On the counting of holomorphic discs in toric Fano manifolds
Cheol-Hyun Cho
备注：
17 pages, 2 figures,v2: rewritten using the language of cyclic A-infinity algebra, v3: added an example of a cyclic cohomology class, published version
Open Gromov-Witten invariants in general are not well-defined. We discuss in detail the enumerative numbers of the Clifford torus $T^2$ in $\CP^2$. For cyclic A-infinity algebras, we show that certain generalized way of counting may be defined up to Hochschild or cyclic boundary elements. In particular we obtain a well-defined function on Hochschild or cyclic homology of a cyclic A-infinity algebra, which has invariance property under cyclic A-infinity homomorphism. We discuss an example of Clifford torus $T^2$ and compute the invariant for a specific cyclic cohomology class.
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 which admit toric degenerations to toric Fano varieties with singularities allowing small resolutions. Examples include (generalized) flag manifolds of type A, and some moduli space of rank two bundles on a genus two curve.
We compute the ring structure of Floer cohomology groups of Lagrangian torus fibers in some toric Fano manifolds continuing the study of \cite{CO}. Related $\AI$-formulas hold for transversal choice of chains. Two different computations are provided: a direct calculation using the classification of holomorphic discs by Oh and the author in \cite{CO}, and another method by using an {\it analogue of divisor equation} in Gromov-Witten invariants to thecase of discs. Floer cohomology rings are shown to be isomorphic to Clifford algebras, whose quadratic forms are given by the Hessians of functions…
In this paper, we give a description of holomorphic multi-vector fields on smooth compact toric varieties, which generalizes Demazure's result of holomorphic vector fields on toric varieties. Based on the result, we compute the Poisson cohomology groups of holomorphic toric Poisson manifolds, i.e., toric varieties endowed with $T$-invariant holomorphic Poisson structures.
We provide an inductive algorithm to compute the bulk-deformed potentials for toric Fano surfaces via wall-crossing techniques and a tropical-holomorphic correspondence theorem for holomorphic discs. As an application of the correspondence theorem, we also prove a big quantum period theorem for toric Fano surfaces which relates the log descendant Gromov-Witten invariants with the oscillatory integrals of the bulk-deformed potentials.
We prove that if there exists a $c_1$-preserving graded ring isomorphism between integral cohomology rings of two Fano Bott manifolds, then they are isomorphic as toric varieties. As a consequence, we give an affirmative answer to McDuff's question on the uniqueness of a toric structure on a Fano Bott manifold.
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 $T^{2π} = e^{-1}$, our computation verifies the folklore that FOOO's obstruction (co)chains correspond to the Landau-Ginzburg superpotentials under the mirror symmetry correspo…
A holomorphic toric Poisson manifold is a nonsingular toric variety equipped with a holomorphic Poisson structure, which is invariant under the torus action. In this paper, we computed the Poisson cohomology groups for all holomorphic toric Poisson structures on $CP^n$, with the stand Poisson structure on $CP^n$ as a special case. We also computed the algebraic and the formal Poisson cohomology groups of holomorphic toric Poisson structures on $C^n$.
The degree zero part of the quantum cohomology algebra of a smooth Fano toric symplectic manifold is determined by the superpotential function, W, of its moment polytope. In particular, this algebra is semisimple, i.e. splits as a product of fields, if and only if all the critical points of W are non-degenerate. In this paper we prove that this non-degeneracy holds for all smooth Fano toric varieties with facet-symmetric duals to moment polytopes.
We construct and apply Strominger-Yau-Zaslow mirror transformations to understand the geometry of the mirror symmetry between toric Fano manifolds and Landau-Ginzburg models.
In this paper we study certain algebraic properties of the quantum homology algebra for the class of symplectic toric Fano manifolds. In particular, we examine the semi-simplicity of the quantum homology algebra, and the more general property of containing a field as a direct summand. Our main result provides an easily-verified sufficient condition for these properties which is independent of the symplectic form. Moreover, we answer two questions of Entov and Polterovich negatively by providing examples of toric Fano manifolds with non semi-simple quantum homology algebra, and others in which …
We study the Lagrangian quantum homology of real parts of Fano toric manifolds of minimal Chern number at least 2, using coefficients in a ring of Laurent polynomials over Z/2Z. We show that these Lagrangians are wide, in the sense that their quantum homology is isomorphic as a module to their classical homology tensored with this ring. Moreover, we show that the quantum homology is isomorphic as a ring to the quantum homology of the ambient symplectic manifold.
In this article, we consider the Floer cohomology (with $\Z_2$ coefficients) between torus fibers and the real Lagrangian in Fano toric manifolds. We first investigate the conditions under which the Floer cohomology is defined, and then develop a combinatorial description of the Floer complex based on the polytope of the toric manifold. We show that if the Floer cohomology is defined, and the Floer cohomology of the torus fiber is non-zero, then the Floer cohomology of the pair is non-zero. We use this result to develop some applications to non-displaceability and the minimum number of interse…
We prove that for a compact toric manifold whose anti-canonical divisor is numerically effective, the Lagrangian Floer superpotential defined by Fukaya-Oh-Ohto-Ono is equal to the superpotential written down by using the toric mirror map under a convergence assumption. This gives a method to compute open Gromov-Witten invariants using mirror symmetry.
In the case of toric varieties, we continue the pursuit of Kontsevich's fundamental insight, Homological Mirror Symmetry, by unifying it with the Mori program. We give a refined conjectural version of Homological Mirror Symmetry relating semi-orthogonal decompositions of the $B$-model on toric varieties to semi-orthogonal decompositions on the $A$-model on the mirror Landau-Ginzburg models. As evidence, we prove a new case of Homological Mirror Symmetry for a toric surface whose anticanonical bundle is not nef, namely a certain blow-up of $¶^2$ at three infinitesimally near points.
In this paper, we compute the open Gromov-Witten invariants for every compact toric surface X which is semi-Fano (i.e. the anticanonical line bundle is nef). Unlike the Fano case, this involves non-trivial obstructions in the corresponding moduli problem. As a consequence, an explicit formula for the Lagrangian Floer superpotential W is obtained, which in turn gives an explicit presentation of the small quantum cohomology ring of X. We also provide a computational verification of the conjectural ring isomorphism between the small quantum cohomology of X and the Jacobian ring of W.
In this paper, using the degeneration formula we obtain a blowup formulae of local Gromov-Witten invariants of Fano surfaces. This formula makes it possible to compute the local Gromov-Witten invariants of non-toric Fano surfaces form toric Fano surface, such as del Pezzo surfaces. This formula also verifed an expectation of Chiang-Klemm-Yau-Zaslow.
For the sake of hyperk{ä}hler SYZ conjecture, finding holomorphic Lagrangian fibrations becomes an important issue. Toric hyperk{ä}hler manifolds are real dimension $4n$ non-compact hyperk{ä}hler manifolds which are quaternion analog of toric varieties. The $n$ dimensional residue circle action on it admitting a hyperk{ä}hler moment map. We use the complex part of this moment map to construct a holomorphic Lagrangian fibration with generic fiber diffeomorphic to $(\mathbb{C}^*)^n$, and study the singular fibers.
This is a short companion paper to arXiv:0810.3470. 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 fibers if the central fiber is a toric Fano variety admitting a small resolution.
In this paper we consider the oscillatory integrals on Lefschetz thimbles in the Landau-Ginzburg model as the mirror of a toric Fano manifold. We show these thimbles represent the same relative homology classes as the characteristic cycles of the corresponding constructible sheaves under the equivalence of \cite{GPS18-2}. Then the oscillatory integrals on such thimbles are the same as the integrals on the characteristic cycles and relate to genus $0$ Gromov-Witten descendant potential for $X$, and this leads to a proof of Gamma II conjecture for toric Fano manifolds.
The mirror of a projective toric manifold $X_Σ$ is given by a Landau-Ginzburg model $(Y,W)$. We introduce a class of Lagrangian submanifolds in $(Y,W)$ and show that, under the SYZ mirror transformation, they can be transformed to torus-invariant hermitian metrics on holomorphic line bundles over $X_Σ$. Through this geometric correspondence, we also identify the mirrors of Hermitian-Einstein metrics, which are given by distinguished Lagrangian sections whose potentials satisfy certain Laplace-type equations.
