Regularisation of Currents with Mass Control and Singular Morse Inequalities

Details Regularisation of Currents with Mass Control and Singular Morse Inequalities Regularisation of Currents with Mass Control and Singular Morse Inequalities

2006-03-31

arxiv.org/abs/math/0603738v1

Mathematics

Complex Variables

44 pages

Scientific paper

Let $X$ be a compact complex, not necessarily K\"ahler, manifold of dimension $n$. We characterise the volume of any holomorphic line bundle $L\to X$ as the supremum of the Monge-Amp\`ere masses $\int_X T_{ac}^n$ over all closed positive currents $T$ in the first Chern class of $L$, where $T_{ac}$ is the absolutely continuous part in the 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 regularisation 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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