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Register a new userLog K-stability of GIT-stable divisors on Fano manifolds
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Authors
Chuyu Zhou
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For a given K-polystable Fano manifold X and a natural number l, we show that there exists a rational number 0 < c < 1 depending only on the dimension of X, such that $Din |-lK_X|$ is GIT-(semi/poly)stable under the action of Aut(X) if and only if the pair $(X, frac{epsilon}{l} D)$ is K-(semi/poly)stable for some rational $0 < {epsilon} < c$.
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We give a lower bound of the $delta$-invariants of ample line bundles in terms of Seshadri constants. As applications, we prove the uniform K-stability of infinitely many families of Fano hypersurfaces of arbitrarily large index, as well as the uniform K-stability of most families of smooth Fano threefolds of Picard number one.
We prove that every projectively normal Fano manifold in $mathbb{P}^{n+r}$ of index $1$, codimension $r$ and dimension $ngeq 10r$ is birationally superrigid and K-stable. This result was previously proved by Zhuang under the complete intersection assumption.
We present some applications of the deformation theory of toric Fano varieties to K-(semi/poly)stability of Fano varieties. First, we present two examples of K-polystable toric Fano 3-fold with obstructed deformations. In one case, the K-moduli spaces and stacks are reducible near the closed point associated to the toric Fano 3-fold, while in the other they are non-reduced near the closed point associated to the toric Fano 3-fold. Second, we study K-stability of the general members of two deformation families of smooth Fano 3-folds by building degenerations to K-polystable toric Fano 3-folds.
We develop a general approach to prove K-stability of Fano varieties. The new theory is used to (a) prove the existence of Kahler-Einstein metrics on all smooth Fano hypersurfaces of Fano index two, (b) to compute the stability thresholds for hypersurfaces at generalized Eckardt points and for cubic surfaces at all points, and (c) to provide a new algebraic proof of Tians criterion for K-stability, amongst other applications.
For (X,L) a polarized toric variety and G a torus of automorphisms of (X,L), denote by Y the GIT quotient X/G. We define a family of fully faithful functors from the category of torus equivariant reflexive sheaves on Y to the category of torus equivariant reflexive sheaves on X. We show, under a genericity assumption on G, that slope stability is preserved by these functors if and only if the pair ((X, L), G) satisfies a combinatorial criterion. As an application, when (X,L) is a polarized toric orbifold of dimension n, we relate stable equivariant reflexive sheaves on (X, L) to stable equivariant reflexive sheaves on certain (n-1)-dimensional weighted projective spaces.
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