Birational Geometry and Derived Categories
Vienna August 1-6, 2012


V. Alexeev - Degenerations of surfaces of general type

I will describe a class of surfaces analogous to hyperelliptic curves for which all stable degenerations can be described explicitly, in a hope that they could be used for higher-dimensional examples of homological mirror symmetry. Based on a joint work with Rita Pardini.

M. Ballard - Exceptional collections on moduli spaces of pointed stable rational curves

Earlier this year, using results of Keel, Manin and Smirnov proved that \bar{M}_{0,n} possesses a full exceptional collection. We provide a different proof of this fact using variation of the GIT quotients, P^1[n]//PGL(2). P^1[n] is the Fulton-MacPherson compactification of the configuration space of n points in P^1. In the process, we also construct full exceptional collections for the Hassett moduli spaces of symmetrically weighted pointed stable rational curves and for the moduli spaces of weighted ordered points on P^1. This is joint work with David Favero and Ludmil Katzarkov.

C. Böhning - On the derived category of the classical Godeaux surface

We construct an exceptional sequence of length 11 on the classical Godeaux surface X which is the Z/5-quotient of the Fermat quintic surface in P^3. This is the maximal possible length of such a sequence on this surface which has Grothendieck group Z^11+Z/5. In particular, the result answers Kuznetsov's Nonvanishing Conjecture, which concerns Hochschild homology of an admissible subcategory, in the negative. The sequence carries a symmetry when interpreted in terms of the root lattice of the simple Lie algebra of type E_8. We also produce explicit nonzero objects in the (right) orthogonal to the exceptional sequence.

P. Cascini - Extending sections in positive characteristic

I will discuss some old and new methods to extend sections of a line bundle on a projective variety defined over an algebraically closed field of positive characteristic.

I. Cheltsov - Projecting Fanos in a nutshell

This is a first part of the talk on projecting Fano varieties. We revise the approach to the study of Fano varieties that is based on their projective geometry. We introduce basic links between Fano varieties and give a new simplified view on their classification and birational transformations. We state several problems related to the issue.

A. Corti - G2 manifolds and algebraic geometry

I explain how to construct many examples of compact 7-dimensional manifolds M with holonomy the exceptional Lie group G2, and associative submanifolds in them, starting from a pair of "asymptically cyclindrical" Calabi--Yau complex 3-folds Y1 and Y2, and glueing Y1*S^1 to Y2*S^1. In some cases it is possible to determine the diffeomorphism type of M. (Work with M Haskins, J Nordstrom, T Pacini.)

D. Deliu - Homological Projective Duality I

C. Diemer - Toric Mirror Symmetry and Birational Geometry

There is a well-known correspondence between the Mori chamber decomposition of a Fano toric variety and certain degenerations of hypersurfaces of its Batyrev mirror, i.e. an identification of Kahler and complex moduli. The compact complex moduli has the structure of a toric stack, a universal family, and a tautological hyperplane section. The one-dimensional strata give topologically interesting pencils of toric hypersurfaces. We consider these pencils from the perspective of (homological) mirror symmetry and their role as mirrors of the Mori program. This is joint with Katzarkov and Gabriel Kerr.

D. Favero - Variation of Geometric Invariant Theory for Derived Categories

Given a quasi-projective algebraic variety, X, with the action of a linear algebraic group, G, there are various (birational) incarnations of the quotient X/G coming from a choice of a G-equivariant ample line bundle. As we vary this choice, there is a semi-orthogonal relationship between the derived categories of the resulting quotients, A and B. Furthermore, if (X,w) is a Landau-Ginzburg model, and w is a G-invariant section of a line bundle on X, then the same holds for "coherent sheaves on" (A,w) and (B,w) (categories of matrix factorizations/categories of singularities/stable derived categories). As a special case, one can reproduce a theorem of Orlov relating categories of coherent sheaves for complete intersections in projective space to the graded category of singularities of the cone, a theorem of Herbst and Walcher demonstrating an equivalence of derived categories between "neighboring" Calabi-Yau complete intersections in toric varieties, and two theorems of Kawamata; one concerning behavior of derived categories of algebraic varieties under simple toroidal flips, the other stating that the derived category of coherent sheaves on any smooth toric variety has a full exceptional collection (in the projective case).

D. Halpern-Leistner - Categorification of Kirwan surjectivity and autoequivalences of derived categories

For a variety X acted on by a reductive group, one can consider the derived category of equivariant coherent sheaves on X, or the derived category of a GIT quotient of X. In this talk I will describe a relationship between these two categories: among other things the category of the GIT quotient can be embedded as a full subcategory of the equivariant category. I will explain how this categorifies several classical theorems on the cohomology of a GIT quotient. I will also apply this theory to produce explicit descriptions of derived autoequivalences of a GIT quotient which conjecturally correspond to monodromy on its complexified Kaehler moduli space.

F. Haiden - Bridgeland stability: a simple example

For the bounded derived category of representations of the A_n quiver, T. Bridgeland's axiomatic approach to stability for triangulated categories can be related, using a classification result of R. Nevanlinna and the notion of a t-structure, to polynomials of degree n+1. I will describe this correspondence in some detail. Part of joint work with L. Katzarkov and M. Kontsevich.

A. Iliev - The period map for prime Fano fourfolds of degree 10

Conics on the Fano fourfold X of degree 10 relate the geometry of X to O'Grady's double EPW sextics. We study the period map from the 24-dimensional moduli space of X to the 20-dimensional period domain of double EPW sextics by means of the birational geometry of X (common with O. Debarre and L. Manivel).

U. Isik - Homological Projective Duality II

A.-S. Kaloghiros - Geography of models and the Sarkisov Program.

The Sarkisov Program studies birational maps between Mori fibre spaces. In this talk, I will describe generators (work of Hacon-McKernan) and relations in the Sarkisov Program. I will show that these correspond to vertices and simplicial loops in a suitable complex .

L. Katzarkov - Moduli Spaces of Landau-Ginzburg Models

G. Kerr - Mirror symmetry of birational transformations

Abstract: In this talk I will discuss a conjectural homological mirror for flips, blow-ups and some Mori fibrations in the toric setting.

T. Logvinenko - Derived Reid's recipe for abelian subgroups of SL3(C)

The classical McKay correspondence is a 1-1 correspondence between non-trivial irreducible representations of a finite subgroup G of SL2(C) and irreducible divisors on the minimal resolution Y of C^2/G. Derived Reid's recipe is a direct generalisation of this to dimension three. We extract from the famous derived equivalence of Bridgeland-King-Reid a correspondence between irreducible representations of G ⊂ SL3(C) and exceptional subvarieties of Y = G-Hilb(C^3). In this talk, I describe joint work with Sabin Cautis and Alastair Craw in which we describe this correspondence completely for any abelian subgroup of SL3(C).

Y.-G. Oh - Lagrangian Floer theory on toric manifolds

P. Pandit - Moduli of objects in linear infinity categories

V. Przyjalkowski - Projecting Fanos in the mirror

This is a second part of the talk on projecting Fano varieties. We discuss basic links as pertrubations of polytopes, smoothings of toric varieties, and their connections with Mirror Symmetry. We are going to show this on a series of examples.

Y. Prokhorov - Subgroups of Cremona groups and Fano varieties

Let Cr(n) be the Cremona group in n variables. We discuss relations between finite subgroups of Cr(n) and (possibly singular) Fano varieties. Most of our results are in dimension 3.

H. Ruddat - Lagrangian skeleta and affine CY hypersurfaces

In a joint work with Zaslow, Treumann, Sibilla we construct a Lagrangian skeleton in the Kato-Nakayama space of the log central fibre of a toric degeneration of an affine Calabi-Yau hypersurface. This gives a Lagrangian model for the homotopy type of the hypersurface. We show that Gross' Hodge to Leray correspondence carries over to the non-compact case

I. Zharkov - Tropical conic bundles.

I will describe a technique how to compute the double cover of the dicriminant of tropical conic bundles with rationality questions in mind. Particular case of cubic threefold will be paid special attention.