On the stability of locally conformal Kähler structures. (English) Zbl 1310.53062

If \(X\) a complex manifold and \(L\) a flat line bundle over \(X\), then we say that \((X,L)\) satisfies the \(\partial\bar{\partial}\)-lemma at degree \((p,q)\) if there is an \(L\)-valued form \(\gamma\) of type \((p-1,q-1)\) such that \(\partial_L\bar{\partial}_L\gamma=\partial_L\alpha\in\Gamma(X,L\otimes\Lambda^{p,q})\), for every \(\bar{\partial}_L\)-closed \(L\)-valued form \(\alpha\) of type \((p-1,q)\). A locally conformal Kähler (LCK) manifold is a complex Hermitian manifold, with a Hermitian form \(\omega\) satisfying \(d\omega=\theta\wedge\omega\), where \(\eta\) is a closed, non-exact 1-form, called the Lee form of the manifold.
In this paper, the author obtains the following criterion for the stability of LCK structures: “If \(L\) is a flat line bundle corresponding to an LCK structure \(\omega_0\) on \(X\) and \((X,L)\) satisfies the \(\partial\bar{\partial}\)-lemma at degree \((1,2)\), then the stability of LCK structures holds, that is, every small deformation \(X_t\) of \(X\) admits an LCK structure \(\omega_t\) where \(L\) is still the corresponding flat line bundle to the deformed structure \(\omega_t\)”. Moreover, the author shows that there are obstructions to the stability of LCK structures under deformations of flat line bundles on Inoue surfaces with \(b_2=0\) and disproves a conjecture of L. Ornea and M. Verbitsky [J. Geom. Phys. 59, No. 3, 295–305 (2009; Zbl 1161.57015)].


53C55 Global differential geometry of Hermitian and Kählerian manifolds
32G05 Deformations of complex structures


Zbl 1161.57015
Full Text: DOI arXiv Euclid


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