Kähler Gradient Ricci Solitons with Large Symmetry

Let $(M,g,J,f)$ be an irreducible non-trivial Kähler gradient Ricci soliton of real dimension $2n$. We show that its group of isometries is of dimension at most $n^2$ and the case of equality is characterized. As a consequence, our framework shows the uniqueness of $U(n)$-invariant Kähler gradient Ricci solitons constructed earlier. There are corollaries regarding the groups of automorphisms or affine transformations and a general version for almost Hermitian GRS. The approach is based on a connection to the geometry of an almost contact metric structure.