Regularization of currents with mass control and singular morse inequalities

Dan Popovici Universit´e Paul Sabatier

Differential Geometry mathscidoc:1609.10082

j. differential geometry, 80, (2), 281-326, 2008
Let X be a compact complex, not necessarily K¨ahler, manifold of dimension n. We characterize the volume of any holomorphic line bundle L → X as the supremum R of the Monge-Amp`ere masses X Tn ac over all closed positive currents T in the first Chern class of L, where Tac is the absolutely continuous part of T in its Lebesgue decomposition. This result, new in the non-K¨ahler context, can be seen as holomorphic Morse inequalities for the cohomology of high tensor powers of line bundles endowed with arbitrarily singular Hermitian metrics. It gives, in particular, a new bigness criterion for line bundles in terms of existence of singular Hermitian metrics satisfying positivity conditions. The proof is based on the construction of a new regularization for closed (1, 1)-currents with a control of the Monge-Amp`ere masses of the approximating sequence. To this end, we prove a potential-theoretic result in one complex variable and study the growth of multiplier ideal sheaves associated with increasingly singular metrics.
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@inproceedings{dan2008regularization,
  title={REGULARIZATION OF CURRENTS WITH MASS CONTROL AND SINGULAR MORSE INEQUALITIES},
  author={Dan Popovici},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160909135523989848739},
  booktitle={j. differential geometry},
  volume={80},
  number={2},
  pages={281-326},
  year={2008},
}
Dan Popovici. REGULARIZATION OF CURRENTS WITH MASS CONTROL AND SINGULAR MORSE INEQUALITIES. 2008. Vol. 80. In j. differential geometry. pp.281-326. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160909135523989848739.
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