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Given an elliptic curve , flat Ek-bundles over  are in one-to-one correspondence with smooth del Pezzo surfaces of degree 9 − k containing  as an anti-canonical curve. This correspondence was generalized to Lie groups of any type. In this article, we show that there is a similar correspondence between del Pezzo surfaces of degree 0 with an Ad-singularity containing  as an anti-canonical curve and Kac–Moody  Ek-bundles over  with k = 8−d. In the degenerate case where surfaces are rational elliptic surfaces, the corresponding  Ek-bundles over  can be reduced to Ek-bundles.

No abstract uploaded!

We obtain a growth estimate for the number of lattice points inside any Q-Gorenstein cone. Our proof uses the result of Futaki-Ono-Wang on Sasaki-Einstein metric for the toric Sasakian manifold associated to the cone, a Yau’s inequality, and the Kawasaki-Riemann-Roch formula for orbifolds.

We give an explicit formula for (T-equivariant) 3-pointed genus zero Gromov-Witten invariants for G/B. We derive it by finding an explicit formula for the Pontryagin product on the equivariant homology of the based loop group ΩK.

No abstract uploaded!

We show that various genus zero Gromov–Witten invariants for flag varieties representing different homology classes are indeed the same. In particular, many of them are classical intersection numbers of Schubert cycles.

We prove that the mirror map is the Strominger–Yau–Zaslow map for every toric Calabi–Yau surface. As a consequence, one obtains an enumerative meaning of the mirror map. This involves computing genus-0 open Gromov–Witten invariants, which is done by relating them with closed Gromov–Witten invariants via compactification and using an earlier computation by Bryan–Leung.

No abstract uploaded!

We study the geometry of the Grassmannians of symplectic subspaces in a symplectic vector space. We construct symplectic twistor spaces by the symplectic quotient construction and use them to describe the symplectic geometry of the symplectic Grassmannians.

We compute certain open Gromov–Witten invariants for toric Calabi–Yau threefolds. The proof relies on a relation for ordinary Gromov–Witten invariants for threefolds under certain birational transformation, and a recent result of Kwokwai Chan.