Effective local finite generation of multiplier ideal sheaves

@article{Popovici2010,
abstract = {Let $\varphi $ be a psh function on a bounded pseudoconvex open set $\Omega \subset \mathbb\{C\}^n$, and let $\{\mathcal\{I\}\}(m\varphi )$ be the associated multiplier ideal sheaves, $m\in \mathbb\{N\}^\{\star \}$. Motivated by global geometric issues, we establish an effective version of the coherence property of $\{\mathcal\{I\}\}(m\varphi )$ as $m\rightarrow +\infty $. Namely, given any $B\Subset \Omega $, we estimate the asymptotic growth rate in $m$ of the number of generators of $\{\mathcal\{I\}\}(m\varphi )_\{|B\}$ over $\{\mathcal\{O\}\}_\{\Omega \}$, as well as the growth of the coefficients of sections in $\Gamma (B, \, \{\mathcal\{I\}\}(m\varphi ))$ 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\rightarrow +\infty $, the decay rate of $\{\mathcal\{I\}\}(m\varphi )$ is proved to be almost linear if the singularities of $\varphi $ are analytic.},
affiliation = {Université Paul Sabatier Institut de mathématiques de Toulouse 118 Route de Narbonne 31062 Toulouse Cedex 4 (France)},
author = {Popovici, Dan},
journal = {Annales de l’institut Fourier},
keywords = {Bergman kernel; closed positive current; $L^2$ estimates; multiplier ideal sheaf; psh function; singular Hermitian metric; Stein manifold; estimates},
language = {eng},
number = {5},
pages = {1561-1594},
publisher = {Association des Annales de l’institut Fourier},
title = {Effective local finite generation of multiplier ideal sheaves},
url = {http://eudml.org/doc/116314},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Popovici, Dan
TI - Effective local finite generation of multiplier ideal sheaves
JO - Annales de l’institut Fourier
PY - 2010
PB - Association des Annales de l’institut Fourier
VL - 60
IS - 5
SP - 1561
EP - 1594
AB - Let $\varphi $ be a psh function on a bounded pseudoconvex open set $\Omega \subset \mathbb{C}^n$, and let ${\mathcal{I}}(m\varphi )$ be the associated multiplier ideal sheaves, $m\in \mathbb{N}^{\star }$. Motivated by global geometric issues, we establish an effective version of the coherence property of ${\mathcal{I}}(m\varphi )$ as $m\rightarrow +\infty $. Namely, given any $B\Subset \Omega $, we estimate the asymptotic growth rate in $m$ of the number of generators of ${\mathcal{I}}(m\varphi )_{|B}$ over ${\mathcal{O}}_{\Omega }$, as well as the growth of the coefficients of sections in $\Gamma (B, \, {\mathcal{I}}(m\varphi ))$ 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\rightarrow +\infty $, the decay rate of ${\mathcal{I}}(m\varphi )$ is proved to be almost linear if the singularities of $\varphi $ are analytic.
LA - eng
KW - Bergman kernel; closed positive current; $L^2$ estimates; multiplier ideal sheaf; psh function; singular Hermitian metric; Stein manifold; estimates
UR - http://eudml.org/doc/116314
ER -