London Mathematical Society - EPSRC Durham Symposium
Interactions of Birational Geometry with other fields

Schedule of Talks
Room for all talks: CG93
(click twice on title to view abstract)

#### 2012-07-02 (Monday)

09:00 - 10:00 Registration at Mathematics Dept (CM201)

10:00 - 11:00 Mihai Paun: Extension of twisted pluricanonical forms: an overview

We will present a few results concerning the extension of twisted pluricanonical forms. In order to make the presentation as "friendly" as possible, we will only highlight the main ideas of the proofs, and try to explain the relevance of the analytic methods in this context.

11:00 - 11:30 Coffee

11:30 - 12:30 Jean-Pierre Demailly: A sharp lower bound for the log canonical threshold of an isolated plurisubharmonic singularity

We give a sharp lower bound for the log canonical threshold of a plurisubharmonic function $\varphi$ with an isolated singularity at $0$ in an open subset of ${\mathbb C}^n$. This threshold can be defined as the supremum of constants $c>0$ such that $e^{-2c\varphi}$ is integrable on a neighborhood of~$0$. We relate $c(\varphi)$ with the intermediate multiplicity numbers $e_j(\varphi)$, defined as the Lelong numbers of $(dd^c\varphi)^j$ at $0$ (so that in particular $e_0(\varphi)=1$). Our main result is that $c(\varphi)\ge\sum e_j(\varphi)/e_{j+1}(\varphi)$, $0\le j\le n-1$. This inequality is shown to be sharp; it~simultaneously improves the classical result $c(\varphi)\ge 1/e_1(\varphi)$ due to Skoda, as well as the lower estimate $c(\varphi)\ge n/e_n(\varphi)^{1/n}$ which has received crucial applications to birational geometry in recent years. The proof consists in a reduction to the toric case, i.e.\ singularities arising from monomial ideals. (joint work with Pham Hoang Hiep)

12:30 - 14:00 Lunch at Grey College

14:00 - 15:00 Simon Donaldson: Projective embeddings of Gromov-Hausdorff limits

(empty)

15:00 - 15:30 Coffee

15:30 - 16:30 Hajime Tsuji: Regularity of the twisted Kähler-Einstein currents and Relative pluricanonical systems of KLT pairs

In this talk, using the Dirichlet problems of complex Monge-Ampère equations, we show that the regularity of the relative twisted Kähler-Einstein currents on a family of KLT pairs. This enables us to generate a complex analytic foliations on the family. We apply this foliations to study relative pluricanonical systems.

17:00 - 18:00 Yuji Odaka: Birational geometry and K-stability

The K-stability of polarised variety (Tian, Donaldson) is a purely algebro-geometric notion of 1, 2 decades old. It is expected to correspond the existence of "canonical" Kahler metric on it, i.e. "canonical" way of regard variety as a differential geometric object, defined after its crowning historical background founded by: (Riemann..), Calabi, Aubin, Yau, Tian and many many other differential geometric works. Having origin of algebro-geometric side rather in (Hilbert-)Mumford's GIT theory, this notion has been unexpectedly coming out its crucial relation with recent MMP-theoritic birational geometry. E.g. discrepancy, log-canonical-threshold (with Y.Sano), birational superrigidity (with T.Okada), MMP with scaling (Li-Xu), KSBA type compact moduli. I'll roughly sketch these and would like to suggest a hard and important problem of judging K-stability explicitly by showing our partial understanding.

19:00 - 20:30 Dinner at Grey College

20:00 - 20:30 Wine Reception at Grey College (JCR)

#### 2012-07-03 (Tuesday)

09:30 - 10:30 Yuri Tschinkel: Birational geometry and arithmetic geometry

10:00 - 10:30 Coffee

11:00 - 12:00 Ekaterina Amerik: Iteration of rational self-maps: non-periodic algebraic points

(empty)

12:30 - 13:30 Lunch at Grey College

13:30 - 14:30 Frederic Campana: Gromov's Oka-Grauert principle and special manifolds

A complex connected manifold X is said to satisfy the h-principle if any continuous map from any Stein manifold S to X is homotopic to some holomorphic map from S to X. The origin lies in the works of Oka and Grauert on classification of line and vector bundles on Stein S. Gromov showed that elliptic' X satisfy this h-principle. 'Elliptic' manifolds in Gromov's sense are C-connected (ie: any two points can be joined by a chain of entire curves. We raise the question whether (non-contractible) X's satisfyng the h-principle are C-connected, and show that such an X: 1. has no holomorphic non null-homotopic map to any Brody hyperbolic space Y. 2. is special' if projective. The link is that, conjecturally, special' projective manifolds are exactly the ones which are C-connected. Specalness' roughly means `opposite' to general type, and is defined by algebro-geometric conditions highly relevant to the birational classificatio of projective manifolds and the LMMP. The cases of quasi-projective (or even quasi-Kaehler) manifolds might be accessible by more refined techniques.

14:30 - 15:00 Coffee

15:00 - 16:00 Atsushi Moriwaki: Birational Arakelov geometry

Roughly speaking, the purpose of birational geometry is studies of big linear series. Birational Arakelov geometry is an arithmetic analogue of the above analyses. In this talk, I will give an overview of the recent developments of birational Arakelov geometry.

16:30 - 17:30 Damiano Testa: The Büchi K3 surface and its rational points

(empty)

19:00 - 20:30 Dinner at Grey College

#### 2012-07-04 (Wednesday)

09:00 - 10:00 Stéphane Lamy: Birational self-maps and piecewise algebraic geometry

If f is a birational selfmap of P^3, the exceptional set Exc(f) is the algebraic set (here a reducible surface) where f is not a local isomorphism. We observe on examples (I will give one) that the exceptional sets of f and of the inverse of f have no reason to be isomorphic. However in a joint work with Julien Sebag we prove that these exceptional sets are always piecewise isomorphic. This is a particular case of a more general conjecture stating that two varieties with the same class in the Grothendieck ring K_0[Var] should be piecewise isomorphic.

10:00 - 10:30 Coffee

10:30 - 11:30 Yuri Prokhorov: Subgroups of Cremona groups and Fano varieties

(empty)

12:00 - 13:00 Igor Dolgachev: Algebraic surfaces with large automorphism group

An automorphism group of a smooth projective algebraic surface acts naturally on the Neron-Severi group of algebraic 2-cycles preserving the intersection form and the canonical class. It is called large if its image $G$ in the group $O$ of isometries of the orthogonal complement of the canonical class is an infinite group of finite index. I will explain a recent result of my joint work with Serge Cantat where we proof that the Picard number of a surface with a large automorphism group is at most 11. In the case of rational surfaces with Picard number $> 11$, the image of the automorphism group is contained in a certain reflection subgroup $W$ of infinite index in $O$. We classify all rational surfaces such that $G$ is of finite index in $W$. In characteristic 0 they are classically known as the Coble or Halphen surfaces.

13:15 - 13:30 Symposium Photograph at Grey College

13:30 - 14:30 Lunch at Grey College

14:30 - 16:00 Tour of Durham Castle

19:00 - 19:30 Pre-dinner drinks

19:30 - 21:00 Conference Dinner

#### 2012-07-05 (Thursday)

09:30 - 10:30 Osamu Fujino: Birational geometry and Hodge theory

(empty)

10:30 - 11:00 Coffee

11:00 - 12:00 Burt Totaro: Symmetric differentials and the fundamental group

Consider a smooth complex projective variety X. Hodge theory shows that sections of exterior powers of the cotangent bundle are related to the topology of X. What about symmetric powers of the cotangent bundle? We discuss the relation between the topology of X and its "symmetric differentials". One interest of these results is that symmetric differentials give information in the direction of "Kobayashi hyperbolicity"; for example, they limit how many rational curves X can contain. This is joint work with Yohan Brunebarbe and Bruno Klingler.

12:30 - 13:30 Lunch at Grey College

13:30 - 14:30 Nick Shepherd-Barron: Compactifications of M_g and A_g

(empty)

14:30 - 15:00 Coffee

15:00 - 16:00 James McKernan: The moduli space of canonical polarised varieties, I

16:30 - 17:30 Chenyang Xu: The moduli space of canonical polarised varieties, II

19:00 - 20:30 Dinner at Grey College

#### 2012-07-06 (Friday)

09:30 - 10:30 Yujiro Kawamata: Birational geometry and derived categories

(empty)

10:30 - 11:00 Coffee

11:00 - 12:00 Ludmil Katzarkov: Stability Hodge Structures and applications

In this talk we will introduce new Hodge theoretic notion and consider some applications to classical geometric questions.

12:30 - 13:30 Lunch at Grey College

13:30 - 14:30 Alexey Bondal: Noncommutative blow-down representation theory

14:30 - 15:00 Coffee

15:00 - 16:00 Nero Budur: Birational geometry and singularities

Singularities are an essential tool in birational geometry, and, viceversa, birational methods are important for understanding singularities. In this talk we will present a few open problems in singularity theory related to birational geometry, and discuss current efforts to solve them.

16:30 - 17:30 Paolo Cascini: Lifting sections in positive characteristic

The Kawamata-Viehweg-Nadel vanishing theorem has played a crucial role in the study of the birational geometry of complex projective varieties, but unfortunately it does not hold in general for varieties defined over a field of positive characteristic. We will discuss a new lifting theorem which hold on any algebraically closed field and some of its applications.

19:00 - 20:30 Dinner at Grey College