Name: Jean-Pierre Demailly
Title: Subadditivity of multiplier ideal sheaves and
asymptotic Zariski decomposition
(joint work with L.~Ein and R.~Lazarsfeld)
Abstract:
We study some basic algebraic properties of the ``multiplier ideal
sheaves'' introduced by A.~Nadel. To every effective ${\bf Q}$-divisor
$D$ on a projective variety $X$ is attached in a canonical way a
coherent sheaf ${\cal I}(D)$ of multiplier ideals in the structure
sheaf ${\cal O}_X$. These ideal sheaves describe in a subtle way
hypersurface singularities, and are also involved in several important
vanishing theorems and results of adjunction theory. More generally,
one can associate a multiplier ideal sheaf with every linear system of
divisors or with every fractional ideal; these cases may even be
considered as special cases of the analytic muliplier ideals attached
to an arbitrary plurisubharmonic function. Our main result is a
subadditivity property: the ideal associated with the sum of two
effective divisors is contained in the product of the corresponding
multiplier ideals. The proof rests upon H.~Esnault and E.~Viehweg's
restriction theorem, combined with a diagonal trick. As an
application, we give a simple and natural proof of Fujita's theorem on
approximate Zariski decomposition of big divisors.