Proceedings of the Japan Academy, Series A, Mathematical Sciences

Potential functions via toric degenerations

Takeo Nishinou, Yuichi Nohara, and Kazushi Ueda

Full-text: Open access


We construct an integrable system on an open subset of a Fano manifold equipped with a toric degeneration, and compute the potential function for its Lagrangian torus fiber if the central fiber is a toric Fano variety admitting a small resolution.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 88, Number 2 (2012), 31-33.

First available in Project Euclid: 2 February 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53D12: Lagrangian submanifolds; Maslov index
Secondary: 53D40: Floer homology and cohomology, symplectic aspects

Toric degeneration Lagrangian torus fibration potential function


Nishinou, Takeo; Nohara, Yuichi; Ueda, Kazushi. Potential functions via toric degenerations. Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), no. 2, 31--33. doi:10.3792/pjaa.88.31.

Export citation


  • D. Auroux, Special Lagrangian fibrations, wall-crossing, and mirror symmetry, in Surveys in differential geometry. Vol. XIII. Geometry, analysis, and algebraic geometry: forty years of the Journal of Differential Geometry, Surv. Differ. Geom., 13, Int. Press, Somerville, MA, 2009, pp. 1–47.
  • C.-H. Cho and Y.-G. Oh, Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds, Asian J. Math. 10 (2006), no. 4, 773–814.
  • K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Lagrangian Floer theory on compact toric manifolds II: Bulk deformations, arXiv:0810.5654.
  • K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Toric degeneration and non-displaceable Lagrangian tori in $S^{2} \times S^{2}$, arXiv:1002.1660.
  • K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Lagrangian intersection Floer theory: anomaly and obstruction, AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI, 2009.
  • K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Lagrangian Floer theory on compact toric manifolds. I, Duke Math. J. 151 (2010), no. 1, 23–174.
  • P. E. Newstead, Stable bundles of rank 2 and odd degree over a curve of genus 2, Topology 7 (1968), 205–215.
  • T. Nishinou, Y. Nohara and K. Ueda, Toric degenerations of Gelfand-Cetlin systems and potential functions, Adv. Math. 224 (2010), no. 2, 648–706.
  • M. S. Narasimhan and S. Ramanan, Moduli of vector bundles on a compact Riemann surface, Ann. of Math. (2) 89 (1969), 14–51.
  • V. Przyjalkowski, Hodge numbers of Fano threefolds via Landau–Ginzburg models, arXiv:0911.5428.
  • P. Seidel, Fukaya categories and Picard-Lefschetz theory, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008.
  • I. Smith, Floer cohomology and pencils of quadrics, arXiv:1006.1099.