Bayer:
Support property and quadratic inequalities |
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Bernardara:
HPD for determinantal varieties and beyond |
In this talk, I will show Homological Projective Duality for determinantal varieties, and some
application such as a derived equivalence between (noncommutative crepant resolutions) of some Calabi-Yau varieties. The main steps in the proof are the construction of a noncommutative resolution of singularities, based on Buchweitz, Leuschke and Van den Bergh's work, and Kuznetsov's Homological Projective Duality for projective bundles. I will also show how these ideas give a HP-dual for G(2,8) which we conjecture to be a noncommutative crepant resolution of singularities of the Pfaffian variety
All these results are obtained in collaboration with M.Bolognesi and D.Faenzi.
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Bridgeland:
Spaces of stability conditions on CY3 quiver categories |
Associated to any quiver with potential is a CY3 triangulated category. These categories provide an amenable class of examples where one can try to calculate spaces of stability conditions. I will review some of the work which has been done on these spaces and hopefully go on to talk a little about DT invariants and cluster varieties. |
Engenhorst:
Maximal green sequences |
Maximal green sequences were introduced as combinatorical counterpart for Donaldson-Thomas invariants for 2-acyclic quivers with potential by B. Keller. A third incarnation are maximal chains in the Hasse quiver of torsions classes. More generally, we introduce maximal green sequences for hearts of bounded t-structures of triangulated categories that can be tilted indefinitely. In the case of preprojective algebras we show that a quiver has a maximal green sequence if and only if it is of Dynkin type. |
Li:
Bridgeland stability conditions on some Picark rank one varieties |
The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds by Bayer, Macri and Stellari provides a systematic way to build Bridgeland stability conditions on smooth 3-fold over C. In particular, when the Picard rank of the 3-fold is 1, one only needs to check Conjecture 5.3 in the paper, and then the stability condition by 'double tilting' is indeed geometric. I will talk about some general pictures for stability conditions on Picard rank one varieties and the proof of Conjecture 5.3 for Picard rank one smooth Fano threefold. |
Piyaratne:
Fourier-Mukai theory and stability conditions on abelian threefolds |
The notion of Fourier-Mukai transform for abelian varieties was introduced by Mukai in early 1980s. Since then Fourier-Mukai theory turned out to be extremely successful in studying stable sheaves and complexes of them, and also their moduli spaces. I will explain how the Fourier-Mukai techniques are useful to show that the conjectural construction proposed by Bayer, Macri and Toda gives rise to Bridgeland stability conditions on abelian threefolds. First we reduce the requirement of the Bogomolov-Gieseker type inequalities to a smaller class of tilt stable objects which are essentially minimal objects of the conjectural stability condition hearts for a given smooth projective threefold. Then we establish the existence of Bogomolov-Gieseker type inequalities for these minimal objects of abelian threefolds by showing certain Fourier-Mukai transforms give equivalences of abelian categories which are double tilts of coherent sheaves. Part of this is a joint work with Antony Maciocia. |
Stellari:
Stability for cubic threefolds and fourfolds |
We investigate the relation between Kuznetsov semiorthogonal decomposition of the bounded derived category of coherent sheaves of smooth cubic threefolds and fourfolds and Bridgeland/tilt stability. We use this to study the geometry of some interesting moduli spaces of stable sheaves on such cubics. In particular, we discuss the cases of aCM bundles and of generalized twisted cubics recently studied by Lehn-Lehn-Sorger-van Straten. This is a joint work (partly) in progress with M. Lahoz and E. Macri'. |
Lee:
Semiorthogonal decompositions of derived categories of Fano varieties |
Derived categories of Fano varieties always have nontrivial semiorthogonal decompositions. I will discuss which categories can be realize as full subcategories of derived categories of Fano varieties. |
Qiu:
Spherical twists actions on
spaces of stability conditions from surfaces |
We study the 3-Calabi-Yau categories associated to unpunctured (decorated) marked surfaces. We show that the spherical twist group is isomorphic to a subgroup of the mapping class group. For instance, when the surface is an annulus, this implies the contractibility of the corresponding space of stability conditions. |
Toda:
Non-commutative virtual structure sheaves |
The moduli spaces of stable sheaves on algebraic varieties admit certain non-commutative structures, which I call quasi NC structures. In this talk, I show that the quasi NC structures on these moduli spaces arise as truncations of smooth NC dg-schemes. This result is used to introduce the notion of NC virtual structure sheaves on the moduli spaces of stable sheaves which admit perfect obstruction theories. |
