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 $t$.