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Abstracts

Asher Auel (Bonn)

A Milnor conjecture for p-adic curves

The Brauer group of a field is an important arithmetic invariant. In 1981, Merkurjev proved that every 2-torsion Brauer class is represented by the Clifford algebra of a quadratic form, establishing part of the Milnor  conjecture. In the 1950s and 60s, Azumaya, Auslander, Goldman, and Grothendieck investigated Brauer groups of rings and schemes. In the 1990s, Parimala, Scharlau, and Sridharan found complete p-adic curves X for which Merkurjev's theorem fails, namely, there exist 2-torsion Brauer classes on X not represented by Clifford algebras of quadratic forms on X. We'll discuss how replacing quadratic forms by line bundle-valued quadratic forms salvages Merkurjev's theorem for these curves.


Jan-Hendrik Bruinier (Darmstadt)

CM Values of Modular Forms

We begin by recalling some basic facts of the theory of complex multiplication. A complex elliptic curve with complex multiplication by the ring of integers of an imaginary quadratic field K is defined over the Hilbert class field H of K. The j-invariant of such an elliptic curve is an algebraic integer which generates H. We report on work of Gross and Zagier on the norms of such singular moduli and on work of Zagier and others on the traces of singular moduli. A different way of viewing this is that we are studying the CM values of the classical j-function, which is a weakly holomorphic modular form of weight zero for the group SL_2(Z). Next we study the CM values of the classical discriminant function, the unique cusp form of weight 12 for SL_2(Z). As an application we consider Colmez' formula for the Faltings height of a CM elliptic curve. Finally, we report on joint work with T. Yang on CM values of Borcherds products on Shimura varieties associated to orthogonal groups. This provides a conceptual approach and a common generalization of our earlier examples.

 

 

Andre Chatzistamatiou (Essen)

Witt rationality of quotient singularities

We report on work in progress (joint with Kay Rülling). The aim is to show that quotient singularities are Witt rational.

 

Barbara Fantechi (Trieste)

Virtual pull-backs on algebraic stacks

a short summary of Fulton's intersection theory.
a short summary of algebraic stacks.
the normal cone (stack) to a DM type morphism
lci Gysin pullback recast in terms of normal cone stacks
virtual pullbacks and virtual fundamental classes
deformation-theoretic construction of obstruction theories
example: Gromov Witten invariants and their properties
a useful trick due to Costello
example: GW degeneration formula
if time allows: virtual structure sheaf and virtual Riemann Roch theorems.

 

Philipp Gross (Düsseldorf)

The resolution property of algebraic surfaces.

We prove that on separated algebraic surfaces every coherent sheaf is aquotient of a locally free sheaf. This class contains many schemes that are neither normal, reduced, quasiprojective or embeddable into toric varieties. Our methods extend to arbitrary two-dimensional schemes that are proper over a noetherian ring.

 

Daniel Greb (Freiburg)

Differential forms on singular varieties

Given an algebraic variety X and a resolution of singularities Z of X with exceptional set E it is a natural (old) question whether, or under which additional assumptions, regular differential forms defined on the smooth part of X extend over E to regular differential forms on Z. 

After discussing examples showing that extension is not possible in general, I will introduce and discuss (log-)canonical singularities and explain the following result: extension (with logarithmic poles) holds for varieties with (log-)canonical singularities. Special emphasis will be put on applications of this result to the birational geometry of higher-dimensional varieties. The talk is based on joint work with Kebekus, Kovács, and Peternell.

 

Thilo Henrich (Bonn)

Simple bundles on genus one curves  and rational solutions of the classical Yang--Baxter equation

Following Polishchuk and Burban, Kreußler, we shall explain how to produce solutions of the classical Yang-Baxter equation by computing certain triple Massey products in the derived category of coherent sheaves on a (singular) projective curve of arithmetic genus one.  I shall discuss the representation theory of the simple vector bundles on degenerations of elliptic curves necessary for the computations of these products, generalizations of this theory on the case of reducible curves and present explicit formulas for solutions of the classical, quantum and associative Yang-Baxter equations depending only on the "topological" invariants of vector bundles.

 

Andreas Höring(Paris)

Minimal Model Program: Theory and Applications

The minimal model program (MMP) is a classification theory for projective varieties which was founded by Kawamata, Koll\'ar, Mori, Shokurov and many other in the 1980s and has recently made some impressive progress due to Hacon and McKernan. The main object of these lectures is  not to give a general introduction to the topic, but rather to explain how techniques coming from the MMP can be applied in different contexts. I will show how singularities of pairs can be used to study linear systems on projective varieties and give an application to integral Hodge classes on Fano manifolds.

 

Donatella Iacono (Rom)

Modern Deformation Theory

In this talk, we will give an overview on deformation theory and relation with moduli spaces. In particular we will focus our attention on differential graded Lie algebras and L-infinity algebras, to get  information on deformation problems.


Franz Kiraly (Ulm)

Wild quotient singularities of surfaces and their regular models

In my talk, I will discuss the concept of quotient singularities and its meaning in the field of models of arithmetic surfaces. In the first part of my talk, I will look at some easy examples of quotient singularities and the regularity criterion of Serre in the tame case. Then I will present some examples and recent results in the wild case of prime order. In the second part, I will show how one can use the theory of quotient singularities to relate minimal regular models of curves over different ground fields to each other.

 

Sven Meinhardt (Bonn)

Lambda-rings and motivic Donaldson-Thomas invariants

The most conceptual approach to (motivic) Donaldson-Thomas invariants is given in terms of Lambda-rings of motives. I will explain the well know notion of Lambda-rings with a very recent example, namely the ring of motives of algebraic stacks. Knowing the precise notion of stacks is not necessary as things are much easier from the motivic point of view. Finally, we will use this structure to define motivic Donaldson-Thomas invariants and to relate them to their classical counterparts.

 

Arvid Perego (Mainz)

The O'Grady moduli spaces

The theory of moduli spaces of sheaves over abelian or K3 surfaces is an important tool to produce examples of irreducible symplectic manifolds. Let S be a projective K3 surface, v a Mukai vector on S, and H a v-generic polarization on S: if v is primitive, the moduli space M_{v}(S,H) is irreducible symplectic, deformation equivalent to an Hilbert scheme of points on S, and its second integral cohomology is Hodge isometric to v^{\perp}. If v=2w, where w is primitive and w^{2}=2, then M_{v}(S,H) admits a symplectic resolution \widetilde{M}_{v}(S,H). A particular case is v=(2,0,-2), example studied by O'Grady, who shows that the symplectic resolution is irreducible symplectic and not deformation equivalent to any previously known example. In a joint work with Antonio Rapagnetta, we show that \widetilde{M}_{v}(S,H) is irreducible symplectic, deformation equivalent to the O'Grady example, and that the second integral cohomology of M_{v}(S,H) is Hodge isometric to v^{\perp}.

 

Joseph Ross (Essen)

The Hilbert-Chow morphism and the incidence divisor

Let P be a smooth projective variety of dimension n, and a,b nonnegative integers such that a+b+1=n. I will explain how to construct in the product of Chow varieties C_a(P) x C_b(P) a Cartier divisor supported on the locus of intersecting cycles, and also what might lead one to do this. The main idea is to define the incidence line bundle on the Hilbert schemes mapping to the Chow varieties, and then show this bundle descends.

 

Kay Rülling (Essen)

Rational points over finite fields for regular models of Hodge type at least 1

 Let X be a regular scheme, which is proper and flat over a DVR of mixed characteristic with finite residue field k. Assume its generic fiber is geometrically connected and has Hodge type at least 1, i.e. the cohomology of its structure sheaf vanishes in all positive degrees. If X has semi-stable reduction, i.e. the special fiber of X is a divisor with normal crossings, then it follows from p-adic Hodge theory, that the number of k-rational points of the special fiber of X is congruent to 1 modulo the cardinality of k; in particular X has a k-rational point. I will explain a proof of this congruence without the semi-stable reduction assumption. This is joint work with Pierre Berthelot and Hélène Esnault.


Timo Schürg (Mainz)

Virtual pullbacks for quasi-smooth derived Deligne-Mumford stacks

After a brief introduction to derived algebraic geometry I will show that for a morphism of quasi-smooth derived Deligne-Mumford stacks virtual pullbacks always exist. This differs from the case of virtually smooth Deligne-Mumford stacks presented in the course of Prof. Fantechi, were an additional compatibility condition had to be imposed. For morphisms of derived stacks this compatibility condition is automatically satisfied.

 

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