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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.
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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.
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 $X$.
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.
We prove a Torelli theorem for the moduli space of semistable parabolic Higgs bundles over a smooth complex projective algebraic curve under the assumption that the parabolic weight system is generic. When the genus is at least two, using this result we also prove a Torelli theorem for the moduli space of semistable parabolic bundles of rank at least two with generic parabolic weights. The key input in the proofs is a method of J.C. Hurtubise, Integrable systems and algebraic surfaces, Duke Math. Jour. 83 (1996), 19--49.
Let $X$ be a compact connected Riemann surface, $D, subset, X$ a reduced effective divisor, $G$ a connected complex reductive affine algebraic group and $H_x, subsetneq, G_x$ a Zariski closed subgroup for every $x, in, D$. A framed principal $G$--bundle is a pair $(E_G,, phi)$, where $E_G$ is a holomorphic principal $G$--bundle on $X$ and $phi$ assigns to each $x, in, D$ a point of the quotient space $(E_G)_x/H_x$. A framed $G$--Higgs bundle is a framed principal $G$--bundle $(E_G,, phi)$ together with a section $theta, in, H^0(X,, text{ad}(E_G)otimes K_Xotimes{mathcal O}_X(D))$ such that $theta(x)$ is compatible with the framing $phi$ for every $x, in, D$. We construct a holomorphic symplectic structure on the moduli space $mathcal{M}_{FH}(G)$ of stable framed $G$--Higgs bundles. Moreover, we prove that the natural morphism from $mathcal{M}_{FH}(G)$ to the moduli space $mathcal{M}_{H}(G)$ of $D$-twisted $G$--Higgs bundles $(E_G,, theta)$ that forgets the framing, is Poisson. These results generalize cite{BLP} where $(G,, {H_x}_{xin D})$ is taken to be $(text{GL}(r,{mathbb C}),, {text{I}_{rtimes r}}_{xin D})$. We also investigate the Hitchin system for $mathcal{M}_{FH}(G)$ and its relationship with that for $mathcal{M}_{H}(G)$.
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