The cotangential and the derived de Rham complex in the h-topology
Tuesday, 2.2.16, 10:15-11:15, Raum 404, Eckerstr. 1
Friday, 5.2.16, 10:00-11:00, Raum 404, Eckerstr. 1
Abundance conjecture for varieties with many differential forms
Friday, 5.2.16, 10:00-11:00, Raum 404, Eckerstr. 1
The abundance conjecture and the existence of good models are the main open problems in the Minimal Model Program in complex algebraic geometry. Even though it is completely proved in dimension 3, almost nothing has been known in higher dimensions. In this talk, I will discuss my recent joint work with Thomas Peternell, where we prove that the abundance conjecture holds on a variety with mild singularities if it has many reflexive differential forms with coeffi cients in pluricanonical bundles, assuming the Minimal Model Program in lower dimensions. Under this assumption, the result has several consequences: for instance, that hermitian semipositive canonical divisors are almost always semiample. When the numerical dimension of the canonical sheaf is 1, our results hold unconditionally in every dimension.
Effective Matsusaka for surfaces in positive characteristic
Friday, 12.2.16, 10:00-11:00, Raum 404, Eckerstr. 1
The problem of determining an effective bound on the multiple which makes an ample divisor D on a smooth variety X very ample is natural and many results are known in characteristic zero. In this talk, based on a joint paper with Gabriele Di Cerbo, I will discuss this problem on surfaces in positive characteristic, giving a complete solution in this setting. \nOur strategy requires an ad hoc study of pathological surfaces, on which Kodaira-type theorems can fail. A Fujita-type theorem and a vanishing result for big and nef divisors on pathological surfaces will also be discussed.