Regularisation of Currents with Mass Control and Singular Morse Inequalities

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.