NCMW - Derived categories and semi-orthogonal decompositions (2024)
Venue: Chennai Mathematical Institute, Siruseri
Dates: 17 Jun 2024 to 29 Jun 2024
Convener(s)
| Name: | Vivek Mohan Mallick | Umesh V Dubey | Sukhendu Mehrotra |
| Mailing Address: | Assistant Professor, IISER Pune Main Building, IISER Campus Dr Homi Bhabha Road Pashan, Pune 411008 |
Reader F, Harish-Chandra Research Institute Chhatnag road, Jhusi Prayagraj 211019 |
Associate Professor, Chennai Mathematical Institute H1, SIPCOT IT Park, Siruseri Kelambakkam 603103 |
| Email: | vmallick at iiserpune.ac.in | umeshdubey at hri.res.in | mehrotra at cmi.ac.in |
This workshop is intended to be an introduction to the study of derived categories of sheaves on an algebraic variety, with the topic of semi-orthogonal decompositions (S.O.D.’s) as the unifying theme. Derived categories encode hidden structures and symmetries of varieties, and therefore, a S.O.D. is an object of intrinsic interest, much as a basis allows one to study a vector space. Further, S.O.D.’s most arise naturally from constructions in algebraic geometry which intertwine their study with that of birational geometry as well as that of algebro-geometric quotients.
Starting with the basic theory and examples of derived categories and their SODs, the workshop shall cover some relatively recent topics, including the work on the derived categories of GIT quotients by Halpern-Leistner and the work on Homological Projective Duality by Kuznetsov. The construction of a GIT quotient depends on certain choices; the parameter space of such choices breaks up into chambers where the GIT quotient does not vary. Crossing the walls dividing these chambers involves certain birational transformations called (generalized) flips of the corresponding quotients. Flips often lead to the derived category of the quotient on one side of the wall being included as a semi-orthogonal component of the derived category of the quotient on the other side. This behaviour serves as a template for the Minimal Model Program in birational geometry and is of considerable interest. Kuznetsov’s Homological Projective Duality is a construction that generalizes the classical projective duality between projective varieties, and leads to geometrically meaningful S.O.D.’s and derived equivalences.
Prerequisites. We shall assume a basic knowledge of Algebraic Geometry, e.g. the contents of Hartshorne’s book (till chapter 3). Basic knowledge of homological algebra. Weibel / Gelfand-Manin’s chapter on Derived categories. Basic knowledge of GIT (eg. Newstead’s notes) and familiarity with the following might help: Caldararu and Weibel for Hochschild homology.