K-theoretic enumerative invariants and q-difference equations

我们得到了两个有趣的结果。第一个是关于q差分方程理论中的合流问题。即证明了非负第一Chern类光滑投影簇的小k -理论j -函数有一个极限q->1，该极限与小上同调j -函数重合。这里的非负第一陈氏类是指切束的第一陈氏类与不可约曲线的同调类的自然配对为非负数。该极限是在k理论j函数中的每个Novikov变量用q-1的适当幂重新标度后得到的。此外，我们期望我们的论点可以推广，从而可以证明大j函数和量子q差分方程的合流性。预计陈氏1级的正性条件是多余的，但消除这一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环面几何的设置。我们能够明确地识别具有一定q振荡积分的Picard秩2的Fano环流形的小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，它可以被视为构建或表述量子k理论中的镜像对称的第一步。

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.