_{1}}$.

The fact that $\mathcal{N}_{\sigma_{H}(k)}(2,1,\tilde{d}_{1},\tilde{d}_{2})$ is torsion free since the flip loci are, follows from the next lemma:

Lemma 3.7**.**

Let $M$ be a complex manifold, and let $\Sigma\subset M$ be a complex submanifold. Let $\tilde{M}$ be the blow-up of $M$ along $\Sigma$. Then

[TABLE]

where $n$ is the rank of $N_{\Sigma/M}$, the normal bundle of $\Sigma$ in $M$.

Proof.

(Lemma 3.7)

Let $E=\mathbb{P}(N_{\Sigma/M})$ be the projectivized normal bundle of $\Sigma$ in $M$, sometimes called exceptional divisor. The result follows from the fact that the additive cohomology of the blow-up $H^{*}(\tilde{M},\mathbb{Z})$, can be expressed as:

[TABLE]

(see for instance Griffiths and Harris [12, Chapter 4.,Section 6.]), and the fact that $H^{*}(E)$ is a free module over $H^{*}(\Sigma)$ via the injective map $\pi^{*}\colon H^{*}(\Sigma)\to H^{*}(E)$ with basis

[TABLE]

where $c\in H^{2}(E)$ is the first Chern class of the tautological line bundle along the fibres of the projective bundle $E\to\Sigma$ (see the general version at Husemoller [21, Chapter 17.,Theorem 2.5.]). ∎

2.2. $(2,1)$-VHS of the form

[TABLE]

By symmetry, similar results can be obtained using the isomorphisms between the $(2,1)$-VHS and the moduli spaces of triples:

[TABLE]

and the dual isomorphisms

[TABLE]

between moduli spaces of triples.

2.3. $(1,1,1)$-VHS of the form

[TABLE]

Finally, we know that

[TABLE]

[TABLE]

where $m_{i}=d_{i+1}-d_{i}+\sigma_{H}$, and so, by Macdonald [23, (12.3)] there is nothing to worry about torsion.

∎

Corollary 3.8**.**

[TABLE]

•  $\boxminus$*

Corollary 3.9**.**

For each $n\geqslant 0$ there is a $k_{0}$, depending on $n$, such that

[TABLE]

is an isomorphism for all $k\geqslant k_{0}$ and for all $j\leqslant n$.  $\boxminus$

By the Universal Coefficient Theorem for Cohomology (see for instance Hatcher [14, Theorem 3.2. and Corollary 3.3.]), we get

Lemma 3.10**.**

For each $n\geqslant 0$ there is a $k_{0}$, depending on $n$, such that

[TABLE]

for all $k\geqslant k_{0}$ and for all $j\leqslant n$.

Proof.

The embedding $i_{k}\colon\mathcal{M}^{k}(r,d)\to\mathcal{M}^{k+1}(r,d)$ is injective, and by Corollary 3.2 we know that $i_{k}^{*}\colon H^{j}(\mathcal{M}^{k},\mathbb{Z})\leftarrow H^{j}(\mathcal{M}^{k+1},\mathbb{Z})$ is surjective for all $k$. Hence, by the Universal Coefficient Theorem, we get that the following diagram

[TABLE]

commutes. By Theorem 3.6 $H^{*}(\mathcal{M}^{k},\mathbb{Z})$ is torsion free, and so, by Corollary 3.9, for all $n\geqslant 0$, there is $k_{0}$, depending on $n$, such that

[TABLE]

for all $k\geqslant k_{0}$ and for all $j\leqslant n$. Hence

[TABLE]

for all $k\geqslant k_{0}$ and for all $j\leqslant n$. ∎

Proposition 3.11**.**

For general rank $r$, denoting $\mathcal{M}^{k}=\mathcal{M}^{k}(r,d)$ for simplicity, and $\mathcal{N}=\mathcal{N}(r,d)$ as the moduli of stable bundles, the following diagram commutes

[TABLE]

Proof.

It is an immediate consequence of the result proved by Bradlow, García-Prada and Gothen [7, Proposition 3.2.] using Morse theory. ∎

Proposition 3.12**.**

For all $k\in\mathbb{N}$, there is an isomorphism between the fundamental group of $\mathcal{M}^{k}$ and the fundamental group of the direct limit of the spaces \big{\{}\mathcal{M}^{k}(r,d)\big{\}}_{k=0}^{\infty}:

[TABLE]

Proof.

Using the generalization of Van Kampen’s Theorem presented by Fulton [9], and using the fact that $\mathcal{M}^{k}\hookrightarrow\mathcal{M}^{k+1}$ are embeddings of Deformation Neighborhood Retracts (DNR), i.e. every $\mathcal{M}^{k}(r,d)$ is the image of a map defined on some open neighborhood of itself and homotopic to the identity (see for instance Hausel and Thaddeus [17, (9.1)]), we can conclude that \displaystyle\pi_{1}\big{(}\lim_{k\to\infty}\mathcal{M}^{k}\big{)}=\lim_{k\to\infty}\pi_{1}\big{(}\mathcal{M}^{k}\big{)}. ∎

Remark 3.13*.*

By Atiyah and Bott [3] we have:

[TABLE]

and hence, by Proposition 3.11 and Proposition 3.12:

[TABLE]

We will need the following version of Hurewicz Theorem, presented by Hatcher [14, Theorem 4.37.] (see also James [22]). Hatcher first mentions that, in the relative case when $(X,A)$ is an $(n-1)$-connected pair of path-connected spaces, the kernel of the Hurewicz map

[TABLE]

contains the elements of the form $[\gamma][f]-[f]$ for $[\gamma]\in\pi_{1}(A)$. Hatcher defines $\pi_{n}^{\prime}(X,A)$ to be the quotient group of $\pi_{n}(X,A)$ obtained by factoring out the subgroup generated by the elements of the form $[\gamma][f]-[f]$, or the normal subgroup generated by such elements in the case $n=2$ when $\pi_{2}(X,A)$ may not be abelian, then $h$ induces a homomorphism $h^{\prime}\colon\pi^{\prime}_{n}(X,A)\to H_{n}(X,A;\mathbb{Z})$. The general form of Hurewicz Theorem presented by Hatcher deals with this homomorphism:

Theorem 3.14**.**

*(Hurewicz Theorem)

If $(X,A)$ is an $(n-1)$-connected pair of path-connected spaces, with $n\geqslant 2$ and $A\neq\emptyset$, then $h^{\prime}\colon\pi^{\prime}_{n}(X,A)\to H_{n}(X,A;\mathbb{Z})$ is an isomorphism and $H_{j}(X,A;\mathbb{Z})=0$ for $j\leqslant n-1$.  $\boxminus$*

Definition 3.15**.**

1. i.

The determinant of a vector bundle $E\to X$ of rank $r$ is a line bundle giving by the exterior power of the vector bundle. It gives a natural map of the form:

[TABLE]

where $\mathcal{N}=\mathcal{N}(r,d)$ is the moduli space of stable bundles $E\to X$ of rank $r$ and degree $d$, and $\mathcal{J}^{d}$ is the Jacobian of $X$. Fixing a line bundle $\Lambda\to X$, $\Lambda\in\mathcal{J}^{d}$, the fibre $\mathcal{N}_{\Lambda}=\mathcal{N}_{\Lambda}(r,d):=\mathrm{det}^{-1}(\Lambda)$ is the moduli space of stable bundles with fixed determinant. 2. ii.

Together with the trace, the determinant allows us to define the map

[TABLE]

and to consider the fibre $\mathcal{M}^{k}_{\Lambda}(r,d):=\zeta^{-1}(\Lambda,0)$ which is the moduli space of $k$-Higgs bundles with fixed determinant and trace zero.

There is an important result of Atiyah and Bott [3] that is relevant to mention here:

Theorem 3.16** (Atiyah and Bott [3, (9.12.)]).**

The moduli space $\mathcal{N}_{\Lambda}(r,d)$ of stable bundles of fixed determinant $\Lambda$, with $\operatorname{GCD}(r,d)=1$, is simply connected.  $\boxminus$

Remark 3.17*.*

1. i.

Some of the results mentioned for the moduli space $\mathcal{M}^{k}(r,d)$ in this section remain valid for the fixed determinant moduli space $\mathcal{M}_{\Lambda}^{k}(r,d)$. For instance, Theorem 3.8 holds true also for fixed determinant:

[TABLE]

(see Hausel and Thaddeus [17]). Nevertheless, Corollary 3.2 does not adapt in a straightforward way, as we shall see in subsection 4.2. 2. ii.

The moduli space $\mathcal{M}_{\Lambda}^{k}(r,d)$ is simply connected because Proposition 3.11 holds also for fixed determinant $k$-Higgs bundles. So, $\pi_{1}(\mathcal{M}_{\Lambda}^{k})$ acts trivially on $\pi_{n}(\mathcal{M}_{\Lambda}^{\infty},\mathcal{M}_{\Lambda}^{k})$.

4 Main Results

4.1 General Results

Here, we will concern the moduli spaces $\mathcal{M}^{k}(r,d)$ of $k$-Higgs bundles, where the results are true under the condition that $\pi_{1}(\mathcal{M}^{k})$ acts trivially on all the higher relative homotopy groups of the pair \big{(}\mathcal{M}^{\infty},\mathcal{M}^{k}\big{)}. However, we do not know if this condition is true or not.

Lemma 4.1**.**

If $\pi_{1}(\mathcal{M}^{k})$ acts trivially on $\pi_{n}(\mathcal{M}^{\infty},\mathcal{M}^{k})$, then for all $n\geqslant 0$ exists $k_{0}$, depending on $n$, such that $\pi_{j}(\mathcal{M}^{\infty},\mathcal{M}^{k})=0$ for all $k\geqslant k_{0}$ and for all $j\leqslant n$.

Proof.

The proof proceeds by induction on $m\in\mathbb{N}$ for $2\leqslant m\leqslant n$. The first induction step is trivial because

[TABLE]

by Proposition 3.11. For $m=2$ we need $\pi_{2}(\mathcal{M}^{\infty},\mathcal{M}^{k})$ to be abelian. Consider the sequence

[TABLE]

where $\pi_{2}(\mathcal{M}^{\infty})\twoheadrightarrow\pi_{2}(\mathcal{M}^{\infty},\mathcal{M}^{k})$ is surjective, $\pi_{1}(\mathcal{M}^{k})\xrightarrow{\ \cong\ }\pi_{1}(\mathcal{M}^{\infty})$ are isomorphic, and hence $\pi_{1}(\mathcal{M}^{\infty},\mathcal{M}^{k})=0$. So, $\pi_{2}(\mathcal{M}^{\infty},\mathcal{M}^{k})$ is a quotient of the abelian group $\pi_{2}(\mathcal{M}^{\infty})$, and so it is also abelian.

Finally, suppose that the statement is true for all $j\leqslant m-1$ for $2\leqslant m\leqslant n$. So, $(\mathcal{M}^{\infty},\mathcal{M}^{k})$ is $(m-1)$-connected, i.e.

[TABLE]

For $m\geqslant 2$, by Hurewicz Theorem 3.14,

[TABLE]

is an isomorphism. By Lemma 3.10, there is an integer $k_{0}$, depending on $m$, such that $H_{m}(\mathcal{M}^{\infty},\mathcal{M}^{k};\mathbb{Z})=0$ for all $k\geqslant k_{0}$. Hence, if $\pi_{1}(\mathcal{M}^{k})$ acts trivially on $\pi_{n}(\mathcal{M}^{\infty},\mathcal{M}^{k})$ for all $n\in\mathbb{N}$ and for all $k\in\mathbb{N}$, then

[TABLE]

finishing the induction process. ∎

Corollary 4.2**.**

If $\pi_{1}(\mathcal{M}^{k})$ acts trivially on $\pi_{n}(\mathcal{M}^{\infty},\mathcal{M}^{k})$, then for all $n\geqslant 0$ exists $k_{0}$,depending on $n$, such that

[TABLE]

for all $k\geqslant k_{0}$ and for all $j\leqslant n-1$.  $\boxminus$

4.2 Fixed determinant case

The main goal here, is to avoid the hypothesis of the trivial action of the fundamental group on the relative homotopy group: \pi_{1}(\mathcal{M}^{k})\rotatebox[origin={c}]{180.0}{\circlearrowright}\pi_{n}(\mathcal{M}^{\infty},\mathcal{M}^{k}). So, we want to get the analogue of Lemma 4.1 for $\mathcal{M}_{\Lambda}^{k}$, the moduli space of $k$-Higgs bundles with fixed determinant, since $\mathcal{M}_{\Lambda}^{k}$ is simply connected. To do that, we will need the analogue of Corollary 3.2, and then the analogue of Lemma 3.10 also for $\mathcal{M}_{\Lambda}^{k}$.

The analogue of Corollary 3.2 for $\mathcal{M}_{\Lambda}^{k}$ is not immediate. Note that the group of $r$-torsion points in the Jacobian:

[TABLE]

acts on $\mathcal{M}_{\Lambda}^{k}(r,d)$ by tensorization:

[TABLE]

Hence, $\Gamma$ acts on $H^{*}(\mathcal{M}_{\Lambda}^{k},\mathbb{Z})$ for all $k$. This cohomology splits in a $\Gamma$-invariant part and in a complement which is called by Hausel and Thaddeus [19] as the “variant part”:

[TABLE]

This decomposition appears in various cohomology calculations, see e.g., Hitchin [20] for rank two, Gothen [10] for rank three, Hausel [16] also for rank two, Bento [4] for the explicit calculations for rank two and rank three, and Hausel and Thaddeus [19] for general rank.

The analogue of Corollary 3.2 for $\mathcal{M}_{\Lambda}^{k}$ will be obtained for each of the pieces in the last direct sum (4.1) separately:

• •

For $H^{*}(\mathcal{M}_{\Lambda}^{k},\mathbb{Z})^{\Gamma}$:

It follows from the corresponding result for $H^{*}(\mathcal{M}^{k},\mathbb{Z})$ because there is a surjection $H^{*}(\mathcal{M}^{k},\mathbb{Z})\twoheadrightarrow H^{*}(\mathcal{M}_{\Lambda}^{k},\mathbb{Z})^{\Gamma}$.

Recall that, for general rank $r$, the moduli space of stable vector bundles corresponds to the first critical submanifold: $F_{0}=f^{-1}(c_{0})=f^{-1}(0)=\mathcal{N}(r,d)$. The group $\Gamma$ acts trivially on $H^{*}(\mathcal{N},\mathbb{Z})$, and there is a surjection

[TABLE]

The reader may see Atiyah and Bott [3, Prop.  9.7.] for the details.

For the rank $r=2$ case, a nontrivial critical submanifold of $\mathcal{M}_{\Lambda}^{k}(2,1)$, is a so-called $(1,1)$-VHS:

[TABLE]

which is a $2^{2g}$-covering with covering group the $2$-torsion points in the Jacobian $\Gamma\cong(\mathbb{Z}_{2})^{2g}$. Hence, the results of Betti numbers presented by Bento [4, Prop. 2.2.3.] let us conclude the following:

Proposition 4.3**.**

The cohomology map

[TABLE]

induced by the $\Gamma$-covering $F_{d_{1}}^{k}(\Lambda)\to\operatorname{Sym}^{m}(X)$ where $m=d_{2}-d_{1}+2g-2+k$, is injective, and its image is the $\Gamma$-invariant subgroup H^{*}\big{(}F_{d_{1}}^{k}(\Lambda),\mathbb{Z}\big{)}^{\Gamma}.  $\boxminus$

Corollary 4.4**.**

There exists a surjection

[TABLE]

•  $\boxminus$*

When $r=3$, the group of $3$-torsion points in the Jacobian looks like $\Gamma\cong(\mathbb{Z}_{3})^{2g}$, and the nontrivial critical submanifolds of $\mathcal{M}^{k}_{\Lambda}(3,d)$ are VHS either of type $(1,2),\ (2,1)$ or $(1,1,1)$, where the cohomology of the $(1,2)$ and $(2,1)$ VHS is invariant under the action of $\Gamma$, and the $(1,1,1)$-VHS is a $3^{2g}$-covering of $\operatorname{Sym}^{m_{1}}(X)\times\operatorname{Sym}^{m_{2}}(X)$ with covering group $\Gamma\cong(\mathbb{Z}_{3})^{2g}$. Hence:

Proposition 4.5**.**

[TABLE]

where

[TABLE]

and

[TABLE]

are the $(1,2)$ and $(2,1)$-VHS of $\mathcal{M}^{k}_{\Lambda}(3,d)$ respectively, with

[TABLE]

Furthermore:

[TABLE]

and the cohomology map

[TABLE]

induced by the $\Gamma$-covering $F_{m_{1}m_{2}}^{k}(\Lambda)\to\operatorname{Sym}^{m_{1}}(X)\times\operatorname{Sym}^{m_{2}}(X)$ where $F^{k}_{m_{1}m_{2}}(\Lambda)=$

[TABLE]

is the $(1,1,1)$-VHS of $\mathcal{M}^{k}_{\Lambda}(3,d)$ with $m_{j}=d_{j+1}-d_{j}+2g-2+k$, is injective, and its image is the $\Gamma$-invariant subgroup H^{*}\big{(}F_{m_{1}m_{2}}^{k}(\Lambda)\big{)}^{\Gamma}.  $\boxminus$

Corollary 4.6**.**

There exists a surjection

[TABLE]

•  $\boxminus$*

The reader may see Bento [4], Gothen [10] and also Hausel and Thaddeus [19], for details. Using the results above, we get:

Lemma 4.7**.**

The induced cohomology homomorphism restricted to the $\Gamma$-invariant cohomology of the moduli spaces of $k$-Higgs bundles with fixed determinant $\Lambda$

[TABLE]

is surjective.

Proof.

It is enough to note that the following diagram

[TABLE]

commutes, where the top arrow is surjective by Corollary 3.2, and the descending arrows are surjective because of Corollary 4.4 and Corollary 4.6. ∎

• •

For $H^{*}(\mathcal{M}_{\Lambda},\mathbb{Z})^{var}$:

First, note that with fixed determinant $\Lambda$ the critical submanifolds of type $(1,1)$ and $(1,1,1)$ are $r^{2g}$-coverings with covering group $\Gamma\cong(\mathbb{Z}_{r})^{2g}$, with $r=2$ or $r=3$ (see Bento [4] Prop. 2.2.1. and Lemma  2.4.4.). Furthermore, when $r=3$ the cohomology of $(1,2)$ and $(2,1)$ critical submanifolds is $\Gamma$-invariant. Then, only the cohomology of $(1,1)$-VHS and $(1,1,1)$-VHS split in the $\Gamma$-invariant part and the variant complement, for rank $r=2$ and $r=3$, respectively. Hence:

[TABLE]

[TABLE]

where d_{k}=\deg\big{(}K\otimes\mathcal{O}_{X}(kp)\big{)}=\deg\big{(}K(kp)\big{)}=2g-2+k, $\frac{1}{2}<d_{1}<\frac{1+d_{k}}{2}$ according to Hitchin [20] for $(1,1)$-VHS in rank two, and $(m_{1},m_{2})\in\Omega_{d_{k}}$ where $M_{j}:=E_{j}^{*}E_{j+1}K(kp)$, $m_{j}:=\mathrm{deg}(M_{j})=d_{j+1}-d_{j}+d_{k}$, and the set of indexes

[TABLE]

for $(1,1,1)$-VHS in rank three is described by Bento [4, Prop. 2.3.9.], Gothen [10, Sec. 3.], Gothen and Zúñiga-Rojas [11, Subsec. 5.1], among others.

There are some results appearing in the work of Bento [4] (Lemma 2.2.4. and Prop. 2.2.5 for $\mathcal{M}^{k}_{\Lambda}(2,1)$ and $F_{d_{1}}^{k}(\Lambda)$ its $(1,1)$-VHS, and Lemma 2.4.4. and Prop. 2.4.5. for $\mathcal{M}^{k}_{\Lambda}(3,d)$ and $F_{m_{1}m_{2}}^{k}(\Lambda)$ its $(1,1,1)$-VHS) where Bento works with Hitchin pairs twisted by a general line bundle $L$ of degree $\deg(L)=d_{L}$, and the following results below correspond to the particular case of $k$-Higgs bundles with $L=K(kp)$, and hence $d_{L}=d_{k}=2g-2+k$:

Lemma 4.8**.**

Let $F_{d_{1}}^{k}(\Lambda)$ be a $(1,1)$-VHS of $\mathcal{M}^{k}_{\Lambda}(2,1)$ and let $m=d_{2}-d_{1}+2g-2+k$. Then

[TABLE]

Proof.

See Bento [4, Prop.  2.2.4.]. ∎

Lemma 4.9**.**

Let $F_{m_{1}m_{2}}^{k}(\Lambda)$ be a $(1,1,1)$-VHS of $\mathcal{M}^{k}_{\Lambda}(3,d)$. Then

[TABLE]

where $m_{j}=d_{j+1}-d_{j}+d_{k}$.

Proof.

See Bento [4, Prop.  2.4.4.]. ∎

Then, in both cases, when $r=2$ and when $r=3$, the cohomology groups with integer coefficients are torsion free:

• –

If $r=2$, we have just one nonzero component, $H^{m}(F_{d_{1}}^{k}(\Lambda),\mathbb{Z})^{var}$ which is the sum of $2^{2g}$ copies of $H^{m}(\operatorname{Sym}^{m}(X),\mathbb{Z})$, since $F_{d_{1}}^{k}(\Lambda)\to\operatorname{Sym}^{m}(X)$ is a $(\mathbb{Z}_{2})^{2g}$-covering.

• –

Similarly, if $r=3$, the nonzero component is H^{m_{1}+m_{2}}\big{(}F_{m_{1}m_{2}}^{k}(\Lambda),\mathbb{Z}\big{)}^{var} which is the sum of $3^{2g}$ copies of H^{m_{1}+m_{2}}\big{(}\operatorname{Sym}^{m_{1}}(X)\times\operatorname{Sym}^{m_{2}}(X),\mathbb{Z}\big{)}, since

[TABLE]

is a $(\mathbb{Z}_{3})^{2g}$-covering.

They are torsion free by Macdonald [23, (12.3)]. The reader may consult Bento [4, Chap. 2] for the details. Hence, we get

Lemma 4.10**.**

Let $i_{k}\colon\mathcal{M}^{k}_{\Lambda}\hookrightarrow\mathcal{M}^{k+1}_{\Lambda}$ be the embedding given by the tensorization map $(E,\Phi^{k})\mapsto(E,\Phi^{k}\otimes s_{p})$ as above mentioned. Then, the induced cohomology homomorphism

[TABLE]

is surjective, restricted to the variant complement.  $\boxminus$

This latter method only works wtih rank $r=2$ or $r=3$, but not in general. The difficulty in calculating $H^{*}(\mathcal{M}_{\Lambda},\mathbb{Z})^{var}$ for general rank is explained also on Hausel and Thaddeus [19].

Finally, we may conclude the following:

Corollary 4.11**.**

Let $i_{k}\colon\mathcal{M}^{k}_{\Lambda}\hookrightarrow\mathcal{M}^{k+1}_{\Lambda}$ be the embedding above mentioned. Then, the induced cohomology homomorphism

[TABLE]

is surjective.

Proof.

It is enough to see that the cohomology of $\mathcal{M}^{k}_{\Lambda}$ splits in the $\Gamma$-invariant part and the variant complement:

[TABLE]

and so, the result follows from Lemma 4.7 and Lemma 4.10. ∎

Lemma 4.12**.**

For all $n$ exists $k_{0}$, depending on $n$, such that

[TABLE]

for all $k\geqslant k_{0}$ and for all $j\leqslant n$.  $\boxminus$

Theorem 4.13**.**

For all $n$ exists $k_{0}$, depending on $n$, such that

[TABLE]

for all $k\geqslant k_{0}$ and for all $j\leqslant n$.

Proof.

The proof is quite similar to the proof of Lemma 4.1, using now Corollary 4.11 and Lemma 4.12, and so, we have a new advantage: $\mathcal{M}^{k}_{\Lambda}$ is simply connected, hence the action \pi_{1}(\mathcal{M}_{\Lambda}^{k})\rotatebox[origin={c}]{180.0}{\circlearrowright}\pi_{n}(\mathcal{M}_{\Lambda}^{\infty},\mathcal{M}_{\Lambda}^{k}) is trivial. ∎

Corollary 4.14**.**

For all $n$ exists $k_{0}$, depending on $n$, such that

[TABLE]

for all $k\geqslant k_{0}$ and for all $j\leqslant n-1$.  $\boxminus$

Acknowledgement

Part of this paper is partially based on my Ph.D. thesis [31] and I would like to thank my supervisor Peter B. Gothen for introducing me to the subject of Higgs bundles, and for all his patience during our illuminating discussions. Mange tak!

I would like to thank André Gama Oliveira for the advice of working with fixed determinant, and so, considering the trivial action \pi_{1}(\mathcal{M}_{\Lambda}^{k})\rotatebox[origin={c}]{180.0}{\circlearrowright}\pi_{n}(\mathcal{M}_{\Lambda}^{\infty},\mathcal{M}_{\Lambda}^{k}). Muito obrigado!

I am grateful to the referee for a very careful reading of my manuscript, specially for pointing out the necessary conditions on the indexes of the critical submanifolds.

I also would like to thank Joseph C. Várilly for his time, listening and reading my results. Thanks for every single advice. Go raibh maith agat!

I benefited from the VBAC-Conference 2014, held in Freie Universität Berlin, and the VBAC-Conference 2016, held in Centre Interfacultaire Bernoulli, both organized by the Vector Bundles and Algebraic Curves research group of the European Research Training Network EAGER.

Finally, I acknowledge the financial support from CIMM, Centro de Investigaciones Matemáticas y Metamatemáticas, here in Costa Rica nowadays as a researcher, through the Project 820-B5-202; and also the financial support from FEDER through Programa Operacional Factores de Competitividade-COMPETE, and FCT, Fundação para a Ciência e a Tecnologia, there in Portugal, through PTDC/MAT-GEO/0675/2012 and PEst-C/MAT/UI0144/2013 with grant reference SFRH/BD/51174/2010, when I was working on my Ph.D. thesis. ¡Pura Vida! Muito obrigado!

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Ronald A. Zúñiga-Rojas

Centro de Investigaciones Matemáticas

y Metamatemáticas CIMM

Universidad de Costa Rica UCR

San José 11501, Costa Rica

e-mail: ronald.zunigarojas@ucr.ac.cr