Algebraic Geometry Seminar
Department of Mathematics
Johns Hopkins University
Spring 2017
Regular meeting time: Tuesdays
4:305:30 (Tea served at 4:00)
Place: Gilman 55
Date  Speaker  Title 
February 7  Amin Gholampour JHU 
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 SeibergWitten invariants) of DurrKabanovOkonek and the stable pair invariants of KoolThomas. 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 CarlssonOkounkov. Our main application of these invariants is in local DonaldsonThomas theory of S. This is a joint work with Artan Sheshmani and ShingTung Yau. 
February 21  Donu Arapura Purdue University 
KodairaSaito 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 7 


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 YuChao 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 11 


April 25  Sam Payne Yale University 
A tropical motivic Fubini theorem with applications to DonaldsonThomas 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 semialgebraic 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 DonaldsonThomas theory. Based on joint work with Johannes Nicaise. 