We prove the generic Torelli theorem for hypersurfaces in $\mathbb {P}^{n}$ of degree $d$ dividing $n+1$, for $d$ sufficiently large. Our proof involves the higher-order study of the variation of Hodge structure along particular one-parameter families of hypersurfaces that we call ‘Schiffer variations.’ We also analyze the case of degree $4$. Combined with Donagi's generic Torelli theorem and results of Cox and Green, this shows that the generic Torelli theorem for hypersurfaces holds with finitely many exceptions.

Given a compact Kähler manifold X, it is shown that pairs of the form $(E,\, D)$ , where E is a trivial holomorphic vector bundle on X, and D is an integrable holomorphic connection on E, produce a neutral Tannakian category. The corresponding pro-algebraic affine group scheme is studied. In particular, it is shown that this pro-algebraic affine group scheme for a compact Riemann surface determines uniquely the isomorphism class of the Riemann surface.

We extend the infinitesimal Torelli theorem for smooth hypersurfaces to nodal hypersurfaces.

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 systemis generic. When the genus is at least two, using this result we also prove a Torelli theoremfor 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.

We give a necessary and sufficient condition in order for a hyperplane arrangement to be of Torelli type, namely that it is recovered as the set of unstable hyperplanes of its Dolgachev sheaf of logarithmic differentials. Decompositions and semistability of non-Torelli arrangements are investigated.

We consider the Prym map from the space of double coverings of a curve of genus g with r branch points to the moduli space of abelian varieties. We prove that 𝒫:ℛg,r→𝒜δg−1+r/2 is generically injective if We also show that a very general Prym variety of dimension at least 4 is not isogenous to a Jacobian.

Fix a $\tau$-connection DL on a line bundle L over $X\times \mathbb{C}$, where X is a compact connected Riemann surface of genus at least three. Let $\mathcal{M}_X(D_L)$ denote the moduli space of all semistable $\tau$-connections of rank n, where $n\geq 2$, with the property that the induced $\tau$-connection on the top exterior product is isomorphic to (L, DL). Let $\mathcal{M}_Y(D_M)$ be the similar moduli space for another Riemann surface Y with genus(Y) = genus(X), where DM is a $\tau$-connection on a line bundle M over $Y\times \mathbb{C}$. We prove that if the variety $\mathcal{M}_X(D_L)$ is isomorphic to $\mathcal{M}_Y(D_M)$, then X is isomorphic to Y.

Let $\mathcal{M}^D_X$ denote the moduli space of all rank n flat connections on X. We prove that $\mathcal{M}^D_X$ determines X up to finitely many Riemann surfaces. For the very general Riemann surface X, the variety $\mathcal{M}^D_X$ determines X.