Give a restriction to ensure a surgery of a balanced manifold is still balanced.

2011-07-11T02:47:33Z

M is a balanced compact complex manifold, if I do a surgery on M and get N. My question is under what appropriate conditions can ensure that N is still balanced?

Answer by xiao for Give a restriction to ensure a surgery of a balanced manifold is still balanced.

2011-07-24T13:12:54Z

Maybe you mean a complex manifold $M$ of dimension $n$ is balanced iff it admits a hermitian metric $\omega$ satisfying $d\omega^{n-1}=0$.

In the paper:Metric properties of manifolds bimeromorphic to compact Kahler spaces. (JDG.v37.1993.95-121), L. Alessandrini and G. Bassanelli had proved the following result:

Let $M$ and $N$ be compact complex manifolds and $f:N\longrightarrow M$ be a modification,then 1) $M$ is balanced $\Longrightarrow N$ is balanced. 2) $N$ is balanced and satisfies a cohomological condition (it 's called B in the above paper)$\Longrightarrow M$ is balanced.

For the details,you should read their paper.

In addition,if you mean "balance" in the Kahler-Einstein problem，the following two papers maybe helpful.

a)S.K. DONALDSON:SCALAR CURVATURE AND PROJECTIVE EMBEDDINGS I

b）CLAUDIO AREZZO AND FRANK PACARD：BLOWING UP AND DESINGULARIZING CONSTANT SCALAR CURVATURE KAHLER MANIFOLDS

Give a restriction to ensure a surgery of a balanced manifold is still balanced. - MathOverflow

Give a restriction to ensure a surgery of a balanced manifold is still balanced.

Answer by xiao for Give a restriction to ensure a surgery of a balanced manifold is still balanced.