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Topology of Hitchin systems and Hodge theory of character varieties: the case \(A_1\). (English) Zbl 1375.14047

Let \(\Sigma\) be a non-singular complex projective curve of genus \(g\geq 2\). Let \(\Sigma^{\mathrm{top}}\) be the underlying topological surface. So \(\Sigma\) consists of \(\Sigma^{\mathrm{top}}\) and a complex structure. A Higgs bundle on \(\Sigma\) is a pair \((E,\phi)\) where \(E\) is a holomorphic vector bundle over \(\Sigma\) and \(\phi\in H^0(\Sigma,\mathrm{End}(E)\otimes\Omega)\), called a Higgs field, is a holomorphic endomorphism of \(E\) twisted by the canonical bundle \(\Omega\) of \(\Sigma\). The slope of \(E\) is the degree of \(E\) divided by the rank of \(E\). The bundle is called stable if the slope of any proper \(\phi\)-invariant subbundle is strictly less than that of \(E\). A Higgs bundle is called polystable if it is the direct sum of stable Higgs bundles with the same slope. Fixing the degree \(d\) and the rank \(n\) determines the Dolbeault moduli space of polystable Higgs bundles \(\mathcal{M}_{\mathrm{Dol}}\).
Let \(\Gamma\) be the universal central extension of \(\pi_1(\Sigma)\), and let \(\mathcal{R}:=\mathrm{Hom}^+(\Gamma,\mathrm{GL}(n,\mathbb{C}))\) be the semisimple representations of \(\Gamma\). Then \(\mathrm{GL}(n,\mathbb{C})\) acts on \(\mathcal{R}\) by conjugation and the quotient \(\mathcal{R}/\mathrm{GL}(n,\mathbb{C})\) is called the twisted character variety; it can be realized as a GIT quotient.
Let \(\mathcal{M}_{\mathrm{B}}\) be the subvariety of \(\mathcal{R}/\mathrm{GL}(n,\mathbb{C})\) consisting of characters that map the central extension of \(\Gamma\) to an element in the center of \(\mathrm{SU}(n)\subset \mathrm{GL}(n,\mathbb{C})\), isomorphic to \(\mathbb{Z}_n\), whose least residue corresponds to the degree \(d\). \(\mathcal{M}_{\mathrm{B}}\) is called the Betti moduli space of \(\Sigma\). The non-Abelian Hodge Theorem states that there is an analytic isomorphism between \(\mathcal{M}_{\mathrm{Dol}}\) and \(\mathcal{M}_{\mathrm{B}}\). Since the Betti moduli space does not depend on the complex structure of \(\Sigma\), but the Dolbeault moduli space does, these moduli spaces are not biholomorphic (thus not algebraically isomorphic). Regardless, this diffeomorphism induces an isomorphism on cohomology: \(H^*(\mathcal{M}_{\mathrm{B}})\cong H^*(\mathcal{M}_{\mathrm{Dol}})\).
We note that the non-abelian Hodge Theorem can be generalized to a setting where \(\mathrm{GL}(n,\mathbb{C})\) is replaced by any connected reductive complex algebraic group \(G\). The paper under review restricts its attention to the cases when the degree \(d=1\) and the rank \(n=2\), but also considers the corresponding moduli spaces for \(G=\mathrm{SL}(2,\mathbb{C})\) and \(G=\mathrm{PGL}(2,\mathbb{C})\).
P. Deligne [Publ. Math., Inst. Hautes Étud. Sci. 44, 5–77 (1974; Zbl 0237.14003)] showed that a complex variety \(X\) admits an increasing weight filtration \(0=W_{-1}\subset W_0\subset\cdots\subset W_{2j}=H^j(X;\mathbb{Q})\), and a decreasing Hodge filtration \(H^j(X;\mathbb{C})=F^{0}\supset\cdots\supset F^{m+1}=0\) such that for all \(0\leq p\leq l\), \[ \mathrm{Gr}^{W\otimes \mathbb{C}}_l:= W_l\otimes\mathbb{C}/W_{l-1}\otimes\mathbb{C}= F^p(\mathrm{Gr}^{W\otimes \mathbb{C}}_l) \oplus\overline{F^{l-p+1}(\mathrm{Gr}^{W\otimes \mathbb{C}}_l)}, \] where \(F^p(\mathrm{Gr}^{W\otimes \mathbb{C}}_l)=(F^p\cap W_l\otimes \mathbb{C}+W_{l-1}\otimes \mathbb{C})/W_{l-1}\otimes \mathbb{C}\). This allows one to define the mixed Hodge numbers for every \(H^j(X;\mathbb{C})\) and subsequently to define the mixed Hodge polynomial for \(X\).
T. Hausel and F. Rodriguez-Villegas [Invent. Math. 174, No. 3, 555–624 (2008; Zbl 1213.14020)] showed what they call the “curious hard Lefschetz theorem”; namely that \(\mathrm{Gr}^W_{\dim\mathcal{M}_{\mathrm{B}}-2k}H^*(\mathcal{M}_{\mathrm{B}})\cong \mathrm{Gr}^W_{\dim\mathcal{M}_{\mathcal{B}}+2k}H^{*+2k}(\mathcal{M}_{\mathrm{B}})\) for all \(k\geq 0\), where the isomorphism is given by the \(k\)-fold iterated cup product with an explicitly given element in \(H^2(\mathcal{M}_{\mathrm{B}})\). A similar theorem holds when \(G=\mathrm{PGL}(2,\mathbb{C})\). This theorem is proved, in part, by determining the mixed Hodge polynomial of \(\mathcal{M}_{\mathrm{B}}\). This is in contrast to the classical hard Lefschetz theorem since the cohomological degree goes up by \(2k\) and the weight type by \(4k\); hence the usage of “curious”.
In the paper under review, the authors state: “The present paper was partly motivated by the desire to give a more conceptual explanation for these curious hard Lefschetz theorems.”
There is a mapping \(\chi:\mathcal{M}_{\mathrm{Dol}}\to \mathcal{A}\), called the Hitchin fibration, where \(\mathcal{A}\) is non-canonically isomorphic to \(\mathbb{C}^{4g-3}\). Let \(\Lambda^s\subset \mathcal{A}\) be an \(s\)-dimensional linear section with respect to an identification of \(\mathcal{A}\) with \(\mathbb{C}^{4g-3}\); then \[ P_pH^d(\mathcal{M}_{\mathrm{Dol}})=\mathrm{Ker}\{H^d(\mathcal{M}_{\mathrm{Dol}})\to H^d(\chi^{-1}(\Lambda^{d-p-1}))\} \] determines the perverse Leray filtration on \(H^*(\mathcal{M}_{\mathrm{Dol}})\).
The main theorem of this paper establishes that, after renumbering, the weight filtration on \(H^*(\mathcal{M}_{\mathrm{B}})\) exactly corresponds to the perverse Leray filtration on \(H^*(\mathcal{M}_{\mathrm{Dol}})\) under the induced map from the non-abelian Hodge Theorem. In particular, for all \(k\), \[ W_{2k}H^*(\mathcal{M}_{\mathrm{B}})=W_{2k+1}H^*(\mathcal{M}_{\mathrm{B}})=P_kH^*(\mathcal{M}_{\mathrm{Dol}}). \]
Using this correspondence, the authors show that the curious hard Lefschetz theorem on \(H^*(\mathcal{M}_{\mathrm{B}})\), which is in terms of the weight filtration, is explained by a general hard Lefschetz theorem applied to \(H^*(\mathcal{M}_{\mathrm{Dol}})\), which is in terms of the perverse Leray filtration.
This is surprising since this implies, as the authors say, that “the weight filtration on \(H^*(\mathcal{M}_{\mathrm{B}})\) keeps track of certain topological properties of the Hitchin map on \(H^*(\mathcal{M}_{\mathrm{Dol}})\)”. But the weight filtration only depends on \(\Sigma^{\mathrm{top}}\) whereas the Hitchin map depends on \(\Sigma\).
The proof of this deep result is lengthy and technical, but the subject matter appears to demand it. Roughly speaking, by using known results about the cohomology rings, the authors show that the place of the multiplicative generators in the perverse Leray filtration of \(H^*(\mathcal{M}_{\mathrm{Dol}})\) is the same as in the weight filtration of \(H^*(\mathcal{M}_{\mathrm{B}})\). The perverse Leray filtration is generally not compatible with cup products, but the authors show that there is a large open subset of \(\mathcal{A}\), called the elliptic locus, where the general linear sections can be chosen to lie entirely. In those cases, the perverse Leray filtration is shown to be compatible with cup products and the result follows. However, not all cases are handled as such, and so they address the exceptions using an ad hoc argument.

MSC:

14D20 Algebraic moduli problems, moduli of vector bundles
53D30 Symplectic structures of moduli spaces
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References:

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