Quasipositivity as an obstruction to sliceness

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For an oriented link L ⊂ S³ = D⁴, let χₛ(L) be the greatest Euler characteristic χ(F) of an oriented 2-manifold F (without closed components) smoothly embedded in D⁴ with boundary L. A knot K is slice if χₛ(K)=1. Realize D⁴ in ² as {(z,w):|z|²+|w|²≤1}. It has been conjectured that, if V is a nonsingular complex plane curve transverse to S³, then χₛ(V∩ S³)=χ(V∩ D⁴). Kronheimer and Mrowka have proved this conjecture in the case that V∩ D⁴ is the Milnor fiber of a singularity. I explain how this seemingly special case implies both the general case and the ``slice-Bennequin inequality'' for braids. As applications, I show that various knots are not slice (e.g., pretzel knots like (-3,5,7); all knots obtained from a positive trefoil O{2,3} by iterated untwisted positive doubling). As a sidelight, I give an optimal counterexample to the ``topologically locally-flat Thom conjecture''.