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.
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
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