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Asymptotically cylindrical Calabi-Yau manifolds

Publication ,  Journal Article
Mark, H; Hein, HJ; Johannes, N
Published in: Journal of Differential Geometry
October 1, 2015

Let M be a complete Ricci-flat Kähler manifold with one end and assume that this end converges at an exponential rate to [0, ∞) x X for some compact connected Ricci-flat manifold X. We begin by proving general structure theorems for M; in particular we show that there is no loss of generality in assuming that M is simply-connected and irreducible with Hol(M) = SU(n), where n is the complex dimension of M. If n > 2 we then show that there exists a projective orbifold M¯ and a divisor D¯ ∈ |-KM¯| with torsion normal bundle such that M is biholomorphic to M¯ \ D¯, thereby settling a long-standing question of Yau in the asymptotically cylindrical setting. We give examples where M¯ is not smooth: the existence of such examples appears not to have been noticed previously. Conversely, for any such pair (M¯, D¯) we give a short and self-contained proof of the existence and uniqueness of exponentially asymptotically cylindrical Calabi-Yau metrics on M¯ \ D¯.

Duke Scholars

Published In

Journal of Differential Geometry

DOI

EISSN

1945-743X

ISSN

0022-040X

Publication Date

October 1, 2015

Volume

101

Issue

2

Start / End Page

213 / 265

Related Subject Headings

  • General Mathematics
  • 4904 Pure mathematics
  • 0101 Pure Mathematics
 

Citation

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Mark, H., Hein, H. J., & Johannes, N. (2015). Asymptotically cylindrical Calabi-Yau manifolds. Journal of Differential Geometry, 101(2), 213–265. https://doi.org/10.4310/jdg/1442364651
Mark, H., H. J. Hein, and N. Johannes. “Asymptotically cylindrical Calabi-Yau manifolds.” Journal of Differential Geometry 101, no. 2 (October 1, 2015): 213–65. https://doi.org/10.4310/jdg/1442364651.
Mark H, Hein HJ, Johannes N. Asymptotically cylindrical Calabi-Yau manifolds. Journal of Differential Geometry. 2015 Oct 1;101(2):213–65.
Mark, H., et al. “Asymptotically cylindrical Calabi-Yau manifolds.” Journal of Differential Geometry, vol. 101, no. 2, Oct. 2015, pp. 213–65. Scopus, doi:10.4310/jdg/1442364651.
Mark H, Hein HJ, Johannes N. Asymptotically cylindrical Calabi-Yau manifolds. Journal of Differential Geometry. 2015 Oct 1;101(2):213–265.

Published In

Journal of Differential Geometry

DOI

EISSN

1945-743X

ISSN

0022-040X

Publication Date

October 1, 2015

Volume

101

Issue

2

Start / End Page

213 / 265

Related Subject Headings

  • General Mathematics
  • 4904 Pure mathematics
  • 0101 Pure Mathematics