Matei Toma

Rationally connected foliations after Bogomolov and McQuillan

This paper is concerned with a sufficient criterion to guarantee that a given foliation on a normal variety has algebraic and rationally connected leaves. Following ideas from a preprint of Bogomolov-McQuillan and using recent works of Langer and Graber-Harris-Starr, we give a clean, short and simple proof of previous results. Apart from a new vanishing theorem for vector bundles in positive characteristic, our proof employs only standard techniques of Mori theory and does not make any reference to the more involved properties of foliations in characteristic p. We apply the result to show that Q-Fano varieties with unstable tangent bundles always admit a sequence of partial rational quotients naturally associated to the Harder-Narasimhan filtration.

On Very Ample Vector Bundles on Curves

We study very ample vector bundles on curves. We first give numerical conditions for the existence of non-special such bundles. Then we prove the inequality \[ h^0(\det E)\ge h^0(E) + {\rm rank}(E)-2 \] over curves of genus at least two. We apply this to prove some special cases of a conjecture on scrolls of small codimension.

Compact moduli spaces for slope-semistable sheaves

We resolve pathological wall-crossing phenomena for moduli spaces of sheaves on higher-dimensional base manifolds. This is achieved by considering slope-semistability with respect to movable curves rather than divisors. Moreover, given a projective n-fold and a curve C that arises as the complete intersection of n-1 very ample divisors, we construct a modular compactification of the moduli space of vector bundles that are slope-stable with respect to C. Our construction generalises the algebro-geometric construction of the Donaldson-Uhlenbeck compactification by Joseph Le Potier and Jun Li. Furthermore, we describe the geometry of the newly construced moduli spaces by relating them to moduli spaces of simple sheaves and to Gieseker-Maruyama moduli spaces.

Holomorphic vector bundles on primary Kodaira surfaces

It is in general unknown which topological complex vector bundles on a non-algebraic surface admit holomorphic structures. We solve this problem for primary Kodaira surfaces by using results of Kani on curves of genus two with elliptic differentials. Some of the corresponding moduli spaces will be smooth compact and holomorphically symplectic.

Variation of Gieseker moduli spaces via quiver GIT

We introduce a notion of stability for sheaves with respect to several polarisations that generalises the usual notion of Gieseker-stability. We prove, under a boundedness assumption, which we show to hold on threefolds or for rank two sheaves on base manifolds of arbitrary dimension, that semistable sheaves have a projective coarse moduli space that depends on a natural stability parameter. We then give two applications of this machinery. First, we show that given a real ample class

On compact complex surfaces of Kähler rank one

\omega \in N^1(X)_\mathbb{R}

Birational models for varieties of Poncelet curves

on a smooth projective threefold

Semi-continuity of Stability for Sheaves and Variation of Gieseker Moduli Spaces

X

An essentially saturated surface not of Kähler type

there exists a projective moduli space of sheaves that are Gieseker-semistable with respect to

TWO-DIMENSIONAL MODULI SPACES OF VECTOR BUNDLES OVER KODAIRA SURFACES

\omega