K-theoretic enumerative invariants and q-difference equations

我们得到了两个有趣的结果。第一个问题与q-差分方程理论中的汇流问题有关。即证明了具有非负第一Chern类的光滑射影簇的小K-理论J-函数在q->1时有一个极限，并且这个极限与小上同调J-函数相一致。这里，非负的第一陈类是指切丛的第一陈类与不可约曲线的同调类的自然配对是非负数。通过对K理论J函数中的每个Novikov变量进行适当的q-1的幂次变换，得到了该极限.此外，我们希望我们的结论可以推广，从而证明大J函数的收敛性和量子q差分方程的收敛性.也可以预期的是，第一个陈省身类的正性条件是多余的，但删除这个条件似乎是一个具有挑战性的问题。我们的第二个结果是在复曲面几何的设置。我们能够明确地确定小J函数的Fano复曲面流形的皮卡德秩2与一定的q-振荡积分。后者是由Givental为了提供量子q-差分方程的解而引入的，它可以被看作是在量子K-理论中构建或形成镜像对称的第一步。

We got two interesting results. The first one is related to the problem of confluence in the theory of q-difference equations. Namely, we proved that the small K-theoretic J-functions of a smooth projectve variety with non-negative first Chern class has a limit as q->1 and this limit coincides with the small cohomological J-function. Here, non-negative first Chern class means that the natural pairing of the 1st Chern class of the tangent bundle and the homology class of an irreducible curve is a non-negative number. The limit is taken after rescaling each Novikov variable in the K-theoretic J-function by an appropriate power of q-1.Moreover, we expect that our argument can be generalized so one can prove the confluence of the big J-function and the confluence of the quantum q-difference equations. It is also expected that the positivity condition of the 1st Chern class is redundant but removing this condition seems to be a challenging problem. Our second result is in the settings of toric geometry. We were able to identify explicitly the small J-function of a Fano toric manifold of Picard rank 2 with a certain q-oscillatory integral. The latter was introduced by Givental in order to provide a solution of the quantum q-difference equations and it can be viewed as a first step towards constructing or fomrulating mirror symmetry in quantum K-theory.