Relatives

Abstract

We say that two Kähler manifolds (finite or infinite dimensional) M ₁ and M ₂ are relatives if they share a complex Kähler submanifold S, i.e. if there exist two Kähler immersions and . Otherwise, we say that M ₁ and M ₂ are not relatives. Further, we say that two Kähler manifolds are strongly not relatives if they are not relatives even when the metric of one of them is rescaled by the multiplication by a positive constant. This terminology has been introduced in Di Scala and Loi (Ann Sc Norm Super Pisa Cl Sci (5) 9(3):495–501, 2010), even if the problem of understanding when two Kähler manifolds share a Kähler submanifold has been firstly considered by Umehara (Tokyo J Math 10(1):203–214, 1987), which solves the case of complex space forms, with holomorphic sectional curvature of different sign and finite dimension, which we summarize in Sect. 6.1. In the remaining part of this chapter we pay particular attention to understanding whether or not a Kähler manifold (M, g) is relative to a projective Kähler manifold, which is by definition a Kähler manifold admitting a Kähler immersion into a finite dimensional complex projective space $\mathbb {C}\mathrm {P}^N$. In Sect. 6.2 we discuss the case when (M, g) is homogeneous while in Sect. 6.3 (M, g) is a Bergman–Hartogs domain .