Movable intersection and bigness criterion

Jian Xiao

In this note, we give a Morse-type bigness criterion for the difference of two pseudo-effective (1,1)-classes by using movable intersections. As an application, we give a Morse-type bigness criterion for the difference of two movable (n−1,n−1)-classes.
Słowa kluczowe: Kahler manifolds, line bundles, Morse inequalities, currents

1. Boucksom S., Coˆnes positifs des vari´et´es complexes compactes, PhD thesis, Universit´e Joseph-Fourier-Grenoble I, 2002.

2. Boucksom S., On the volume of a line bundle, Internat. J. Math., 13(10) (2002), 1043– 1063.

3. Boucksom S., Divisorial Zariski decompositions on compact complex manifolds, Ann. Sci. ´ Ecole Norm. Sup. (4), 37(1) (2004), 45–76.

4. Boucksom S., Demailly J.-P., P˘aun M., Peternell T., The pseudo-effective cone of a compact Ka¨hler manifold and varieties of negative Kodaira dimension, J. Algebraic Geom., 22(2) (2013), 201–248.

5. Boucksom S., Eyssidieux P., Guedj V., Zeriahi A., Monge–Amp`ere equations in big cohomology classes, Acta Math., 205(2) (2010), 199–262.

6. Boucksom S., Favre C., Jonsson M., Differentiability of volumes of divisors and a problem of Teissier, J. Algebraic Geom., 18(2) (2009), 279–308.

7. Chiose I., The ka¨hler rank of compact complex manifolds, The Journal of Geometric Analysis, 26(1) (2016), 603–615.

8. Demailly J.-P., Champs magn´etiques et in´egalit´es de Morse pour la d00-cohomologie, Ann. Inst. Fourier (Grenoble), 35(4) (1985), 189–229.

9. Demailly J.-P., Regularization of closed positive currents and intersection theory, J. Algebraic Geom., 1(3) (1992), 361–409.

10. Demailly J.-P., Holomorphic Morse inequalities and the Green-Griffiths-Lang conjecture, Pure Appl. Math. Q., 7(4), Special Issue: In memory of Eckart Viehweg (2011), 1165– 1207. 11. Demailly J.-P., Complex analytic and differential geometry. Online book available at, Institut Fourier, Grenoble, 2012.

12. Demailly J.-P., Pa˘un M., Numerical characterization of the Ka¨hler cone of a compact Ka¨hler manifold, Ann. of Math. (2), 159(3) (2004), 1247–1274.

13. Deng Y., Transcendental Morse inequality and generalized Okounkov bodies, arXiv:1503.00112, 2015.

14. Diverio S., Merker J., Rousseau E., Effective algebraic degeneracy, Invent. Math., 180(1) (2010), 161–223.

15. Lehmann B., Xiao J., Positivity functions for curves on algebraic varieties, arXiv:1607.05337, 2016.

16. Nakayama N., Zariski-decomposition and abundance, volume 14 of MSJ Memoirs, Mathematical Society of Japan, Tokyo, 2004.

17. Popovici D., Sufficient bigness criterion for differences of two nef classes, Math. Ann., 364(1-2) (2016), 649–655.

18. Principato M., Mobile product and zariski decomposition, arXiv:1301.1477, 2013.

19. Siu Y. T., An effective Matsusaka big theorem, Ann. Inst. Fourier (Grenoble), 43(5) (1993), 1387–1405.

20. Trapani S., Numerical criteria for the positivity of the difference of ample divisors, Math. Z., 219(3) (1995), 387–401.

21. Trapani S., Divisorial Zariski decomposition and algebraic Morse inequalities, Proc. Amer. Math. Soc., 139(1) (2011), 177–183.

22. Witt Nystro¨m D., Duality between the pseudoeffective and the movable cone on a projective manifold, with an appendix by S. Boucksom, arXiv:1602.03778, 2016.

23. Xiao J., Weak transcendental holomorphic Morse inequalities on compact Ka¨hler manifolds, Ann. Inst. Fourier (Grenoble), 65(3) (2015), 1367–1379.

Czasopismo ukazuje się w sposób ciągły on-line.
Pierwotną formą czasopisma jest wersja elektroniczna.