Moments and Large Geometric Monodromy for Linearly Constrained Kloosterman Families

Abstract

We determine the connected geometric monodromy of the top-weight sheaves attached to linearly constrained Kloosterman families in every dimension n>= 4 and every characteristic p>max(2,n). The quantitative input is a fourth-moment identity whose auxiliary geometry depends on the moment degree rather than on the ambient family dimension. After an exact decoupling, the generic stratum is controlled by plane sections of the Cayley cubic, and a projective-ray argument gives the required power saving uniformly in each fixed dimension. On the geometric side, a local Fourier-transform calculation and Rojas-Leon's multiplicative-convolution formula produce a nonidentity unipotent boundary element. The fourth moment, reductivity, infinitude, and Larsen's alternative then give G_geom^0(W_n) = SL_r(n) without finite-group classification or numerical exceptional-case certificates. For comparison, we retain the independent Guralnick-Tiep route, including its sixth-moment and exact finite computations, but it is not used in the proof of the main theorem.