(The seminar webpage is here.)
Alexei Oblomkov (U Mass Amherst)
October 4, 2013
Plane curve singularities, knot homology and Hilbert scheme of points on plane
I will present a conjectural formula for the Poincare polynomial of the Hilbert scheme of points on a planar curve (joint with Rasmussen and Shende). The formula is written in terms of the Khovanov-Rozansky invariants of the links of the singularities of the curve. In the case of toric curve the Poincare polynomial also could be written in terms of equivariant Euler characteristic of some sheaf of some particular equivariant sheaf on (joint with Yun). If time permits I will also discuss cohomology ring of the compactified Jacobian of the toric curve and the conjectural description of the ring for general curve singularity (joint with Yun).
Picard Groups of Moduli Spaces of K3 Surfaces
Polarized K3 surfaces of genus can be thought of as families of canonical curves. As such, their moduli space has similar properties to . For instance, both are unirational for low values of g, and both have discrete Picard group. In this talk, we will use the explicit unirationality of in the range of Mukai models to compute its Picard number, and verify the Noether-Lefschetz conjecture for genus up to 10. This is joint work with Zhiyuan Li and Zhiyu Tian.
Lefschetz for local Picard groups
A classical theorem of Lefschetz asserts that non-trivial line bundles on a smooth projective variety of dimension remain non-trivial upon restriction to an ample divisor. In SGA2, Grothendieck recast this result in purely local algebraic terms. Answering a question raised recently by Koll\’ar, we will explain how this local reformulation remains true under milder hypotheses than those imposed in SGA2. Our approach relies on a vanishing theorem in characteristic p, and formal geometry over certain very large (non-noetherian) schemes. This is joint work with Johan de Jong.
The projectors of the decomposition theorem are absolute Hodge
I report on joint work with Luca Migliorini at Bologna. If you have a map of complex projective manifolds, then the rational cohomology of the domain splits into a direct sum of pieces in a way dictated by the singularities of the map. By Poincaré duality, the corresponding projections can be viewed as cohomology classes (projectors) on the self-product of the domain. These projectors are Hodge classes, i.e. rational and of type (p,p) for the Hodge decomposition. Take the same situation after application of an automorphism of the ground field of complex numbers. The new projectors are of course Hodge classes. On the other hand, you can also transplant, using the field automorphism, the old projectors into the new situation and it is not clear that the new projectors and the transplants of the old projectors coincide. We prove they do, thus proving that the projectors are absolute Hodge classes, i.e. their being of Hodge type survives the totally discontinuous process of a field automorphism. We also prove that these projectors are motivated in the sense of Andre.
Conormal varieties and the Temperley-Lieb algebra
Each permutation pi has an associated determinantal variety called its “matrix Schubert variety”. If one degenerates these determinants to monomials, the components of the resulting scheme are naturally indexed by reduced words for pi as a product of simple reflections ([K-Miller 2005]). It is particularly notable that unlike in most limits of this sort, the resulting scheme has no multiplicities on its components.
I’ll describe the “conormal variety” to a variety, which shows up in many contexts (e.g. projective duality), and extend this degeneration idea to the conormal varieties of the matrix Schubert varieties. The limit scheme now includes the conormal variety to the original limit, and to its projective dual, but also some fundamentally new components which appear with multiplicity. Then I’ll state some conjectures, in particular that the components are naturally indexed by words in the generators of the Temperley-Lieb algebra, from which one can also predict the multiplicities.
This work is joint with Paul Zinn-Justin.
Geometric Igusa integrals appear as important technical tools in the study of rational and integral points on algebraic varieties. I will describe some of these applications (joint work with A. Chambert-Loir).
Nick Katz (Princeton)
January 31, 2014
Equidistribution questions arising from universal extensions
The Galois group of a stable homotopy theory
To a “stable homotopy theory” (a presentable, symmetric monoidal stable ∞-category), we naturally associate a category of finite ́etale algebra objects and, using Grothendieck’s categorical machine, a profinite group that we call the Galois group. This construction builds on, and generalizes, ideas of many authors, and includes the ́etale fundamental group of algebraic geometry as a special case. We calculate the Galois groups in several examples, both in settings of rational and p-adic homotopy and in “chromatic” stable homotopy theories. For instance, we show that the Galois group of the periodic E∞-algebra of topological modular forms is trivial, and, extending work of Baker and Richter, that the Galois group of K(n)-local stable homotopy theory is an extended version of the Morava stabilizer group.
The Euclidean Distance Degree of an Algebraic Variety
The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. The Euclidean distance degree is the number of critical points of this optimization problem. We focus on varieties encountered in engineering applications, and we discuss exact computational methods. Our running example is the Eckart-Young Theorem which states that the nearest point map for low rank matrices is given by the singular value decomposition. This is joint work with Jan Draisma, Emil Horobet, Giorgio Ottaviani, Rekha Thomas.
Sato-Tate groups of abelian surfaces
There are a number of theorems and conjectures in arithmetic geometry which control the structure of an abelian variety over a number field in terms of its reductions to various finite fields. Here we consider a rather crude question along these lines: how much structure can be read off by computing the zeta functions of these reductions and retaining only the statistical behavior of these zeta functions as one averages over primes? For elliptic curves, this is known (under appropriate conjectures, which are known in many cases) to distinguish whether or not the curve has complex multiplication, and if so whether this happens over the base field or an extension field. We describe the corresponding picture for genus 2 curves: the punchline is that there are 52 different possible cases, corresponding to Galois module structures on real endomorphism algebras. These cases will be illustrated with some pretty pictures! Joint work with Francesc Fite, Victor Rotger, and Andrew Sutherland.
Homology classes in algebraic varieties: nef, effective, and prime
The homology group of an algebraic variety is an abelian group equipped with several additional structures. It contains the set of primes, the homology classes of subvarieties. It contains the semigroup of effective classes, the nonnegative linear combinations of primes. It contains the semigroup of nef classes, the classes which intersect all primes of complementary dimension nonnegatively. We will see how these subsets look like in a particular algebraic variety, the one associated to the polytope `permutahedron’. The semigroups in this case have tractable structures, while the distribution of primes is more mysterious and related to some deep combinatorial conjectures on matroids.
Special cycles on Shimura varieties of orthogonal type
Let be a connected smooth Shimura variety. There are many special algebraic cycles coming from sub-Shimura variety of the same type in all codimensions, called special cycles on . A Hodge type question is whether these special cycles exhaust all the low degree cohomology classes of . In this talk, I will discuss relation between this question and Arthur’s theory and briefly talk about the work of Beregon-Millson-Moeglin in this direction.
-curves on quasi-projective varieties
The theory of stable log maps was developed recently for studying the degeneration of Gromov-Witten invariants. In this talk, I will introduce another interesting aspect of stable log maps as a useful tool for investigating -curves on quasi-projective varieties, which are the analogue of rational curves on proper varieties. At least two applications of -curves will be discussed in this talk. For classical birational geometry, the -curves can be used to produce very free rational curves on general Fano complete intersections in projective spaces. On the arithmetic side, -connectedness gives a general frame work for the existence of integral points over function field of curves. This is a joint work in progress with Yi Zhu.
Double ramification cycles and tautological relations
Double ramification cycles parametrize curves that admit maps to the projective line with specified ramification over zero and infinity. They can be extended to the moduli space of stable curves by using the virtual class in relative Gromov-Witten theory. I will describe a conjectural formula for these extensions in terms of tautological classes. The formula is motivated by a connection with recent joint work with Pandharipande and Zvonkine on Witten’s r-spin class, and it comes with a family of tautological relations which extend relations studied by Grushevsky and Zakharov.
Negative answers to some positivity questions
I will explain how the classically-studied action of Cremona transformations on configurations of points in projective space can be used to construct several counterexamples in birational geometry: nefness is not an open condition in families, Zariski decompositions do not always exist in dimension 3, and a variety can have infinitely many Fourier-Mukai partners. As time permits, I’ll discuss some related examples on Calabi-Yau threefolds.