The Sheffield Algebraic Geometry Seminar

Product-quotient surfaces form an interesting class of surfaces of general type. We explore the stability manifold of these surfaces. In the cases when the group action is not free, we construct non-geometric stability conditions.

Nicaise--Shinder showed that stable rationality specializes in smooth projective families by studying the motivic volume. We discuss an analogue of the motivic volume for algebraic cycles and its applications.

Nicaise--Shinder showed that stable rationality specializes in smooth projective families by studying the motivic volume. We discuss an analogue of the motivic volume for algebraic cycles and its applications.

I will talk about our work with Julia Schneider and Evgeny Shinder about canonical, or atomic, semi-orthogonal decompositions for derived categories of surfaces. I will explain motivations, construction (including examples), and (time permitting) some applications of atomic decompositions.  (If you have been at a talk about the same work at the Sheffield-LMS symposium this summer, please note that this time I will concentrate on different aspects).    

Gromov—Witten invariants of a target variety X count maps from algebraic curves to X, passing through certain loci. If X is equipped with a normal-crossings boundary divisor D, we can also count maps from curves to X which have prescribed contact orders with the components of D; these are log Gromov—Witten invariants, and arise in degeneration formulas for classical Gromov-Witten invariants. Classical Gromov—Witten invariants form a cohomological field theory - this is a collection of relations between the high- and low-genus invariants of a fixed target X. In this talk we will explain how to extend (some of) these structures to log Gromov-Witten invariants. This is joint work with Leo Herr and Pim Spelier.

We study the support property for stability conditions glued along semiorthogonal decompositions. The main tool is an abelian analogue of the dg-gluing construction of Kuznetsov–Lunts that gives an explicit description of glued hearts. We establish the support property under phase bounds that include those of Collins–Polishchuk. Applications include augmented curves studied by Alexeev and Kuznetsov. We also discuss how the construction behaves under mutations of semiorthogonal decompositions. This is joint work with Fabian Haiden.

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