Algebraic Geometry Seminar Spring 2012
The seminar meets on Fridays at 2:25 pm in Van Vleck B215.
The schedule for the previous semester is here.
|January 27||Sukhendu Mehrotra (Madison)||Generalized deformations of K3 surfaces||(local)|
|February 3||Travis Schedler (MIT)||Symplectic resolutions and Poisson-de Rham homology||Andrei|
|February 10||Matthew Ballard (UW-Madison)||Variation of GIT for gauged Landau-Ginzburg models||(local)|
|February 17||Arend Bayer (UConn)||Projectivity and birational geometry of Bridgeland moduli spaces||Andrei|
|February 24||Laurentiu Maxim (UW-Madison)||Characteristic classes of Hilbert schemes of points via symmetric products||local|
|March 2||Marti Lahoz (Bonn)||Effective Iitaka fibrations of varieties of maximal Albanese dimension||Sukhendu|
|March 9||Shilin Yu (Penn State)||Coherent analytic sheaves over formal neighborhoods||Andrei|
|March 16||Weizhe Zheng (Columbia)||Fixed point sets and mod l cohomology algebra||Tonghai|
|March 23||Ryan Grady (Notre Dame)||Twisted differential operators as observables in QFT||Andrei|
|March 28 (!! Room 901)||Zheng Hua (Kansas)||Donaldson-Thomas theory and HOMFLY polynomials of algebraic knots||Andrei|
|April 27||Ursula Whitcher (UW-Eau Claire)||K3 surfaces with S_4 symmetry||Matt|
|May 11||Michael Penkava (UW-Eau Claire)||Ribbon Graphs, Grothendieck's dessins d' enfants', and the Poincare Polynomial||David|
|May 15 (!! Room 901)||David Favero (University of Vienna)||Variation of Geometric Invariant Theory for Derived Categories||Matt|
Generalized deformations of K3 surfaces
Symplectic resolutions and Poisson-de Rham homology
Abstract: A symplectic resolution is a resolution of singularities of a singular variety by a symplectic algebraic variety. Examples include symmetric powers of Kleinian (or du Val) singularities, resolved by Hilbert schemes of the minimal resolutions of Kleinian singularities, and the Springer resolution of the nilpotent cone of semisimple Lie algebras. Based on joint work with P. Etingof, I define a new homology theory on the singular variety, called Poisson-de Rham homology, which conjecturally coincides with the de Rham cohomology of the symplectic resolution. Its definition is based on "derived solutions" of Hamiltonian flow, using the algebraic theory of D-modules. I will give applications to the representation theory of noncommutative deformations of the algebra of functions of the singular variety. In the examples above, these are the spherical symplectic reflection algebras and finite W-algebras (modulo their center).
Variation of GIT for gauged Landau-Ginzburg models
Abstract: Let X be a variety equipped with a G-action and G-invariant regular function, w. The GIT quotient X//G depends on the additional data of a G-linearized line bundle. As one varies the G-linearized line bundle, X//G changes in a very controlled manner. We will discuss how the category of matrix factorizations, mf(X//G,w), changes as the G-linearized line bundle varies. We will focus on the case where G is toroidal. In this case, we show that, as one travels through a wall in the GIT cone, semi-orthogonal components coming from the wall are either added or subtracted. The talk is based on joint work with D. Favero (U Wien) and L. Katzarkov (U Wien and U Miami).
Projectivity and birational geometry of Bridgeland moduli spaces
I will present a construction of a nef divisor for every moduli space of Bridgeland stable complexes on an algebraic variety. In the case of K3 surfaces, we can use it to prove projectivity of the moduli space, generalizing a result of Minamide, Yanagida and Yoshioka. It's dependence on the stability condition gives a systematic explanation for the compatibility of wall-crossing of the moduli space with its birational transformations; this had first been observed in examples by Arcara-Bertram. This is based on joint work with Emanuele Macrì.
Characteristic classes of Hilbert schemes of points via symmetric products
I will explain a formula for the generating series of (the push-forward under the Hilbert-Chow morphism of) homology characteristic classes of the Hilbert schemes of points for a smooth quasi-projective variety of arbitrary pure dimension. The result is based on a geometric construction of a motivic exponentiation generalizing the notion of motivic power structure, as well as a formula for the generating series of homology characteristic classes of symmetric products.
Effective Iitaka fibrations of varieties of maximal Albanese dimension
Let X be a variety of maximal Albanese dimension. Chen and Hacon proved that if X has positive holomorphic Euler-characteristic, then its tricanonical map is birational onto its image; in particular, X is of general type. When the Euler characteristic is not positive, we use generic vanishing techniques to prove that the tetracanonical map of X induces the Iitaka fibration. Moreover, if X is of general type, then the tricanonical map is already birational. If time permits, I will also construct examples showing that these results are optimal.
This is a joint work with Zhi Jiang and Sofia Tirabassi.
Coherent analytic sheaves over formal neighborhoods
The infinitesimal behavior of an embedded submanifold in a complex manifold is encoded by its (holomorphic) formal neighborhood. For example, it governs all the infinitesimal deformations of the submanifold. We will define a notion of Dolbeault differential graded algebra (dga) associated to such formal neighborhood, which plays the same role as the Dolbeault complex of an ordinary complex manifold. An explicit description of this Dolbeault dga is available at least when the ambient manifold is Kahler, which generalizes a result of Mikhail Kapranov about the special case of diagonal embedding. Then we will discuss the notion of 'cohesive modules' over a dga invented by Jonathan Block. He showed that, for the Dolbeault dga of a compact complex manifold, such cohesive modules are essentially the same as coherent analytic sheaves over the manifold. We show that the same result also holds for Dolbeault dga of formal neighborhoods.
Title: Fixed point sets and mod l cohomology algebra Abstract: A theory of Quillen relates equivariant cohomology algebras to the connected components of the fixed point sets under elementary abelian l-groups. In this talk, I will give an analogue of Quillen's theory for quotient stacks, with coefficients. I will also discuss Smith's theory, finiteness, and localization in the algebraic setting. This is a joint work with Luc Illusie.
Twisted differential operators as observables in QFT
We discuss Chern-Simons type theories in perturbative quantum field theory. The observables of such a theory has the structure of a factorization algebra. We recover the Rees algebra of differential operators on a space X from the one-dimensional theory with target T*X. We also consider twists of these theories which lead to twisted differential operators. If time allows, we will sketch dimensional reduction from 3 to 2 real dimensions. We will not assume any familiarity with quantum field theory in this talk.
Donaldson-Thomas theory and HOMFLY polynomials of algebraic knots
Oblomkov and Shende conjectured that the HOMFLY polynomial of a planar curve singularity is equal to certain generating series of Hilbert scheme of points of the curve. I will introduce a Calabi-Yau threefold such that OS conjecture can possibly be intepreted as a question about existence of wallcrossing formulas of Donaldson-Thomas type invariants in the derived category of this CY 3-fold. This is a joint work with Diaconescu and Soibelman.
K3 surfaces with S_4 symmetry
The compact complex surfaces known as K3 surfaces may be considered as the higher-dimensional analogues of the elliptic curves loved by number theorists or as lower-dimensional analogues of the Calabi-Yau manifolds loved by string theorists. Hypersurfaces in toric varieties offer a rich source of examples of K3 surfaces. We characterize families of highly symmetric K3 surfaces using toric varieties, symmetric polyhedra, and differential equations.
Ribbon Graphs, Grothendieck's dessins d' enfants', and the Poincare Polynomial
In this talk I will first describe how to glue together a Riemann surface using the data of a metric ribbon graph, and show that metric ribbon graphs whose edge lengths are rationally commensurate give rise to Riemann surfaces defined over the algebraic closure of the rationals, using Belyi's theorem, which I then will relate to Grothendieck's Child's Drawings, or dessins d' enfants. The Poincare polynomial of the ribbon graph complex will be defined and shown to satisfy an Eynard-Orantin recursion formula. This polynomial arises from a Laplace transform of a formula which counts Grothendieck's Child's Drawings, and contains information about the orbifold Euler characteristics of the moduli space of Riemann surfaces of genus g with n marked points, as well as the Gromov Witten invariants, and recovers the Virasoro constraint condition or the DVV formula, due to Dijkgraaf-Verlinde-Verlinde, Witten and Kontsevich.
Variation of Geometric Invariant Theory for Derived Categories
Given a quasi-projective algebraic variety X, with the action of a linear algebraic group G, there are various (birational) incarnations of the quotient X/G coming from a choice of a G-equivariant ample line bundle. As we vary this choice, there is a semi-orthogonal relationship between the derived categories of the resulting quotients, A and B. Furthermore, if (X, w) is a Landau-Ginzburg model, and w is a G-invariant section of a line bundle on X, then the same holds for ”coherent sheaves on” (A, w) and B, w) (categories of matrix factorizations/categories of singularities/stable derived categories). As a special case, one can reproduce a theorem of Orlov relating categories of coherent sheaves for complete intersections in projective space to the graded category of singularities of the cone, a theorem of Herbst and Walcher demonstrating an equivalence of derived categories between ”neighboring” Calabi-Yau complete intersections in toric varieties, and two theorems of Kawamata; one concerning behavior of derived categories of algebraic varieties under simple toroidal ﬂips, the other stating that the derived category of coherent sheaves on any smooth toric variety has a full exceptional collection (in the projective case). If time permits, I will also discuss the relationship with Kuznetsov’s homological projective duality.