Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-05-17T12:11:01.628Z Has data issue: false hasContentIssue false
  • English
  • Français

Degeneration of algebraic manifolds and the spectrum of Laplacian

Published online by Cambridge University Press:  22 January 2016

Ken-Ichi Yoshikawa
Ken-Ichi Yoshikawa*
Affiliation:
Nagoya University, Chikusa-ku, Nagoya 464-01, Japan, yosikawa@math.nagoya-u.ac.jp
Rights & Permissions [Opens in a new window]

Abstract.

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We shall show that the spectrum of Laplacian depends continuously on the parameter for one parameter degeneration of projective algebraic manifolds.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 146 , June 1997 , pp. 83 - 129
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1997

References

[A] Akutagawa, A., Convergence for Yamabe metrics of positive scalar curvature with integral bounds on curvature, preprint (1994).Google Scholar
[B-K-N] Bando, S., Kasue, A. and Nakajima, H., On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth, Invent. Math., 97 (1989), 313–349.Google Scholar
[C-K-S] Carlen, E. A., Kusuoka, S. and Stroock, P. W., Upper bounds for symmetric Markov transition functions, Ann. Inst. H. Poincaré, 23 (1987), 245–287.Google Scholar
[C] Chavel, I., Eigenvalues in Riemannian Geometry, Academic Press, New York, 1984.Google Scholar
[C-F] Chavel, I. and Feldman, E. A., Spectra of domains in compact manifolds, J. Funct. Anal, 30 (1978), 198–222.Google Scholar
[C-L] Cheng, S.-Y. and Li, P., Heat kernel estimates and lower bound of eigenvalues, Comment. Math. Helv., 56 (1981), 327–338.Google Scholar
[C-L-Y] Cheng, S.-Y., Li, P. and Yau, S.-T., Heat equations on minimal submanifolds and their applications, Amer. J. Math., 106 (1984), 1033–1065.Google Scholar
[CI] Clemens, C. H., Degeneration of Kähler manifolds, Duke Math. J., 44 (1977), 215–290.CrossRefGoogle Scholar
[D] Davis, E. B., Heat kernels and spectral theory, Cambridge Univ. Press, Cambridge, 1989.Google Scholar
[F] Fujiki, A., Closedness of the Duady spaces of compact Kähler spaces, Publ. RIMS, Kyoto Univ., 14 (1978), 1–52.Google Scholar
[Fk] Fukaya, K., Collapsing Riemannian manifolds and eigenvalues of the Laplace operator, Invent. Math., 87 (1987), 517–547.Google Scholar
[G-H] Griffiths, P. and Harris, J., Principles of algebraic geometry, John Wiley, New York, 1978.Google Scholar
[G-R] Grauert, H. and Remmert, R., Coherent analytic sheaves, Springer, Berlin, Heidelberg, New York, 1984.CrossRefGoogle Scholar
[G-T] Gilberg, D. and Trudinger, N. S., Elliptic partial differential equations of second order (2nd ed.), Springer, Berlin, Heidelberg, New York, 1983.Google Scholar
[G] Gray, A., Tubes, Addison-Wesley, New York, 1990.Google Scholar
[Gr1] Gromov, M., Spectral geometry of semi-algebraic sets, Ann. Inst. Fourier, 42 (1992), 249–274.Google Scholar
[Gr2] Gromov, M., Metric invariants of Kähler manifolds, preprint (1992).Google Scholar
[H] Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. of Math., 79 (1964), 109–326.CrossRefGoogle Scholar
[J-W] Ji, L. and Wentworth, R., Spectral convergence on degenerating surfaces, Duke Math. J., 66 (1992), 469–501.CrossRefGoogle Scholar
[K-K1] Kasue, A. and Kumura, H., Spectral convergence of Riemannian manifolds, Tôhoku Math. J., 46 (1994), 147–179.Google Scholar
[K-K2] Kasue, A. and Kumura, H., Spectral convergence of Riemannian manifolds, II, Tohoku Math. J., 48 (1996), 71–120.Google Scholar
[L-T] Li, P. and Tian, G., On the heat kernel of the Bergmann metric on algebraic varieties, J. Amer. Math. Soc, 8 (1995), 857–877.Google Scholar
[M] Matsumura, H., Commutative ring theory, Cambridjge University Press, 1986.Google Scholar
[N] Nakajima, H., Hausdorff convergence of Einstein 4-manifolds, J. Fac. Sci. Univ. Tokyo, 35 (1988), 411–424.Google Scholar
[Na] Nakano, S., Several complex variables, (in Japanese), Asakura shoten Publ., Tokyo, 1981.Google Scholar
[Y1] Yoshikawa, K., Conic degeneration of Riemannian manifolds and the spectral zeta function, preprint (1993).Google Scholar
[Y2] Yoshikawa, K., Degeneration of complex manifolds and the spectrum of Laplace operator, (in Japanese), Master thesis at Kyoto University (1992).Google Scholar