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BY-NC-ND 4.0 license Open Access Published by De Gruyter Open Access July 20, 2017

Regularization of closed positive currents and intersection theory

Michel Méo EMAIL logo
From the journal Complex Manifolds

Abstract

We prove the existence of a closed regularization of the integration current associated to an effective analytic cycle, with a bounded negative part. By means of the King formula, we are reduced to regularize a closed differential form with L1loc coefficients, which by extension has a test value on any positive current with the same support as the cycle. As a consequence, the restriction of a closed positive current to a closed analytic submanifold is well defined as a closed positive current. Lastly, given a closed smooth differential (qʹ, qʹ)-form on a closed analytic submanifold, we prove the existence of a closed (qʹ, qʹ)-current having a restriction equal to that differential form. After blowing up we deal with the case of a hypersurface and then the extension current is obtained as a solution of a linear differential equation of order 1.

MSC 2010: 14C17; 32C30; 32J25

References

[1] Andersson M., A generalized Poincaré-Lelong formula, Math. Scand. 101, 2007, no. 2, pp. 195-218.10.7146/math.scand.a-15040Search in Google Scholar

[2] Andersson M., Wulcan E., Residue currents with prescribed annihilator ideals, Ann. Sci. École Norm. Sup. (4) 40, 2007, no. 6, pp. 985-1007.10.1016/j.ansens.2007.11.001Search in Google Scholar

[3] Baum P., Fulton W., MacPherson R., Riemann-Roch for singular varieties, Inst. Hautes Études Sci. Publ. Math. 45, 1975, pp. 101-145.10.1007/BF02684299Search in Google Scholar

[4] Bismut J.-M., Gillet H., Soulé C., Complex immersions and Arakelov geometry, in: The Grothendieck Festschrift, Vol. I, Progress in Math. 86, Birkhäuser, 1990, pp. 249-331.10.1007/978-0-8176-4574-8_8Search in Google Scholar

[5] Blel M., Sur le cône tangent à un courant positif fermé, J. Math. Pures Appl. (9) 72, 1993, no. 6, pp. 517-536.Search in Google Scholar

[6] Bost J.-B., Gillet H., Soulé C., Heights of projective varieties and positive Green forms, J. Amer. Math. Soc. 7, 1994, pp. 903-1027.10.1090/S0894-0347-1994-1260106-XSearch in Google Scholar

[7] Bott R., Chern S.-S., Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections, Acta Math. 114, 1965, pp. 71-112.10.1007/BF02391818Search in Google Scholar

[8] Bott R., Chern S.-S., Some formulas related to complex transgression, in: Essays on topology and related topics, Mémoires dédiés à G. De Rham, Springer, 1970, pp. 48-57.10.1007/978-3-642-49197-9_5Search in Google Scholar

[9] Demailly J.-P., Estimations L2 pour l’opérateur @ d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählerienne complète, Ann. Sci. École Norm. Sup. 15, 1982, pp. 457-511.10.24033/asens.1434Search in Google Scholar

[10] Demailly J.-P., Monge-Ampère operators, Lelong numbers and intersection theory, in: Complex analysis and geometry, The Univer. Series in Math., Plenum Press, 1993, pp. 115-193.10.1007/978-1-4757-9771-8_4Search in Google Scholar

[11] Demailly J.-P., Regularization of closed positive currents of type .1; 1/ by the flow of a Chern connection, in: Contributions to complex analysis and analytic geometry, Aspects Math. E26, Vieweg, 1994, pp. 105-126.10.1007/978-3-663-14196-9_4Search in Google Scholar

[12] Dieudonné J., Eléments d’analyse, Tome 2, 1968.Search in Google Scholar

[13] Dinh T.-C., Sibony N., Regularization of currents and entropy, Ann. Sci. École Norm. Sup. (4) 37, 2004, no. 6, pp. 959-971.10.1016/j.ansens.2004.09.002Search in Google Scholar

[14] Fornaess J.E., Sibony N., Oka’s inequality for currents and applications, Math. Ann. 301, 1995, pp. 399-419.10.1007/BF01446636Search in Google Scholar

[15] Gillet H., Soulé C., An arithmetic Riemann-Roch theorem, Invent. Math. 110, 1992, no. 3, pp. 473-543.10.1007/BF01231343Search in Google Scholar

[16] Griffiths P., Harris J., Principles of algebraic geometry, 1978.Search in Google Scholar

[17] King J.R., A residue formula for complex subvarieties, in: Proc. Carolina conf. on holomorphic mappings and minimal surfaces, Univ. of North Carolina, Chapel Hill, 1970, pp. 43-56.Search in Google Scholar

[18] Méo M., Courants résidus et formule de King, Ark. Mat. 44, 2006, no. 1, pp. 149-165.10.1007/s11512-005-0003-4Search in Google Scholar

[19] Poly J.-B., Sur l’homologie des courants à support dans un ensemble semi-analytique, Mém. Soc. Math. Fr. 38, 1974, pp. 35-43.10.24033/msmf.150Search in Google Scholar

[20] Raisonnier J., Formes de Chern et résidus raffinés de J.R. King, Bull. Sci. Math. (2) 102, 1978, pp. 145-154.Search in Google Scholar

[21] Rossi H., Picard variety of an isolated singular point, Rice Univ. Studies 54, 1968, pp. 63-73.Search in Google Scholar

[22] Soulé C., Lectures on Arakelov geometry. With the collaboration of D. Abramovich, J.-F. Burnol and J. Kramer. Cambridge Sudies in Advanced Mathematics 33, 1992.10.1017/CBO9780511623950Search in Google Scholar

[23] Weil A., Introduction à l’étude des variétés kähleriennes, Actualités Sci. Indust. 1267, 1958.Search in Google Scholar

[24] Yger A., Résidus, courants résiduels et courants de Green, in: Géométrie complexe (Paris, 1992), Actualités Sci. Indust. 1438, 1996, pp. 123-147.Search in Google Scholar

Received: 2016-12-23
Accepted: 2017-6-26
Published Online: 2017-7-20
Published in Print: 2017-2-23

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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