## Divisorial Zariski decompositions on compact complex manifolds.(English)Zbl 1054.32010

Summary: Using currents with minimal singularities, we introduce pointwise minimal multiplicities for a real pseudo-effective $$(1,1)$$-cohomology class $$\alpha$$ on a compact complex manifold $$X$$,which are the local obstructions to the numerical effectivity of $$\alpha$$. The negative part of $$\alpha$$ is then defined as the real effective divisor $$N(\alpha)$$ whose multiplicity along a prime divisor $$D$$ is just the generic multiplicity of $$\alpha$$ along $$D$$, and we get in that way a divisorial Zariski decomposition of $$\alpha$$ into the sum of a class $$Z(\alpha)$$ which is nef in codimension 1 and the class of its negative part $$N(\alpha)$$, which is an exceptional divisor in the sense that it is very rigidly embedded in $$X$$. The positive parts $$Z(\alpha)$$ generate a modified nef cone, and the pseudo-effective cone is shown to be locally polyhedral away from the modified nef cone, with extremal rays generated by exceptional divisors. We then treat the case of a surface and a hyper-Kähler manifold in some detail. Using the intersection form (respectively the Beauville–Bogomolov form), we characterize the modified nef cone and the exceptional divisors. The divisorial Zariski decomposition is orthogonal, and is thus a rational decomposition, which fact accounts for the usual existence statement of a Zariski decomposition on a projective surface, which is thus extended to the hyper-Kähler case. Finally, we explain how the divisorial Zariski decomposition of (the first Chern class of) a big line bundle on a projective manifold can be characterized in terms of the asymptotics of the linear series $$|kL|$$ as $$k\to\infty$$.

### MSC:

 32J27 Compact Kähler manifolds: generalizations, classification 32J25 Transcendental methods of algebraic geometry (complex-analytic aspects) 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry 14C20 Divisors, linear systems, invertible sheaves
Full Text:

### References:

 [1] Boucksom S. , Le cône kählérien d’une variété hyperkählérienne , C. R. Acad. Sci. Paris Sér. I Math. 333 ( 2001 ) 935 - 938 . MR 1873811 | Zbl 1068.32014 · Zbl 1068.32014 [2] Boucksom S. , On the volume of a line bundle , math.AG/0201031 . arXiv · Zbl 1101.14008 [3] Cutkosky S.D. , Zariski decomposition of divisors on algebraic varieties , Duke Math. J. 53 ( 1986 ) 149 - 156 . Article | MR 835801 | Zbl 0604.14002 · Zbl 0604.14002 [4] Demailly J.-P. , Estimations L 2 pour l’opérateur $$\overline{\partial}$$ d’un fibré vectoriel holomorphe semi-positif au dessus d’une variété kählérienne complète , Ann. Sci. École Norm. Sup. 15 ( 1982 ) 457 - 511 . Numdam | Zbl 0507.32021 · Zbl 0507.32021 [5] Demailly J.-P. , Regularization of closed positive currents and intersection theory , J. Algebraic Geom. 1 ( 1992 ) 361 - 409 . MR 1158622 | Zbl 0777.32016 · Zbl 0777.32016 [6] Demailly J.-P. , Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials , Proc. Symp. Pure Math. 62 ( 2 ) ( 1997 ). MR 1492539 | Zbl 0919.32014 · Zbl 0919.32014 [7] Demailly J.-P. , Paun M. , Numerical characterization of the Kähler cone of a compact Kähler manifold , math.AG/0105176 . arXiv · Zbl 1064.32019 [8] Demailly J.-P. , Peternell T. , Schneider M. , Compact complex manifolds with numerically effective tangent bundles , J. Algebraic Geom. 3 ( 1994 ) 295 - 345 . MR 1257325 | Zbl 0827.14027 · Zbl 0827.14027 [9] Demailly J.-P. , Ein L. , Lazarsfeld R. , A subadditivity property of multiplier ideals , math.AG/0002035 . arXiv · Zbl 1077.14516 [10] Demailly J.-P. , Peternell T. , Schneider M. , Pseudoeffective line bundles on compact Kähler manifolds , math.AG/0006205 . arXiv · Zbl 1111.32302 [11] Fujita T. , On Zariski problem , Proc. Japan Acad., Ser. A 55 ( 1979 ) 106 - 110 . Article | MR 531454 | Zbl 0444.14026 · Zbl 0444.14026 [12] Fujita T. , Remarks on quasi-polarized varieties , Nagoya Math. J. 115 ( 1989 ) 105 - 123 . Article | MR 1018086 | Zbl 0699.14002 · Zbl 0699.14002 [13] Hartshorne R. , Algebraic Geometry , GTM , vol. 52 , Springer-Verlag , 1977 . MR 463157 | Zbl 0367.14001 · Zbl 0367.14001 [14] Huybrechts D. , The Kähler cone of a compact hyperkähler manifold , math.AG/9909109 . arXiv · Zbl 1023.14015 [15] Lamari A. , Courants kählériens et surfaces compactes , Ann. Inst. Fourier 49 ( 1999 ) 249 - 263 . Numdam | MR 1688140 | Zbl 0926.32026 · Zbl 0926.32026 [16] Nakayama N., Zariski decomposition and abundance, preprint RIMS. MR 2104208 [17] Paun M. , Sur l’effectivité numérique des images inverses de fibrés en droites , Math. Ann. 310 ( 1998 ) 411 - 421 . MR 1612321 | Zbl 1023.32014 · Zbl 1023.32014 [18] Siu Y.T. , Analyticity of sets associated to Lelong numbers and the extension of closed positive currents , Invent. Math. 27 ( 1974 ) 53 - 156 . MR 352516 | Zbl 0289.32003 · Zbl 0289.32003 [19] Zariski O. , The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface , Ann. of Math. 76 ( 2 ) ( 1962 ) 560 - 615 . MR 141668 | Zbl 0124.37001 · Zbl 0124.37001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.