A solution to the Yau-Tian-Donaldson Conjecture through Special Fujita Approximations

Abstract

We show that any big line bundle on a smooth projective variety admits a special Fujita approximation: the volume and the first Riemann-Roch coefficient are both approximated by those of ample $\mathbb{Q}$-line bundles on higher models. Exploiting previous works by Boucksom, Jonsson and Li, we solve the Boucksom-Jonsson Regularization Conjecture on the Non-Archimedean entropy functional. As a main consequence, we obtain a solution to the (uniform version of the) Yau-Tian-Donaldson Conjecture: a polarized smooth projective variety $(X,L)$ admits a cscK metric if and only if it is ${\mathrm{Aut}}^{\circ }(X,L)$-uniformly $K$-stable. This extends the known Yau-Tian-Donaldson correspondence for smooth Fano varieties.

25 pages, no figures. Improved version: proofs of the main results simplified, corrected typos and new Appendix by S. Boucksom added (non-Archimedean point of view and proofs of the main results)