Algebraic Geometry Seminar
Department of Mathematics
Princeton University
Spring 2012 Lectures
Regular meeting time: Tuesdays
4:30-5:30 (Tea served at 3:30)
Place: Fine 322
| Date | Speaker | Title |
| Feb 7 |
Andrei Căldăraru University of Wisconsin SPECIAL TIME: 3:30-4:30 SPECIAL ROOM: Fine 224 |
The Hodge theorem as a derived self-intersection The Hodge theorem is one of the most important results in complex geometry. It asserts that for a complex projective variety X the topological invariants H^*(X, C) can be refined to new ones that reflect the complex structure. The traditional statement and proof of the Hodge theorem are analytic. Given the multiple applications of the Hodge theorem in algebraic geometry, for many years it has been a major challenge to eliminate this analytic aspect and to obtain a purely algebraic proof of the Hodge theorem. An algebraic formulation of the Hodge theorem has been known since Grothendieck's work in the early 1970's. However, the first purely algebraic (and very surprising) proof was obtained only in 1991 by Deligne and Illusie, using methods involving reduction to characteristic p. In my talk I shall try to explain their ideas, and how recent developments in the field of derived algebraic geometry make their proof more geometric, by allowing us to realize the Deligne-Illusie main result as a formality result for the derived self-intersection of a subvariety of a twisted space. |
| Feb 14 |
Aise Johan de Jong Columbia University |
The stacks project This is a talk about the stacks project, which is located on the web at http://math.columbia.edu/algebraic_geometry/stacks-git The stacks project is a long term open source, collaborative project documenting and developing theory on algebraic stacks. I will spend a bit of time talking about what it is, who it is for, what its goals are and how it is supposed to work. |
| Feb 21 |
Ryan Kinser Northeastern University |
Geometrically characterizing representation type of
finite-dimensional algebras Given a finite-dimensional algebra A, the set of A-modules of a fixed dimension d can be viewed as a variety. This variety carries a group action whose orbits correspond to isomorphism classes of A-modules. A natural problem is to characterize various properties of an algebra A in terms of its module varieties. For example, if A is assumed to have global dimension one, then it is not difficult to show that A has finitely many indecomposable modules (up to isomorphism) if and only if all of its module varieties have a dense orbit, which is also if and only if all weight spaces of semi-invariants in the coordinate rings of its module varieties have dimension one. Our goal is to generalize these statements (with modification) to higher global dimension. After explaining the background, we present counterexamples to the naive generalizations, along with plausible modifications and cases where these modifications are correct. (Joint work with Calin Chindris, Piotr Dowbor, and Jerzy Weyman) |
| Feb 28 |
Mina Teicher Bar-Ilan University, IAS |
Braid Group Techniques for fundamental groups of surfaces,
The K3 example |
| Mar 6 |
Yu-Han Liu Princeton University |
Tensor products on triangulated and abelian categories |
| Mar 13 |
Karl Schwede Penn State University |
A Frobenius variant of Seshadri constants I will define a new variant of the Seshadri constant for ample line bundles in positive characteristic. We will then explore how lower bounds for this constant imply the global generation and/or very ampleness of the corresponding adjoint line bundle. As a consequence, we will deduce that the criterion for global generation and very ampleness of adjoint line bundles in terms of usual Seshadri constants holds also in positive characteristic (even though we may lack the usual vanishing theorems). This is joint work with Mircea Mustata. |
| Mar 20 |
Spring Recess |
|
| Mar 27 |
Burt Totaro DPMMS, Cambridge Univ. |
The integral Hodge conjecture for 3-folds The Hodge conjecture predicts which rational homology classes on a smooth complex projective variety can be represented by linear combinations of complex subvarieties. In other words, it is about the difference between topology and algebraic geometry. The integral Hodge conjecture, the analogous conjecture for integral homology classes, is false in general. We discuss negative results and some new positive results on the integral Hodge conjecture for 3-folds. |
| Apr 3 |
Lev Borisov Rutgers, The State University of New Jersey |
Elliptic functions and equations of modular curves I will talk about an old paper joint with Paul Gunnells and Sorin Popescu that explicitly describes modular curves for congruence subgroups $\Gamma_1(p)$ of $SL_2(Z)$ as intersections of quadrics in a projective space. I will aim to keep the talk accessible to graduate students. |
| Apr 10 |
Daniel Erman University of Michigan |
Categorified Duality in Boij-Soederberg Theory The central idea in Boij-Soederberg Theory is that there is a connection between sheaf cohomology on projective space and free resolutions over the polynomial ring. I'll describe the construction of a duality pairing that provides a new foundation for this theory, and that greatly extends the reach of the theory. This is joint work with David Eisenbud. |
| Apr 17 |
Dima Arinkin Univ. of North Carolina, IAS |
Autoduality of Jacobians for singular curves Let C be a (smooth projective algebraic) curve. It is well known that the Jacobian J of C is a principally polarized abelian variety. In other words, J is self-dual in the sense that J is identified with the space of topologically trivial line bundles on itself. Suppose now that C is singular. The Jacobian J of C parametrizes topologically trivial line bundles on C; it is an algebraic group that is no longer compact. By considering torsion-free sheaves instead of line bundles, one obtains a natural singular compactification J' of J. In this talk, I consider (projective) curves C with planar singularities. The main result is that J' is self-dual: J' is identified with a space of torsion-free sheaves on itself. This identification also provides an auto-equivalence of the coherent derived category of J' (the Fourier-Mukai transform). The autoduality naturally fits into the framework of the geometric Langlands conjecture; I hope to sketch this relation in my talk. |
| Apr 24 |
Pramod N. Achar Lousiana State University |
Perverse coherent sheaves on the nilpotent cone in
positive
characteristic In the context of geometric Langlands duality, it is a general principle that the "topological" aspects (e.g., intersection cohomology, perverse sheaves) of a given group G should correspond to "algebraic" aspects (e.g., representations, coherent sheaves) of its dual group G'. An archetypal instance of this idea is the "geometric Satake isomorphism" of Ginzburg and Mirkovic-Vilonen, but by now there are many results asserting an equivalence (often derived) between a topological category associated to G and an algebraic one associated to G'. In this talk, I will try to explain a few examples of this phenomenon, which can give rise to surprising kinds of objects: coherent sheaves (in characteristic 0) that behave as though they were perverse, or vice versa. I will also say a few words about the titular objects, which don't (yet?) come from such an equivalence. |
| May 1 |
Bhargav Bhatt University of Michigan |
Comparison theorems in p-adic Hodge theory A basic theorem in Hodge theory is the isomorphism between de Rham and Betti cohomology for complex manifolds; this follows directly from the Poincare lemma. The p-adic analogue of this comparison lies deeper, and was the subject of a series of extremely influential conjectures made by Fontaine in the early 80s (which have since been established by various mathematicians). In my talk, I will first discuss the geometric motivation behind Fontaine's conjectures, and then explain a simple new proof based on general principles in derived algebraic geometry --- specifically, derived de Rham cohomology --- and some classical geometry with curve fibrations. This work builds on ideas of Beilinson who proved the de Rham comparison conjecture this way. |
| May 8 |
Reading Period | |
Other seminars in this department
Future lectures
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For more information about this seminar, contact yuliu@math.princeton.edu and/or kftucker@math.princeton.edu.