5.21 Plurisubharmonic functions and singularitiesEDB5A2485B58

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A function whose complex Hessian is bounded below by a smooth form becomes plurisubharmonic after a local smooth correction. This brings the local theory of plurisubharmonic functions to a compact complex manifold, provided the smooth remainder left by the correction is kept when potentials are glued, when their Hessians are multiplied and when one integrates by parts. Normalization by the supremum then controls the resulting families globally, giving compactness, uniform exponential integrability and capacity estimates on the whole manifold. Products of Hessians require more care than the potentials themselves: without a common bound, convergence of potentials in $L^1$, or even in capacity, does not prevent their Monge–Ampère measures from concentrating.

To keep track of singularities under these operations, we compare potentials in two ways: by domination up to an additive constant, which leads to ordinary singularity types and their envelopes, and by the local integrability of holomorphic functions against exponential weights, which leads to multiplier ideals. Lelong numbers, analytic approximation and modifications are the main tools for computing the latter.