Complex supermanifolds with many unipotent automorphisms

2016/07/23 by Kalus, Matthias
#54H15 #58A50 #58H15 #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1607.06947

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∈ \mathcal V_{\mathcal M, 0}⁽²⁾. Plenitude of unipotent automorphisms is understood as follows: the presheaf of common kernels of the operators [X,⋅] for X∈ \mathcal V_{\mathcal M, 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, 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.

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