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Complex Manifolds

Ed. by Fino, Anna Maria


CiteScore 2017: 0.39

SCImago Journal Rank (SJR) 2017: 0.260
Source Normalized Impact per Paper (SNIP) 2017: 0.660

Mathematical Citation Quotient (MCQ) 2017: 0.18

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Online
ISSN
2300-7443
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Toric extremal Kähler-Ricci solitons are Kähler-Einstein

Simone Calamai / David Petrecca
  • Corresponding author
  • Institut für Differentialgeometrie - Leibniz Universität Hannover, Welfengarten 1, Hanover, Germany
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Published Online: 2017-12-22 | DOI: https://doi.org/10.1515/coma-2017-0012

Abstract

In this short note, we prove that a Calabi extremal Kähler-Ricci soliton on a compact toric Kähler manifold is Einstein. This settles for the class of toric manifolds a general problem stated by the authors that they solved only under some curvature assumptions.

Keywords: Extremal Kähler metrics; Kähler-Ricci solitons; Einstein manifolds; Toric manifolds

MSC 2010: 53C25; 53C55; 58D19

References

  • [1] M. Abreu, Kähler geometry of toric manifolds in symplectic coordinates, Symplectic and contact topology: interactions and perspectives (Toronto, ON/Montreal, QC, 2001), Fields Inst. Commun., vol. 35, Amer. Math. Soc., Providence, RI, 2003, pp. 1-24.Google Scholar

  • [2] S. Calamai and D. Petrecca, On Calabi extremal Kähler-Ricci solitons, Proc. Amer. Math. Soc. 144 (2016), no. 2, 813-821. MR 3430856Web of ScienceGoogle Scholar

  • [3] B. Chow et al., The Ricci Flow: Techniques and Applications: Geometric Aspects, Mathematical surveys and monographs, vol. 135, American Mathematical Society, 2007.Google Scholar

  • [4] S. K. Donaldson, Kähler geometry on toric manifolds, and some other manifolds with large symmetry, Handbook of geometric analysis. No. 1, Adv. Lect. Math. (ALM), vol. 7, Int. Press, Somerville, MA, 2008, pp. 29-75. MR 2483362Google Scholar

  • [5] F. Podestà and A. Spiro, Kähler-Ricci solitons on homogeneous toric bundles, J. Reine Angew. Math. 642 (2010), 109-127. MR 2658183Google Scholar

About the article

Received: 2017-08-21

Accepted: 2017-11-23

Published Online: 2017-12-22

Published in Print: 2017-12-20


Citation Information: Complex Manifolds, Volume 4, Issue 1, Pages 179–182, ISSN (Online) 2300-7443, DOI: https://doi.org/10.1515/coma-2017-0012.

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© 2017. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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