Suppose that $f:X\to C$ is a general Jacobian elliptic surface over ${\mathbb {C}}$ of irregularity $q$ and positive geometric genus $h$. Assume that $10 h>12(q-1)$, that $h>0$ and let $\overline {\mathcal {E}\ell \ell }$ denote the stack of generalized elliptic curves. (1) The moduli stack $\mathcal {JE}$ of such surfaces is smooth at the point $X$ and its tangent space $T$ there is naturally a direct sum of lines $(v_a)_{a\in Z}$, where $Z\subset C$ is the ramification locus of the classifying morphism $\phi :C\to \overline {\mathcal {E}\ell \ell }$ that corresponds to $X\to C$. (2) For each $a\in Z$ the map $\overline {\nabla }_{v_a}:H^{2,0}(X)\to H^{1,1}_{\rm prim}(X)$ defined by the derivative $per_*$ of the period map $per$ is of rank one. Its image is a line ${\mathbb {C}}[\eta _a]$ and its kernel is $H^0(X,\Omega ^2_X(-E_a))$, where $E_a=f^{-1}(a)$. (3) The classes $[\eta _a]$ form an orthogonal basis of $H^{1,1}_{\rm prim}(X)$ and $[\eta _a]$ is represented by a meromorphic $2$-form $\eta _a$ in $H^0(X,\Omega ^2_X(2E_a))$ of the second kind. (4) We prove a local Schottky theorem; that is, we give a description of $per_*$ in terms of a certain additional structure on the vector bundles that are involved. Assume further that $8h>10(q-1)$ and that $h\ge q+3$. (5) Given the period point $per(X)$ of $X$ that classifies the Hodge structure on the primitive cohomology $H^2_{\rm prim}(X)$ and the image of $T$ under $per_*$ we recover $Z$ as a subset of ${\mathbb {P}}^{h-1}$ and then, by quadratic interpolation, the curve $C$. (6) We prove a generic Torelli theorem for these surfaces. Everything relies on the construction, via certain kinds of Schiffer variations of curves, of certain variations of $X$ for which $per_*$ can be calculated. (In an earlier version of this paper we used variations constructed by Fay. However, Schiffer variations are slightly more powerful.)

We extend classical results of Perego and Rapagnetta on moduli spaces of sheaves of type OG10 to moduli spaces of Bridgeland semistable objects on the Kuznetsov component of a cubic fourfold. In particular, we determine the period of this class of varieties and use it to understand when they become birational to moduli spaces of sheaves on a K3 surface.

Let $\pi \colon \mathcal {X}\to B$ be a family whose general fibre $X_b$ is a $(d_1,\,\ldots,\,d_a)$-polarization on a general abelian variety, where $1\leq d_i\leq 2$, $i=1,\,\ldots,\,a$ and $a\geq 4$. We show that the fibres are in the same birational class if all the $(m,\,0)$-forms on $X_b$ are liftable to $(m,\,0)$-forms on $\mathcal {X}$, where $m=1$ and $m=a-1$. Actually, we show a general criteria to establish whether the fibres of certain families belong to the same birational class.

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 study irreducible holomorphic symplectic manifolds deformation equivalent to Hilbert schemes of points on a $K3$ surface and admitting a non-symplectic involution. We classify the possible discriminant quadratic forms of the invariant and coinvariant lattice for the action of the involution on cohomology and explicitly describe the lattices in the cases where the invariant lattice has small rank. We also give a modular description of all $d$-dimensional families of manifolds of $K3^{[n]}$-type with a non-symplectic involution for $d\geqslant 19$ and $n\leqslant 5$ and provide examples arising as moduli spaces of twisted sheaves on a $K3$ surface.

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.

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.