Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userBirational superrigidity and K-stability of projectively normal Fano manifolds of index one
107
0
0.0
(
0
)
Authors
Fumiaki Suzuki
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
We prove birational superrigidity of every hypersurface of degree N in P^N with singular locus of dimension s, under the assumption that N is at least 2s+8 and it has only quadratic singularities of rank at least N-s. Combined with the results of I. A. Cheltsov and T. de Fernex, this completes the list of birationally superrigid singular hypersurfaces with only ordinary double points except in dimension 4 and 6. Further we impose an additional condition on the base locus of a birational map to a Mori fiber space. Then we prove conditional birational superrigidity of certain smooth Fano hypersurfaces of index larger or equal to 2, and birational superrigidity of smooth Fano complete intersections of index 1 in weak form.
For every integer $a geq 2$, we relate the K-stability of hypersurfaces in the weighted projective space $mathbb{P}(1,1,a,a)$ of degree $2a$ with the GIT stability of binary forms of degree $2a$. Moreover, we prove that such a hypersurface is K-polystable and not K-stable if it is quasi-smooth.
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 show that the K-moduli spaces of log Fano pairs $(mathbb{P}^3, cS)$ where $S$ is a quartic surface interpolate between the GIT moduli space of quartic surfaces and the Baily-Borel compactification of moduli of quartic K3 surfaces as $c$ varies in the interval $(0,1)$. We completely describe the wall crossings of these K-moduli spaces. As the main application, we verify Laza-OGradys prediction on the Hassett-Keel-Looijenga program for quartic K3 surfaces. We also obtain the K-moduli compactification of quartic double solids, and classify all Gorenstein canonical Fano degenerations of $mathbb{P}^3$.
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.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
