In this note, we study the SYZ mirror construction for a toric Calabi-Yau manifold using instanton corrections coming from Woodward's quasimap Floer theory [40] instead of Fukaya-Oh-Ohta-Ono's Lagrangian Floer theory [18, 19, 20, 21]. We show that the resulting SYZ mirror coincides with the one written down via physical means [33,30,29] (as expected). Date: February 12, 2018. 2010 Mathematics Subject Classification. 53D37, 14J33 (primary) and 53D40, 53D45, 14N35 (secondary). Key words and phrases. Mirror symmetry, SYZ conjecture, quasimap, Lagrangian Floer theory, Calabi-Yau manifold, toric variety. 45. F. Ziltener, A quantum Kirwan map: bubbling and Fredholm theory for symplectic vortices over the plane, Mem.

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.Comment: v4: final version, to appear in Adv. Math.; v3: 38 pages, substantially revised and expanded, the role of tropical geometry clarified in the newly added Subsection 2.

For a toric Calabi-Yau (CY) orbifold X whose underlying toric variety is semiprojective, we construct and study a non-toric Lagrangian torus fibration on X , which we call the Gross fibration. We apply the Strominger-Yau-Zaslow (SYZ) recipe to the Gross fibration of X to construct its mirror with the instanton corrections coming from genus 0 open orbifold Gromov-Witten (GW) invariants, which are virtual counts of holomorphic orbi-disks in X bounded by fibers of the Gross fibration.We explicitly evaluate all these invariants by first proving an open/closed equality and then employing the toric mirror theorem for suitable toric (partial) compactifications of X . Our calculations are then applied to (1) prove a conjecture of Gross-Siebert on a relation between genus 0 open orbifold GW invariants and mirror maps of X -this is called the open mirror theorem, which leads to an enumerative meaning of mirror maps, and (2) demonstrate how open (orbifold) GW invariants for toric CY orbifolds change under toric crepant resolutions -an open analogue of Ruan's crepant resolution conjecture. arXiv:1306.0437v4 [math.SG] 1 Apr 20151 As explained in [18, Section 5.2], to prove the original stronger form of the conjecture, we need integral cycles whose periods have specific logarithmic terms. Such cycles have been constructed by Doran and Kerr in [33, Section 5.3 and Theorem 5.1] when X is the total space of the canonical line bundle K Y over a toric del Pezzo surface Y . Doran suggested to us that it should not be difficult to extend their construction to general toric CY varieties. Hence the stronger form of the conjecture should follow from Corollary 1.7 and their construction; cf. the discussion in [34, Section 4]. In the recent paper [74], Ruddat and Siebert gave yet another construction of such integral cycles by tropical methods. Though they worked only in the compact CY case, Ruddat pointed out that the method can be generalized to handle the toric CY case as well.

Let X be a compact toric Kähler manifold with −K X nef. Let L ⊂ X be a regular fiber of the moment map of the Hamiltonian torus action on X. Fukaya-Oh-Ohta-Ono [12] defined open Gromov-Witten (GW) invariants of X as virtual counts of holomorphic discs with Lagrangian boundary condition L. We prove a formula which equates such open GW invariants with closed GW invariants of certain X-bundles over P 1 used to construct the Seidel representations [31,29] for X. We apply this formula and degeneration techniques to explicitly calculate all these open GW invariants. This yields a formula for the disc potential of X, an enumerative meaning of mirror maps, and a description of the inverse of the ring isomorphism of Fukaya-Oh-Ohta-Ono [15].

We prove that the inverse of a mirror map for a toric Calabi-Yau manifold of the form K Y , where Y is a compact toric Fano manifold, can be expressed in terms of generating functions of genus 0 open Gromov-Witten invariants defined by Fukaya-Oh-Ohta-Ono [FOOO10]. Such a relation between mirror maps and disk counting invariants was first conjectured by Gross and Siebert [GS11a, Conjecture 0.2 and Remark 5.1] as part of their program, and was later formulated in terms of Fukaya-Oh-Ohta-Ono's invariants in the toric Calabi-Yau case in [CLL12, Conjecture 1.1].v i = (w i , 1) ∈ N