Polystable Poisson structures on Kähler–Einstein Fano manifolds admit constant-scalar-curvature symplectic generalized Kähler metrics for small enough Poisson tensors, settling the semiclassical YTD conjecture on P².
A solution to the Yau-Tian-Donaldson Conjecture through Special Fujita Approximations
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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.
fields
math.DG 2years
2026 2representative citing papers
Gromov-Hausdorff convergence of non-collapsed polarized cscK surfaces is realized as Hilbert scheme convergence, with Bergman kernel estimates enabling Zariski openness of cscK metrics in smooth polarized families.
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Poisson K-stability and the semiclassical Yau--Tian--Donaldson correspondence
Polystable Poisson structures on Kähler–Einstein Fano manifolds admit constant-scalar-curvature symplectic generalized Kähler metrics for small enough Poisson tensors, settling the semiclassical YTD conjecture on P².
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On the geometry of non-collapsed polarized cscK surfaces
Gromov-Hausdorff convergence of non-collapsed polarized cscK surfaces is realized as Hilbert scheme convergence, with Bergman kernel estimates enabling Zariski openness of cscK metrics in smooth polarized families.