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Weighted Bott-Chern and Dolbeault cohomology for LCK-manifolds with potential. (English) Zbl 1390.32019

Let \(M\) be a locally conformal Kähler (LCK) manifold with universal covering \(\widetilde{M}\), i.e., \(M\) is a complex manifold with a Kähler structure on \(\widetilde{M}\) and with a deck transform group acting on \(\widetilde{M}\) by holomorphic homothetic maps and which is hence naturally equipped with a Hermitian form \(\omega\) (called the Lee form) such that \(d\omega = \omega \wedge \theta\) for some closed 1-form \(\theta\) with values in a local system \(L\) (called conformal weight bundle). The main result, Theorem 3.2, establishes generic vanishing properties for the weighted Dolbeault cohomology and serves in particular as the basis for the proof of an analogue of the \(dd^c\)-lemma on Kähler manifolds, in the setting of \(L\)-valued (Morse-Novikov) cohomology: Let \((M,\omega,\theta)\) be a compact LCK manifold, and let \(L\) its weight bundle. Then there exists a countable discrete subset \(S\) of the complex plane \(\mathbb C\) such that, for each \(\alpha\in\mathbb C\setminus S\), if \(L_\alpha\) is the flat line bundle on \(M\) corresponding to \(\alpha\cdot \theta\), then \(H^q(M,\Omega_M^p \otimes L_\alpha) =0\) for all \(q\in\mathbb N\). As a consequence one can prove the following geometrically far reaching result (in particular a true \(dd^c\)-conjecture version in Vaisman manifolds with coefficients in a sufficiently large power of \(L\)): If \(M\) is an LCK manifold with proper potential, then \(H^{p,q}_{BC}(M,L\alpha)=0\) for all \(\alpha\in\mathbb C\setminus S\) (Bott-Chern cohomology).

MSC:

32Q15 Kähler manifolds
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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