Program
ABSTRACTS
LECTURE SERIES
Mark Andrea de Cataldo: "SLn - Higgs bundles and their intersection cohomology"
Abstract: Higgs bundles have played a central role in many important developments in Algebraic Geometry and in the Langlands Program.
I will give a general introduction to the moduli space of Higgs bundles for the general and special linear groups over a compact Riemann surface. I will then focus on discussing ongoing work concerning the dependence of the intersection cohomology groups of these moduli spaces on the degree of the Higgs bundles. This will require an introduction to the decomposition theorem and to support theorems. If time allows, I will present a rather concrete description of the local systems that appear in connection with the Hitchin morphism exiting these moduli spaces.
Alexandru Suciu: "Cohomology jump loci in geometry and topology"
Abstract: The cohomology jumping loci of a space come in two basic flavors: the characteristic varieties, which are the jump loci for homology with coefficients in rank 1 local systems, and the resonance varieties, which are the jump loci for the homology of cochain complexes arising from multiplication by degree 1 classes in the cohomology ring. I will explain the algebraic notions underlying these constructions and will present some structural results, such as the influence of formality, the interplay between resonance and duality, connections to tropical geometry, and the relationship to the finiteness properties of spaces and groups. Along the way, I will illustrate the general theory with a number of examples, mostly drawn from complex algebraic geometry, singularity theory, and low-dimensional topology.
Botong Wang: "Structures of the cohomology jump loci"
Abstract: In the 1980’s Green and Lazarsfeld proved two important theorems about the coherent cohomology jump loci: the generic vanishing theorem and the structure theorem. In the topological setting, the structure theorem says that every irreducible component of the cohomology jump loci of a quasi-projective variety is a torsion translated subtorus, and the generic vanishing theorem says that the dimension of the cohomology jump loci can be bounded using invariants of the Albanese map. I will give a survey of these results, and some further generalizations such as the propagation properties.
CONTRIBUTED TALKS
Ananyo Dan: "Hodge conjecture for singular varieties"
Abstract: In this talk I will discuss a cohomological version of the Hodge conjecture for singular varieties. I will give a sufficient condition in terms of Mumford-Tate groups for a variety to satisfy the singular Hodge conjecture. If time allows I will give explicit examples of such varieties. This is joint work with Inder Kaur.
Aleksandar Milivojevic: "Formality and non-zero degree maps" (slides)
Abstract: In the mid 70’s, Deligne-Griffiths-Morgan-Sullivan demonstrated a strong topological condition a closed manifold would have to satisfy if it were to carry a Kähler complex structure. Namely, the manifold would have to be formal, in the sense of its de Rham algebra of forms being weakly equivalent to its cohomology. In particular, there can be no non-trivial Massey products on a compact Kähler manifold. The salient underlying property of compact Kähler manifolds which implies formality is preserved under surjective holomorphic maps (non-zero degree maps in the holomorphic setting). It turns out (joint work with Jonas Stelzig and Leopold Zoller), formality itself is preserved under non-zero degree continuous maps of spaces satisfying Poincaré duality on their rational cohomology. I will explain the key components of this argument and show how one can then apply it to various situations. Using this result we can recover several seemingly disparate results in this area: the formality of singular complex projective varieties satisfying rational Poincaré duality, the formality of closed manifolds with sufficiently large first Betti number and a non-negative Ricci curvature metric, descent of formality to a subfield, and some more.
Gabriel Ribeiro: "Cartier duality, character sheaves, and generic vanishing".
Abstract: Since Green and Lazarsfeld's seminal work, generic vanishing theorems have influenced many fields ranging from birational geometry to analytic number theory. In this talk, I will present a new generic vanishing theorem for holonomic D-modules, that may shed new light on both of these worlds. Following ideas of Laumon, I'll explain how "character sheaves" play the role of topologically trivial line bundles and construct their moduli space based on a stacky version of Cartier duality.
Vasily Rogov: "The BNS sets of Kähler and quasi-projective groups"
Abstract: The BNS (Bieri-Neumann-Strebel) set of a finitely generated group is a certain canonical subset of the space of additive real-valued characters on the group. It is a subtle invariant of the group which is tightly connected to very different questions in group theory (actions on real trees, finite-generativity of the derived subgroup, presentation of a group as HNN-extension...). In 2010 Thomas Delzant found an elegant description of the BNS set of the fundamental group of a compact Kähler manifold in terms of holomorphic fibrations over hyperbolic orbicurves. Combining it with Arapura's results on Green-Lazarsfeld sets, he deduced that virtually solvable Kähler groups are virtually nilpotent. I am going to discuss the main ideas behind the proofs of Delzant's theorems and explain, how to generalise them in the case of fundamental groups of smooth quasi-projective varieties and quasi-Kähler manifolds. This generalisation uses ideas from real semialgebraic geometry.
Connor Simpson: "Polymatroid Schubert varieties"
Abstract: We introduce a new family of singular algebraic varieties, "polymatroid Schubert varieties", associated to arrangements of linear subspaces. Each polymatroid Schubert variety X carries an action of an additive group G_a^n, which partitions the variety into finitely many algebraic cells. The poset of cells, and their homology classes, are both governed by the combinatorics of the subspace arrangement used to define X. Polymatroid Schubert varieties generalize the "matroid Schubert varieties" associated to hyperplane arrangements; however, many basic questions about polymatroid Schubert varieties are more difficult than for matroid Schubert varieties, and hence remain open. Several such questions will be posed.
Carolina Tamborini: "Hodge theory and projective structures on compact Riemann surfaces"
Abstract: A projective structure on a compact Riemann surface is an equivalence class of projective atlases, i.e. an equivalence class of coverings by holomorphic coordinate charts such that the transition functions are all Moebius transformations. Any compact Riemann surface admits two canonical projective structures: one coming from uniformization's theorem, and one from Hodge theory. These yield two (different) families of projective structures over the moduli space Mg of compact Riemann surfaces. We wish to compare them and give a characterization of the Hodge theoretic family.
Sridhar Venkatesh: "Hodge modules on toric varieties"
Abstract: The intersection cohomology complex IC_X on a toric variety X has been well studied starting with the works of Stanley and Fieseler, and more recently, the works of de Cataldo-Migliorini-Mustata and Saito. However, it has a richer structure as a Hodge module (denoted IC^H_X) in the sense of Saito’s theory, and so we have the graded de Rham complexes gr_k(DR(IC^H_X)), which are complexes of coherent sheaves carrying significant information about X. In this talk, I will describe the generating function of the cohomology sheaves of gr_k(DR(IC^H_X)) and give a precise formula relating it with the stalks of the perverse sheaf IC_X (in particular, this implies that the generating function depends only on the combinatorial data of the toric variety). Time permitting, I will also show that the generating function can be computed explicitly in an algorithmic way. This is joint work with Hyunsuk Kim.