Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds

By Cheol-Hyun Cho, Yong-Geun Oh. 2003

259 citations recorded by the source corpus.

In this paper, we first provide an explicit description of {\it all} holomorphic discs (``disc instantons'') attached to Lagrangian torus fibers of arbitrary compact toric manifolds, and prove their Fredholm regularity. Using this, we compute Fukaya-Oh-Ohta-Ono's (FOOO's) obstruction (co)chains and the Floer cohomology of Lagrangian torus fibers of Fano toric manifolds. In particular specializing to the formal parameter $T^{2\pi} = e^{-1}$, our computation verifies the folklore that FOOO's obstruction (co)chains correspond to the Landau-Ginzburg superpotentials under the mirror symmetry correspondence, and also proves the prediction made by K. Hori about the Floer cohomology of Lagrangian torus fibers of Fano toric manifolds. The latter states that the Floer cohomology (for the parameter value $T^{2\pi} = e^{-1}$) of all the fibers vanish except at a finite number, the Euler characteristic of the toric manifold, of base points in the momentum polytope that are critical points of the superpotential of the Landau-Ginzburg mirror to the toric manifold. In the latter cases, we also prove that the Floer cohomology of the corresponding fiber is isomorphic to its singular cohomology. We also introduce a restricted version of the Floer cohomology of Lagrangian submanifolds, which is a priori more flexible to define in general, and which we call the {\it adapted Floer cohomology}. We then prove that the adapted Floer cohomology of any non-singular torus fiber of Fano toric manifolds is well-defined, invariant under the Hamiltonian isotopy and isomorphic to the Bott-Morse Floer cohomology of the fiber.

Subfields: Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/018f609a7d93b0d65b5f04f6dfc3af84a037d6e2

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

Citations in the corpus

Toric degenerations of Gelfand-Cetlin systems and potential functions (2008, methodology): “The first problem is we have to care about the singularity of the toric variety ([6] deals with smooth toric manifolds), but this point is not very problematic for the calculation of the potential functions, for which we need to consider only those disks with Maslov index two.”
Quantum Structures for Lagrangian Submanifolds (2007, background): “In [22, 21] there are some examples of non-monotone tori which satisfy similar dichotomy type properties (the argument used there is different from the one here).”
Mirror symmetry and T-duality in the complement of an anticanonical divisor (2007, methodology): “Then, ignoring convergence issues, we can tentatively make the following definition (see also [10], §12.”
T-Duality and Homological Mirror Symmetry of Toric Varieties (2008, result): “We also mention some complementary work which studies the A-model for toric varieties [11,15,16] and varieties with effective anticanonical divisors [5], and the relation to the Landau– Ginzburg mirror—especially Chan and Leung [10], which employs similar T-duality reasoning.”
The wall-crossing formula and Lagrangian mutations (2017, methodology): “When X is toric, Table 1 shows the standard toric potential which has been computed by Cho and Oh [9].”
Generating the Fukaya categories of Hamiltonian 𝐺-manifolds (2015, methodology): “The following is an immediate corollary our main theorem coupled with of computations of Floer cohomology due to Cho-Oh [16] and Fukaya-Oh-OhtaOno [28]:”
Quasi-states and symplectic intersections (2004, result): “Our interest in this example is due to the fact that in this case the obstructions to displaceability of L coming from the Lagrangian Floer homology do vanish according to a result by Cho and Oh [15].”
On the Fukaya category of a Fano hypersurface in projective space (2013, background): “, [7]) necessarily has a critical point at the origin, so the Floer cohomology does not vanish.”
The monotone wrapped Fukaya category and the open-closed string map (2012, methodology): “We assume the reader has some familiarity with the relevant literature (Cho-Oh [17], Auroux [12], Fukaya-Oh-OhtaOno [22]), and we refer to [36, Appendix] for details.”
Gauged Floer Theory Of Toric Moment Fibers (2010, methodology): “The argument is the same as in Cho-Oh [12], Fukaya et al [21, Theorem 13.”
Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces (2012, background): “These discs are easily seen to be regular, by reduction to the toric case [10]; their symplectic area is λ (by definition of the moment map μX , see the beginning of §4.”