Lundi 23 janvier 2012
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, January 23th in the conference room of the institute.
Organizer in Strasbourg : G. Pacienza
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Lundi 23 janvier 2012
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10:30
Kiwamu Watanabe, Tokyo
Lengths of chains of minimal rational curves on Fano manifolds
We consider a natural question how many minimal rational curves are needed to join two general points on a Fano manifold of Picard number 1. As an application, we give a better bound on the degree of Fano 5-folds of Picard number 1. If we have time, I also talk about a bound of the minimal length under mild conditions on the variety of minimal rational tangents. -
11:45
Frédéric Campana, Nancy
An alternative proof of Miyaoka's semi positivity theorem.
This result claims that $X$ smooth projective complex is uniruled if and only if its cotangent bundle is not generically semi-positive (ie: has a quotient of negative degree when restricted to a Mehta-Ramanathan curve relative to some polarisation of $X$). The original proof mixes char0 and char p>0 arguments in an intricated way. We show in char0 that the canonical bundle of $X$ is pseudo-effective iff its cotangent bundle is generically semi-positive. The very simple proof rests on Bogomolov-McQuillan's criterion for the algebraicity of leaves of a foliation, together with orbifold versions of the weak-positivity of direct image sheaves of powers of the canonical bundle, `a la Viehweg'. From Miyaoka-Mori's uniruledness criterion one recovers the original statement. The motivation for this new proof is that it extends immediately to the orbifold setting, crucial in the birational classification. This is a joint work with M. Paun -
14:30
Arvid Perego, Nancy
Moduli spaces of sheaves over abelian or projective K3 surfaces
Up to deformation, every known example of irreducible symplectic manifold is contructed from a moduli space of semistable sheaves on an abelian or projective K3 surface. If S is projective K3 surface, v is a primitive Mukai vector and H is a v-generic polarization, then the moduli space M_{v} of H-semistable sheaves on S with Mukai vector v is an irreducible symplectic manifold which is deformation equivalent to a Hilbert scheme of points on S. In this talk, I will present a joint work with A. Rapagnetta, in which we show that if v is no longer primitive, and M_{v} is singular and has a symplectic resolution, then this is an irreducible symplectic manifold which is deformation equivalent to the 10-dimensional O'Grady example. Similar results hold true when S is abelian. -
16:00
Alena Pirutka, Strasbourg
On some birational invariants for varieties over finite fields.
We will prove birational invariance of some cohomology groups coming from the Bloch-Ogus spectral sequence for varieties over finite fields. This uses the theory of cycle modules of Rost and the Kato conjecture. One of these invariants is related to the cokernel of the cycle class map for 1-cycles and we will explain an application to some local-global principle for 0-cycles.