Complex supermanifolds with many unipotent automorphisms

TL;DR

This paper investigates unipotent automorphisms of complex supermanifolds, their classification, and how they influence the structure and deformation properties of supermanifolds, providing new insights into their automorphism groups and classification.

Contribution

It introduces the concept of strictly t-nildominated supermanifolds, linking unipotent automorphisms to supermanifold classification and deformation theory, and provides examples of non-split supermanifolds.

Findings

Strictly t-nildominated supermanifolds are classified up to degree t errors.

Unipotent automorphisms are induced by even degree-increasing vector fields.

Examples show non-split supermanifolds can deform in degrees lower than t.

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 $t$.