K\"ahler submanifolds in Iwasawa manifolds

TL;DR

This paper investigates the structure of K"ahler submanifolds within Iwasawa manifolds, revealing that complex curves lie in tori and classifying K"ahler surfaces as either abelian or elliptic surfaces, with additional properties of complex tori.

Contribution

It classifies K"ahler submanifolds in Iwasawa manifolds and demonstrates that complex tori in these manifolds have complex multiplication.

Findings

Complex curves are contained in holomorphic tori.

K"ahler surfaces are either abelian or non-projective elliptic surfaces.

Complex tori in Iwasawa manifolds carry complex multiplication.

Abstract

Iwasawa manifold is a compact complex homogeneous manifold isomorphic to a quotient of the group of complex unipotent $3 \times 3$ matrices by a cocompact lattice. We prove that any compact complex curve in an Iwasawa manifold is contained in a holomorphic torus. We also prove that any K\"ahler surface in an Iwasawa manifold is either an abelian surface or a non-projective isotrivial elliptic surface of Kodaira dimension one. In the Appendix we show that any complex torus in an Iwasawa manifold carries complex multiplication.