Lundi 14 novembre 2011
IRMA
The "Joint Seminar in Algebraic and Complex Geometry" is a research seminar, organized by the research groups in Freiburg, Nancy and Strasbourg. The seminar meets roughly twice per semester in Strasbourg, for a full day. There are about four talks per meeting, both by invited guests and by speakers from the organizing universities. We aim to leave ample room for discussions and for a friendly chat.
The talks are open for everyone. Contact one of the organizers if you are interested in attending the meeting. We have some (very limited) funds that might help to support travel for some junior participants.
The seminar will meet in Strasbourg, November 14th in the conference room of the institute.
Organizer in Strasbourg : G. Pacienza
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Lundi 14 novembre 2011
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10:30
Daniel Greb, Freiburg
Singular spaces with trivial canonical class
The Beauville-Bogomolov decomposition theorem asserts that any compact Kähler manifold with trivial canonical bundle is finitely covered by the product of a compact complex torus, simply connected Calabi-Yau manifolds, and simply connected irreducible holomorphic symplectic manifolds. The decomposition of the non-toral part of the étale cover corresponds to a decomposition of the tangent bundle into a direct sum whose summands are integrable and stable with respect to any polarization. Building on recent extension theorems for differential forms on singular spaces, we prove an analogous decomposition theorem for the tangent sheaf of a projective variety with canonical singularities and numerically trivial canonical class. This is joint work with Stefan Kebekus and Thomas Peternell. -
11:45
Pierre-Emmanuel Chaput, Nancy
Classical-quantum principle for the cohomology and the K-theory of minuscule spaces.
Minuscule spaces form a class of homogeneous spaces containing Grassmannians. We will see that for such a space G/P there exists
an auxiliary homogeneous space G/Q such that the Gromov-Witten invariants for G/P (which count rational curves) equal the classical intersection numbers on G/Q.
A version of this result concerns intersection numbers in K-theory. These two results are corollaries of geometric results on G/P, such as the rationality
of the locus of rational curves passing through 3 fixed points. -
14:30
Nicolas Perrin, Bonn
Finitness of Quantum K-theory for homogeneous spaces
For a Fano variety with Picard rank one, and in particular for rational homogeneous spaces with Picard rank one, the quantum cohomology product is -- by definition -- a polynomial in the quantum parameter q. For the quantum K-theoretical product, this is not obvious any more. In this talk I shall explain, for rational homogeneous spaces, how the rational connectedness of some subvarieties of the moduli space of stable maps imply that the quantum K-theoretical product is polynomial in q. -
16:00
Luc Pirio, Rennes
Extremal projective varieties n-covered by curves of a fixed degree
We study extremal projective varieties that are `n-covered' by curves of a fixed degree d. In the first part of the talk, I will present general results on such varieties. In particular, I will state their classification when d is distinct from 2n-3. In the second part, I will focus on the more intersesting case when d=2n-3. In particular, for n=3, I will explain that there are natural correspondences between the following objects: -- nondegenerate r-dimensional projective varieties in P^{2r+1} that are 3-covered by twisted cubics; -- r-dimensional complex Jordan algebra of rank 3; -- quadro-quadric Cremona transformations of P^{r-1}. If time allows, I will deduce from these equivalences general structure theorems for quadro-quadric Cremona transformations or for extremal varieties 3-covered by cubics. (Talk based on joint works with J.-M. Trépreau on one hand and with F. Russo on the ot her hand).