Mitglieder

Lehre

Forschung / SFB 647

Vorträge/Veranstaltungen

Kontakt

Impressum

Seminar: Algebraische Geometrie

Seminartermine:

18.10.2005	Lutz Hille (Hamburg, Bielefeld)
25.10.2005	SFB
01.11.2005	Michele Bolognesi (Montpellier)
08.11.2005	Stefano Chiantese (1)
15.11.2005	SFB
22.11.2005	Stefano Chiantese (2)
29.11.2005	Diane Maclagan (Rutgers, NJ)
06.12.2005	SFB
13.12.2005	Andras Szenes (Budapest)
03. 01.2006	Robert Vollmert (1) [aequivariante Schnittheorie]
10. 01.2006	SFB
17. 01.2006	Robert Vollmert (2)
24. 01.2006	Lorenz Wotzlaw (HU Berlin)
31. 01.2006	SFB
07. 02.2006	Thomas Zink (Bielefeld)
14. 02.2006	Alexej Bondal (Steklov, Moskau)
14. 03.2006	Davis Doherty (Seattle)
14. 03.2006	Zachary Treisman (Seattle)

Abstracts:

Zusammenfassung:
It is well known that the threefold $\mathcal{B}\subset\mathbb{P}^4$ known as the B\"{u}rkhardt quartic is deeply related to moduli spaces of abelian surfaces with some 3-level structure, in particular to $\mathcal{A}_2(3)$ and $\mathcal{A}_2(3,6)$. We show another such relation with $\mathcal{A}_2(3)^-$, a moduli space that we introduce, parametrizing principally polarized abelian surfaces with a symmetric theta structure and the choice of an odd theta characteristic. We shall build the arithmetic group that defines $\mathcal{A}_2(3)^-$ as a quotient of the Siegel half-space and prove that it is birational to $\mathbb{P}^3$ thus giving a moduli interpretation to the classical unirationalization of $\B$ given by Weddle quartics. We also investigate the relation between these quartics, Kummer surfaces and the Igusa quartic.

Zusammenfassung:
In this talk I review some basic notions about topological string. One of its versions (the A-type) is interesting from a mathematical point of view since it counts holomorphic maps from curves to a Calabi-Yau manifold. I will also introduce the recently discovered generalized notion of topological string, which is closely related to generalized complex geometry introduced by Hitchin.

Die Schnitttheorie untersucht den einer glatten Varietät zugeordneten Chow-Ring, der das Schnittverhalten von Untervarietäten beschreibt. Inhalt der äquivarianten Schnitttheorie ist eine Verallgemeinerung auf Varietäten mit algebraischer Gruppenwirkung.
Im zweiten Teil des Vortrags wird der äquivariante Chow-Ring einer glatten Varietät mit Gruppenwirkung definiert. Weiterhin stellen wir einige Beispiele und Anwendungen vor.

If $X=V(F)$ is a smooth hypersurface in $\mathbb{P}^n$, the Jacobian ring of $F$ calculates the graded parts of the Deligne-Hodge-structure on the primitive cohomology:

\[ Gr^{p-1}_FH_0^{n-1}(X)\stackrel{\sim}{\longrightarrow} (\mathbb{C}[x_0,\ldots,x_n]/ \langle\partial_0F,\ldots,\partial_nF\rangle)_{(n-p)\text{deg}(X)-n-1} \]

This is a well-known result of P. Griffiths, which extends to case of (quasi-) smooth complete intersections in toric varieties. It allows to represent Hodge-theoretical objects related to families of such varieties, like Higgs-bundles and Yukawa-couplings. Among the applications are global Torelli theorems and curve-counting in mirror symmetry.
The aim of the talk is to relax the smoothness condition and give a generalized ``Griffiths calculus'' for intersection cohomology of (families of) hypersurfaces in $\mathbb{P}^n$ with isolated singularities. The proof will use the theory of $D$-modules, namely mixed Hodge modules of M. Saito. A special focus will be on nodal threefolds in $\mathbb{P}^4$; here intersection cohomology is isomorphic to ordinary cohomology of a small resolution.