## Lefschetz theorems for the integral leaves of a holomorphic one-form.(English)Zbl 0802.58004

The classic Lefschetz theorems concern intersections of subvarieties of projective space with hyperplanes. This paper is about some similar theorems concerning the integral leaves of holomorphic or real harmonic one-forms. Let $$X$$ be a smooth projective variety, and let $$\alpha^ h \in H^ 0 (X,\Omega^ 1_ X)$$ be a holomorphic one-form on $$X$$. We will look at $$\alpha = \alpha^ h$$ or the real part $$\alpha = {\mathfrak R} \alpha^ h$$. Let $$Y$$ be any connected covering space of $$X$$ such that the pullback of $$\alpha$$ is exact, and let $$g:Y \to \mathbb{C}$$ (resp. $$\mathbb{R})$$ denote an integral of $$\alpha$$. It is well defined up to addition of a constant. We may think of the fibers $$g^{-1} (v)$$ as intersections of $$Y$$ with linear hyperplanes in a vector space (this is precise if we take $$Y$$ to be the covering $$Z$$ defined below). We obtain some theorems about connectivity of the pairs $$(Y,g^{-1} (v))$$, analogues of the classical Lefschetz theorems.

### MSC:

 58A10 Differential forms in global analysis 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 57R70 Critical points and critical submanifolds in differential topology 58E20 Harmonic maps, etc.
Full Text:

### References:

 [1] K. Corlette , C. Simpson : Classification of rank two local systems on a smooth quasiprojective variety . In preparation (1991). [2] C.H. Clemens : Degeneration of Kähler manifolds . Duke Math. J. 44 (1977), 215-290. · Zbl 0353.14005 [3] M. Gromov , R. Schoen : Preprint on harmonic mappings to simplicial complexes; and lecture (Schoen) at University of Chicago, February 1991. [4] S. Lefschetz : L’Analysis Situs et la Géométrie Algébrique . Gauthier-Villars, Paris (1924). · JFM 50.0663.01 [5] J. Milnor : Singular Points of Complex Hypersurfaces . Annals of Math. Studies 61, Princeton (1961). · Zbl 0184.48405
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.