Construction of projective special Kähler manifolds

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 $$\mathrm {SU}(n,1)/\mathrm {S}(\mathrm {U}(n)\mathrm {U}(1))$$ SU ( n , 1 ) / S ( U ( n ) U ( 1 ) ) . We use this characterisation to classify 4-dimensional projective special Kähler Lie groups.