Dan Popovici, U. Warwick, UK

Singular Morse inequalities

We prove singular Morse inequalities estimating the asymptotic growth of the cohomology groups of high tensor powers of singular Hermitian holomorphic line bundles twisted by the corresponding multiplier ideal sheaves over a compact complex manifold. The main step in the proof is the construction of a new regularisation of closed almost positive (1, 1)-currents with controlled Monge-Ampère masses. To this end, we prove two results describing the asymptotic growth of multiplier ideal sheaves associated with increasingly singular metrics: almost linear growth and an effective version of their coherence property. The asymptotics of Bergman kernels associated with singular metrics will play an important part.