Smooth deformations of singular contractions of class VII surfaces
We consider normal compact surfaces $Y$ obtained from a minimal class VII surface $X$ by contraction of a cycle $C$ of $r$ rational curves with $C^2<0$. Our main result states that, if the obtained cusp is smoothable, then $Y$ is globally smoothable. The proof is based on a vanishing theorem for $H^2(\Theta_Y)$. If $r<b_2(X)$ any smooth small deformation of $Y$ is rational, and if $r=b_2(X)$ (i.e. when $X$ is a half-Inoue surface) any smooth small deformation of $Y$ is an Enriques surface. The condition "the cusp is smoothable" in our main theorem can be checked in terms of the intersection numbers of the cycle, using the Looijenga conjecture (which has recently become a theorem). Therefore this is a "decidable" condition. We prove that this condition is always satisfied if $r<b_2(X)\leq 11$. Therefore the singular surface $Y$ obtained by contracting a cycle $C$ of $r$ rational curves in a minimal class VII surface $X$ with $r<b_2(X)\leq 11$ is always smoothable by rational surfaces. The statement holds even for unknown class VII surfaces.