Algebraic Geometry Seminar
Department of Mathematics
Johns Hopkins University
Regular meeting time: Tuesdays
4:30-5:30 (Tea served at 4:00)
Place: Gilman 55
|February 7||Amin Gholampour
|Virtual fundamental class for nested Hilbert schemes on surfaces
We construct natural virtual fundamental classes for nested Hilbert schemes (of points and curves) on a nonsingular projective surface S. This allows us to define new invariants of S that recover some of the known important cases such as Poincare invariants (algebraic Seiberg-Witten invariants) of Durr-Kabanov-Okonek and the stable pair invariants of Kool-Thomas. In the case of the nested Hilbert scheme of points, we can express some of our invariants in terms of integrals over the products of Hilbert scheme of points on S, and relate them to the vertex operator formulas found by Carlsson-Okounkov. Our main application of these invariants is in local Donaldson-Thomas theory of S. This is a joint work with Artan Sheshmani and Shing-Tung Yau.
|February 21||Donu Arapura
|Kodaira-Saito vanishing via Higgs bundles in positive characteristic
In 1990, Saito gave a strong generalization of Kodaira’s vanishing theorem using his theory of mixed Hodge modules. I want to explain the statement in the special case of a variation of Hodge structure on the complement of a divisor with normal crossings. I will describe a proof using characteristic p methods.
|March 31||Tommaso de Fernex
University of Utah
|Towards a link theoretic characterization of smoothness
A theorem of Mumford states that, on complex surfaces, any normal isolated singularity whose link is diffeomorphic to a sphere is actually a smooth point. While this property fails in higher dimensions, McLean asks whether the contact structure that the link inherits from its embedding in the variety may suffice to characterize smooth points among normal isolated singularities. He proves that this is the case in dimension 3. In joint work with Yu-Chao Tu, we use techniques from birational geometry to extend McLean's result to a large class of higher dimensional singularities. We also introduce a more refined invariant of the link using CR geometry, and conjecture that this invariant is strong enough to characterize smoothness in full generality.
|April 25||Sam Payne
|A tropical motivic Fubini theorem with applications to Donaldson-Thomas theory
I will present a new tool for the calculation of Denef and Loeser’s motivic nearby fiber and motivic Milnor fiber: a motivic Fubini theorem for the tropicalization map. Our method uses Hrushovski and Kazhdan’s theory of motivic volumes of semi-algebraic sets as a substitute for computations with motivic Igusa zeta functions, and allows us to sidestep, in some cases, the absence of an explicit resolution of singularities. Applications include the solution to a conjecture of Davison and Meinhardt on motivic nearby fibers of weighted homogeneous polynomials, and a very short and conceptual new proof of the integral identity conjecture of Kontsevich and Soibelman, first proved by Lê Quy Thuong. Both of these conjectures emerged in the context of motivic Donaldson-Thomas theory.
Based on joint work with Johannes Nicaise.