Cheol-Hyun Cho

Holomorphic discs, spin structures, and Floer cohomology of the Clifford torus

We compute the Bott-Morse Floer cohomology of the Clifford torus in ℙ n with all possible spin structures. Each spin structure is known to determine an orientation of the moduli space of holomorphic discs, and we analyze the change of orientation according to the change of spin structure of the Clifford torus. Also, we classify all holomorphic discs with boundary lying on the Clifford torus by establishing a Maslov index formula for such discs. As a result, we show that in odd dimensions, there exist two spin structures which give nonvanishing Floer cohomology of the Clifford torus, and in even dimensions, there is only one such spin structure. When the Floer cohomology is nonvanishing, it is isomorphic to the singular cohomology of the torus (with a Novikov ring as its coefficients). As a corollary, we prove that any Hamiltonian deformation of the Clifford torus intersects with it at least at 2 n distinct intersection points when the intersection is transversal. We also compute the Floer cohomology of the Clifford torus with flat line bundles on it and verify the prediction made by Hori using a mirror symmetry calculation.

Non-displaceable Lagrangian submanifolds and Floer cohomology with non-unitary line bundle

Abstract We show that in many examples the non-displaceability of Lagrangian submanifolds by Hamiltonian isotopy can be proved via Lagrangian Floer cohomology with non-unitary line bundle. The examples include all monotone Lagrangian torus fibers in a toric Fano manifold (which was also proven by Entov and Polterovich via the theory of symplectic quasi-states) and some non-monotone Lagrangian torus fibers. We also extend the results by Oh and the author about the computations of Floer cohomology of Lagrangian torus fibers to the case of all toric Fano manifolds, removing the convexity assumption in the previous work.

COUNTING REAL J-HOLOMORPHIC DISCS AND SPHERES IN DIMENSION FOUR AND SIX

We provide another proof that the signed count of the real J-holomorphic spheres (or J- holomorphic discs) passing through a generic real configuration of k points is independent of the choice of the real configuration and the choice of J, if the dimension of the Lagrangian submanifold L (fixed point set of involution) is two or three, and also if we assume L is orient able and relatively spin. We also assume that M is strongly semi-positive. This theorem was first proved by Welschinger in a more general setting, and we provide more natural approach using the signed degree of an evaluation map.

Gross fibrations, SYZ mirror symmetry, and open Gromov–Witten invariants for toric Calabi–Yau orbifolds

Given a toric Calabi-Yau orbifold X whose underlying toric variety is semi- projective, we construct and study a non-toric Lagrangian torus bration on X , which we call the Gross bration. We apply the Strominger-Yau-Zaslow recipe to the Gross bration of (a toric modication of) X to construct its instanton-corrected mirror, where the instanton corrections come from genus 0 open orbifold Gromov-Witten invariants, which are virtual counts of holomorphic orbi-disks inX bounded by bers of the Gross bration. We explicitly evaluate all these invariants by rst proving an open/closed equality and then employing the toric mirror theorem for suitable toric (parital) compactications of X. Our calculations are then applied to (1) prove a conjecture of Gross-Siebert on a relation between genus 0 open orbifold Gromov- Witten invariants and mirror maps ofX { this is called the open mirror theorem, which leads to an enumerative meaning of mirror maps, and (2) demonstrate how open (orbifold) Gromov-Witten invariants for toric Calabi-Yau orb- ifolds change under toric crepant resolutions { this is an open analogue of Ruans crepant resolution conjecture.

Lagrangian Floer Superpotentials and Crepant Resolutions for Toric Orbifolds

We investigate the relationship between the Lagrangian Floer superpotentials for a toric orbifold and its toric crepant resolutions. More specifically, we study an open string version of the crepant resolution conjecture (CRC) which states that the Lagrangian Floer superpotential of a Gorenstein toric orbifold

ON ORBIFOLD EMBEDDINGS

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Orbifold Holomorphic Discs and Crepant Resolutions

X and that of its toric crepant resolution Y coincide after analytic continuation of quantum parameters and a change of variables. Relating this conjecture with the closed CRC, we find that the change of variable formula which appears in closed CRC can be explained by relations between open (orbifold) Gromov-Witten invariants. We also discover a geometric explanation (in terms of virtual counting of stable orbi-discs) for the specialization of quantum parameters to roots of unity which appears in Ruan’s original CRC (Gromov-Witten theory of spin curves and orbifolds, contemp math, Amer. Math. Soc., Providence, RI, pp 117–126, 2006). We prove the open CRC for the weighted projective spaces