Triangulated categories of mixed motives
Monday, 27.10.08, 11:15-12:15, Raum 404, Eckerstr. 1
Lecture 1 will deal with the explicit construction\nof the categories DM(S) for a general scheme S\nand its basic properties.
Localization of mixed motives
Tuesday, 28.10.08, 11:15-12:15, Raum 403, Eckerstr. 1
Lecture 2 will be about localization in DM(S):\nif i : Z --> S is a closed immersion with complementary\nopen immersion j : U --> S, for any object M of DM(S),\nthere is a distinguished triangle of shape\nj!j^*(M) --> M --> ii^(M) --> j_!j^*(M)[1]\n(this holds with rational coefficients in general,\nand with integral coefficients if S and Z are smooth\nover a common base). Using Ayoub's thesis, localization\nallows to produce the six Grothendieck operations\non the categories DM(X) (with rational coefficients).\nThe aim of the talk is to give an idea of the proof\nof localization (which is quite tricky).\nEven if I suspect there won't be enough time to\nmention this during the talk, I mention here some other\nnice consequences: localization property also implies that\nrational motivic cohomology of regular schemes defined\nfrom DM coincide with motivic cohomology in the sense\nof Beilinson (using Adams operations on algebraic K-theory),\nby reducing to the case of a field, which is already known.\nThis and a version of Riemann-Roch in turns implies an absolute\npurity theorem in DM. These consequences have their\nrole to play in lecture 3.
l-adic completion of etale motives
Wednesday, 29.10.08, 14:15-15:15, Raum 403, Eckerstr. 1
Lectures 3 will be about the étale version DMet(X)\nof DM(X), which has nice properties (at least for schemes X\nwhich are of finite type over a an excellent regular scheme\nof Krull dimension less or equal to 1):\n- it coincides with DM(X) up to torsion\n- localization is true in DMet(X) (even with integral\ncoefficients)\n- the absolute purity theorem is true in DMet(X)\n- the torsion part of DMet(X) is essentially\nthe category of torsion l-adic sheaves\n- using the l-adic completion in DMet(X), one constructs\na nice l-adic realization from the rational version of DM(X)\ninto smooth Ql-sheaves which commutes with the\nGrothendieck six operations.
Linear Algebraic Groups and A^1-Homotopy Theory
Friday, 14.11.08, 11:15-12:15, Raum 403, Eckerstr. 1
In this talk, I will explain how elementary matrix factorizations lead to a computation of the A^1-homotopy groups of linear algebraic groups and homogeneous spaces. Besides a better understanding of A^1-homotopy groups, these results have implications for questions on stabilization of group homology and algebraic K-theory. I will try my very best to make this talk locally accessible to anybody. There will be a couple of follow-up talks concerning model categories in general and A^1-homotopy in particular which will provide a more detailed introduction to this subject.
An Introduction to Model Categories
Friday, 21.11.08, 11:15-12:15, Raum 403, Eckerstr. 1
Cancelled. This talk will be rescheduled. I apologize for any inconveniences.
tba
Friday, 5.12.08, 11:15-12:15, Raum 403, Eckerstr. 1
Garben auf affinen Schubert Varietäten, modulare Darstellungen und Lusztig's Vermutung
Friday, 12.12.08, 11:15-12:15, Raum 403, Eckerstr. 1
George Lusztig stellte 1980 eine Vermutung über die Charaktere einfacher Darstellungen von algebraischen Gruppen über einem Körper positiver Charakteristik auf und formulierte 1990 ein Programm, das zu einem Beweis in fast allen Charakteristiken führte. Die Ausnahmecharakteristiken sind jedoch bis heute unbekannt. Im Vortrag möchte ich einen neuen, deutlich einfacheren Beweis vorstellen, der einen Teil der Vermutung für alle relevanten Charakteristiken liefert und zudem eine obere Grenze für die Ausnahmen. Das wesentliche Hilfsmittel hierzu ist die Theorie von Garben auf Bruhatgraphen.
Stable couples on elliptic K3 surfaces
Friday, 19.12.08, 11:15-12:15, Raum 403, Eckerstr. 1
We consider a smooth K3 elliptic surface S with a section and we investigate the behaviour of moduli spaces of pairs on it. For a suitable choice of the framing, we get a finite family of moduli spaces related by wall crossing phenomena giving rise to birational maps. In a particular case, this allows to recover an isomorphism (described by Friedman with different techniques) between a moduli space of rank two coherent sheaves on S and the Hilbert scheme.
Rational curves and asymptotic base locus of some divisors
Friday, 9.1.09, 11:15-12:15, Raum 403, Eckerstr. 1
I will present two joint works with Gianluca Pacienza and also Sébastien Boucksom for one of them.\n\nThe geometry of the rational curves on a variety is a fundamental tool for the classification of complex projective varieties.\nRecently, two notions of base locus attached to divisors/complex line bundles have been discovered and studied by Nakayama, Boucksom, Nakamaye and others.\nOne of them, the non-nef locus, is as suggest his name, empty if and only if the divisor (or the first chern of the line bundle) is nef. That is, it has non-negative instersection on every curve in the variety. We will show how this base loci are related to the geometry of rational\ncurves on a variety. Namely, I will give a survey of the proof of the two following statements:\nIf X is a projective variety with nef anticanonical bundle (dual of the canonical bundle) then every non-nef base locus of a divisor is uniruled. It is a generalization of a result of Takayama who proved this result in the case of varieties with (numerically) trivial anticanonical bundle.\n\nIf X is big then X is rationnally connected modulo the non-nef locus of -KX : two points of X are connected by a chain of curves which are either rational or in the non-nef locus of -KX . This a generalization of the well-known result of the rational connectedness of Fano varieties. The proof uses technics introduced by Zhang and\nHacon-McKernan for the proof of the rational connectedness of weak log Fano varieties and the proof of the Shokurov conjecture.
Del Pezzo surfaces, unprojections, and Calabi threefolds
Friday, 16.1.09, 11:15-12:15, Raum 403, Eckerstr. 1
A geometric transition between two Calabi-Yau threefolds is the composition of a degeneration a contraction and a smoothing. In my talk I would like to describe geometric transitions for which the contraction is an unprojection contracting a del Pezzo surface to a point. I shall show how using such constructions one can obtain new families of well described Calabi-Yau threefolds.
Explicit Beilinson-Bernstein theorem for arithmetic D-modules
Friday, 23.1.09, 11:15-12:15, Raum 403, Eckerstr. 1
In the years 80's, Beilinson-Bernstein and Brylinski-Kashiwara proved that there is an equivalence of categories between the D-modules on the flag variety of a semi-simple algebraic group over the field of complex numbers and modules over the Lie algebra of the group, with some condition on the action of the center of the envelopping algebra. This result lead to important results in the theory of Lie Algebra representation. We will explain here an arithmetic version of this equivalence of category by considering flag varieties in car. p>0 and arithmetic D-modules of Berthelot attached to these varieties.