Fundamental solutions and complex cotangent line fields

Abstract

We consider a fundamental solution for the $\overline{\partial}$-operator on a complex $n$-manifold, which is given by an $(n,n-1)$-form of the Cauchy–Leray type $\Theta=\theta\wedge(\overline{\partial}\theta)^{n-1}$, 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 $E\subset T^{*}_{(1,0)}M$, 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 $\Theta\neq0$, 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.