K-theoretic enumerative invariants and q-difference equations

我们得到了两个有趣的结果。第一个与q-差分方程理论中的汇合问题有关。也就是说，我们证明了具有非负第一陈类的光滑射影簇的小 K 理论 J 函数具有 q->1 的极限，并且该极限与小上同调 J 函数一致。这里，非负第一陈氏类是指切丛的第一陈氏类与不可约曲线的同调类的自然配对是非负数。该极限是在通过适当的 q-1 幂重新调整 K 理论 J 函数中的每个诺维科夫变量后得出的。此外，我们期望我们的论证可以推广，以便可以证明大 J 函数的汇合和量子 q 差分方程的汇合。还预计第一陈省级的积极条件是多余的，但消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复曲面几何的设置。我们能够明确地识别具有特定 q 振荡积分的皮卡德 2 阶 Fano 复曲面流形的小 J 函数。后者是由 Givental 引入的，目的是提供量子 q 差分方程的解，它可以被视为在量子 K 理论中构造或公式化镜像对称性的第一步。

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.