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Register a new userWallcrossing and Cohomology of The Moduli Space of Hitchin Pairs
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A conjectural recursive relation for the Poincare polynomial of the Hitchin moduli space is derived from wallcrossing in the refined local Donaldson-Thomas theory of a curve. A doubly refined generalization of this theory is also conjectured and shown to similarly determine the Hodge polynomial of the same moduli space.
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Let $X$ be a compact Riemann surface $X$ of genus at--least two. Fix a holomorphic line bundle $L$ over $X$. Let $mathcal M$ be the moduli space of Hitchin pairs $(E ,phiin H^0(End(E)otimes L))$ over $X$ of rank $r$ and fixed determinant of degree $d$. We prove that, for some numerical conditions, $mathcal M$ is irreducible, and that the isomorphism class of the variety $mathcal M$ uniquely determines the isomorphism class of the Riemann surface $X$.
Fix integers $ggeq 3$ and $rgeq 2$, with $rgeq 3$ if $g=3$. Given a compact connected Riemann surface $X$ of genus $g$, let $MDH(X)$ denote the corresponding $text{SL}(r, {mathbb C})$ Deligne--Hitchin moduli space. We prove that the complex analytic space $MDH(X)$ determines (up to an isomorphism) the unordered pair ${X, overline{X}}$, where $overline{X}$ is the Riemann surface defined by the opposite almost complex structure on $X$.
We provide a construction of the moduli spaces of framed Hitchin pairs and their master spaces. These objects have come to interest as algebra
We compute and compare the (intersection) cohomology of various natural geometric compactifications of the moduli space of cubic threefolds: the GIT compactification and its Kirwan blowup, as well as the Baily-Borel and toroidal compactifications of the ball quotient model, due to Allcock-Carlson-Toledo. Our starting point is Kirwans method. We then follow by investigating the behavior of the cohomology under the birational maps relating the various models, using the decomposition theorem in different ways, and via a detailed study of the boundary of the ball quotient model. As an easy illustration of our methods, the simpler case of the moduli of cubic surfaces is discussed in an appendix.
Generalized Donaldson-Thomas invariants corresponding to local D6-D2-D0 configurations are defined applying the formalism of Joyce and Song to ADHM sheaves on curves. A wallcrossing formula for invariants of D6-rank two is proven and shown to agree with the wallcrossing formula of Kontsevich and Soibelman. Using this result, the asymptotic D6-rank two invariants of (-1,-1) and (0,-2) local rational curves are computed in terms of the D6-rank one invariants.
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