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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

   

 

The Calabi-Yau equation on 4-manifolds over 2-tori


Authors: A. Fino, Y.Y. Li, S. Salamon and L. Vezzoni
Journal: Trans. Amer. Math. Soc. 365 (2013), 1551-1575
MSC (2010): Primary 53C25, 35J60, 53D35
Published electronically: October 1, 2012
MathSciNet review: 3003274
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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.


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Additional Information

A. Fino
Affiliation: Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italia
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

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05692-3
Received by editor(s): April 11, 2011
Received by editor(s) in revised form: August 6, 2011
Published electronically: October 1, 2012
Article copyright: © Copyright 2012 American Mathematical Society