Holomorphic Line Bundles on Projective Toric Manifolds from Lagrangian Sections of Their Mirrors by Syz Transformations
نویسنده
چکیده
The mirror of a projective toric manifold XΣ is given by a LandauGinzburg 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 XΣ. 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.
منابع مشابه
Lagrangian Sections and Holomorphic U(1)-connections
It was conjectured in [SYZ] that Calabi-Yau spaces can be often fibered by special Lagrangian tori and their mirrors can be constructed by dualizing these tori. It was further suggested by Vafa in [V] that the holomorphic vector bundles on a Calabi-Yau n-fold M correspond to the Lagrangian submanifolds in the mirror M̌ and the stable vector bundles correspond to the special Lagrangian submanifol...
متن کاملRiemannian Geometry over different Normed Division Algebras
We develop a unifed theory to study geometry of manifolds with different holonomy groups. They are classified by (1) real, complex, quaternion or octonion number they are defined over and (2) being special or not. Specialty is an orientation with respect to the corresponding normed algebra A. For example, special Riemannian A-manifolds are oriented Riemannian, Calabi-Yau, Hyperkähler and G2-man...
متن کاملHyperkähler SYZ conjecture and semipositive line bundles
LetM be a compact, holomorphic symplectic Kähler manifold, and L a non-trivial line bundle admitting a metric of semi-positive curvature. We show that some power of L is effective. This result is related to the hyperkähler SYZ conjecture, which states that such a manifold admits a holomorphic Lagrangian fibration, if L is not big.
متن کاملOpen Gromov-Witten invariants and SYZ under local conifold transitions
For a local non-toric Calabi–Yau manifold which arises as a smoothing of a toric Gorenstein singularity, this paper derives the open Gromov–Witten invariants of a generic fiber of the special Lagrangian fibration constructed by Gross and thereby constructs its Strominger-YauZaslow (SYZ) mirror. Moreover, it proves that the SYZ mirrors and disk potentials vary smoothly under conifold transitions...
متن کاملMirror Symmetry for Toric Fano Manifolds via Syz Transformations
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.
متن کامل