These are the revised notes of my lectures at the 1995 Santa Cruz summer institute. Changes from the Jan.26,1996 version: Major: 7.1--2; Medium: 3.7, 3.11, 5.3.3, 7.9.2, 9.8.

Singularities of pairs

— LetX be a compact complex manifold and let T be a closed positive current of bidegree (1, 1) on X . It is shown that T is the weak limit of a sequence (Tk) of smooth closed real (1, 1)-currents with small negative part. The negative part of the Tk ’s can be bounded in terms of the Lelong numbers of T , once a lower bound for the curvature of the tangent bundle TX is known. Moreover, Kiselman’s procedure for killing Lelong numbers of a plurisubharmonic function is extended to manifolds by an alternative method based on Hormander’s L estimates for ∂. These results are then applied to derive various results concerning divisors or intersection theory in the context of analytic geometry. Especially, we obtain a relation between effective and numerically effective divisors on arbitrary compact manifolds, and we show that every manifold X in the Fujiki class C with nef tangent bundle is Kahler. If D is an effective divisor in a Kahler manifold, we also obtain a general self-intersection inequality giving a bound of the degrees of the constant multiplicity strata of D, in terms of a polynomial in the cohomology class {D} ∈ H(X, IR).

/pdf/regularization-of-closed-positive-currents-and-intersection-4qcgwrc6m5.pdf

Regularization of closed positive currents and Intersection Theory

We prove that a holomorphic line bundle on a projective manifold is pseudo-effective if and only if its degree on any member of a covering family of curves is non-negative. This is a consequence of a duality statement between the cone of pseudo-effective divisors and the cone of " movable curves " , which is obtained from a general theory of movable intersections and approximate Zariski decomposition for closed positive (1, 1)-currents. As a corollary, a projective manifold has a pseudo-effective canonical bundle if and only if it is not uniruled. We also prove that a 4-fold with a canonical bundle which is pseudo-effective and of numerical class zero in restriction to curves of a good covering family, has non-negative Kodaira dimension.

/pdf/the-pseudo-effective-cone-of-a-compact-kahler-manifold-and-53afhv29x2.pdf

The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension

We introduce complex singularity exponents of plurisubharmonic functions and prove a general semi-continuity result for them. This concept contains as a special case several similar concepts which have been considered e.g. by Arnold and Varchenko, mostly for the study of hypersurface singularities. The plurisubharmonic version is somehow based on a reduction to the algebraic case, but it also takes into account more quantitative informations of great interest for complex analysis and complex differential geometry. We give as an application a new derivation of criteria for the existence of Kahler–Einstein metrics on certain Fano orbifolds, following Nadel's original ideas (but with a drastic simplication in the technique, once the semi-continuity result is taken for granted). In this way, three new examples of rigid Kahler–Einstein Del Pezzo surfaces with quotient singularities are obtained.

/pdf/semi-continuity-of-complex-singularity-exponents-and-kahler-d6ljq73f8n.pdf

Semi-continuity of complex singularity exponents and Kähler–Einstein metrics on Fano orbifolds

We prove that a holomorphic line bundle on a projective manifold is pseudo-effective if and only if its degree on any member of a covering family of curves is non-negative. This is a consequence of a duality statement between the cone of pseudo-effective divisors and the cone of ``movable curves'', which is obtained from a general theory of movable intersections and approximate Zariski decomposition for closed positive (1,1)-currents. As a corollary, a projective manifold has a pseudo-effective canonical bundle if and only if it is is not uniruled. We also prove that a 4-fold with a canonical bundle which is pseudo-effective and of numerical class zero in restriction to curves of a covering family, has non negative Kodaira dimension.

The pseudo-effective cone of a compact K\"ahler manifold and varieties of negative Kodaira dimension

The notion of a singular hermitian metric on a holomorphic line bundle is introduced as a tool for the study of various algebraic questions. One of the main interests of such metrics is the corresponding L2 vanishing theorem for Open image in new window cohomology, which gives a useful criterion for the existence of sections. In this context, numerically effective line bundles and line bundles with maximum Kodaira dimension are characterized by means of positivity properties of the curvature in the sense of currents. The coefficients of isolated logarithmic poles of a plurisubharmonic singular metric are shown to have a simple interpretation in terms of the constant e of Seshadri’s ampleness criterion. Finally, we use singular metrics and approximations of the curvature current to prove a new asymptotic estimate for the dimension of cohomology groups with values in high multiples O(kL) of a line bundle L with maximum Kodaira dimension.

https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/bayreuth.pdf

Jean-Pierre Damailly

Papers

Singular hermitian metrics on positive line bundles