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.
Hisamoto, Stability and coercivity for toric polarizations , arXiv preprint arXiv:1610.07998 (2016)
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abstract
We introduce uniform K-stability and its relationship with the coercivity property of the K-energy functional, for general polarized manifolds. Since the automorphism groups are not necessarily finite, size of the norm measuring uniformity should be reduced with respect to the group action. About this point we explain that it is enough to take the reduced norm for a single sub-torus, actually the center, in the cscK problem. Our main theorem then describes the slope of the reduced J-functional along any torus-equivariant test configuration. In the toric case it is shown that the uniform stability is indeed equivalent to the coercivity of the K-energy. In the Fano manifolds case existence of the KE metric implies the uniform stability.
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Relative uniform K-stability over models implies existence of extremal metrics on polarised smooth complex projective varieties.
A survey summarizing that existence of coupled Kähler-Einstein metrics and weighted solitons on Fano manifolds equates to generalized K-polystability conditions.
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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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Relative uniform $K$-stability over models implies existence of extremal metrics
Relative uniform K-stability over models implies existence of extremal metrics on polarised smooth complex projective varieties.
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On coupled K\"ahler-Einstein metrics and weighted solitons on Fano manifolds
A survey summarizing that existence of coupled Kähler-Einstein metrics and weighted solitons on Fano manifolds equates to generalized K-polystability conditions.