18,947 research outputs found
Enumerative Geometry of Del Pezzo Surfaces
We prove an equivalence between the superpotential defined via tropical
geometry and Lagrangian Floer theory for special Lagrangian torus fibres in del
Pezzo surfaces constructed by Collins-Jacob-Lin. We also include some explicit
calculations for the projective plane, which confirm some folklore conjecture
in this case.Comment: 42 pages, 1 figrure. Comments are welcom
Bulk-deformed potentials for toric Fano surfaces, wall-crossing and period
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.Comment: 44 pages, 9 figures, comments are welcom
Special Lagrangian submanifolds of log Calabi-Yau manifolds
We study the existence of special Lagrangian submanifolds of log Calabi-Yau manifolds equipped with the complete Ricci-flat Kähler metric constructed by Tian-Yau. We prove that if X is a Tian-Yau manifold, and if the compact Calabi-Yau manifold at infinty admits a single special Lagrangian, then X admits infinitely many disjoint special Lagrangians. In complex dimension 2, we prove that if Y is a del Pezzo surface, or a rational elliptic surface, and D∈|−KY| is a smooth divisor with D2=d, then X=Y∖D admits a special Lagrangian torus fibration, as conjectured by Strominger-Yau-Zaslow and Auroux. In fact, we show that X admits twin special Lagrangian fibrations, confirming a prediction of Leung-Yau. In the special case that Y is a rational elliptic surface, or Y=ℙ2 we identify the singular fibers for generic data, thereby confirming two conjectures of Auroux. Finally, we prove that after a hyper-Kähler rotation, X can be compactified to the complement of a Kodaira type Id fiber appearing as a singular fiber in a rational elliptic surface πˇ:Yˇ→ℙ1.https://arxiv.org/abs/1904.08363First author draf
Decomposition of Lagrangian classes on K3 surfaces
We study the decomposability of a Lagrangian homology class on a K3 surface into a sum of classes represented by special Lagrangian submanifolds, and develop criteria for it in terms of lattice theory. As a result, we prove the decomposability on an arbitrary K3 surface with respect to the Kähler classes in dense subsets of the Kähler cone. Using the same technique, we show that the Kähler classes on a K3 surface which admit a special Lagrangian fibration form a dense subset also. This implies that there are infinitely many special Lagrangian 3-tori in any log Calabi-Yau 3-fold.https://arxiv.org/abs/2001.00202Othe
Special Lagrangian submanifolds of log Calabi-Yau manifolds
We study the existence of special Lagrangian submanifolds of log Calabi-Yau
manifolds equipped with the complete Ricci-flat K\"ahler metric constructed by
Tian-Yau. We prove that if is a Tian-Yau manifold, and if the compact
Calabi-Yau manifold at infinty admits a single special Lagrangian, then
admits infinitely many disjoint special Lagrangians. In complex dimension ,
we prove that if is a del Pezzo surface, or a rational elliptic surface,
and is a smooth divisor with , then
admits a special Lagrangian torus fibration, as conjectured by
Strominger-Yau-Zaslow and Auroux. In fact, we show that admits twin special
Lagrangian fibrations, confirming a prediction of Leung-Yau. In the special
case that is a rational elliptic surface, or we identify
the singular fibers for generic data, thereby confirming two conjectures of
Auroux. Finally, we prove that after a hyper-K\"ahler rotation, can be
compactified to the complement of a Kodaira type fiber appearing as a
singular fiber in a rational elliptic surface .Comment: 70 pages. Updates and improvements. To appear in Duke Mathematical
Journa
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