Room 515, Cosmology Building, National Taiwan University + Zoom, Physical+Online Seminar
(實體+線上演講 台灣大學次震宇宙館515研討室+ Zoom)
Finite Group Actions and Duality on Picard Stacks
Guillermo Gallego (Freie Universität Berlin)
Abstract
Picard stacks are commutative group objects in the 2-category of stacks. Examples include commutative group schemes and abelian varieties, but also classifying stacks BG, for G a commutative group scheme. More generally, for such G over a scheme X, the stack of G-torsors Bun(G/X) is also a Picard stack. There is a natural notion of duality inside Picard stacks, given by considering maps to BGm, where Gm denotes the multiplicative group. This duality generalizes both Cartier duality and the duality of abelian varieties. There is also a notion of Fourier-Mukai transform associated a Picard stack, which gives an equivalence of categories when the Picard stack is "good". A particularly interesting class of good Picard stacks is formed by Beilinson 1-motives, which are objects essentially of the form M x A x BG, for M a finitely generated abelian group, A an abelian variety, and G a commutative group scheme.
In this talk, we review this theory, and consider the actions of finite groups on Picard stacks. This leads us to construct the appropriate notions of "invariants" and "coinvariants" on Picard stacks, which are linked to group cohomology and homology, respectively. This allows us to recover the SYZ mirror symmetry over the generic locus of the Hitchin fibration developed by Hausel-Thaddeus, Donagi-Pantev and Chen-Zhu and give a slight generalization of their results, which allows us to explore similar structures in other fibrations related to the Hitchin fibration.
Organizer: Pedro Núñez (NTU)
