Mukai bundles on curves, K3 surfaces and Fano threefolds
by Arend Bayer
Abstract: A central result in the theory of Fano threefolds of index one and their derived categories is the existence of certain simple rigid vector bundles. Unfortunately, the proofs in the literature contain gaps. I will present work joint with Alexander Kuznetsk and Emanuele Macrì where we address this gap, and extend the result to finite characteristic. The proof is based on extending spherical bundles on K3 hyperplane sections, which in turn relies on uniqueness of vector bundles on curves satisfying a Brill-Noether condition.
Twistors, clusters and stability conditions
by Tom Bridgeland
Abstract: Given an ADE quiver Q, I will explain how to construct a twistor space. This is a complex manifold with a map to CP^1 whose fibre over 0 is the stability space of the CY3 Ginzburg algebra of Q, quotiented by spherical twists, and whose general fibre is an etale cover of the cluster Poisson variety of Q. This is joint work with Helge Ruddat.
Stability conditions on free abelian quotients
by Hannah Dell
Abstract: The space of Bridgeland stability conditions on a given triangulated category is a complex manifold, which gives us a way to extract geometry from homological algebra. An explicit description is only known in a few cases.
In this talk, I will discuss two approaches to studying stability conditions on derived categories of surfaces that are free quotients by finite abelian groups. One method is via Le Potier functions, which characterise the existence of slope-semistable sheaves. The second method uses Deligne’s notion of group actions on triangulated categories to describe a connected component of so-called geometric stability conditions inside the stability manifold of these free abelian quotients when the cover has finite Albanese morphism. A consequence of this is a disproof of the expectation that surfaces with irregularity 0 always admit a wall of the geometric chamber. (arXiv:2307.00815)
A Thurston compactification for the space of Bridgeland stability conditions
by Naoki Koseki
Abstract: Motivated by an analogue between the theory of Bridgeland stability conditions and the classical Teichmuller theory, Bapat–Deopurkar–Licata proposed a way to construct a Thurston type compactification of the space of Bridgeland stability conditions. In this talk, I will review their proposal, and report a recent progress in the cases of smooth projective curves and K3 surfaces. This is based on a joint work with K.Kikuta(Osaka) and G.Ouchi(Nagoya).
Intersections of tropical psi classes in genus zero, with non-trivial valuations
by Rohini Ramadas
Abstract: Psi classes are tautological divisor classes on moduli spaces of stable curves. In genus zero they determine birational morphisms to projective space. Tropical psi classes were defined by Mikhalkin as subfans of M_{0,n}-trop. Their stable intersections were computed by Kerber and Markwig, and shown to be equivalent to the corresponding algebro-geometric intersections. I will present joint work with Sean Griffin, Jake Levinson and Rob Silversmith, in which we expand the notion of tropical psi classes by considering tropicalizations of carefully chosen effective representatives defined over non-trivially valued fields. We study their intersections in this modified setting, and establish several attractive features. I will contrast with the Kerber-Markwig tropical intersections, and describe how the two versions encode dual information.
Towards homological mirror symmetry for log del Pezzo surfaces.
by Franco Rota
Abstract: The homological mirror symmetry conjecture predicts a duality, expressed in terms of categorical equivalences, between the complex geometry of a variety X (the B side) and the symplectic geometry of its mirror object Y (the A side). Motivated by this, we study a series of singular surfaces (called log del Pezzo). I will describe the category arising in the B side, using the McKay correspondence and explicit birational geometry. If time permits, I will discuss some results on the A side, in the special case of a smooth degree 2 del Pezzo surface. The description of the B side is joint with Giulia Gugiatti, while the work (in progress) on the A side is in collaboration with Giulia Gugiatti and Matt Habermann.
by Arend Bayer
Abstract: A central result in the theory of Fano threefolds of index one and their derived categories is the existence of certain simple rigid vector bundles. Unfortunately, the proofs in the literature contain gaps. I will present work joint with Alexander Kuznetsk and Emanuele Macrì where we address this gap, and extend the result to finite characteristic. The proof is based on extending spherical bundles on K3 hyperplane sections, which in turn relies on uniqueness of vector bundles on curves satisfying a Brill-Noether condition.
Twistors, clusters and stability conditions
by Tom Bridgeland
Abstract: Given an ADE quiver Q, I will explain how to construct a twistor space. This is a complex manifold with a map to CP^1 whose fibre over 0 is the stability space of the CY3 Ginzburg algebra of Q, quotiented by spherical twists, and whose general fibre is an etale cover of the cluster Poisson variety of Q. This is joint work with Helge Ruddat.
Stability conditions on free abelian quotients
by Hannah Dell
Abstract: The space of Bridgeland stability conditions on a given triangulated category is a complex manifold, which gives us a way to extract geometry from homological algebra. An explicit description is only known in a few cases.
In this talk, I will discuss two approaches to studying stability conditions on derived categories of surfaces that are free quotients by finite abelian groups. One method is via Le Potier functions, which characterise the existence of slope-semistable sheaves. The second method uses Deligne’s notion of group actions on triangulated categories to describe a connected component of so-called geometric stability conditions inside the stability manifold of these free abelian quotients when the cover has finite Albanese morphism. A consequence of this is a disproof of the expectation that surfaces with irregularity 0 always admit a wall of the geometric chamber. (arXiv:2307.00815)
A Thurston compactification for the space of Bridgeland stability conditions
by Naoki Koseki
Abstract: Motivated by an analogue between the theory of Bridgeland stability conditions and the classical Teichmuller theory, Bapat–Deopurkar–Licata proposed a way to construct a Thurston type compactification of the space of Bridgeland stability conditions. In this talk, I will review their proposal, and report a recent progress in the cases of smooth projective curves and K3 surfaces. This is based on a joint work with K.Kikuta(Osaka) and G.Ouchi(Nagoya).
Intersections of tropical psi classes in genus zero, with non-trivial valuations
by Rohini Ramadas
Abstract: Psi classes are tautological divisor classes on moduli spaces of stable curves. In genus zero they determine birational morphisms to projective space. Tropical psi classes were defined by Mikhalkin as subfans of M_{0,n}-trop. Their stable intersections were computed by Kerber and Markwig, and shown to be equivalent to the corresponding algebro-geometric intersections. I will present joint work with Sean Griffin, Jake Levinson and Rob Silversmith, in which we expand the notion of tropical psi classes by considering tropicalizations of carefully chosen effective representatives defined over non-trivially valued fields. We study their intersections in this modified setting, and establish several attractive features. I will contrast with the Kerber-Markwig tropical intersections, and describe how the two versions encode dual information.
Towards homological mirror symmetry for log del Pezzo surfaces.
by Franco Rota
Abstract: The homological mirror symmetry conjecture predicts a duality, expressed in terms of categorical equivalences, between the complex geometry of a variety X (the B side) and the symplectic geometry of its mirror object Y (the A side). Motivated by this, we study a series of singular surfaces (called log del Pezzo). I will describe the category arising in the B side, using the McKay correspondence and explicit birational geometry. If time permits, I will discuss some results on the A side, in the special case of a smooth degree 2 del Pezzo surface. The description of the B side is joint with Giulia Gugiatti, while the work (in progress) on the A side is in collaboration with Giulia Gugiatti and Matt Habermann.