Effective local finite generation of multiplier ideal sheaves

International audienceLet phi be a psh function on a bounded pseudoconvex open set Omega subset of C-n, and let I(m phi) be the associated multiplier ideal sheaves, m is an element of N*. Motivated by global geometric issues, we establish an effective version of the coherence property of I (m phi) as m -> +infinity. Namely, given any B (sic) Omega, we estimate the asymptotic growth rate in in of the number of generators of I (m phi)(vertical bar B) over O-Omega, as well as the growth of the coefficients of sections in Gamma(B, I(m phi)) with respect to finitely many generators globally defined on Omega. Our approach relies on proving asymptotic integral estimates for Bergman kernels associated with singular weights. These estimates extend to the singular case previous estimates obtained by Lindholm and Berndtsson for Bergman kernels with smooth weights and are of independent interest. In the final section, we estimate asymptotically the additivity defect of multiplier ideal sheaves. As m -> +infinity, the decay rate of I(m phi) is proved to be almost linear if the singularities of phi are analytic