| 概要 | Seidel and Thomas introduced some years ago a notion of a
spherical object in the derived category D(X) of a smooth projective
variety X. Such objects induce, in a simple way, auto-equivalences of D(X)
called 'spherical twists'. In a sense, they are mirror-symmetric analogues
of Lagrangian spheres on a symplectic manifold and the induced
auto-equivalences mirror the Dehn twists associated with the latter. We
generalise this notion to the relative context by explaining what does it
mean for an object of D(Z x X) to be _spherical over Z_ for any two
separated schemes Z and X of finite type. Such objects induce naturally
two
auto-equivalences: one of D(X), called "the twist", and the other of D(Z)
called "the co-twist". For objects of D(Z x X) which are orthogonal over Z
we give a simple cohomological criterion for being spherical over Z which
resembles the original definition of Seidel and Thomas. This is a joint
work with Rina Anno (UMass/Amherst). |