Holography and the representation theory of residue families
Friday, 28.4.17, 10:15-11:15, Raum 404, Eckerstr. 1
We shall introduce the concept of holography principle in\nconformal differential geometry. A prominent role in the \nanalysis is played by the residue family operators, and their\nrepresentation theoretical interpretation will be explained.
Quotients of the unit ball by unipotent discrete groups
Friday, 12.5.17, 10:15-11:15, Raum 404, Eckerstr. 1
We study actions of unipotent discrete groups on the unit ball in C^n and give a criterion which permits to decide when the associated quotient manifold is Stein.
Volumes of open surfaces
Friday, 26.5.17, 10:15-11:15, Raum 404, Eckerstr. 1
A volume of an open surface measures the rate of growth for\nthe number ofpluricanonical sections with simple poles at infinity. By Alexeev and Mori, there exists an absolute minimum for the set of positive volumes, with an explicit -- but unrealistically small -- bound. I will explain a related conjecture due to Kollár and some existing examples. Then I will explain a new candidate for the surface of the smallest volume, found in a joint work with Wenfei Liu.
Two-block Springer fibers and Springer representations in type D
Friday, 16.6.17, 10:15-11:15, Raum 404, Eckerstr. 1
We explain how to construct an explicit topological model for\nevery two-block Springer fiber of type D. These so-called topological\nSpringer fibers are homeomorphic to their corresponding algebro-geometric\nSpringer fiber. They are defined combinatorially using cup diagrams which\nappear in the context of finding closed formulas for parabolic\nKazhdan-Lusztig polynomials of type D with respect to a maximal parabolic\nof type A. As an application it is discussed how the topological Springer\nfibers can be used to reconstruct the famous Springer representation in an\nelementary and combinatorial way.
Algebraic models of the euclidean plane
Friday, 23.6.17, 10:15-11:15, Raum 404, Eckerstr. 1
A fake euclidean plane is a real algebraic surface whose complexification has the rational homology type of the plane and whose real locus is diffeomorphic to the euclidean plane, but which is not isomorphic as a real algebraic surface to the affine plane. In this talk, I will give elements of classification of such surfaces up to biregular isomorphisms of real algebraic varieties as well as up to birational diffeomorphisms, that is, algebraic birational maps whose restrictions to the real loci are diffeomorphisms. (Joint work with F. Mangolte (Angers) and J. Blanc (Basel)).
The Mumford-Tate conjecture for products of K3 surfaces
Friday, 30.6.17, 10:15-11:15, Raum 404, Eckerstr. 1
The Mumford-Tate conjecture relates the Hodge structure on the singular cohomology of an algebraic variety (over a number field) with the Galois representation on the etale cohomology of that variety. In this talk we explain a new technique that allows us to prove this conjecture for products of K3 surfaces. Along the way we also prove that the system of l-adic realisations of an abelian motive form a compatible system.\n
A decomposition theorem for the pushforwards of pluricanonical bundles to abelian varieties
Friday, 7.7.17, 10:15-11:15, Raum 404, Eckerstr. 1
I will describe a direct-sum decomposition in pull-backs of ample sheaves for the pushforwards of pluricanonical bundles via morphisms from smooth projective varieties to an abelian varieties. The techniques to proving this decomposition rely on generic vanishing theory, and the use of semipositive singular hermitian metrics. Time permitting, I will provide an application of the above decomposition towards the global generation and very ampleness properties of pluricanonical divisors defined on singular varieties of general type. The talk is based on a recent joint work with M. Popa and C. Schnell.
On a theorem of Campana and Paun
Friday, 14.7.17, 10:15-11:15, Raum 404, Eckerstr. 1
Let X be a smooth projective variety over the complex numbers, D a divisor with normal crossings, and consider the bundle of log one-forms 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 one-forms 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.\n\n
Representation theory in stable derivators and tilting bimodules
Thursday, 20.7.17, 14:15-15:15, Raum 403, Eckerstr. 1
I will discuss some classical concepts and results from the representation theory of finite dimensional hereditary algebras (reflection functors, Coxeter functors, Serre duality) and their incarnation in an arbitrary stable derivator. I will also show how to represent such functors and relations among them in terms of spectral tilting bimodules (these are rather small diagrams of spectra, in the sense of topology, with very favorable properties).\n\n
An Extension Theorem for differential forms on 4-dimensional GIT-quotients
Friday, 21.7.17, 10:15-11:15, Raum 404, Eckerstr. 1
Some consequences from Hodge theory in representation theory
Wednesday, 26.7.17, 12:00-13:00, Raum 119, Eckerstraße 1
Motivic Hodge modules and the Decomposition Theorem
Friday, 28.7.17, 10:15-11:15, Raum 404, Eckerstr. 1
Let \(p: X \bto S\) be a proper morphism of complex varieties. If we regard the higher direct image \(\boperatorname{R}^ip_{\bast} \bmathbf{Q}\) of the constant analytic sheaf \(\bmathbf{Q}\) as an \(S\)-parametrized family of mixed Hodge structures via the identifications \((\boperatorname{R}^ip_{\bast} \bmathbf{Q})_s = \boperatorname{H}^i(X_s(\bmathbf{C}),\bmathbf{Q})\), then M. Saito's theory of Hodge modules provides a categorical framework for studying such families and their functoriality. In this talk, we will explore an alternative framework for this inspired by \(\bmathbf{A}^1\)-homotopy theory. As an application, we sketch a proof of the Decomposition Theorem that avoids many of the subtleties of Saito's proof.