Related papers: A global Torelli theorem for hyperkaehler manifold…

We define Hitchin's moduli space for a principal bundle $P$, whose structure group is a compact semisimple Lie group $K$, over a compact non-orientable Riemannian manifold $M$. We use the Donaldson-Corlette correspondence, which identifies…

By restricting to (a linear subspace of) an affine chart in projective space, a complex stably rational or unirational manifold of dimension $m$ is meromorphically dominable by $\mathbb C^m$, i.e., admits a meromorphic dominating map from…

A classical example of Mumford gives a generically non-reduced component of the Hilbert scheme of smooth curves in the projective 3-space such that a general element of the component is contained in a smooth cubic hypersurface in the…

The transcendental Hodge lattice of a projective manifold $M$ is the smallest Hodge substructure in $p$-th cohomology which contains all holomorphic $p$-forms. We prove that the direct sum of all transcendental Hodge lattices has a natural…

The Hodge decomposition is well-known for compact manifolds. The result has been extended by Kodaira to include non-compact manifolds and $L^2$ forms. We further extend the Hodge decomposition to the Sobolev space $H^1$ for general…

This paper has 3 principal goals: (1) to survey what is know about mapping class and Torelli groups of simply connected compact Kaehler manifolds, (2) supplement these results, and (3) present a list of questions and open problems to…

We survey crystalline cohomology, crystals, and formal group laws with an emphasis on geometry. We apply these concepts to K3 surfaces, and especially to supersingular K3 surfaces. In particular, we discuss stratifications of the moduli…

The structure of topological quantum field theories on the compact Kahler manifold is interpreted. The BRST transformations of fields are derived from universal bundle and the observables are found from the second Chern class of universal…

We develop a theory of twistor spaces for supersingular K3 surfaces, extending the analogy between supersingular K3 surfaces and complex analytic K3 surfaces. Our twistor spaces are obtained as relative moduli spaces of twisted sheaves on…

In this paper we realize the moduli spaces of cubic fourfolds with specified automorphism groups as arithmetic quotients of complex hyperbolic balls or type IV symmetric domains, and study their compactifications. Our results mainly depend…

We present the simplest non-abelian version of Seiberg-Witten theory: Quaternionic monopoles. These monopoles are associated with Spin^h(4)-structures on 4-manifolds and form finite-dimensional moduli spaces. On a Kahler surface the…

It is shown that a compact $n$-dimensional K\"ahler manifold with $\frac{n}{2}$-positive Calabi curvature operator has the rational cohomology of complex projective space. For even $n,$ this is sharp in the sense that the complex quadric…

Building on a recent joint paper with Sturmfels, here we argue that the combinatorics of matroids is intimately related to the geometry and topology of toric hyperkaehler varieties. We show that just like toric varieties occupy a central…

We establish a global Torelli theorem for the complete family of Calabi-Yau threefolds arising from cyclic triple covers of $\mathbb P^3$ branched along stable hyperplane arrangements.

Mumford constructed a family of abelian fourfolds with special stucture not characterized by endomorphism ring. Galluzzi showed that the weight 2 Hodge structure of such a variety decomposes into Hodge substructures via the action of…

A hypercomplex manifold is a manifold equipped with three complex structures I, J, K satisfying the quaternionic relations. Let M be a 4-dimensional compact smooth manifold equipped with a hypercomplex structure, and E be a vector bundle on…

Any minimal model of a projective Hyperkaehler manifold is a projective Hyperkaehler manifold. As a consequence, moduli spaces of sheaves on a k3 that don't admit a symplectic resolution are not birational to Hyperkaehler manifolds.

Let M be a compact locally conformal hyperkaehler manifold. We prove a version of Kodaira-Nakano vanishing theorem for M. This is used to show that M admits no holomorphic differential forms, and the cohomology of the structure sheaf…

We develop a general theory for irreducible homogeneous spaces $M= G/H$, in relation to the nullity $\nu$ of their curvature tensor. We construct natural invariant (different and increasing) distributions associated with the nullity, that…

We treat two quite different problems related to changes of complex structures on K\"ahler manifolds by using global geometric method. First, by using operators from Hodge theory on compact K\"ahler manifold, we present a closed explicit…