Delta invariants of weighted hypersurfaces

We give a lower bound for the delta invariant of the fundamental divisor of a quasi-smooth weighted hypersurface. As a consequence, we prove K-stability of a large class of quasi-smooth Fano hypersurfaces of index 1 and of all smooth Fano weighted hypersurfaces of index 1 and 2. The proofs are based on the Abban–Zhuang method and on the study of linear systems on flags of weighted hypersurfaces.

💡 Research Summary

The paper studies delta invariants of quasi‑smooth weighted hypersurfaces and uses these invariants to establish K‑stability for broad families of weighted Fano hypersurfaces. After recalling the Yau–Tian–Donaldson conjecture and recent breakthroughs linking the existence of Kähler–Einstein metrics to K‑stability via the delta invariant, the authors focus on hypersurfaces in a weighted projective space (\mathbb{P}(a_0,\dots ,a_{n+1})).

The main theorem (Theorem 1.1, restated as Theorem 5.2) asserts that if a weight (a_r>1) divides the degree (d) of the hypersurface, then the delta invariant of the fundamental divisor satisfies
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