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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.
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References:

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