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.