Isotrivial elliptic K3 surfaces and Lagrangian fibrations

A fibration is said to be isotrivial if all of its smooth fibres are isomorphic to a single fixed variety. We classify the elliptic K3 surfaces that are isotrivial, and use them to construct Lagrangian fibrations that are isotrivial. We then modify the construction to produce new examples of holomorphic symplectic orbifolds, that also admit isotrivial Lagrangian fibrations.Comment: 17 page

Moduli spaces of sheaves on K3 surfaces

In this survey article we describe moduli spaces of simple, stable, and semistable sheaves on K3 surfaces, following the work of Mukai, O'Grady, Huybrechts, Yoshioka, and others. We also describe some recent developments, including applications to the study of Chow rings of K3 surfaces, determination of the ample and nef cones of irreducible holomorphic symplectic manifolds, and moduli spaces of Bridgeland stable complexes of sheaves.Comment: 24 pages, to appear in the Journal of Geometry and Physics special issue: proceedings of the "Workshop on Instanton Counting: Moduli Spaces, Representation Theory and Integrable Systems" (Leiden, 16-20 June 2014

Derived equivalence of holomorphic symplectic manifolds

We use twisted Fourier-Mukai transforms to study the relation between an abelian fibration on a holomorphic symplectic manifold and its dual fibration. Our reasoning leads to an equivalence between the derived category of coherent sheaves on one space and the derived category of twisted sheaves on the other space.Comment: 19 pages, to appear in the proceedings of the Workshop on algebraic structures and moduli spaces, CRM Montreal, July 200

Perturbative expansion of Chern-Simons theory

An overview of the perturbative expansion of the Chern--Simons path integral is given. The main goal is to describe how trivalent graphs appear: as they already occur in the perturbative expansion of an analogous finite-dimensional integral, we discuss this case in detail.Comment: This is the version published by Geometry & Topology Monographs on 22 April 200