Grassmannian twists on the derived category via spherical functors (Q2864127)

This paper discusses some examples of ``twisted'' autoequivalences on certain Calabi-Yau spaces constructed from Grassmannians. These twisted equivalences are endofunctors that can be thought of as mirror to symplectic monodromies.NEWLINENEWLINEConsider \(G=\mathrm{Gr}(r,V)\) to be the Grassmannian of \(r\)-dimensional subspaces of a vector space \(V\) and let \(S\) be the tautological vector bundle on \(G\). The total space \(X:=\text{Tot}(\text{Hom}(V,S))\) is a Calabi-Yau variety and is stratified by the rank of the tautological map \(f:V\rightarrow p^*S\), where \(p\) is the projection \(p:X\rightarrow G\). The main result of this paper is, for \(r=2\), a description of an autoequivalence of the derived category \(D^b(X)\) as a twist around the big stratum \(B\), where \(B\) is the locus where the rank of \(f\) is not full.NEWLINENEWLINEThis construction naturally extends the work of Seidel-Thomas which gives a similar description for the case \(r=1\). In this paper, in order to obtain the main result, the author constructs a desingularization of the big stratum \(B\), which unlike the case \(r=1\) is singular, however the desingularization has a nice geometric description. This allows one then to define the twisted functor that defines the autoequivalence of \(D^b(X)\).