D-equivalence conjecture for Hyperkähler varieties

• Applicant: Dr. Franco Giovenzana
• Subject Area: Mathematics
• Term: since 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 509501007

A notable conjecture in algebraic geometry named after the mathematicians Bondal and Orlov predicts that if two smooth algebraic projective varieties with trivial canonical bundles are birational, then they are derived equivalent, that means there exists an equivalence between their derived categories. The present project addresses the instance of this conjecture for the specific class of varieties with trivial canonical bundles, called Hyperkähler varieties. These can be thought of as one of the fundamental building blocks for varieties with trivial canonical bundle thanks to the decomposition theorem of Beauville and Bogomolov and they occupy a central role in algebraic geometry, because they are a very rich source of geometry.Various partial results of this conjecture are already available in the literature, they comprise specific elementary birational transformations between Hyperkähler varieties (called Mukai flops) and, thanks to a recent work of Halpern-Leistner, any birational map of a particular class of varieties, namely all the birational models of any moduli space of Gieseker stable sheaves on some K3 surface. A basic result about derived categories says that any derived equivalence between two smooth projective varieties X and Y has actually a particular form which goes under the name of Fourier-Mukai functor and is defined from a kernel, which is an object in the derived category of the product X x Y. The first goal of the project is then to compute explicitly these Fourier-Mukai kernels for the derived equivalences between birational models of Gieseker moduli spaces of sheaves which were introduced by Halpern-Leistner.Despite a solid theory for Hyperkähler varieties has been developed, a classification is still missing and up to now all the known examples belong to 2 infinite series of deformation families (punctual Hilbert schemes of K3 surfaces and generalized Kummer varieties) or to 2 other sporadic examples named after O'Grady. The second goal of the current project is to prove the Bondal-Orlov conjecture for the known examples of Hyperkähler varieties, starting from the deformation families of the punctual Hilbert schemes of a K3. A preliminary step in this direction is to extend Halpern-Leistner's result by means of deformation theory, where we can take advantage of some recent advances by Bakker, Guenancia and Lehn.Ever since the results on standard flops by Bondal and Orlov, it is expected that the so-called flop-flop equivalence defined for a birational transformation of a variety X gives a non-trivial derived auto-equivalence of X. This phenomenon has been completely exploited for a Mukai flop of a Hyperkähler variety by Addington, Donovan and Meachan, we aim to carry over this analysis for the case of a general birational transformation of Hyperkähler varieties, starting from the stratified Mukai flops appearing in the works of Bayer and Macrì on the birational models of moduli spaces of sheaves.

• DFG Programme: WBP Fellowship
• International Connection: France
• Host: Professor Dr. Emanuele Macri