  We study polynomial growth holomorphic functions and forms on complete
gradient shrinking Ricci solitons. By relating to the spectral data of the
$f$-Laplacian, we show that the dimension of the space of polynomial growth
holomorphic functions or holomorphic $(p,0)$-forms are finite. In particular, a
sharp dimension estimate for the space of linear growth holomorphic functions
was obtained. Under some additional curvature assumption, we prove a sharp
estimate for the frequency of polynomial growth holomorphic functions, which
was used to obtain dimension upper bound as a power function of the polynomial
order.
