Related papers: A virtual Kawasaki formula

When studying a graded module $M$ over the Cox ring of a smooth projective toric variety $X$, there are two standard types of resolutions commonly used to glean information: free resolutions of $M$ and vector bundle resolutions of its…

The so-called Hitchin-Kobayashi correspondence, proved by Donaldson, Uhlenbeck and Yau, establishes that an indecomposable holomorphic vector bundle over a compact Kahler manifold admits a Hermitian-Einstein metric if and only if the bundle…

On a threefold with trivial canonical bundle, Kuranishi theory gives an algebro-geometry construction of the (local analytic) Hilbert scheme of curves at a smooth holomorphic curve as a gradient scheme, that is, the zero-scheme of the…

We develop a theory of complex Kuranishi structures on projective schemes. These are sufficiently rigid to be equivalent to weak perfect obstruction theories, but sufficiently flexible to admit global complex Kuranishi charts. We apply the…

Recently, Oh and Thomas constructed algebraic virtual cycles for moduli spaces of sheaves on Calabi-Yau 4-folds. The purpose of this paper is to provide a virtual pullback formula between these Oh-Thomas virtual cycles. We find a natural…

Using $L^2$-methods for the $\bar\partial$-equation we prove that the Ohsawa-Takegoshi extension theorem also holds for holomorphic sections of a vector bundle, over compact K\"ahler manifolds. We then proceed to show that the conditions…

In this paper we study the holomorphic Euler characteristics of determinant line bundles on moduli spaces of rank 2 semistable sheaves on an algebraic surface X, which can be viewed as $K$-theoretic versions of the Donaldson invariants. In…

In this paper we will introduce a new notion of geometric structures defined by systems of closed differential forms in term of the Clifford algebra of the direct sum of the tangent bundle and the cotangent bundle on a manifold. We develop…

Let $G$ be an algebraic group and let $X$ be a smooth $G$-variety with two orbits: an open orbit and a a closed orbit of codimension $1$. We give an algebraic description of the category of $G$-equivariant vector bundles on $X$ under a mild…

I conjecture that index formulas for $K$-theory classes on the moduli of holomorphic $G$-bundles over a compact Riemann surface $\Sigma$ are controlled, in a precise way, by Frobenius algebra deformations of the Verlinde algebra of $G$. The…

In this note, we discuss the concept of pseudoeffective vector bundle and also introduce pseudoeffective torsion-free sheaves over compact K\"ahler manifolds. We show that a pseudoeffective reflexive sheaf over a compact K\"ahler manifold…

Given a compact K\"ahler manifold, Geometric Invariant Theory is applied to construct analytic GIT-quotients that are local models for a classifying space of (poly)stable holomorphic vector bundles containing the coarse moduli space of…

Let $M=P(E)$ be the complex manifold underlying the total space of the projectivization of a holomorphic vector bundle $E \to \Sigma$ over a compact complex curve $\Sigma$ of genus $\ge 2$. Building on ideas of Fujiki, we prove that $M$…

The deformation of a variety $X$ to the normal cone of a subvariety $Y$ is a classical construction in algebraic geometry. In this paper we study the case when $(X,\omega)$ is a compact K\"ahler manifold and $Y$ is a submanifold. The…

This paper presents the theory of holomorphic vector valued modular forms from a geometric perspective. More precisely, we define certain holomorphic vector bundles on the modular orbifold of generalized elliptic curves whose sections are…

We investigate the formal principle for holomorphic line bundles on neighborhoods of an analytic subset of a complex manifold mainly in the case where it can be realized as an open subset of a compact K\"ahler manifold. Our approach…

Given a compact complex manifold X of dimension n, we define a bimeromorphic invariant $\kappa_+(X)$ as the maximum p for which there is a saturated line subsheaf L of the sheaf of holomorphic p forms whose Kodaira dimension $\kappa (L)$…

We conjecture the equality of the numerical and Kodaira dimensions $\nu_1^*(X)$ and $\kappa_1^*(X)$ for the cotangent bundle of compact K\"ahler manifolds $X$, generalising the classical case of the canonical bundle. We show or reduce it to…

Let $S$ be an irreducible smooth projective surface defined over an algebraically closed field $k$. For a positive integer $d$, let ${\rm Hilb}^d(S)$ be the Hilbert scheme parametrizing the zero-dimensional subschemes of $S$ of length $d$.…

We prove that the forgetful morphism from the moduli space of orthogonal bundles to the moduli space of vector bundles over a smooth curve is an embedding. Our proof relies on an explicit description of a set of generators for the…