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Register a new userIntersection cohomology of the moduli space of Higgs bundles on a genus 2 curve
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Camilla Felisetti
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Let $C$ be a smooth projective curve of genus $2$. Following a method by O Grady, we construct a semismall desingularization $tilde{mathcal{M}}_{Dol}^G$ of the moduli space $mathcal{M}_{Dol}^G$ of semistable $G$-Higgs bundles of degree 0 for $G=GL(2,mathbb{C}), SL(2,mathbb{C})$. By the decomposition theorem by Beilinson, Bernstein, Deligne one can write the cohomology of $tilde{mathcal{M}}_{Dol}^G$ as a direct sum of the intersection cohomology of $mathcal{M}_{Dol}^G$ plus other summands supported on the singular locus. We use this splitting to compute the intersection cohomology of $mathcal{M}_{Dol}^G$ and prove that the mixed Hodge structure on it is actually pure, in analogy with what happens to ordinary cohomology in the smooth case of coprime rank and degree.
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We explore the cohomological structure for the (possibly singular) moduli of $mathrm{SL}_n$-Higgs bundles for arbitrary degree on a genus g curve with respect to an effective divisor of degree >2g-2. We prove a support theorem for the $mathrm{SL}_n$-Hitchin fibration extending de Cataldos support theorem in the nonsingular case, and a version of the Hausel-Thaddeus topological mirror symmetry conjecture for intersection cohomology. This implies a generalization of the Harder-Narasimhan theorem concerning semistable vector bundles for any degree. Our main tool is an Ng^{o}-type support inequality established recently which works for possibly singular ambient spaces and intersection cohomology complexes.
Let $X$ be a compact connected Riemann surface and $D$ an effective divisor on $X$. Let ${mathcal N}_H(r,d)$ denote the moduli space of $D$-twisted stable Higgs bundles (a special class of Hitchin pairs) on $X$ of rank $r$ and degree $d$. It is known that ${mathcal N}_H(r,d)$ has a natural holomorphic Poisson structure which is in fact symplectic if and only if $D$ is the zero divisor. We prove that ${mathcal N}_H(r,d)$ admits a natural enhancement to a holomorphic symplectic manifold which is called here ${mathcal M}_H(r,d)$. This ${mathcal M}_H(r,d)$ is constructed by trivializing, over $D$, the restriction of the vector bundles underlying the $D$-twisted Higgs bundles; such objects are called here as framed Higgs bundles. We also investigate the symplectic structure on the moduli space ${mathcal M}_H(r,d)$ of framed Higgs bundles as well as the Hitchin system associated to it.
The moduli space of stable vector bundles on a Riemann surface is smooth when the rank and degree are coprime, and is diffeomorphic to the space of unitary connections of central constant curvature. A classic result of Newstead and Atiyah-Bott asserts that its rational cohomology ring is generated by the universal classes, that is, by the Kunneth components of the Chern classes of the universal bundle. This paper studies the larger, non-compact moduli space of Higgs bundles, as introduced by Hitchin and Simpson, with values in the canonical bundle K. This is diffeomorphic to the space of all connections of central constant curvature, whether unitary or not. The main result of the paper is that, in the rank 2 case, the rational cohomology ring of this space is again generated by universal classes. The spaces of Higgs bundles with values in K(n) for n > 0 turn out to be essential to the story. Indeed, we show that their direct limit has the homotopy type of the classifying space of the gauge group, and hence has cohomology generated by universal classes. A companion paper treats the problem of finding relations between these generators in the rank 2 case.
The moduli space of stable bundles of rank 2 and degree 1 on a Riemann surface has rational cohomology generated by the so-called universal classes. The work of Baranovsky, King-Newstead, Siebert-Tian and Zagier provided a complete set of relations between these classes, expressed in terms of a recursion in the genus. This paper accomplishes the same thing for the non-compact moduli spaces of Higgs bundles, in the sense of Hitchin and Simpson. There are many more independent relations than for stable bundles, but in a sense the answer is simpler, since the formulas are completely explicit, not recursive. The results of Kirwan on equivariant cohomology for holomorphic circle actions are of key importance. Together, Parts I and II describe the cohomology rings of spaces of rank 2 Higgs bundles at essentially the same level of detail as is known for stable bundles.
The moduli space of Higgs bundles has two stratifications. The Bialynicki-Birula stratification comes from the action of the non-zero complex numbers by multiplication on the Higgs field, and the Shatz stratification arises from the Harder-Narasimhan type of the vector bundle underlying a Higgs bundle. While these two stratification coincide in the case of rank two Higgs bundles, this is not the case in higher rank. In this paper we analyze the relation between the two stratifications for the moduli space of rank three Higgs bundles.
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