Schedule and abstracts

Zaslow, Gräfnitz, Zhou and I interpret the q-refined theta function ϑ1 of a log Calabi-Yau surface (ℙ,E) as a natural q-refinement of the open mirror map, defined by quantum periods of mirror curves for outer Aganagic-Vafa branes on a local Calabi-Yau Kℙ . The series coefficients are all-genus logarithmic two-point invariants, directly extending the relation found in earlier work. Yet we find an explicit discrepancy at higher genus in the relation to open Gromov-Witten invariants of the Aganagic-Vafa brane. Using a degeneration argument, we express the difference in terms of relative invariants of an elliptic curve. With π: ℙ^→ℙ the toric blow up of a point, we use the Topological Vertex [AKMV] to show a correspondence between open invariants of Kℙ and closed invariants of Kℙ^ generalizing a variant of Chan et al's work to arbitrary genus and winding. We also equate winding-1, open-BPS invariants with closed Gopakumar-Vafa invariants. 

Weighted blow-ups are a class of binational transformations that appears naturally in the study of moduli spaces and of resolution of singularities. The title of this talk is a quote of my advisor, Dan Abramovich, in relation to the ubiquity of this technique when trying to understand these morphisms or proving statements about them. 

I will give an introduction on weighted blow-up and deformations to the normal cone. I will then give examples on how this technique is used when proving results about intersection theory in the joint work with Stephen Obinna.

We prove that a very general hypersurface of degree d > 3 and dimension at most (d+1)2^{d-4} does not admit a decomposition of the diagonal. In particular it is neither stably nor retract rational nor A^1-connected. This improves earlier works of Schreieder (2019) and Moe (2023). Joint work with Stefan Schreieder.

I will give an overview of the recent techniques used to prove strong generation statements for triangulated categories associated to algebro-geometric objects. In particular, we will see Neeman's proof of the Bondal – Van den Bergh conjecture, which states that strong generation of the category of perfect complexes is Zariski local. I will then discuss its generalisation to mild noncommutative schemes, i.e. schemes equipped with noncommutative structure sheaves, and how this can be extended to other Grothendieck topologies. The talk is based on joint work with Pat Lank and Kabeer Manali Rahul.

We will explore necessary conditions for a vector bundle to be self-dual, with a particular focus on those of rank at most 3. For instance, we will show that on a K3 surface or an abelian surface, any slope-stable self-dual vector bundle of rank 3 must have an even second Chern class. Joint work with Ryo Yamagishi.

I will report on the ongoing project to construct a perverse schober, a poor man’s perverse sheaf of triangulated categories, in the context of the classical two-dimensional McKay correspondence for G in SL_2(C) The braid group of the corresponding ADE type acts on the derived category D(Y) of the minimal resolution Y of C^2/G by spherical twists in the exceptional curves. Since this braid group is the fundamental group of the open stratum of h/W, the quotient of the ADE Cartan algebra by the Weil group action, its action on D(Y) can be thought of as a local system of triangulated categories with the fiber D(Y) on this open stratum. A perverse schober extends this structure to the higher codimension strata.

We actually construct a W-equivariant schober on h by using an instance of the McKay correspondence – the root hyperplane arrangement in h coincides with the wall-and-chamber structure in the stability space Theta for the GIT construction of Y as the moduli space of theta-stable G-constellations. We can thus make use of the techniques of Halpern-Leistner – Sam and Spenko – Van den Bergh for the quasi-symmetric reductive group action on a vector space, though in our case the group acts on a very singular subvariety of a quasi-symmetric vector space.

Our work is motivated by wanting to eventually tackle dim=3 case, where h/W picture no longer exists and the GIT action for G-constellation moduli is no longer quasi-symmetric. However, it might still be possible to construct a schober on the GIT stability space, neatly packaging up all the Craw - Ishii GIT wall-crossing equivalences and more. This is a joint work with Arman Sarikyan (LIMS).