Blow-ups in generalized K\"ahler geometry

TL;DR

This paper develops a blow-up theory for generalized K"ahler manifolds, identifying conditions for blow-ups and deformations of structures, with applications to Lie groups and explicit examples.

Contribution

It introduces a blow-up procedure for generalized K"ahler manifolds, extending the theory and providing methods to deform degenerate structures into non-degenerate ones.

Findings

Bi-Hermitian structure lifts to blow-up as degenerate structure

Deformation techniques transform degenerate structures into non-degenerate ones

Explicit example on product of circles and 3-spheres

Abstract

We continue the study of blow-ups in generalized complex geometry with the blow-up theory for generalized K\"ahler manifolds. The natural candidates for submanifolds to be blown-up are those which are generalized Poisson for one of the two generalized complex structures and can be blown up in a generalized complex manner. We show that the bi-Hermitian structure underlying the generalized K\"ahler pair lifts to a degenerate bi-Hermitian structure on this blow-up. Then, using a deformation procedure based on potentials in K\"ahler geometry, we identify two concrete situations in which one can deform the degenerate structure on the blow-up into a non-degenerate one. We end with an investigation of generalized K\"ahler Lie groups and give a concrete example on $(S^1)^n\times (S^3)^m$, for $n+m$ even.