Algebra and Algebraic Geometry Seminar Fall 2024

The seminar normally meets 2:30-3:30pm on Fridays, in the room Van Vleck B131.

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Fall 2024 Schedule

date	speaker	title	host/link to talk
September 27	Joshua Mundinger (Madison)	Hochschild homology and the HKR spectral sequence	local
October 4	Dima Arinkin (Madison)	Derived category of the stacky compactified Jacobian	local
November 15	Yunfeng Jiang (Kansas)	TBA	Andrei/Ruobing
December 13	Thomas Brazelton (Harvard)	TBA	Andrei/Josh

Abstracts

Joshua Mundinger

Hochschild homology and the HKR spectral sequence

Hochschild homology of an algebraic variety carries the Hochschild-Konstant-Rosenberg (HKR) filtration. In characteristic zero, this filtration is split, yielding the HKR decomposition of Hochschild homology. In characteristic p, this filtration does not split, giving rise to the HKR spectral sequence. We describe the first nonzero differential of this spectral sequence. Our description is related to the Atiyah class.

Dima Arinkin

Derived category of the stacky compactified Jacobian

Abstract: The Jacobian of a smooth projective curve is an abelian variety which is identified with its own dual. This implies that its derived category carries a non-trivial auto-equivalence - the Fourier-Mukai transform. When the curve has planar singularities, the Jacobian is no longer compact (and, in particular, not an abelian variety), but it turns out that the Fourier-Mukai transform still exists, provided we compactify the Jacobian. The transform can be viewed as the `classical limit' of the geometric Langlands correspondence.

In this talk, I will explore what happens when the curve becomes reducible. From the point of view of the geometric Langlands conjecture, it is important to work with the compactified Jacobian viewed as a stack (rather than the corresponding moduli space). In my talk I will show that this also leads to certain issues, and in fact that the most general version of the statement is inconsistent, while more conservative versions are true.