Effective Local Finite Generation of Multiplier Ideal Sheaves

Let $\phi$ be a psh function on a bounded pseudoconvex open set $\Omega \subset \C^n$, and let ${\cal I}(\phi)$ be the associated multiplier ideal sheaf. Motivated by resolution of singularities issues, we establish an effective version of the coherence property of ${\cal I}(m\phi)$ as $m\to +\infty$. Namely, we estimate the order of growth in $m$ of the number of generators needed to engender ${\cal I}(m\phi)$ on a fixed compact subset, as well as the growth of the coefficients featuring in the decomposition of local sections as linear combinations over ${\cal O}_{\Omega}$ of finitely many generators. The main idea is to use Toeplitz concentration operators involving Bergman kernels associated with singular weights. Our approach relies on asymptotic integral estimates of singularly weighted Bergman kernels of independent interest. In the second part of the paper, we estimate the additivity defect of multiplier ideal sheaves already known to be subadditive by a result of Demailly, Ein, and Lazarsfeld. This implies that the decay rate of ${\cal I}(m\phi)$ is not far from being linear if the singularities of $\phi$ are reasonable.