Topology of Algebraic Varieties (TAV)

The construction problem for Hodge numbers

Stefan Schreieder
University of Bonn
October 8, 2014
What are the possible Hodge numbers of a smooth complex projective variety? We construct enough varieties to show that many of the Hodge numbers can take all possible values satisfying the constraints given by Hodge theory. For example, there are varieties such that a Hodge number \(h^{p,0}\) is big and the intermediate Hodge numbers \(h^{i,p-i}\) are small.

Two counterexamples arising from infinite sequences of flops

John Lesieutre
Member, School of Mathematics
October 7, 2014
I will explain how infinite sequences of flops give rise to some interesting phenomena: first, an infinite set of smooth projective varieties that have equivalent derived categories but are not isomorphic; second, a pseudoeffective divisor for which the asymptotic multiplicity along a certain subvariety is infinite, in the relative setting.

Chow rings and modified diagonals

Kieran O'Grady
Sapienza - Università di Roma; Member, School of Mathematics
October 7, 2014
Beauville and Voisin proved that decomposable cycles (intersections of divisors) on a projective K3 surface span a 1-dimensional subspace of the (infinite-dimensional) group of 0-cycles modulo rational equivalence. I will address the following question: what is the rank of the group of decomposable 0-cycles of a smooth projective variety? Beauville and Voisin also proved a refinement of the result mentioned above, namely a decomposition (modulo rational equivalence) of the small diagonal in the cube of a K3.

Tropical currents

June Huh
Princeton University; Veblen Fellow, School of Mathematics
September 30, 2014
I will outline a construction of "tropical current", a positive closed current associated to a tropical variety. I will state basic properties of tropical currents, and discuss how tropical currents are related to a version of Hodge conjecture for positive currents. This is an ongoing joint work with Farhad Babaee.

Generic K3 categories and Hodge theory

Daniel Huybrechts
University of Bonn
September 16, 2014
In this talk I will focus on two examples of K3 categories: bounded derived categories of (twisted) coherent sheaves and K3 categories associated with smooth cubic fourfolds. The group of autoequivalences of the former has been intensively studied over the years (work by Mukai, Orlov, Bridgeland and others), whereas the investigation of the latter has only just began. As a motivation, I shall recall Mukai's classification of finite groups of automorphisms of K3 surfaces and its more recent derived version which involves the Leech lattice.

Hodge theory and derived categories of cubic fourfolds

Richard Thomas
Imperial College London
September 16, 2014
Cubic fourfolds behave in many ways like K3 surfaces. Certain cubics - conjecturally, the ones that are rational - have specific K3s associated to them geometrically. Hassett has studied cubics with K3s associated to them at the level of Hodge theory, and Kuznetsov has studied cubics with K3s associated to them at the level of derived categories.

I will explain all this via some pretty explicit examples, and then I will explain joint work with Addington showing that these 2 notions of having an associated K3 surface coincide generically.