Abstracts for Seminar Algebraic Geometry (SAG)

We construct the mirror dual of the cubic threefold using a duality of Landau Ginzburg models that has
been developed in a joint work with Mark Gross and Ludmil Katzarkov for the study of mirror duals
of varieties of general type. We show that our five-dimensional model is equivalent to the three-dimensional
ones from the literature and study its geometry in detail, verify a conjecture of Katzarkov, discuss the
cohomology of the sheaf of vanishing cycles, homological mirror symmetry and Orlov's theorem.

The Beauville-Bogomolov decomposition theorem asserts that any compact Kähler manifold
with trivial canonical bundle is finitely covered by the product of a compact complex torus,
simply connected Calabi-Yau manifolds, and simply connected irreducible holomorphic
symplectic manifolds. The decomposition of the étale cover corresponds to a decomposition
of the tangent bundle into a direct sum, whose summands are integrable and stable with
respect to any polarization. Building on recent extension theorems for differential forms on

Whenever a variety X lifts from characteristic p to characteristic zero, say over the Witt ring, then many classical results over the complex numbers hold for X, and certain "characteristic p pathologies" cannot occur, simply because one can reduce modulo p (I will discuss this in examples). But then, lifting results are difficult, and generally, varieties do not lift. However, in many situations, it is possible or easier to lift a birational model of X, maybe even one that has "mild" singularities (again, I will give examples).