A Koszul-Type Splitting Theorem for Complex Supermanifolds

Abstract

We study the splitting problem for complex supermanifolds using formal exponential maps and Ehresmann connections. We show that every formal exponential map induces a splitting of the underlying supermanifold. Furthermore, we establish that every Ehresmann connection determines a linearization of the infinite jet space, extending the corresponding construction for ordinary manifolds to supermanifolds. For Ehresmann connections preserving the zero section, we construct a formal exponential map in two ways: via the linearization of the infinite jet space and directly from the connection operator. We prove that both constructions yield the same map and induce the same splitting, thereby generalizing Koszul's splitting theorem from linear connections to zero-section-preserving Ehresmann connections. Finally, we provide examples showing that, without this assumption, the two constructions may yield different splittings, or one or both of them may fail to induce a splitting.