Geometry of compact complex manifolds with maximal torus action

Yu. M. Ustinovsky

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebraic topology, convex polytopes, and related topics, Tome 286 (2014), pp. 219-230

Résumé

We study the geometry of compact complex manifolds $M$ equipped with a maximal action of a torus $T=(S^1)^k$. We present two equivalent constructions that allow one to build any such manifold on the basis of special combinatorial data given by a simplicial fan $\Sigma$ and a complex subgroup $H\subset T_\mathbb C=(\mathbb C^*)^k$. On every manifold $M$ we define a canonical holomorphic foliation $\mathcal F$ and, under additional restrictions on the combinatorial data, construct a transverse Kähler form $\omega _\mathcal F$. As an application of these constructions, we extend some results on the geometry of moment–angle manifolds to the case of manifolds $M$.