Given a variety Y with a rectangular Lefschetz decomposition of its derived category, we consider a degree n cyclic cover X→Y ramified over a divisor Z⊂Y. We construct semiorthogonal decompositions of Db(X) and Db(Z) with distinguished components AX and AZ and prove the equivariant category of AX (with respect to an action of the nth roots of unity) admits a semiorthogonal decomposition into n−1n−1 copies of AZ. As examples, we consider quartic double solids, Gushel–Mukai varieties, and cyclic cubic hypersurfaces.
We describe the (equivariant) intersection cohomology of certain moduli spaces ("framed Uhlenbeck spaces") together with some structures on them (such as e.g., the Poincare pairing) in terms of representation theory of some vertex operator algebras (" 1V-Algebras").
We prove that every quasi-smooth weighted Fano threefold hypersurface in the 95 families of Fletcher and Reid is birationally rigid. © 2016 American Mathematical Society.