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We describe recent progress on QH*(G/P) with special emphasis on its functoriality property, quantum Pieri rule and their applications.

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Given any Kodaira curve C in a complex surface X, we construct a simply-laced affine Lie algebra bundle E over X. When pg(X) = 0, we construct deformations of holomorphic structures on E such that the new bundle is trivial over any ADE curve C inside C and therefore descends to the singular surface obtained by contracting C.

The Cox rings of del Pezzo surfaces are closely related to the Lie groups En. In this paper, we generalize the definition of Cox rings to G-surfaces defined by us earlier, where the Lie groups G = An,Dn or En. We show that the Cox ring of a G-surface S is closely related to an irreducible representation V of G, and is generated by degree one elements. The Proj of the Cox ring of S is a sub-variety of the orbit of the highest weight vector in V , and both are closed sub-varieties of P(V ) defined by quadratic equations. The GIT quotient of the Spec of such a Cox ring by a natural torus action is considered.

Based on the Brieskorn-Slodowy-Grothendieck diagram, we write the holomorphic structures (or filtrations) of the ADE Lie algebra bundles over the corresponding type ADE flag varieties, over the cotangent bundles of these flag varieties, and over the corresponding type ADE singular surfaces. The main tool is the cohomology of line bundles over flag varieties and their cotangent bundles.

In G2 manifolds, 3-dimensional associative submanifolds (instantons) play a role similar to J-holomorphic curves in symplectic geometry. In [21], instantons in G2 manifolds were constructed from regular J-holomorphic curves in coassociative submanifolds. In this exposition paper, after reviewing the background of G2 geometry, we explain the main ingredients in the proofs in [21]. We also construct new examples of instantons.

Given a compact complex surface X with an ADE singularity and pg=0, we construct ADE bundles over X and its minimal resolution Y. Furthermore, we describe their minuscule representation bundles in terms of configurations of (reducible) (−1)-curves in Y.

For two nearby disjoint coassociative submanifolds C and C′ in a G2-manifold, we construct thin instantons with boundaries lying on C and C′ from regular J-holomorphic curves in C. We explain their relationship with the Seiberg-Witten invariants for C.

No abstract uploaded!

No abstract uploaded!