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Spring 2012 Fundamental solutions and complex cotangent line fields
Sidney M. Webster
Illinois J. Math. 56(1): 251-263 (Spring 2012). DOI: 10.1215/ijm/1380287471


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.


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Sidney M. Webster. "Fundamental solutions and complex cotangent line fields." Illinois J. Math. 56 (1) 251 - 263, Spring 2012.


Published: Spring 2012
First available in Project Euclid: 27 September 2013

zbMATH: 1283.32002
MathSciNet: MR3117029
Digital Object Identifier: 10.1215/ijm/1380287471

Primary: 32V40
Secondary: 32N05

Rights: Copyright © 2012 University of Illinois at Urbana-Champaign


Vol.56 • No. 1 • Spring 2012
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