The GIT stability of polarized varieties via discrepancy

Odaka, Yuji

doi:10.4007/annals.2013.177.2.6

Cited by 91 publications

(112 citation statements)

References 26 publications

(40 reference statements)

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“…In what follows we shall tacitly identify

\overset{T}{true}

with

double-struckT

and

\overset{G}{true}

with

double-struckG

. Definition A smooth

double-struckT

‐compatible Kähler test configuration

(X, A, ρ, T)

for

(X, α, T)

is called trivial if it is given by

({scriptX}_{0} = X \times {double-struckP}^{1}, {scriptA}_{0} = π_{X}^{*} α + π_{P^{1}}^{*} false[ω_{FS} false], T)

and

C^{★}

‐action

ρ_{0} false(τ false) false(x, z false) = false(x, τ z false)

for any

τ \in {double-struckC}^{★}

and

false(x, z false) \in X \times {double-struckP}^{1}

; product if it is given by

({scriptX}_{prod}, {scriptA}_{prod}, ρ_{prod}, T)

where

X_{prod}

is the compactification (in the sense of , see also [, Example 2.8; , p. 12–13]) of

X \times C

with

C^{★}

‐action

ρ_{prod} false(τ false) false(x <...>

…”

Section: The (Vw)‐futaki Invariant Of a Smooth Test Configurationmentioning

confidence: 99%

Kähler metrics with constant weighted scalar curvature and weighted K‐stability

Lahdili

2019

Proc. London Math. Soc.

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We introduce a notion of a Kähler metric with constant weighted scalar curvature on a compact Kähler manifold X, depending on a fixed real torus double-struckT in the reduced group of automorphisms of X, and two smooth (weight) functions v>0 and normalw, defined on the momentum image (with respect to a given Kähler class α on X) of X in the dual Lie algebra of double-struckT. We show that a number of known results obstructing the existence of constant scalar curvature Kähler (cscK) metrics can be extended to the weighted setting. In particular, we introduce a functional Mnormalv,normalw on the space of double-struckT‐invariant Kähler metrics in α, extending the Mabuchi energy in the cscK case, and show that if α is Hodge, then constant weighted scalar curvature metrics in α are minima of Mnormalv,normalw. We define a (v,w)‐weighted Futaki invariant of a double-struckT‐compatible smooth Kähler test configuration associated to (X,α,T), and show that the boundedness from below of the (v,w)‐weighted Mabuchi functional Mnormalv,normalw implies a suitable notion of a (v,w)‐weighted K‐semistability.

show abstract

“…In what follows we shall tacitly identify

\overset{T}{true}

with

double-struckT

and

\overset{G}{true}

with

double-struckG

. Definition A smooth

double-struckT

‐compatible Kähler test configuration

(X, A, ρ, T)

for

(X, α, T)

is called trivial if it is given by

({scriptX}_{0} = X \times {double-struckP}^{1}, {scriptA}_{0} = π_{X}^{*} α + π_{P^{1}}^{*} false[ω_{FS} false], T)

and

C^{★}

‐action

ρ_{0} false(τ false) false(x, z false) = false(x, τ z false)

for any

τ \in {double-struckC}^{★}

and

false(x, z false) \in X \times {double-struckP}^{1}

; product if it is given by

({scriptX}_{prod}, {scriptA}_{prod}, ρ_{prod}, T)

where

X_{prod}

is the compactification (in the sense of , see also [, Example 2.8; , p. 12–13]) of

X \times C

with

C^{★}

‐action

ρ_{prod} false(τ false) false(x <...>

…”

Section: The (Vw)‐futaki Invariant Of a Smooth Test Configurationmentioning

confidence: 99%

Kähler metrics with constant weighted scalar curvature and weighted K‐stability

Lahdili

2019

Proc. London Math. Soc.

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show abstract

“…See [26] for a discussion and [49] for a recent example. Theorem 5.2.1 is an instance of this principle, and it works especially nicely because hyperplanes B i are linear subvarieties.…”

Section: Semi-log Canonical Singularities and Gitmentioning

confidence: 99%

Moduli of Weighted Hyperplane Arrangements

Alexeev

2015

Advanced Courses in Mathematics - CRM Barcelona

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“…Roughly speaking, assuming the non-semi-log-canonicity of X, Odaka [Odaka11] proved that we can construct "destabilizing test configuration" by using the semi-logcanonical model of X. We refer to [Odaka11] for more details.…”

Section: Corollary 11 (Inversion Of Adjunction) Let (X D + δ) Be Amentioning

confidence: 99%

“…In [Odaka11], the first named author proved K-semi-stability implies semi-log canonicity, assuming the existence of semi-log-canonical models. Since (1.2) verifies this assumption, the following theorem now becomes unconditional.…”

Section: Corollary 11 (Inversion Of Adjunction) Let (X D + δ) Be Amentioning

confidence: 99%