Locally conformal Kähler manifolds with potential

Abstract

A locally conformally Kähler (LCK) manifold M is one which is covered by a Kähler manifold M with the deck transformation group acting conformally on M. If M admits a holomorphic flow, acting on M conformally, it is called a Vaisman manifold. Neither the class of LCK manifolds nor that of Vaisman manifolds is stable under small deformations. We define a new class of LCK-manifolds, called LCK manifolds with potential, which is closed under small deformations. All Vaisman manifolds are LCK with potential. We show that an LCK-manifold with potential admits a covering which can be compactified to a Stein variety by adding one point. This is used to show that any LCK manifold M with potential, dim M ≥ 3, can be embedded into a Hopf manifold, thus improving similar results for Vaisman manifolds Ornea and Verbitsky

Morse–Novikov cohomology of locally conformally Kähler manifolds

“…It is known ( [GO], [Ve2]) that all diagonal Hopf manifolds are Vaisman. In [OV2], we have shown that any Vaisman manifold admits a holomorphic immersion into a diagonal Hopf manifold, and in [OV3] we proved that for dim C M 3 there exists an embedding into a diagonal Hopf manifold. In fact, the assumption dim C M 3 is not needed, because in [Be], all 2-dimensional Vaisman manifolds were classified, and embeddability of those into H A can be easily checked using the same arguments as in [OV3].…”

“…[OV3]) to the deck group being isomorphic to Z (a condition satisfied by compact Vaisman manifolds).…”

“…The main property of LCK manifolds with potential is that they satisfy an embedding theorem similar to the Kodaira-Nakano one in Kähler geometry: OV3]) Any compact LCK manifold with potential of complex dimension at least 3 can be holomorphically embedded in a Hopf manifold. Moreover, a compact Vaisman manifold of complex dimension at least 3 can be holomorphically embedded in a diagonal Hopf manifold.…”

Deformation of Sasakian metrics

“…by the Hodge decomposition theorem of El Kacimi Alaoui (see Theorem 6.7), the proof is done by (38) and (42). by the complex conjugation.…”

“…Belgun [5] showed that both of structures are not stable under small deformations of complex structures. Note that Ornea and Verbitsky [38] discovered the stability of another class of geometric structure under small deformations of complex structures, which they call locally conformal Kähler manifolds with potential. Theorem 1.1.…”