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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1$\begingroup$ You may get more answers if you give a bit more background to your question. See mathoverflow.net/howtoask#motivation $\endgroup$– j.c.Commented Jul 11, 2011 at 14:24
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3$\begingroup$ 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). $\endgroup$– Joel FineCommented Jul 12, 2011 at 8:32
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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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1$\begingroup$ Let $f:M′→M$ be a meromorphic surjective map of compact complex manifolds with$ M′$ Kähler and n-dimensional. Then M is $(n−1)$-Kähler. (In particular, Moishezon manifolds are $(n−1)$-Kähler. A compact complex manifold M is called p-Kähler if there is a strictly weakly positive smooth closed (p,p)-form. In particular, 1-Kähler is equivalent to Kähler, and $(n−1)$-Kähler to balanced manifold. The balanced manifold property is also equivalent to the existence of a Hermitian metric whose Kähler form satisfies $dω^{n−1}=0, n=\dim M$ $\endgroup$– user21574Commented Jul 24, 2017 at 15:33
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1$\begingroup$ Every Kähler metric is balanced and balanced metrics are invariant under proper modifications. $\endgroup$– user21574Commented Jul 24, 2017 at 15:35
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1$\begingroup$ A compact connected complex manifold $X$ is balanced if it admits a mapping onto a complex curve such that: (1) The smooth fibers are balanced. (2) Each fiber, as a codimension two current on X, is not the $(n−1,n−1)$-component of a boundary (in the sense of currents). $\endgroup$– user21574Commented Jul 24, 2017 at 17:50
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1$\begingroup$ A compact complex manifold X admits a balanced metric if and only if every closed $(2n−2)$-dimensional current, whose $(n−1,n−1)$-component is positive (and nonzero), represents a nontrivial $\mathbb C$-homology class. See M. L. Michelsohn, On the existence of special metrics in complex geometry, Acta Math.Volume 149 (1982), 261-295 $\endgroup$– user21574Commented Jul 24, 2017 at 17:50