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Register a new userTorelli theorem for the Deligne--Hitchin moduli space
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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 $X$.
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Let X be an irreducible smooth complex projective curve of genus g>2, and let x be a fixed point. A framed bundle is a pair (E,phi), where E is a vector bundle over X, of rank r and degree d, and phi:E_xto C^r is a non-zero homomorphism. There is a notion of (semi)stability for framed bundles depending on a parameter tau>0, which gives rise to the moduli space of tau-semistable framed bundles M^tau. We prove a Torelli theorem for M^tau, for tau>0 small enough, meaning, the isomorphism class of the one-pointed curve (X,x), and also the integer r, are uniquely determined by the isomorphism class of the variety M^tau.
In this article we extend the proof given by Biswas and Gomez of a Torelli theorem for the moduli space of Higgs bundles with fixed determinant, to the parabolic situation.
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.
In this paper, we generalize the construction of Deligne-Hitchin twistor space by gluing two certain Hodge moduli spaces. We investigate such generalized Deligne-Hitchin twistor space as a complex analytic manifold, more precisely, we show it admits a global smooth trivialization such that the induced product metric is balanced, and it carries a semistable holomorphic tangent bundle. Moreover, we study the automorphism groups of the Hodge moduli spaces and the generalized Deligne-Hitchin twistor space.
Given two semistable, non potentially isotrivial elliptic surfaces over a curve $C$ defined over a field of characteristic zero or finitely generated over its prime field, we show that any compatible family of effective isometries of the N{e}ron-Severi lattices of the base changed elliptic surfaces for all finite separable maps $Bto C$ arises from an isomorphism of the elliptic surfaces. Without the effectivity hypothesis, we show that the two elliptic surfaces are isomorphic. We also determine the group of universal automorphisms of a semistable elliptic surface. In particular, this includes showing that the Picard-Lefschetz transformations corresponding to an irreducible component of a singular fibre, can be extended as universal isometries. In the process, we get a family of homomorphisms of the affine Weyl group associated to $tilde{A}_{n-1}$ to that of $tilde{A}_{dn-1}$, indexed by natural numbers $d$, which are closed under composition.
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