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Pseudoholomorphic tori in the Kodaira–Thurston manifold

Published online by Cambridge University Press:  16 July 2015

Jonathan David Evans
Department of Mathematics, University College London, Gower Street, LondonWC1E 6BT, UK email
Jarek Kędra
University of Aberdeen, UK University of Szczecin, Poland email
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The Kodaira–Thurston manifold is a quotient of a nilpotent Lie group by a cocompact lattice. We compute the family Gromov–Witten invariants which count pseudoholomorphic tori in the Kodaira–Thurston manifold. For a fixed symplectic form the Gromov–Witten invariant is trivial so we consider the twistor family of left-invariant symplectic forms which are orthogonal for some fixed metric on the Lie algebra. This family defines a loop in the space of symplectic forms. This is the first example of a genus one family Gromov–Witten computation for a non-Kähler manifold.

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© The Authors 2015 


Benson, C. and Gordon, C. S., Kähler and symplectic structures on nilmanifolds, Topology 27 (1988), 513518.CrossRefGoogle Scholar
Bershadsky, M., Cecotti, S., Ooguri, H. and Vafa, C., Holomorphic anomalies in topological field theories, Nuclear Phys. B 405 (1993), 279304 (with an appendix by S. Katz).CrossRefGoogle Scholar
Bryan, J. and Leung, N. C., The enumerative geometry of K3 surfaces and modular forms, J. Amer. Math. Soc. 13 (2000), 371410.CrossRefGoogle Scholar
Buse, O., Relative family Gromov–Witten invariants and symplectomorphisms, Pacific J. Math. 218 (2005), 315342.CrossRefGoogle Scholar
Candelas, P., de la Ossa, X. C., Green, P. S. and Parkes, L., A pair of Calabi–Yau manifolds as an exactly soluble superconformal theory, Nuclear Phys. B 359 (1991), 2174.CrossRefGoogle Scholar
Cordero, L. A., Fernandez, M. and de Leon, M., Examples of compact non-Kähler almost Kähler manifolds, Proc. Amer. Math. Soc. 95 (1985), 280296.Google Scholar
Dickson, L. E., History of the theory of numbers. Volume I: divisibility and primality, Carnegie Institute of Washington, Publication No. 256, vol. I (Chelsea Publishing Company, New York, 1952).Google Scholar
Gilbarg, D. and Trudinger, N. S., Elliptic partial differential equations of second order, Grundlehren der mathematischen Wissenschaften, vol. 224 (Springer, 2001).Google Scholar
Gromov, M., Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307347.CrossRefGoogle Scholar
Gromov, M., Response to steele prize for a seminal contribution to research, Notices Amer. Math. Soc. 44 (1997), 342345.Google Scholar
Kędra, J., Restrictions on symplectic fibrations, Differential Geom. Appl. 21 (2004), 93112 (with an appendix ‘Simple examples of nontrivial Gromov–Witten invariants’ written jointly with Kaoru Ono).CrossRefGoogle Scholar
Kleiman, S. L., Problem 15: Rigorous foundation of Schubert’s enumerative calculus, in Mathematical developments arising from Hilbert problems, Proceedings and Symposia in Pure Mathematics, vol. XXVIII (American Mathematical Society, Providence, RI, 1976), 445482.CrossRefGoogle Scholar
Klemm, A., Maulik, D., Pandharipande, R. and Scheidegger, E., Noether–Lefschetz theory and the Yau–Zaslow conjecture, J. Amer. Math. Soc. 23 (2010), 10131040.CrossRefGoogle Scholar
, H. V. and Ono, K., Parameterized Gromov–Witten invariants and topology of symplectomorphism groups, Adv. Stud. Pure Math. 52 (2008), 5175.Google Scholar
Lee, J., Family Gromov–Witten invariants for Kähler surfaces, Duke Math. J. 123 (2004), 209233.CrossRefGoogle Scholar
Lee, J., Counting curves in elliptic surfaces by symplectic methods, Comm. Anal. Geom. 14 (2006), 107134.CrossRefGoogle Scholar
Lee, J. and Leung, N. C., Yau–Zaslow formula on K3 surfaces for non-primitive classes, Geom. Topol. 9 (2005), 19772012.CrossRefGoogle Scholar
Lee, J. and Parker, T., Symplectic gluing and family Gromov–Witten invariants, in Geometry and topology of manifolds, Fields Institute Communications, vol. 47 (American Mathematical Society, Providence, RI, 2005), 147172.Google Scholar
Lu, P., A rigorous definition of fiberwise quantum cohomology and equivariant quantum cohomology, Comm. Anal. Geom. 6 (1998), 511588.CrossRefGoogle Scholar
Maulik, D. and Pandharipande, R., Gromov–Witten theory and Noether–Lefschetz theory, in A celebration of algebraic geometry, Clay Mathematics Proceedings, vol. 18 (American Mathematical Society, Providence, RI, 2013), 469507.Google Scholar
McDuff, D. and Salamon, D., J-holomorphic Curves and Symplectic Topology, American Mathematical Society Colloquium Publications, vol. 52 (American Mathematical Society, Providence, RI, 2004).Google Scholar
Ruan, Y. and Tian, G., A mathematical theory of quantum cohomology, J. Differential Geom. 42 (1995), 259367.CrossRefGoogle Scholar
Ruan, Y. and Tian, G., Higher genus symplectic invariants and sigma models coupled with gravity, Invent. Math. 130 (1997), 455516.CrossRefGoogle Scholar
Seidel, P., On the group of symplectic automorphisms of CPm ×CPn, Amer. Math. Soc. Transl. Ser. 2 196 (1999), 237250.Google Scholar
Thurston, W., Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55 (1976), 467468.Google Scholar
Vinberg, E. B., Gorbatsevich, V. V. and Shvartsman, O. V., Discrete subgroups of Lie groups, in Lie groups and Lie algebras II, Encyclopædia of Mathematical Sciences, vol. 21 (Springer, 2000).Google Scholar
Witten, E., Topological sigma models, Comm. Math. Phys. 118 (1988), 411449.CrossRefGoogle Scholar
Zinger, A., The reduced genus 1 Gromov–Witten invariants of Calabi–Yau hypersurfaces, J. Amer. Math. Soc. 22 (2009), 691737.CrossRefGoogle Scholar