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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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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