Semiorthogonal Decompositions on Total Spaces of Tautological Bundles

Student: Pirozhkov Dmitrii

Supervisor: Aleksandr Kuznetsov

Educational Programme: Mathematics (Master)

Final Grade: 10

Year of Graduation: 2019

Let U be the tautological subbundle on the Grassmannian Gr(k, n). There is a natural morphism Tot(U) -> A^n. Using it, we give a semiorthogonal decomposition for the bounded derived category Dbcoh(Tot(U)) into several exceptional objects and several copies of Dbcoh(A^n). We also prove a global version of this result: given a vector bundle E with a regular section s consider a subvariety of the relative Grassmannian Gr(k, E) of those subspaces which contain the value of s. The derived category of this subvariety admits a similar decomposition into copies of the base and the zero locus of s. This may be viewed as a generalization of the blow-up formula of Orlov, which is the case k = 1.

Full text (added June 11, 2019)

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