Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces

Abstract

We consider mirror symmetry for (essentially arbitrary) hypersurfaces in (possibly noncompact) toric varieties from the perspective of the Strominger-Yau-Zaslow (SYZ) conjecture. Given a hypersurface H in a toric variety V we construct a Landau-Ginzburg model which is SYZ mirror to the blowup of V × C along H × 0, under a positivity assumption. This construction also yields SYZ mirrors to affine conic bundles, as well as a Landau-Ginzburg model which can be naturally viewed as a mirror to H. The main applications concern affine hypersurfaces of general type, for which our results provide a geometric basis for various mirror symmetry statements that appear in the recent literature. We also obtain analogous results for complete intersections.

Non-Kähler SYZ Mirror Symmetry

“…The Gross-Siebert program [GS11] gave a sophisticated algebraic formulation of the SYZ construction. Moreover, the SYZ program was successfully carried out in various situations for Kähler manifolds [LYZ00, Leu05,Aur07,CO06,CL10,FOOO10,CLL12,AAK]. This paper explores the SYZ approach for non-Kähler Calabi-Yau manifolds, i.e.…”

Koszul duality via suspending Lefschetz fibrations

“…In order to get the Legendrian frontal description of the Milnor fibers M p,q,r , we start with a Lefschetz fibration on T p,q,r constructed by Keating in [38], whose smooth fiber is symplectomorphic to a 3-punctured torus. This fibration induces naturally a Lefschetz fibration on the stabilization M p,q,r , whose smooth fiber is symplectomorphic to a four-dimensional D 4 Milnor fiber. Using an algorithm due to Casals-Murphy [13], this Lefschetz fibration on M p,q,r can then be translated to produce a Legendrian frontal description of M p,q,r which involves both 2-handles and 3-handles.…”

“…Correspondingly, its mirror is Floer theoretically trivial over . The Landau–Ginzburg model has already been studied in [, Example 2.4], where it is equivalently interpreted as a one‐dimensional Landau–Ginzburg model . In fact, let be a properly embedded arc which connects to itself by passing around the origin, then which shows that is mirror to the skyscraper sheaf at the origin of .…”

“…The Landau–Ginzburg model has been studied prominently in the literature (see, for example, ). Its mirror is given by the pair‐of‐pants .…”

“…Together with Seidel-Solomon's inductive argument based on Lefschetz fibration techniques [62], this implies that the total space M p,q,r also admits a quasi-dilation. However, this argument does not apply when K = Z/2, since in this case Koszul duality does not hold for D 4 Milnor fibers (see [24,Theorem 14] for details).…”