Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-05-28T23:30:16.140Z Has data issue: false hasContentIssue false

Toric Degenerations, Tropical Curve, and Gromov–Witten Invariants of Fano Manifolds

Published online by Cambridge University Press:  20 November 2018

Takeo Nishinou*
Department of Mathematics, Tohoku University, Sendai, Miyagi, Japan. email:
Rights & Permissions [Opens in a new window]


Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we give a tropical method for computing Gromov–Witten type invariants of Fano manifolds of special type. This method applies to those Fano manifolds that admit toric degenerations to toric Fano varieties with singularities allowing small resolutions. Examples include (generalized) flag manifolds of type $\text{A}$ and some moduli space of rank two bundles on a genus two curve.

Research Article
Copyright © Canadian Mathematical Society 2015


[1] Cho, C.-H. and Oh, Y.-G., Floer cohomology and disc instantons of Lagrangian torus fibers inFano toric manifolds. Asian J. Math. 10(2006), no. 4, 773814. http://dx.doi.Org/10.4310/AJM.2OO6.v10.n4.a10 Google Scholar
[2] Cox, D. and Katz, S., Mirror symmetry and algebraic geometry. Mathematical Surveys and Monographs, 68, American Mathematical Society, Providence, RI, 1999.Google Scholar
[3] Goldman, W., Invariant functions on Lie groups and Hamiltonian flows of surface group representations. Invent. Math. 85(1986), no. 2, 263302.http://dx.doi.Org/10.1007/BF01389091 Google Scholar
[4] Gromov, M. Pseudo holomorphiccurves in symplectic manifolds. Invent. Math. 82(1985), no. 2, 307347.Google Scholar
[5] Jeffrey, L. C. and Weitsman, J., Bohr–Sommerfeld orbits in the moduli space of flat connections and the Verlinde dimension formula. Comm. Math. Phys. 150(1992), no. 3, 593630.http://dx.doi.Org/10.1007/BF02096964 Google Scholar
[6] Kogan, M., and Miller, E., Toric degeneration of Schubert varieties and Gelfand–Tsetlin polytopes. Adv. Math. 193(2005), no. 1, 117. http://dx.doi.Org/10.1016/j.aim.2004.03.017 Google Scholar
[7] McDuff, D. and Salamon, D. J–holomorphic curves and symplectic topology. American Mathematical Society Colloquium Publications, 52, American Mathematical Society, Providence, RI, 2012.Google Scholar
[8] Mikhalkin, G., Enumerative tropical algebraic geometry in ℝ2. J. Amer. Math. Soc. 18(2005), no. 2, 313377. Google Scholar
[9] Narasimhan, M. S. and Ramanan, S., Moduli of vector bundles on a compact Riemann surface. Ann. c Math. 89(1969), 1451. Google Scholar
[10] Newstead, P.E., Stable bundles of rank 2 and odd degree over a curve of genus 2. Topology 7(1968), 205215. http://dx.doi.Org/10.1016/0040-9383(68)90001-3 Google Scholar
[11] Nishinou, T., Disc counting on toric varieties via tropical curves. Amer. J. Math. 134(2012), no. 6, 14231472. http://dx.doi.Org/10.1353/ajm.2O12.0043 Google Scholar
[12] Nishinou, T., Correspondence theorems for tropical curves. Google Scholar
[13] Nishinou, T., Nohara, Y., and Ueda, K., Toric degenerations of Gelfand–Cetlin systems andpotential functions. Adv. Math. 224(2010), no. 2, 648706.http://dx.doi.Org/10.1016/j.aim.2009.12.012 Google Scholar
[14] Nishinou, T. and Siebert, B., Toric degenerations of toric varieties and tropical curves. Duke Math. J. 135(2006), no. 1, 151. Google Scholar
[15] –D. Ruan, W., Lagrangian torus fibration of quintic Calabi–Yau hypersurfaces. II. Technical results on gradient flow construction. J. Symplectic Geom. 1(2002), no. 3, 435521. Google Scholar