Unirationality of moduli spaces of special cubic fourfolds and K3 surfaces

We provide explicit descriptions of the generic members of Hassett's divisors $\mathcal C_d$ for relevant $18\leq d\leq 38$ and for $d=44$. In doing so, we prove that $\mathcal C_d$ is unirational for $18\leq d\leq 38,d=44$. As a corollary, we prove that the moduli space $\mathcal N_{d}$ of polarized K3 surfaces of degree $d$ is unirational for $d=14,26,38$. The case $d=26$ is entirely new, while the other two cases have been previously proven by Mukai.Comment: 13 pages, 2 tables. Script for the computer calculations used are provided on the author's websit

Moduli of sheaves and the Chow group of K3 surfaces

Let X be a projective complex K3 surface. Beauville and Voisin singled out a 0-cycle c_X on X of degree 1: it is represented by any point lying on a rational curve in X. Huybrechts proved that the second Chern class of a rigid simple vector-bundle on X is a multiple of the Beauville-Voisin class c_X if certain hypotheses hold and he conjectured that the additional hypotheses are unnecessary. We believe that the following generalization of Huybrechts' conjecture holds. Let M and N be moduli spaces of stable pure sheaves on X (with fixed cohomological Chern characters) and suppose that they have the same dimension: then the set whose elements are second Chern classes of sheaves parametrized by the closure of M (in the corresponding moduli spaces of semistable sheaves) is equal to the set whose elements are second Chern classes of sheaves parametrized by the closure of N after a translation by a suitable multiple of c_X (so that degrees match). We will prove that the above statement holds under some additional assumptions.Comment: Deleted a footnote and replaced it by a sentence in the main body of the pape

The Hodge numbers of O'Grady 10 via Ng\^o strings

We determine the Hodge numbers of the hyper-K\"ahler manifold known as O'Grady 10 by studying some related modular Lagrangian fibrations by means of a refinement of the Ng\^o Support Theorem.Comment: Revised and final version to appear in Jour. Math. Pur. et App

Automorphisms and autoequivalences of generic analytic K3 surfaces

This is a systematic exposition of recent results which completely describe the group of automorphisms and the group of autoequivalences of generic analytic K3 surfaces. These groups, hard to determine in the algebraic case, admit a good description for generic analytic K3 surfaces, and are in fact seen to be closely interrelated.Comment: 34 pages. Minor corrections. Final version to appear in J. Geom. Phy

Singularities of moduli spaces of sheaves on K3 surfaces and Nakajima quiver varieties

The aim of this paper is to study the singularities of certain moduli spaces of sheaves on K3 surfaces by means of Nakajima quiver varieties. The singularities in question arise from the choice of a non--generic polarization, with respect to which we consider stability, and admit natural symplectic resolutions corresponding to choices of general polarizations. For sheaves that are pure of dimension one, we show that these moduli spaces are, locally around a singular point, isomorphic to a quiver variety and that, via this isomorphism, the natural symplectic resolutions correspond to variations of GIT quotients of the quiver variety.Comment: 40 pages; final version; As pointed out to us by Z. Zhang, we prove quadraticity and not formality of the Kuranishi family. Quadraticity is all we need for our main theorem. The current version reflects this correction. A few other improvements in exposition and correction of typo

Fourier Mukai Transforms and Applications to String Theory

We give an introductory review of Fourier-Mukai transforms and their application to various aspects of moduli problems, string theory and mirror symmetry. We develop the necessary mathematical background for Fourier-Mukai transforms such as aspects of derived categories and integral functors as well as their relative version which becomes important for making precise the notion of fiberwise T-duality on elliptic Calabi-Yau threefolds. We discuss various applications of the Fourier-Mukai transform to D-branes on Calabi-Yau manifolds as well as homological mirror symmetry and the construction of vector bundles for heterotic string theory.Comment: 52 pages. To appear in Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. Minor changes, reference of conjecture in section 7.5 changed, references update