Give a restriction to ensure a surgery of a balanced manifold is still balanced. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T10:01:01Z http://mathoverflow.net/feeds/question/69980 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/69980/give-a-restriction-to-ensure-a-surgery-of-a-balanced-manifold-is-still-balanced Give a restriction to ensure a surgery of a balanced manifold is still balanced. Whitacre 2011-07-11T02:47:33Z 2011-08-21T17:28:24Z <p>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?</p> http://mathoverflow.net/questions/69980/give-a-restriction-to-ensure-a-surgery-of-a-balanced-manifold-is-still-balanced/71125#71125 Answer by xiao for Give a restriction to ensure a surgery of a balanced manifold is still balanced. xiao 2011-07-24T13:12:54Z 2011-07-24T14:04:13Z <p>Maybe you mean a complex manifold $M$ of dimension $n$ is balanced iff it admits a hermitian metric $\omega$ satisfying $d\omega^{n-1}=0$.</p> <p>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:</p> <p>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.</p> <p>For the details,you should read their paper.</p> <p>In addition,if you mean "balance" in the Kahler-Einstein problem，the following two papers maybe helpful.</p> <p>a)S.K. DONALDSON:SCALAR CURVATURE AND PROJECTIVE EMBEDDINGS I </p> <p>b）CLAUDIO AREZZO AND FRANK PACARD：BLOWING UP AND DESINGULARIZING CONSTANT SCALAR CURVATURE KAHLER MANIFOLDS</p>