|TITLE||Derived Categories of Projectivization of Coherent Sheaves and Flops|
The derived categories of coherent sheaves on algebraic varieties, introduced by Grothendieck and Verdier in late 1950s, are subjects of active research in algebraic geometry. In this talk we start with reviewing derived categories of varieties, with focus on semiorthogonal decompositions (SOD), which is a way to break derived categories into smaller building blocks. Then we concentrate on the main topic of this talk: how derived categories behave under geometric operations. More concretely, first we review Orlov's series of results on (i) projective bundles, (ii) generalized universal hyperplanes, and (iii) blowing-ups along smooth centers. Then we discuss how to generalize (i) and (ii) to projectivization of a coherent sheaf (which locally admits two-step resolutions) and show how this leads to derived equivalences of an interesting class of flops. Our approach is based ``chess game” method of mutations of SODs, which is a very powerful method to compare different subcategories, and can be applied to many situations. If time allows we will also discuss more on the ``chess game" method, also how to generalize (iii) to blowing-ups along non-smooth locally complete intersection centers, and the relationships between these formulae. The talk is based on joint works with Prof. Conan Leung.