In this paper, we extend our previous work to construct (0, 2) Toda-like
mirrors to A/2-twisted theories on more general spaces, as part of a program of understanding (0,2) mirror symmetry. Specifically, we propose (0, 2)
mirrors to GLSMs on toric del Pezzo surfaces and Hirzebruch surfaces with
deformations of the tangent bundle. We check the results by comparing correlation functions, global symmetries, as well as geometric blowdowns with the
corresponding (0, 2) Toda-like mirrors. We also briefly discuss Grassmannian
Let the vector bundle E be a deformation of the tangent bundle over the Grassmannian G(k, n). We compute the ring structure of sheaf cohomology valued in exterior
powers of E, also known as the polymology. This is the first part of a project studying
the quantum sheaf cohomology of Grassmannians with deformations of the tangent bundle,
a generalization of ordinary quantum cohomology rings of Grassmannians. A companion
physics paper  describes physical aspects of the theory, including a conjecture for the
quantum sheaf cohomology ring, and numerous examples.
Taiwang DengYau Mathematical Sciences Center, Tsinghua University, Haidian District, Beijing, 100084, ChinaBin XuYau Mathematical Sciences Center and Department of Mathematics, Tsinghua University, Beijing, China
In this article we study a conjecture of Geiss-Leclerc-Schröer, which is an analogue of a classical conjecture of Lusztig in the Weyl group case. It concerns the relation between canonical basis and semi-canonical basis through the characteristic cycles. We formulate an approach to this conjecture and prove it for type quiver. In the general type A case, we reduce the conjecture to show that certain nearby cycles have vanishing Euler characteristic.
In this work, we show that for a certain class of threefolds in positive characteristics, rational-chain-connectivity is equivalent to supersingularity. The same result is known for K3 surfaces with elliptic fibrations. And there are examples of threefolds that are both supersingular and rationally chain connected.
Tautological systems developed in [8, 9] are Picard-Fuchs type systems to study period integrals of complete intersections in Fano varieties. We generalize tautological systems to zero loci of global sections of vector bundles. In particular, we obtain similar criterion as in [8, 9] about holonomicity and regularity of the systems. We also prove solution rank formulas and geometric realizations of solutions following the work on hypersurfaces in homogeneous varieties .
Moonshine, the Monster, and Related Topics: Joint Research Conference on Moonshine, the Monster, and Related Topics, June 18-23, 1994, Mount Holyoke College, South Hadley, Massachusetts, 193, 237, 1996
1. INDEX THEORY, ELLIPTIC CURVES AND LOOP GROUPS One can look at elliptic genus from several different points of view; from index theory, from representation theory of Kac-Moody affine Lie algebras or from the theory of elliptic functions and modular forms. Each of them shows us some quite different interesting features of ellitic genus. On the other hand we can also combine the forces of these three different mathematical fields to derive many interesting results in topology such as rigidity, divisibility and vanishing of topological invariants..
The mirror theorem is generalized to any smooth projective variety X. That is, a fundamental relation between the Gromov-Witten invariants of X and Gromov-Witten invariants of complete intersections Y in X is established.
Tautological systems are Picard-Fuchs type systems arising from varieties with large symmetry. In this survey, we discuss recent
progress on the study of tautological systems. This includes tautological systems for vector bundles, a new construction of Jacobian rings for homogenous vector bundles, and relations between period integrals and zeta functions.