Complex supermanifolds with many unipotent automorphisms

Abstract

An automorphism on a complex supermanifold $\mathcal M$ is called unipotent if it reduces to the identity on the associated graded supermanifold $gr(\mathcal M)$. These automorphisms are close to be complementary to those responsible for homogeneity of a supermanifold. In analogy, their study yields results on the classification of supermanifolds. Unipotent automorphisms are induced by even global degree increasing vector fields $X\in \mathcal V_{\mathcal M,\bar 0}^{(2)}$. Plenitude of unipotent automorphisms is understood as follows: the presheaf of common kernels of the operators $[X,\cdot]$ for $X\in \mathcal V_{\mathcal M,\bar 0}^{(2)}$, on superderivations vanishes up to errors of a fixed degree $t$ and higher. The isomorphy class of such strictly $t$-nildominated supermanifolds is determined up to errors of degree $t$ and higher by $\mathcal V_{\mathcal M,\bar 0}^{(2)}$ and $gr(\mathcal M)$. An example shows that a strictly $t$-nildominated supermanifold can be non-split, deformed already in degrees lower than $t$.