Montag, 18.10.10, 10:30-11:30, Institut de Recherche Mathématique Avancée 7 rue René Descartes 67084 Strasbourg Cedex
The monodromy conjecture for nondegenerate surface singularities.
Montag, 18.10.10, 11:45-12:45, Institut de Recherche Mathématique Avancée 7 rue René Descartes 67084 Strasbourg Cedex
The monodromy conjecture predicts a relation between the geometry and the topology of singularities. In particular, it says that a pole s0 of the local topological zeta function in 0 associated to a hypersurface induces an eigenvalue of monodromy e^{2i pi s0} at a point of the hypersurface in the neighbourhood of 0. When the singularity is given by a polynomial that is nondegenerate with respect to its Newton polyhedron, then one can express the local topological zeta function and the zeta function of monodromy in terms of the Newton polyhedron. We analyze these formulas for surface singularities: we provide a set of monodromy eigenvalues and a set of false candidate poles. In this way we obtain a proof for the monodromy conjecture for nondegenerate surface singularities.
Higher dimensional tautological inequalities and applications
Montag, 18.10.10, 14:30-15:30, Institut de Recherche Mathématique Avancée 7 rue René Descartes 67084 Strasbourg Cedex
We study the degeneracy of holomorphic mappings tangent to holomorphic foliations on projective manifolds. Using Ahlfors currents in higher dimension, we obtain several degeneracy statements. Joint work with C. Gasbarri and G. Pacienza.
On the Kaehler rank of compact complex surfaces
Montag, 18.10.10, 16:00-17:00, Institut de Recherche Mathématique Avancée 7 rue René Descartes 67084 Strasbourg Cedex
The Kaehler rank of compact complex surfaces was introduced by Harvey and Lawson in their 1983 paper on Kähler manifolds as a measure of (non-)kählerianity. It was not clear though whether the Kaehler rank was a birational invariant. The purpose of this talk is to show that it is one. This will follow from a partial classification of surfaces of Kähler rank 1.
K-trivial fibrations
Freitag, 22.10.10, 11:15-12:15, Raum 125, Eckerstr. 1
Kodaira obtained a formula expressing the canonical divisor\nof an elliptic surface in terms of the canonical divisor of the base curve, and two extra terms, measuring the variation of the generic fiber in its moduli space and the singularities of the degenerate fibers.\n In this talk I will discuss an extension of Kodaira's formula to arbitrary families of varieties with trivial canonical class. Such a formula may be used to reduce the study of varieties of intermediate Kodaira dimension\nto varieties of elliptic and hyperbolic type.\n
Antikanonische Divisoren und Hodgeklassen auf Fanomannigfaltigkeiten
Freitag, 29.10.10, 11:15-12:15, Raum 125, Eckerstr. 1
Transzendenzeigenschaften rationaler Integrale
Freitag, 12.11.10, 10:15-11:15, Raum 125, Eckerstr. 1
Frobenius push-forwards
Freitag, 19.11.10, 11:15-12:15, Raum 127, Eckerstr. 1
If \(X\) is a scheme of characteristic \(p>0\), we have the absolute\nFrobenius morphism \(F: X\to X\). For any coherent sheaf \(\mathcal{F}\) on \(X\)\nwe can then try to understand the sheaf \(F_* \mathcal{F}\) (the easiest example,\n\(F_* \mathcal{O}\) on \(\mathbb{P}1\), is already nontrivial).\nI will provide state-of-the-art results concerning these Frobenius\npush-forwards\non several classes of varieties and show their important applications.\nThe second part of the talk will be devoted to the case when \(X\) is\na smooth quadric hypersurface in \(\mathbb{P}^n\).\n
Freitag, 26.11.10, 11:15-12:15, Raum 125, Eckerstr. 1
Topologische Stringtheorie
Montag, 29.11.10, 16:00-17:00, Raum 125, Eckerstr. 1
Arithmetic properties of volumes of divisors
Freitag, 3.12.10, 11:15-12:15, Raum 125, Eckerstr. 1
Base change for the relative canonical sheaf in families of normal varieties
Freitag, 17.12.10, 11:15-12:15, Raum 127, Eckerstr. 1
In higher dimensional algebraic geometry one of the most important invariant\nof a variety is the canonical morphism, i.e., the morphism determined by\nsome high enough reflexive power of the canonical sheaf. For this reason, in\nhigher dimensional moduli theory, it is crucial to understand the base\nchange behavior of the relative canonical sheaf, and of its reflexive\npowers. This talk will focus on the base change behavior of the relative\ncanonical sheaf itself. It has been known for a while that it is compatible\nwith base change if the fibers are Cohen-Macaulay. Recently, compatibility\nhas been proven if the fibers are Du Bois. I will present a statement\nthat underscores the importance of these results by showing that this\ncompatibility does not hold generally in families of normal varieties, not\neven over smooth bases.
Nakayama-Zariski decomposition, finite generation and other stories
Freitag, 14.1.11, 11:00-12:00, Raum 127, Eckerstr. 1
In this talk, I will introduce the (b-divisorial) Nakayama-Zariski decomposition, and will show how, under the hypothesis of finite generation, it can be used to recover Shokurov's polytopes, to describe the relationship between different end products of the MMP on a klt pair (X, D). If time permits, I will show how these results can be used to give a different proof of the Sarkisov program.
Topology of Hitchin systems and Hodge theory of character varieties I
Freitag, 28.1.11, 11:00-12:00, Raum 127, Eckerstr. 1
Topology of Hitchin systems and Hodge theory of character varieties II
Freitag, 4.2.11, 11:00-12:00, Raum 127, Eckerstr. 1
Syzygies of curves on K3 surfaces (joint work with G. Farkas)
Freitag, 11.2.11, 11:00-12:00, Raum 127, Eckerstr. 1
Green's conjecture predicts that the shape of the Betti tables of canonical curves are completely determined by the Clifford indices. We present a proof of Green's conjecture for any smooth curve on an arbitrary K3 surface. This result has a particular interest, due to Green's hyperplane section theorem. Our result implies that the shapes of Betti tables of projective K3 surfaces are determined by the Clifford indices of corresponding hyperplane sections.\n
Group homology of SL_4 over polynomial rings
Freitag, 11.2.11, 11:00-12:00, Raum 127, Eckerstr. 1