## Fundamental solutions and complex cotangent line fields.(English)Zbl 1283.32002

Author’s 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.

### MSC:

 32V40 Real submanifolds in complex manifolds 32A26 Integral representations, constructed kernels (e.g., Cauchy, Fantappiè-type kernels) 53A55 Differential invariants (local theory), geometric objects
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