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On deformation of sheaves
I. V. Artamkin
Abstract:
Let
$X$ be an algebraic variety over an algebraically closed field
$k$,
$\mathscr F$ a sheaf on
$X$,
$A$ and
$\widetilde A$ commutative Artinian
$k$-algebras,
$A=\widetilde A/I$, where
$I$ is a one-dimensional ideal,
$\mathscr E$ a deformation of
$\mathscr F$ with base
$\operatorname{Spec}A$, and $\operatorname{Ob}(\mathscr E,A,\widetilde A)\in\operatorname{Ext}^2(\mathscr F,\mathscr F)$ the obstruction to the extension of the deformation to
$\operatorname{Spec}\widetilde A$. The author constructs natural trace maps $\operatorname{tr}^i\colon\operatorname{Ext}^i(\mathscr F,\mathscr F)\to H^i(\mathscr O_X)$ and proves that if
$\operatorname{Pic}X$ is nonsingular then $\operatorname{tr}^2(\operatorname{Ob}(\mathscr E,A,\widetilde A))=0$. As a consequence, a universal deformation of a simple sheaf
$\mathscr F$ on
$X$ with nonsingular
$\operatorname{Pic}X$ exists if the map
$\operatorname{tr}^2$ is injective or, in the case
$\operatorname{rk}\mathscr F\ne0$, and $\operatorname{char}k\nmid\operatorname{rk}\mathscr F$, $\operatorname{Ext}^2(\mathscr F,\mathscr F)=H^2(\mathscr O_X)$.
Bibliography: 3 titles.
UDC:
512.7
MSC: 14D15,
14F05 Received: 23.10.1986
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