Camilla Felisetti
Title: Parabolic bundles and intersection cohomology of moduli of vector bundles
Abstract: Intersection cohomology is a topological notion adapted to the description of singular topological spaces, and the Decomposition Theorem for algebraic maps is a key tool in the subject. Motivated by the work of Mozgovoy and Reineke, in joint work with Andras Szenes and Olga Trapeznikova, we give a complete description of the intersection cohomology of the moduli space of vector bundles of any rank via a detailed analysis of the Decomposition Theorem applied to a certain map from parabolic bundles. We also give a new formula for the intersection Betti numbers of these moduli spaces, which has a clear geometric meaning.
Roberto Fringuelli
Title: Decomposition Theorem for the logarithmic Hitchin fibration
Abstract: I will report on an ongoing project with Mark Andrea de Cataldo, Andres Fernandez-Herrero and Mirko Mauri on the intersection cohomology of the moduli space of semistable logarithmic G-Higgs bundles on a smooth curve, where G is a reductive group. For any given degree d in the fundamental group of G, we exhibit a uniform description of the decomposition theorem for the corresponding Hitchin fibration of degree d logarithmic G-Higgs bundles.
Andres Fernandez Herrero
Title: Projectivity of the moduli of branchvarieties
Abstract: The moduli of (reduced) equidimensional subvarieties of projective space is often not complete. In 2010, Alexeev and Knutson introduced a compactification called the moduli of branchvarieties. This is a proper Deligne-Mumford stack parameterizing equidimensional varieties equipped with a finite morphism to projective space. In principle, this is a great parameter space to carry out GIT constructions of moduli of varieties. However, there is one caveat to this: Alexeev and Knutson left as an open problem whether their proper DM stack is projective. In this talk, I will explain a proof of projectivity obtained in joint work with Dan Halpern-Leistner, Trevor Jones and Ritvik Ramkumar.
Tamas Hausel
Title: Ringifying intersection cohomology
Abstract: Although there is no natural ring structure on intersection cohomology, in the case of affine and finite Schubert varieties also at their singular points, partly conjecturally, we ringify intersection cohomology using big algebras and quantum big algebras.
Jochen Heinloth
Title: On the moment measure conjecture
Abstract: Stability conditions help to cut out subspaces of a moduli problem that admit proper moduli spaces. In many cases these stability conditions are in the end determined by the datum of a line bundle on the original
moduli problem.
As we also know geometric criteria that decide whether an open substack of a moduli problem admits a proper quotient, it seems natural to ask, whether there are other methods to find such open substacks.
The moment measure conjecture Bialynicki-Birula and Sommese gave a conjecturally complete classification of such open subsets for quotients of torus actions on smooth projective varieties, which in particular include quotients that are not projective and thus cannot be constructed through GIT. I would like to explain how this can be understood by looking at other cohomological degrees of the corresponding quotient stack. (This is joint work with X. Zhang.)
Vicky Hoskins
Title: When is the motive of a moduli space of Higgs bundles on a curve
generated by the motive of the curve?
Abstract: I will explain how the (rational Voevodsky) motive of the moduli space of Higgs bundles on a curve (with coprime invariants) is generated by the motive of the curve, and how this fails for moduli spaces of SL-Higgs bundles on a general curve. I will then explain how the motive of the SL-Higgs moduli space is generated by motives of certain covers of the curve. For GL-Higgs bundles, our argument involves Hitchin's scaling action, geometric ideas of Garcia-Prada, Heinloth and Schmitt, together with motivic descriptions of small maps following work of de Cataldo and Migliorini. This talk is based on joint work with S. Pepin Lehalleur, and partially also with L. Fu.
Matthew Huynh
Title: The Hitchin morphism for certain surfaces fibered over a curve
Abstract: Let X be a smooth projective variety, and let G be a reductive group. We will study G-Higgs bundles (twisted by the bundle of 1-forms and satisfying the integrability condition) on X, and the Hitchin morphism from the moduli stack of these objects to the Hitchin base. The Chen-Ngô Conjecture predicts that the image of this morphism is the "space of spectral data", which one can think of as the locus where there exists a correspondence between a fiber of the Hitchin morphism h^{-1}(b), and a stack of objects on a covering X_b --> X.
We will prove the conjecture for any reductive group when X is a ruled surface or nonisotrivial elliptic fibration with reduced fibers. Moreover, when G is a classical group, (SL_n, SO_n, Sp_2n), then we show that the Dolbeault moduli space of semiharmonic G-Higgs bundles surjects onto the space of spectral data.
Andrés Ibáñez Núñez
Title: Component lattices and cohomology of symmetric stacks
Abstract: We give an explicit form of the decomposition theorem for the map from a stack X to its good moduli space under the assumption that tangent spaces are orthogonally symmetric and that X is either smooth, (-1)-shifted symplectic or 0-shifted symplectic, with a different coefficient sheaf in each case. Examples of such stacks include moduli of G-bundles and G-Higgs bundles on a curve, and sheaves on Calabi-Yau threefolds. We will discuss the main ideas involved in the proof, including the theory of component lattices, which allows to both formulate the statement and reduce it to the local case. Among the applications, we give a version of Hausel-Thaddeus conjecture for general semisimple groups. This is joint work over two different projects with Chenjing Bu, Ben Davison, Daniel Halpern-Leistner, Tasuki Kinjo and Tudor Pădurariu.
Davesh Maulik
Title: Overview of Mark's work
Mirko Mauri
Title: Boundedness results for some fibred K-trivial varietie
Abstract: We show that certain fibered varieties with trivial canonical bundle are bounded. In particular, there are only finitely many deformation classes of hyperkähler varieties of a fixed dimension, deformation equivalent to one admitting a Lagrangian fibration. This is a work in progress, joint with Engel, Filipazzi, Greer and Svaldi.
Li Li
Title: Nakajima's quiver varieties, quantum cluster algebras, and determinantal ideals
Abstract: Nakajima’s affine graded quiver varieties are rich geometric objects that support the study of many problems in algebraic geometry and representation theory. In this talk, I will focus on the quiver varieties associated with bipartite quivers and explain how these varieties can be used to study the support conjecture for triangular bases in some quantum cluster algebras. I will also present an interesting combinatorial structure of the initial ideals of the ideals defining these quiver varieties, which generalize similar results for determinantal ideals.
Siqing Zhang
Title: Etale homotopy in prime characteristic
Abstract: The etale fundamental group of a prime characteristic variety can be quite unwieldy, but how about higher homotopy groups? In joint work with Runjie Hu, we show that the l-adic etale homotopy type of a simply connected variety in characteristic p is the same as that of a simply connected complex variety. In this passage from char. p to 0, smoothness is typically lost. However, if we start with a smooth projective variety in char p, then we still have a mod-l Poincare space, and various notions from surgery theory make sense. We then connect the Frobenius action on the char p variety to Sullivan’s abelianized Galois action on the l-adic normal invariants set.
Zili Zhang
Title: The P=W phenomemon for cluster varieties
Abstract: The cohomology of even-dimensional full rank Louise-type cluster varieties satisfy the curious hard Lefschetz property. Since curious hard Lefschetz property is rarely observed and strongly related to the P=W identity, we conjecture that there exists a P=W type identity for cluster varieties. In this talk, we will study the cohomology cluster varieties in a more general setting and establish a conjectural P=W type identity. We will also provide various low-dimensional cluster varieties as evidence toward the P=W conjecture for cluster varieties.