  We prove that a $n$-dimensional, $4 \leq n \leq 6$, compact gradient
shrinking Ricci soliton satisfying a $L^{n/2}$-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.
