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We prove that if positive integer p-surgery along a knot K \subset S^3produces an L-space and it bounds a sharp 4-manifold, then the knot genus obeysthe bound 2g(K) -1 \leq p - \sqrt{3p+1}. Moreover, there exists an infinitefamily of pairs (K_n,p_n) attaining this bound, where K_n denotes an n-folditerated cable of the unknot and p_n \to \infty. In particular, the statedbound applies when the knot surgery produces a lens space or a connected sumthereof. Combined with work of Gordon-Luecke, Hoffman, and Matignon-Sayari, itfollows that if surgery along a knot produces a connected sum of lens spaces,then the knot is either a torus knot or a cable thereof, confirming the cablingconjecture in this case.

L-space surgeries, genus bounds, and the cabling conjecture