Related papers: A global Torelli theorem for hyperkaehler manifold…

In 1972, E. P. Senkin generalized the celebrated theorem of A. V. Pogorelov on unique determination of compact convex surfaces by their intrinsic metrics in the Euclidean 3-space $E^3$ to higher dimensional Euclidean spaces $E^{n+1}$ under…

The Torelli group of a compact non-orientable Klein surface is the subgroup of the modular group consisting of the mapping classes that act trivially on the first homology group of the surface. We prove that if a surface has genus at least…

We present a classification of compact Kaehler manifolds admitting a hamiltonian 2-form (which were classified locally in part I of this work). This involves two components of independent interest. The first is the notion of a rigid…

A mapping class group of an oriented manifold is a quotient of its diffeomorphism group by the isotopies. In the published version of "Mapping class group and a global Torelli theorem for hyperkahler manifolds" I made an error based on a…

We give sharp lower bounds for the degree of the syzygies involving the partial derivatives of a homogeneous polynomial defining an even dimensional nodal hypersurface. This implies the validity of formulas due to M. Saito, L. Wotzlaw and…

This text contributes to the foundations of the theory of global Berkovich spaces, that is to say Berkovich spaces over Banach rings with nice properties such as $\mathbf{Z}$, rings of integers of number fields, discrete valuation rings,…

Let $(M, \partial M)$ be a compact 3-manifold with boundary which admits a complete, convex co-compact hyperbolic metric. For each hyperbolic metric $g$ on $M$ such that $\dr M$ is smooth and strictly convex, the induced metric on $\dr M$…

We introduce in this paper the notion of Hodge similarities of transcendental lattices of hyperk\"ahler manifolds and investigate the Hodge conjecture for these Hodge morphisms. Studying K3 surfaces with a symplectic automorphism, we prove…

The Euler-Korteweg equations are a modification of the Euler equations that takes into account capillary effects. In the general case they form a quasi-linear system that can be recast as a degenerate Schr\"odinger type equation. Local…

We show that there is a good notion of irreducible sympelectic varieties of $\mathrm{K3}^{[n]}$-type over an arbitrary field of characteristic zero or $p > n + 1$. Then we construct mixed characteristic moduli spaces for these varieties.…

We study the Hodge structure of elliptic surfaces which are canonically defined from bielliptic curves of genus three. We prove that the period map for the second cohomology has one dimensional fibers, and the period map for the total…

The disk complex of a surface in a 3-manifold is used to define its {\it topological index}. Surfaces with well-defined topological index are shown to generalize well-known classes, such as incompressible, strongly irreducible, and critical…

We study rank $1$ flat bundles over solvmanifolds whose cohomologies are non-trivial. By using Hodge theoretical properties for all topologically trivial rank $1$ flat bundles, we represent the structure theorem of K\"ahler solvmanifolds as…

We prove that there are no unexpected universal integral linear relations and congruences between Hodge, Betti and Chern numbers of compact complex manifolds and determine the linear combinations of such numbers which are bimeromorphic or…

Given a complete non-compact surface embedded in R^3, we consider the Dirichlet Laplacian in a layer of constant width about the surface. Using an intrinsic approach to the layer geometry, we generalise the spectral results of an original…

A classical theorem of Frankel for compact K\"ahler manifolds states that a K\"ahler S^1-action is Hamiltonian if and only if it has fixed points. We prove a metatheorem which says that when Hodge theory holds on non-compact manifolds, then…

We develop a geometric invariant Littlewood-Paley theory for arbitrary tensors on a compact 2 dimensional manifold. We show that all the important features of the classical LP theory survive with estimates which depend only on very limited…

We construct non-geometric compactifications by using the F-theory dual of the heterotic string compactified on a two-torus, together with a close connection between Siegel modular forms of genus two and the equations of certain K3…

We discuss two generalizations of the collar lemma. The first is the stable neighborhood theorem which says that a (not necessarily simple) closed geodesic in a hyperbolic surface has a \lq\lq stable neighborhood\rq\rq whose width only…

We first prove an isomorphism between the moduli space of smooth cubic threefolds and the moduli space of hyperkaehler fourfolds of K3^{[2]}-type with a non-symplectic automorphism of order three, whose invariant lattice has rank one and is…