Stability of the Poincare bundle

Let X be an irreducible smooth projective curve, of genus at least two, over an algebraically closed field k. Let $mathcal{M}^d_G$ denote the moduli stack of principal G-bundles over X of fixed topological type $d in pi_1(G)$, where G is any almost simple affine algebraic group over k. We prove that the universal bundle over $X times mathcal{M}^d_G$ is stable with respect to any polarization on $X times mathcal{M}^d_G$. A similar result is proved for the Poincare adjoint bundle over $X times M_G^{d, rs}$, where $M_G^{d, rs}$ is the coarse moduli space of regularly stable principal G-bundles over X of fixed topological type d.

Integrable systems and Torelli theorems for the moduli spaces of parabolic bundles and parabolic Higgs bundles

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.

Torelli theorem for the moduli space of parabolic Higgs bundles

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.

Torelli theorem for the Deligne--Hitchin moduli space

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 $X$.

Torelli theorem for the moduli space of framed bundles

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.