On deformation of sheaves

I. V. Artamkin

Izvestiya. Mathematics , Tome 32 (1989) no. 3, pp. 663-668

Résumé

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.