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

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?

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You may get more answers if you give a bit more background to your question. See mathoverflow.net/howtoask#motivation – jc Jul 11 2011 at 14:24
There are at least two totally different definitions of "balanced" for a complex manifold. Also, what do you mean by "surgery"? There are many different things you could mean, most of which do not keep your manifold complex (at least not in a natural way). – Joel Fine Jul 12 2011 at 8:32

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

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