arXiv subjects

Jean-Pierre Demailly

Publications and source records attributed to Jean-Pierre Demailly.

42 records · Page 3Linked to original sources

arXiv2000

A Subadditivity Property of Multiplier Ideals

Given an effective Q-divisor D on a smooth complex variety, one can associate to D its multiplier ideal sheaf J(D), which measures in a somewhat subtle way the singularities of D. Because of their strong vanishing properties, these ideals have come to play an increasingly important role in higher dimensional geometry. We prove that for two effective Q-divisors D and E, one has the "subadditivity" relation: J(D + E) \subseteq J(D) . J(E) . (We also establish several natural variants, including the analogous statement for the analytic multiplier ideals associated to plurisubharmonic functions.) As an application, we give a new proof of a theorem of Fujita concerning the volume of a big linear series on a projective variety. The first section of the paper contains an overview of the construction and basic properties of multiplier ideals from an algebro-geometric perspective, as well as a discussion of the relation between some asymptotic algebraic constructions and their analytic counterparts.

math.AG↗

arXiv2000

L^2 Cohomology on coverings of a compact Kähler manifold

Andreotti-Vesentini, Ohsawa, Gromov, Kollár, among others, have observed that Hodge theory could be extended to (non compact) Kähler complete manifolds, within the L^2 framework. Also, many vanishing theorems on projective or Kähler manifolds rely on the Kodaira-Bochner-Nakano identity, and thus possess natural L^2 versions. Our goal is to define canonical L^2 cohomology groups on any unramified covering of an analytic variety X, with values in a coherent analytic sheaf on X. This cohomology shares all usual properties of standard coherent sheaf cohomology (especially, exact sequences, spectral sequences, vanishing theorems,...). These properties are obtained by incorporating the information provided by L^2 estimates in the standard proofs, with suitable adaptations. L^2 cohomology should provide a comfortable and efficient framework for the study the geometry of coverings, by providing a relevant fonctorial formalism. When the base variety is compact and the covering is a Galois covering of group $Γ$, there is a $Γ$ action on cohomology groups and a related concept of $Γ$-dimension. In that case, we prove that the $Γ$-dimension of cohomology groups is finite, and we extend Atiyah's L^2 index theorem to the case of arbitrary coherent sheaves. Finally, if the base manifold is projective, L^2 analogues of the usual vanishing theorems (Kodaira-Serre, Kawamata-Viehweg,...) are valid. Similar constructions and results have also been studied by P. Eyssidieux.

math.AG↗

arXiv1999

Hyperbolicity of Generic Surfaces of High Degree in Projective 3-Space

The main goal of this work is to prove that a very generic surface of degree at least 21 in complex projective 3-dimensional space is hyperbolic in the sense of Kobayashi. This means that every entire holomorphic map $f:{\Bbb C} \to X$ to the surface is constant. In 1970, Kobayashi conjectured more generally that a (very) generic hypersurface of sufficiently high degree in projective space is hyperbolic (here, the terminology "very generic" refers to complements of countable unions of proper algebraic subsets). Our technique follows the stream of ideas initiated by Green and Griffiths in 1979, which consists in considering jet differentials and their associated base loci. However, a key ingredient is the use of a different kind of jet bundles, namely the "Semple jet bundles" previously studied by the first named author (Santa Cruz Summer School, July 1995, Proc. Symposia Pure Math., Vol. 62.2, 1997). The base locus calculation is achieved through a sequence of Riemann-Roch formulas combined with a suitable generic vanishing theorem for order 2-jets. Our method covers the case of surfaces of general type with Picard group ${\Bbb Z}$ and $(13+12θ_2) c_1^2 - 9 c_2 > 0$, where $θ_2$ is what we call the "2-jet threshold" (the 2-jet threshold turns out to be bounded below by -1/6 for surfaces in ${\Bbb P}^3$). The final conclusion is obtained by using very recent results of McQuillan on holomorphic foliations.

math.AG↗

arXiv1996

The algebraic dimension of compact complex threefolds with vanishing second Betti number

We investigate compact complex manifolds of dimension three and second Betti number $b_2(X) = 0$. We are interested in the algebraic dimension $a(X)$, which is by definition the transcendence degree of the field of meromorphic functions over the field of complex numbers. The topological Euler characteristic $χ_{\mathrm{ top}}(X) $ equals the third Chern class $c_3(X)$ by a theorem of Hopf. Our main result is that, if $X$ is a compact 3-dimensional complex manifold with $b_2(X) = 0$ and $a(X) > 0$, then $c_3(X) = χ_{\rm top}(X) = 0$, that is, we either have $b_1(X) = 0, \ b_3(X) = 2$ or $b_1(X) = 1, \ b_3(X) = 0.$

math.AG↗

arXiv1995

$L^2$ vanishing theorems for positive line bundles and adjunction theory, Lecture Notes of a CIME course on "Transcendental Methods of Algebraic Geometry" (Cetraro, Italy, July 1994)

The notes start with an elementary introduction to a few important analytic techniques of algebraic geometry: closed positive currents, $L^2$ estimates for the $\dbar$-operator on positive vector bundles, Nadel's vanishing theorem for multiplier ideal sheaves (a generalization of the well-known Kawamata-Viehweg vanishing theorem). Applications to adjoint line bundles are then discussed. T.~Fujita conjectured in 1987 that $K_X+(n+2)L$ is very ample for every ample line bundle $L$ on a non singular projective variety $X$ with $\dim X=n$. The answer is known only for $n\le 2$ (I.~Reider, 1988). In the last years, various bounds have been obtained for integers $m$ such that $2K_X+mL$ is very ample (by J.~Kollár, L.~Ein-R.~Lazarsfeld, Y.T.~Siu and the author, among others). Two approaches are discussed: an analytic approach via Monge-Ampère equations and current theory, and a more algebraic one (due to Siu) via multiplier ideal sheaves and Riemann-Roch. Finally, an effective version of the big Matsusaka theorem is derived, in the form of an explicit bound for an integer $m$ such that $mL$ is very ample, depending only on $L^n$ and $L^{n-1}\cdot K_X$; these bounds improve Siu's results (1993), and essentially contain the optimal bounds obtained by Fernandez del Busto for the surface case.

alg-geom↗

arXiv1993

Algebraic approximations of holomorphic maps from Stein domains to projective manifolds

It is shown that every holomorphic map $f$ from a Runge domain $Ω$ of an affine algebraic variety $S$ into a projective algebraic manifold $X$ is a uniform limit of Nash algebraic maps $f_ν$ defined over an exhausting sequence of relatively compact open sets $Ω_ν$ in $Ω$. A relative version is also given: If there is an algebraic subvariety $A$ (not necessarily reduced) in $S$ such that the restriction of $f$ to $A\capΩ$ is algebraic, then $f_ν$ can be taken to coincide with $f$ on $A\capΩ_ν$. The main application of these results, when $Ω$ is the unit disk, is to show that the Kobayashi pseudodistance and the Kobayashi-Royden infinitesimal metric of a quasi-projective algebraic manifold $Z$ are computable solely in terms of the closed algebraic curves in $Z$. Similarly, the $p$-dimensional Eisenman metric of a quasi-projective algebraic manifold can be computed in terms of the Eisenman volumes of its $p$-dimensional algebraic subvarieties. Another question addressed in the paper is whether the approximations $f_ν$ can be taken to have their images contained in affine Zariski open subsets of $X$. By using complex analytic methods (pluricomplex potential theory and Hörmander's $L^2$ estimates), we show that this is the case if $f$ is an embedding (with $\dim S<\dim X$) and if there is an ample line bundle $L$ on $X$ such that

alg-geom↗