Miniworkshop in Birational Geometry and Hodge Theory
Northwestern University
May 1314, 2017
Harris Hall L07
All are welcome to attend. Information about registration and funding can be found
here.
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Preliminary Schedule:
Saturday, May 13


9.30am  10.00am

Breakfast

10.00am  11.00am

Robert Lazarsfeld (Stony Brook University)
Ran's theorem on secant lines and measures of irrationality  Abstract
I'll start by presenting a simplified proof of a theorem of Ran's to the effect that the (n+2)secant lines to a smooth variety of dimension n sweep out a variety of dimension at most (n+1). Then I'll explain how this comes up in connection with bounding the "degree of irrationality" of a very general smooth hypersurface of large degree in projective space. This is joint work with Bastianelli, De Poi, Ein and Ullery.

11.30am  12.30pm

Mircea Mustata (University of Michigan)
A vanishing theorem for rational singularities  Abstract
Given a variety Z with rational singularities, and a log resolution
Y of Z, with exceptional divisor E, we conjecture the vanishing of the (n1)th higher direct image of the sheaf of differential forms on Y, with log poles along E. I will discuss a proof in the case of isolated singularities. This is joint work with Sebastian Olano and Mihnea Popa.

2.30pm  3.30pm

Sándor Kovács (University of Washington)
Superrational singularities  Abstract
Superrational singularities are a resolutionfree analogue of rational singularities. The main result discussed in this talk is that superrational singularities turn out to be equivalent to pseudorational singularities, and when resolution of singularities exists, then also to rational singularities. As applications of the main result, I will answer several open questions about the higher direct images of structure sheaves and dualizing sheaves, prove that CohenMacaulay klt singularities are superrational in arbitrary characteristic, and generalize Esnault's result on the existence of rational points on smooth Fano varieties to mildly singular log Fano varieties.

4.00pm  5.00pm

Christopher Hacon (University of Utah)
Birational characterization of abelian varieties and ordinary abelian varieties in characteristic p>0
 Abstract

Sunday, May 14


9.00am  9.30am

Breakfast

9.30am  10.30am

Christian Schnell (Stony Brook University)
On a theorem of Campana and Paun  Abstract
Let X be a smooth projective variety over the complex numbers, D a divisor with normal crossings, and consider the bundle of log oneforms on (X, D). I will explain a slightly simplified proof for the following theorem by Campana and Paun: If some tensor power of the bundle of log oneforms on (X, D) contains a subsheaf with big determinant, then (X, D) is of log general type. This result is a key step in the proof of Viehweg's hyperbolicity conjecture.

10.50am  11.50pm

Mihai Paun (UIC)
Algebraic fiber spaces and curvature of higher direct images  Abstract
We will report on a few
recent results obtained in collaboration with
Bo Berndtsson and Xu Wang.

1.40pm  2.40pm

Chenyang Xu (BICMR Beijing)
Stability on valuations of a KLT singularity  Abstract
In higher dimensional geometry, it has been known that from many perspectives a log terminal singularity is a local analogue of Fano varieties. Many statements of Fano varieties have a counterpart for log terminal singularities. One central topic on the geometry of a Fano variety is its stability which for instance reflects whether the Fano variety carries a canonical metric. In this talk, we will discuss a recent joint work with Chi Li (some part still in progress) in which we want to establish a local stability theory of a fixed log terminal singularity. Inspired by the study from differential geometry, (e.g. metric tangent cone, SasakianEinstein metric), for any log terminal singularity, we investigate the valuation which has the minimal normalized volume. Our goal is to prove various properties of this valuation which enable us to canonically degenerate the singularity to a Tsingularity (with a torus action) with "stability". If time permits, I will also discuss some applications.

3.00pm  4.00pm

János Kollár (Princeton University)
Deformations of stable varieties  Abstract
The talk will discuss, mostly through examples, the deformation theory of
singularities that occur on stable varieties.

Organizer:
Supported by the Nemmers Fund and the Northwestern University Department of Mathematics.