## 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)].

### MSC:

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

Zbl 1161.57015
Full Text:

### References:

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