Locally conformally Kähler solvmanifolds: a survey

Abstract

A Hermitian structure on a manifold is called locally conformally Kähler (LCK) if it locally admits a conformal change which is Kähler. In this survey we review recent results of invariant LCK structures on solvmanifolds and present original results regarding the canonical bundle of solvmanifolds equipped with a Vaisman structure, that is, a LCK structure whose associated Lee form is parallel.2010 Mathematics Subject Classification. 53B35, 53A30, 22E25.

Locally conformally balanced metrics on almost abelian Lie algebras

“…. , e 2n } satisfying the structure equations de 1 = a e 1 ∧ e 2n , de 2 = − a 2 e 2 ∧ e 2n + e 3 ∧ e 2n , de 3 = −e 2 ∧ e 2n − a 2 e 3 ∧ e 2n , de 2n = 0, de 2i = c e 2i+1 ∧ e 2n , de 2i+1 = −c e 2i ∧ e 2n , i = 2, . .…”

“…• If m = 3, g admits a hyperkähler structure if and only if it is isomorphic to one among 12R = (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (f 2,12 , −f 1,12 , f 4,12 , −f 3,12 , 0, 0, 0, 0, 0, 0, 0, 0), (f 2,12 , −f 1,12 , f 4,12 , −f 3,12 , pf 6,12 , −pf 5,12 , pf 8,12 , −pf 7,12 , 0, 0, 0, 0), p = 0, while it admits a non-hyperkähler LCHK structure if and only if it is isomorphic to (f 1,12 , f 2,12 , f 3,12 , f 4,12 , f 5,12 , f 6,12 , f 7,12 , f 8,12 , f 9,12 , f 10,12 , f 11,12 , 0), (f 1,12 , f 2,12 , f 3,12 , f 4,12 , f 5,12 , f 6,12 , f 7,12 , f 8,12 + pf 9,12 , −pf 8,12 + f 9,12 , f 10,12 + pf 11,12 , − pf 10,12 + f 11,12 , 0), p = 0, (f 1,12 , f 2,12 , f 3,12 , f 4,12 +pf 5,12 , −pf 4,12 +f 5,12 , f 6,12 +pf 7,12 , −pf 6,12 +f 7,12 , f 8,12 +qf 9,12 , − qf 8,12 + f 9,12 , f 10,12 + qf 11,12 , −qf 10,12 + f 11,12 , 0), pq = 0.…”

“…If α is parallel with respect to the Levi-Civita connection, the LCK metric is called Vaisman. For general results about LCK metrics, we refer the reader to [12,25,27,3].…”