Christian Lehn

Twisted cubics on singular cubic fourfolds—On Starr’s fibration

We show that the Hilbert scheme compactification of the total space of Starr’s fibration on the space of twisted cubics on a cubic hypersurface in

The complex geometry of a hypothetical complex structure on S 6

{\mathbb P}^5

Twisted cubics on cubic fourfolds

P5 not containing a plane admits a contraction to a singular projective symplectic variety of dimension eight which has a crepant resolution deformation equivalent to the symplectic eightfold constructed from twisted cubics on a smooth cubic fourfold. This yields another proof that the symplectic eightfold and the Hilbert scheme of four points on a K3 surface are deformation equivalent. As a byproduct we obtain similar results for the variety of lines on a singular cubic fourfold.

Lagrangian fibrations on hyperkähler manifolds – On a question of Beauville

In answer to the strong form of a?question posed by Beauville, we give a?short geometric proof that any hyperk?hler fourfold containing a?Lagrangian subtorus? admits a?holomorphic Lagrangian fibration with fibre?.

Base Manifolds for Lagrangian Fibrations on Hyperkähler Manifolds

This article is a survey about or introduction to certain aspects of the complex geometry of a hypothetical complex structure on the six-sphere. We discuss a result of Peternell--Campana--Demailly on the algebraic dimension of a hypothetical complex six-sphere and give some examples. We also give an overview over an application of Huckleberry--Kebekus--Peternell on the group of biholomorphisms of such a complex six-sphere.

DEFORMATIONS OF LAGRANGIAN SUBVARIETIES OF HOLOMORPHIC SYMPLECTIC MANIFOLDS

If

A global Torelli theorem for singular symplectic varieties

(G,V)

Normal crossing singularities and Hodge theory over Artin rings

is a polar representation with Cartan subspace