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Napier, Terrence; Ramachandran, Mohan

\(L^2\) Castelnuovo-de Franchis, the cup product lemma, and filtered ends of Kähler manifolds. (English) Zbl 1173.32010

J. Topol. Anal. 1, No. 1, 29-64 (2009).
New approaches to the proof of the \(L^2\) Castelnuovo-de Franchis theorem and to the cup product lemma are developed. In particular a version of the \(L^2\) Castelnuovo-de Franchis theorem is obtained in which only one of the holomorphic 1-forms needs to be in \(L^2\) for \((X,g)\) being a connected complete Kähler manifold with bounded geometry. For the cup product lemma it is assumed that only one of the forms belongs to \(L^{\infty}\) instead of in \(L^2\) and the other form does not need to have exact real part. These results allow to prove the existence of a surjective proper holomorphic mapping onto a Riemann surface \(\phi:X \to S\), where \(X\) denotes a connected complete Kähler manifold with bounded geometry. Further generalizations are also treated so as applications to filtered ends of Kähler manifolds.
Reviewer: Gabriela Paola Ovando (Freiburg)

Cited in 3 Documents

MSC:

32Q15 Kähler manifolds

Keywords:

connected complete Kähler manifolds; Riemann surface; holomorphic convexity
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References:

[1] DOI: 10.1090/surv/044 · doi:10.1090/surv/044
[2] D. Arapura, Current Topics in Complex Algebraic Geometry, Math. Sci. Res. Inst. Publ. 28 (Cambridge Univ. Press, Berkeley, CA, 1995) pp. 1–16.
[3] Arapura D., Duke Math. J. 64 pp 477–
[4] DOI: 10.1007/978-3-642-96754-2 · doi:10.1007/978-3-642-96754-2
[5] Beauville A., London Math. Soc. Lect. Note Series 68, in: Complex Algebraic Surfaces (1983)
[6] DOI: 10.1007/BF02698844 · Zbl 0695.58010 · doi:10.1007/BF02698844
[7] DOI: 10.1007/BF01245076 · Zbl 0743.32025 · doi:10.1007/BF01245076
[8] DOI: 10.1007/978-94-009-2366-9 · doi:10.1007/978-94-009-2366-9
[9] Colţoiu M., J. Reine Angew. Math. 412 pp 108–
[10] DOI: 10.1007/BF02393214 · JFM 41.0492.02 · doi:10.1007/BF02393214
[11] DOI: 10.1007/3-7643-7447-0_3 · doi:10.1007/3-7643-7447-0_3
[12] Demailly J.-P., Ann. Sci. École Norm. Sup. 15 pp 457–
[13] DOI: 10.1007/BF02570874 · Zbl 0682.32017 · doi:10.1007/BF02570874
[14] DOI: 10.1007/s00209-006-0958-2 · Zbl 1106.32014 · doi:10.1007/s00209-006-0958-2
[15] DOI: 10.2307/1969703 · Zbl 0055.40301 · doi:10.2307/1969703
[16] H. Grauert and O. Riemenschneider, Problems in Analysis, A Symposium in Honor of S. Bochner (Princeton Univ. Press, 1970) pp. 61–79.
[17] DOI: 10.5802/aif.549 · Zbl 0307.31003 · doi:10.5802/aif.549
[18] DOI: 10.1090/S0273-0979-99-00776-4 · Zbl 0927.58019 · doi:10.1090/S0273-0979-99-00776-4
[19] Gromov M., C. R. Acad. Sci. Paris 308 pp 67–
[20] Gromov M., J. Diff. Geom. 33 pp 263– · Zbl 0719.53042 · doi:10.4310/jdg/1214446039
[21] DOI: 10.1007/BF02699433 · Zbl 0896.58024 · doi:10.1007/BF02699433
[22] DOI: 10.1090/S0002-9939-03-07119-3 · Zbl 1023.32005 · doi:10.1090/S0002-9939-03-07119-3
[23] J. Jost and S.T. Yau, Algebraic Geometry and Related Topics, Conf. Proc. Lect. Notes Alg. Geom. I (Internat. Press, Inchon, 1993) pp. 169–193.
[24] DOI: 10.2307/2374964 · Zbl 0824.14008 · doi:10.2307/2374964
[25] DOI: 10.4310/CAG.2000.v8.n1.a1 · Zbl 0978.32024 · doi:10.4310/CAG.2000.v8.n1.a1
[26] DOI: 10.1016/0022-4049(89)90014-5 · Zbl 0691.20036 · doi:10.1016/0022-4049(89)90014-5
[27] DOI: 10.1007/BF01234432 · Zbl 0695.53052 · doi:10.1007/BF01234432
[28] Li P., J. Diff. Geom. 35 pp 359– · Zbl 0768.53018 · doi:10.4310/jdg/1214448079
[29] DOI: 10.1515/9783110908961.261 · doi:10.1515/9783110908961.261
[30] Nakai M., J. Fac. Sci. Univ. Tokyo, Sect. I 17 pp 101–
[31] DOI: 10.3792/pja/1195523234 · Zbl 0197.08604 · doi:10.3792/pja/1195523234
[32] DOI: 10.2977/prims/1195192175 · Zbl 0298.32019 · doi:10.2977/prims/1195192175
[33] DOI: 10.1007/BF01897052 · Zbl 0860.53045 · doi:10.1007/BF01897052
[34] DOI: 10.5802/aif.1602 · Zbl 0904.32008 · doi:10.5802/aif.1602
[35] DOI: 10.1007/PL00001677 · Zbl 1003.32005 · doi:10.1007/PL00001677
[36] DOI: 10.1090/conm/394/07444 · doi:10.1090/conm/394/07444
[37] DOI: 10.2977/prims/1195181418 · Zbl 0568.32008 · doi:10.2977/prims/1195181418
[38] DOI: 10.1007/BF02063212 · Zbl 0153.15401 · doi:10.1007/BF02063212
[39] DOI: 10.1007/978-1-4684-8038-2 · doi:10.1007/978-1-4684-8038-2
[40] DOI: 10.1007/978-3-642-48269-4 · doi:10.1007/978-3-642-48269-4
[41] DOI: 10.1007/978-1-4615-8126-0 · doi:10.1007/978-1-4615-8126-0
[42] DOI: 10.1090/pspum/053/1141208 · doi:10.1090/pspum/053/1141208
[43] Simpson C., Comp. Math. 87 pp 99–
[44] Siu Y.-T., J. Diff. Geom. 17 pp 55– · Zbl 0497.32025 · doi:10.4310/jdg/1214436700
[45] DOI: 10.1007/978-1-4899-6664-3_5 · doi:10.1007/978-1-4899-6664-3_5
[46] DOI: 10.2307/2373216 · Zbl 0144.33901 · doi:10.2307/2373216
[47] DOI: 10.1007/BF01114919 · Zbl 0164.09302 · doi:10.1007/BF01114919
[48] DOI: 10.1007/BFb0096228 · doi:10.1007/BFb0096228
[49] Tworzewski P., Ann. Polon. Math. 42 pp 387–
[50] DOI: 10.1090/pspum/041/740887 · doi:10.1090/pspum/041/740887
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