Construction of projective special Kähler manifolds

Abstract

In this paper we present an intrinsic characterisation of projective special Kähler manifolds in terms of a symmetric tensor satisfying certain differential and algebraic conditions. We show that this tensor vanishes precisely when the structure is locally isomorphic to a standard projective special Kähler structure on $\mathrm{SU}(n,1)/\mathrm{S}(\mathrm{U}(n)\mathrm{U}(1))$. We use this characterisation to classify 4-dimensional projective special Kähler Lie groups.

46 pages, 5 tables. v2:Prop.7.9 and Def.7.10 replaced with Sec.8; Cor.7.7 (and proofs relying on it) fixed by requiring H^2(M)=0 (Cor.7.9); proof of Theo.10.2 simplified and fixed by adding abelian case; proofs in Sec.3 replaced with references; Rmk.7.5 added; ref. added and fixed; typos fixed; presentation improved. v3:Table 1 added; ref. added and updated; typos fixed; presentation improved