Formal deformations and their categorical general fibre

Abstract

Abstract. We study the general fibre of a formal deformation over the formal disk of a projective variety from the view point of abelian and derived categories. The abelian category of coherent sheaves of the general fibre is constructed directly from the formal deformation and is shown to be linear over the field of Laurent series. The various candidates for the derived category of the general fibre are compared.If the variety is a surface with trivial canonical bundle, we show that the derived category of the general fibre is again a linear triangulated category with a Serre functor given by the square of the shift functor. The paper is a companion to [9], where the results are applied to Fourier-Mukai equivalences of K3 surfaces. Mathematics Subject Classification (2010). 18E30, 14D15.

Hyperkähler Manifolds and Sheaves

“…Note also that the deformation X → Spf(C[[t]]) used in [13] is the formal neighbourhood of a very generic twistor space X (α) → P 1 and thus highly nonalgebraic. In particular, the generic fibre X K does not exist as a projective variety.…”

“…Instead of working with rigid analytic varieties, [13] makes only use of Coh(X K ) and its derived category which can both be constructed directly as quotients of Coh(X ) respectively D b (X ) without ever defining X K .…”

“…The general fibre of a twistor space is a rigid analytic variety. In [13] its category of sheaves was studied, and somehow identified with the variety. As a geometric object or as a category, it should be viewed as naturally associated to the Ricci-flat metric.…”

Notes on formal deformations of abelian categories

“…Let us also mention that a very complete treatment of deformations of algebraic varieties as ringed spaces (including their derived categories) over k [[t]] has been given in [9].…”

*-Quantizations of Fourier–Mukai Transforms

“…This leads to strong geometric statements such as the variational derived Torelli theorem [HMS09,HMS08].…”