The Tibetan Image of Confucius

Revue d’Etudes Tibétaines Number 12, March 200

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 $X$ 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 $D\in |-K_{Y}|$ is a smooth divisor with $D^2=d$, then $X= Y\backslash D$ 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 $Y= \mathbb{P}^2$ we identify the singular fibers for generic data, thereby confirming two conjectures of Auroux. Finally, we prove that after a hyper-K\"ahler rotation, $X$ can be compactified to the complement of a Kodaira type $I_{d}$ fiber appearing as a singular fiber in a rational elliptic surface $\check{\pi}: \check{Y}\rightarrow \mathbb{P}^1$.Comment: 70 pages. Updates and improvements. To appear in Duke Mathematical Journa