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Register a new userOn moduli spaces of Hitchin pairs
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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 $X$.
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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 smooth projective curve of genus $ggeq 2$ over the complex numbers. A holomorphic triple $(E_1,E_2,phi)$ on $X$ consists of two holomorphic vector bundles $E_1$ and $E_2$ over $X$ and a holomorphic map $phi:E_2 to E_1$. There is a concept of stability for triples which depends on a real parameter $sigma$. In this paper, we determine the Hodge polynomials of the moduli spaces of $sigma$-stable triples with $rk(E_1)=3$, $rk(E_2)=1$, using the theory of mixed Hodge structures. This gives in particular the Poincare polynomials of these moduli spaces. As a byproduct, we recover the Hodge polynomial of the moduli space of odd degree rank 3 stable vector bundles.
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