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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The Calabi–Yau equation on 4-manifolds over 2-tori
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by A. Fino, Y.Y. Li, S. Salamon and L. Vezzoni PDF
Trans. Amer. Math. Soc. 365 (2013), 1551-1575 Request permission

Abstract:

This paper pursues the study of the Calabi–Yau equation on certain symplectic non-Kähler 4-manifolds, building on a key example of Tosatti and Weinkove in which more general theory had proved less effective. Symplectic 4-manifolds admitting a 2-torus fibration over a 2-torus base $\mathbb {T}^2$ are modelled on one of three solvable Lie groups. Having assigned an invariant almost-Kähler structure and a volume form that effectively varies only on $\mathbb {T}^2$, one seeks a symplectic form with this volume. Our approach simplifies the previous analysis of the problem and establishes the existence of solutions in various other cases.
References
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Additional Information
  • A. Fino
  • Affiliation: Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italia
  • MR Author ID: 363840
  • ORCID: 0000-0003-0048-2970
  • Email: annamaria.fino@unito.it
  • Y.Y. Li
  • Affiliation: Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey 08854
  • Email: yyli@math.rutgers.edu
  • S. Salamon
  • Affiliation: Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italia – and – Department of Mathematics, King’s College London, Strand, London, WC2R 2LS, United Kingdom
  • Email: simon.salamon@kcl.ac.uk
  • L. Vezzoni
  • Affiliation: Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italia
  • Email: luigi.vezzoni@unito.it
  • Received by editor(s): April 11, 2011
  • Received by editor(s) in revised form: August 6, 2011
  • Published electronically: October 1, 2012
  • © Copyright 2012 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 365 (2013), 1551-1575
  • MSC (2010): Primary 53C25, 35J60, 53D35
  • DOI: https://doi.org/10.1090/S0002-9947-2012-05692-3
  • MathSciNet review: 3003274