Fundamental solutions and complex cotangent line fields

Abstract

We consider a fundamental solution for the $\bar{\partial }$-operator on a complex $n$-manifold, which is given by an $(n,n-1)$-form of the Cauchy–Leray type $\mathrm{\Theta }=\theta \wedge (\bar{\partial }\theta {)}^{n-1}$, where $\theta$ is a suitable $(1,0)$-form. On the open submanifold $M^n$ where $\theta$ is smooth and nonzero, its multiples generate a complex line sub-bundle $E\subset T_{(1,0)}^{\ast }M$, which we assume to satisfy a certain integrability condition. To such an $E$ we attach a global holomorphic invariant, in the form of a complex Godbillon–Vey $\partial$-cohomology class, provided a certain primary obstruction class vanishes. If $\theta$ is also Levi nondegenerate, in that $\mathrm{\Theta }\ne 0$, then it determines an invariant connection on the hyperplane bundle given by $\theta =0$. This provides $\theta$ formally with a complete system of local holomorphic invariants.