Effective local finite generation of multiplier ideal sheaves

Popovici, Dan

Annales de l'Institut Fourier, Tome 60 (2010) no. 5, pp. 1561-1594.

Soit $\varphi$ une fonction psh sur un ouvert pseudo-convexe borné $\Omega \subset {ℂ}^n$ et soit $ℐ\left(m\varphi \right)$ les faisceaux d’idéaux multiplicateurs associés, $m\in {\mathbb{N}}^{☆}$. Motivé par des considérations de géométrie globale, nous donnons une version effective de la propriété de cohérence de $ℐ\left(m\varphi \right)$ lorsque $m\to +\infty$. Étant donné $B⋐\Omega$, nous estimons la croissance asymptotique en $m$ du nombre de générateurs du ${𝒪}_{\Omega }$-module $ℐ{\left(m\varphi \right)}_{|B}$, ainsi que la croissance des coefficients des sections de $\Gamma (B,ℐ(m\varphi \left)\right)$ par rapport à un nombre fini de générateurs globalement définis sur $\Omega$. Notre approche consiste à démontrer des estimations intégrales asymptotiques pour des noyaux de Bergman associés à des poids singuliers. Ces estimations généralisent au cas singulier des estimations obtenues antérieurement par Lindholm et Berndtsson pour des noyaux de Bergman à poids lisses et présentent un intérêt propre. Nous donnons également des estimations asymptotiques pour le défaut d’additivité des faisceaux d’idéaux multiplicateurs. Nous montrons que lorsque $m\to +\infty$ le taux de décroissance de $ℐ\left(m\varphi \right)$ est presque linéaire si les singularités de $\varphi$ sont analytiques.

Let $\varphi$ be a psh function on a bounded pseudoconvex open set $\Omega \subset {ℂ}^n$, and let $ℐ\left(m\varphi \right)$ be the associated multiplier ideal sheaves, $m\in {\mathbb{N}}^{☆}$. Motivated by global geometric issues, we establish an effective version of the coherence property of $ℐ\left(m\varphi \right)$ as $m\to +\infty$. Namely, given any $B⋐\Omega$, we estimate the asymptotic growth rate in $m$ of the number of generators of $ℐ{\left(m\varphi \right)}_{|B}$ over ${𝒪}_{\Omega }$, as well as the growth of the coefficients of sections in $\Gamma (B,ℐ(m\varphi \left)\right)$ 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 $m\to +\infty$, the decay rate of $ℐ\left(m\varphi \right)$ is proved to be almost linear if the singularities of $\varphi$ are analytic.

Classification : 32C35, 32U05, 32A36
Keywords: Bergman kernel, closed positive current, $L^2$ estimates, multiplier ideal sheaf, psh function, singular Hermitian metric, Stein manifold
Mot clés : courant positif fermé, estimations $L^2$, faisceau d’idéaux multiplicateurs, fonction psh, métrique hermitienne singuliére, noyau de Bergman, variété de Stein

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