Kähler--Ricci Shrinkers with Constant Scalar Curvature Are Rigid

Let $(M^{2m},g,f,J)$ be a complete gradient Kähler–Ricci shrinker satisfying $$\mathrm{Ric}+{\mathrm{\nabla }}^{2}f=\frac{1}{2}g\mathrm{.}$$ We prove that if the scalar curvature is constant, then the soliton is rigid. More precisely, there exists an integer $k\in \{0,\dots ,m\}$ such that $$R\equiv k$$ and $$(M^{2m},g,J)\cong (N^{2k}\times {\mathbb{C}}^{m-k},g_N+g_{\mathrm{E}\mathrm{u}\mathrm{c}},J_N\oplus J_0),$$ where $(N^{2k},g_N,J_N)$ is Kähler–Einstein and $${\mathrm{Ric}}_{g_N}=\frac{1}{2}{g}_N\mathrm{.}$$ As part of the proof, we establish a Riemannian rigidity criterion: for a complete nonsteady gradient Ricci soliton with constant scalar curvature, the condition ${\mathcal{L}}_{\mathrm{\nabla }f}\mathrm{Ric}=0$ implies radial flatness and hence rigidity.