CscK metrics on birational models of projective varieties

We prove that every complex projective variety is birational to a smooth projective manifold admitting a constant scalar curvature K\"ahler (cscK) metric. For any smooth projective variety, a birational cscK model is obtained by resolving the codimension two base locus of a general Lefschetz pencil in a sufficiently positive linear system. The cscK polarization is given explicitly, producing a cscK model also when the initial variety is unstable. In dimension two this proves the folklore conjecture that the blowup of any complex projective surface in enough points admits cscK metrics.