On higher dimensional complex Plateau problem

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• 作者：Rong Du ; Yun Gao ; Stephen Yau
• 刊名：Mathematische Zeitschrift
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：282
• 期：1-2
• 页码：389-403
• 全文大小：465 KB
• 参考文献：1.Alexeev, V.: Two two-dimensional terminations. Duke Math. J. 69(3), 527–545 (1993)MathSciNet CrossRef
  2.Andreotti, A., Grauert, H.: Théorèmes de finitude pour la cohomologie des espaces complexes. Bull. Soc. Math. Fr. 90, 193–259 (1962)MathSciNet
  3.Brenton, L.: On singular complex surfaces with negative canonical bundle, with applications to singular compactifications of \({\mathbb{C}}^2\) and to \(3\) -dimensional rational singularities. Math. Ann. 248(2), 117–124 (1980)MathSciNet CrossRef
  4.Clemens, H., Kollár, J., Mori, S.: Higher-dimensional complex geometry. Astérisque, vol. 166 (1988)
  5.Cutkosky, S.: Elementary contractions of Gorenstein threefolds. Math. Ann. 280(3), 521–525 (1988)MathSciNet CrossRef
  6.Du, R., Gao, Y.: New invariants for complex manifolds and rational singularities. Pacific J. Math. 269(1), 73–97 (2014)MathSciNet CrossRef
  7.Du, R., Gao, Y.: Some remarks on Yau’s conjecture and complex Plateau problem. Methods Appl. Anal. 21(3), 357–364 (2014)MathSciNet
  8.Du, R., Luk, H.S., Yau, S.S.-T.: New invariants for complex manifolds and isolated singularities. Commun. Anal. Geom. 19(5), 991–1021 (2011)MathSciNet CrossRef
  9.Du, R., Yau, S.S.-T.: Kohn-Rossi cohomology and its application to the complex Plateau problem, III. J. Differ. Geom. 90(2), 251–266 (2012)MathSciNet
  10.Ein, L., Mustaţǎ, M.: Inversion of adjunction for local complete intersection varieties. Am. J. Math. 126(6), 1355–1365 (2004)CrossRef
  11.Ein, L., Mustaţǎ, M., Yasuda, T.: Jet schemes, log discrepancies and inversion of adjunction. Inven. Math. 153(3), 519–535 (2003)CrossRef
  12.Esnault, H., Viehweg, E.: Lectures on Vanishing Theorems. DMV Seminar, 20. Birkhäuser Verlag, Basel (1992)CrossRef
  13.Greuel, G.-M.: Dualität in der lokalen Kohomologie isolierter Singularitäten. Math. Ann. 250, 157–173 (1980)MathSciNet CrossRef
  14.Harvey, R., Lawson, B.: On boundaries of complex analytic varieties I. Ann. Math. 102, 233–290 (1975)MathSciNet CrossRef
  15.Harvey, R. and Lawson, B.: Addendum to Theorem 10.4 of [HL]. arXiv:​math/​0002195
  16.Kollár, J., Mori, S.: Birational geometry of algebraic varieties. In: Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998, With the collaboration of C. H. Clemens and A. Corti, translated from the 1998 Japanese original. MR 1658959 (2000b:14018)
  17.Luk, H.S., Yau, S.S.-T.: Counterexample to boundary regularity of a strongly pseudoconvex CR manifold: an addendum to the paper of Harvey–Lawson. Ann. Math. 148, 1153–1154 (1998). MR 1670081MathSciNet CrossRef
  18.Markushevich, D.: Minimal discrepancy for a terminal cDV singularity is 1. J. Math. Sci. Univ. Tokyo 3(2), 445–456 (1996)MathSciNet
  19.Mori, S.: Shigefumi threefolds whose canonical bundles are not numerically effective. Ann. Math. 2 116(1), 133–176 (1982)CrossRef
  20.Reid, M.: Canonical 3-folds. Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, pp. 273-310, Sijthoff & Noordhoff, Alphen aan den Rijn-Germantown, Md., (1980)
  21.Reid, M.: Minimal models of canonical 3-folds. Algebraic varieties and analytic varieties (Tokyo, 1981), 131C180, Adv. Stud. Pure Math., vol. 1, North-Holland, Amsterdam, (1983)
  22.Shokurov, V.V.: Problems about Fano varieties. In: Birational Geometry of Algebraic Varieties. Open Problems-Katata, pp. 30-32 (1988)
  23.Siu, Y.-T.: Analytic sheaves of local cohomology. Trans. AMS 148, 347–366 (1970)MathSciNet CrossRef
  24.Straten, D.V., Steenbrink, J.: Extendability of holomorphic differential forms near isolated hypersurface singularities. Abh. Math. Sem. Univ. Hamburg 55, 97–110 (1985)MathSciNet CrossRef
  25.Tanaka, N.: A differentail geometry study on strongly pseudoconvex manifolds. In: Lecture in Mathematics. Kyoto University, 9, Kinokuniya Bookstroe Co. Ltd, (1975)
  26.Yau, S.S.-T.: Kohn-Rossi cohomology and its application to the complex Plateau problem, I. Ann. Math. 113, 67–110 (1981). MR 0604043CrossRef
  27.Yau, S.S.-T.: Various numerical invariants for isolated singularities. Am. J. Math. 104(5), 1063–1110 (1982)CrossRef
  
• 作者单位：Rong Du (1)
  Yun Gao (2)
  Stephen Yau (3)
  
  1. Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Rm. 312, Math. Bldg, No. 500, Dongchuan Road, Shanghai, 200241, People’s Republic of China
  2. Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China
  3. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1823

文摘

Let X be a compact connected strongly pseudoconvex CR manifold of real dimension \(2n-1\) in \({\mathbb {C}}^{N}\). It has been an interesting question to find an intrinsic smoothness criteria for the complex Plateau problem. For \(n\ge 3\) and \(N=n+1\), Yau found a necessary and sufficient condition for the interior regularity of the Harvey–Lawson solution to the complex Plateau problem by means of Kohn–Rossi cohomology groups on X in 1981. For \(n=2\) and \(N\ge n+1\), the first and third authors introduced a new CR invariant \(g^{(1,1)}(X)\) of X. The vanishing of this invariant will give the interior regularity of the Harvey–Lawson solution up to normalization. For \(n\ge 3\) and \(N>n+1\), the problem still remains open. In this paper, we generalize the invariant \(g^{(1,1)}(X)\) to higher dimension as \(g^{(\Lambda ^n 1)}(X)\) and show that if \(g^{(\Lambda ^n 1)}(X)=0\), then the interior has at most finite number of rational singularities. In particular, if X is Calabi–Yau of real dimension 5, then the vanishing of this invariant is equivalent to give the interior regularity up to normalization. Rong Du: The Research Sponsored by the National Natural Science Foundation of China (Grant No. 11471116), Science and Technology Commission of Shanghai Municipality (Grant No. 13dz2260400) and Shanghai Pujiang Program (Grant No. 12PJ1402400). Yun Gao: The Research Sponsored by the National Natural Science Foundation of China (Grant Nos. 11271250, 11271251) and SMC program of Shanghai Jiao Tong University. Stephen Yau: The Research supported by Tsingua Start up fund. All the authors are supported by China NSF (Grant No. 11531007).