Integral pinched shrinking Ricci solitons

We prove that a $n$-dimensional, $4\le n\le 6$, compact gradient shrinking Ricci soliton satisfying a $L^{n\mathrm{/}2}$-pinching condition is isometric to a quotient of the round ${\mathbb{S}}^n$. The proof relies mainly on sharp algebraic curvature estimates, the Yamabe-Sobolev inequality and an improved rigidity result for integral pinched Einstein metrics.