# Trimester Seminar

**Venue:** HIM lecture hall, Poppelsdorfer Allee 45

Organizers: Nicolas Addington, Nathan Broomhead, Dennis Eriksson, Vladimir Lazic, Margherita Lelli-Chiesa, Francesco Polizzi

## Monday, January 20

16:30 - 18:00 Welcome Meeting

Abstract: Welcome meeting with the participants.

## Thursday, February 20

12:00 - 13:00 Sonke Rollenske: Stable Gorenstein surfaces with K_{X}^{^2}=1

Abstract: The moduli space of surfaces of general type admits a modular compactification, the moduli space of stable surfaces. Unlike in the case of curves, the geometry of stable surfaces is far from understood.

I will present partial results on the classification of Gorenstein stable surfaces with K_{X}^{^2}=1. This is joint work with Marco Franciosi and Rita Pardini.

## Thursday, March 6

10:30 - 12:00 Nicolas Addington: Spherical functors and P^{n}-functors

## Friday, March 7

13:30 - 15:00 Will Donovan: Window shifts

## Thursday, March 20

9:30 - 10:30 Marco Franciosi: Green's conjecture for singular curve

Abstract: In this talk I will explain some recent results obtainend with E. Tenni on the ananlysis of syszygies of canonically embedded singular curves. I am going to show an analysis of the relations among syzygies of a canonically embedded binary curve (i.e. curves consisting of two rational components) and rational normal scrolls containing such curve, which as a corollary yields a new proof of Green's conjecture.

## Friday, March 21

11:00 - 12:00 Gilberto Bini: Calabi-Yau varieties - not only threefolds

Abstract: In this talk we will describe new Calabi-Yau fourfolds containing a del Pezzo surface of degree 6. This is joint work with Matteo Penegini.

13:30 - 14:30 Remke Kloosterman: Hypersurfaces with defect

Abstract: Let X ⊂ P^{4} be a hypersurface of degree d, with at worst isolated semi-quasihomogeneous hypersurface singularities (e.g. ADE-singularities). We show that if X has defect, i.e., if h^{4}(X)>h^{2}(X) holds then the sum of the Milnor numbers of the singularities is at least (d-1)^{2} and that if equality holds then X contains a plane and has precisely (d-1)^{2} nodes. In the second half of the talk we explain how this result can be extended to the case of three-dimensional complete intersections in P^{n}.

## Thursday, March 27

10:30 - 12:00 Thomas Eckl: Numerical dimensions and Abundance Conjecture

Abstract: We discuss different notions of the numerical dimension of a pseudoeffective divisor, their relations, and how they can be used to reformulate the Abundance Conjecture.

## Thursday, April 3

10:30 - 12:00 Junyan Cao: Ohsawa-Takegoshi extension theorem for Kähler manifolds

Abstract: In this talk, we first prove a version of the Ohsawa-Takegoshi extension theorem valid for on arbitrary Kähler manifolds, and for holomorphic line bundles equipped with possibly singular metrics. As an application, we generalise Berndtsson and Paun's result about the pseudo-effectivity of the relative canonical bundles to arbitrary compact Kähler families.

## Tuesday, April 8

15:00 - 16:00 Nicolas Addington: Derived categories and motives

16:30 - 17:30 Stefano Urbinati: Minkowski decomposition of Okounkov bodies on Toric varieties

Abstract: We prove that for smooth projective toric varieties, the Okounkov body of a T-invariant pseudo-effective divisor with respect to a T-invariant flag decomposes as a finite Minkowski sum of indecomposable polytopes. We prove that these indecomposable polytopes form a finite Minkowski base and that they correspond to the rays in the secondary fan.

This work is in collaboration with P. Pokora and D. Schmitz.

## Thursday, April 10

10:30 - 12:00 John Christian Ottem: Ample subschemes and two conjectures of Hartshorne

Abstract: The talk will survey geometric properties of subvarieties with various positivity properties. We also discuss related conjectures of Hartshorne and Peternell about subvarieties with ample normal bundle.

## Friday, April 18

10:30 - 12:00 Olivier Benoist: Hodge conjecture for K3 surfaces