Speaker: Sergey Galkin
Title: Exceptional collections on punctual Hilbert schemes of surfaces, and categorical zeta-functions
Date: Feb 25, 2013, 14:00 - 17:00
Place: IPMU
Abstract:
A punctual Hilbert scheme of a projective plane have a full exceptional collection in its bounded derived category of coherent sheaves D(S^[n]). Punctual Hilbert schemes of dual abelian surfaces are derived equivalent. Let S be a surface with semi-orthogonal decomposition (SOD) of D(S) into r admissible subcategories; then D(S^{[n]}) has a SOD into binomial(n+r-1,r-1) admissible subcategories.
These and many other similar theorems are made easy thanks to idea of categorical symmetric powers by Ganter-Kapranov and descent theory by Elagin, together with the derived McKay correspondence by Bridgeland-King-Reid-Haiman-Bezrukavnikov-Kaledin.
Zeta-function (ZF) of a category is defined as the Moebius inverse transform of the generating series of categorical symmetric powers (the latter can be called the naive zf or categorical eta-function), it naturally takes values in the power series over Bondal-Larsen-Lunts's ring of pre-triangulated categories modulo SODs
and ZF is multiplicative with respect to SODs. ZF of D(X) conjecturally equals to the evaluation of Kapranov's "motivic" ZF of a manifold X. In dimensions less than 3 it is true thanks to a theorem of Goettsche and the facts above. Categorical ZF is related to Kontsevich's nc-motive ZF in a similar way as Kapranov's ZF is related to the usual "Chow" ZF. Non-commutative first Weil conjecture says that nc-motive ZF is rational. Conjectures of Denef-Loeser would also imply that categorical ZF is rational, on the contrast any example of irrational categorical ZF of a manifold would prove a conjecture of Lunts. Categorical ZF are proved to be rational for categories with a FEC, curves, rational threefolds, etc. We don't know if they are rational for rationally-connected threefolds or uniruled surfaces.
All above is a work in progress by Evgeny Shinder and the speaker.