The moduli space of Fano manifolds with Kähler–Ricci solitons

Inoue, Eiji

doi:10.1016/j.aim.2019.106841

Cited by 22 publications

(26 citation statements)

References 78 publications

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“…This is the case when

X

is a smooth Fano manifold,

α = 2 π c_{1} false(X false)

corresponds to the anti‐canonical polarization,

T \subset {Aut}_{red} false(X false)

is a maximal torus with momentum image

normalP

, and

v false(p false) = w false(p false) = e^{⟨ ξ, p ⟩}

for some

ξ \in t

. It was shown recently in that a solutions of with

{normalw}_{ext} false(p false) = 2 false(⟨ ξ, p ⟩ + c false)

(for some real constant

c

) corresponds to a Kähler metric

ω \in α

which is a gradient Kähler–Ricci soliton with respect to

ξ

, that is, satisfies

Ric false(ω false) - ω = - \frac{1}{2} {scriptL}_{J ξ} ω,

where

Ric (ω)

is the Ricci form of

ω

. Thus, the theory of gradient Kähler–Ricci solitons (see, for example, ) fits in to our setting too.…”

Section: Examplesmentioning

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.

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“…This is the case when

X

is a smooth Fano manifold,

α = 2 π c_{1} false(X false)

corresponds to the anti‐canonical polarization,

T \subset {Aut}_{red} false(X false)

is a maximal torus with momentum image

normalP

, and

v false(p false) = w false(p false) = e^{⟨ ξ, p ⟩}

for some

ξ \in t

. It was shown recently in that a solutions of with

{normalw}_{ext} false(p false) = 2 false(⟨ ξ, p ⟩ + c false)

(for some real constant

c

) corresponds to a Kähler metric

ω \in α

which is a gradient Kähler–Ricci soliton with respect to

ξ

, that is, satisfies

Ric false(ω false) - ω = - \frac{1}{2} {scriptL}_{J ξ} ω,

where

Ric (ω)

is the Ricci form of

ω

. Thus, the theory of gradient Kähler–Ricci solitons (see, for example, ) fits in to our setting too.…”

Section: Examplesmentioning

confidence: 99%

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

Lahdili

2019

Proc. London Math. Soc.

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“…After establishing this theorem, we can apply it to study the deformation space of Eiji Inoue [10], which is the same as the definition of the kernel space of second order variation of Perelman's entropy [16]. We will prove: Theorem 0.2.…”

Section: Introductionmentioning

confidence: 95%

Lower bound of modified $K$-energy on a Fano manifold with degeneration for Kähler-Ricci solitons

Zhang¹

2021

Preprint

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“…For polarised Kähler manifolds, there is an algebraic notion of stability, K-stability, which is conjecturally equivalent by the Yau-Tian-Donaldson conjecture to the existence of cscK metrics [38,36,13]. It has been proved that it is possible to form an analytic moduli space of cscK manifolds [18,5]; see also [26].…”

Section: Scal(ωmentioning

confidence: 99%

“…We will later need to choose ω B appropriately, to produce cscK and extremal metrics on Y . To do so, we need the following definitions from the moduli theory of cscK manifolds [18,26,5]. In [5] it is shown that there exists a complex space M which is the moduli space of polarised cscK manifolds and that there exists a Kähler metric on M called the Weil-Petersson metric.…”

Section: Extremal Metrics On the Total Spacementioning

confidence: 99%

Optimal Symplectic Connections and Deformations of Holomorphic Submersions

Ortu¹

2022

Preprint

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We give a general construction of extremal Kähler metrics on the total space of certain holomorphic submersions, extending results of Dervan-Sektnan, Fine, and Hong. We consider submersions whose fibres admit a degeneration to Kähler manifolds with constant scalar curvature, in a way that is compatible with the fibration structure. Thus we allow fibres that are K-semistable, rather than K-polystable; this is crucial to moduli theory. On these fibrations we phrase a partial differential equation whose solutions, called optimal symplectic connections, represent a canonical choice of a relatively Kähler metric. We expect this to be the most general construction of a canonical relatively Kähler metric provided all input is smooth. We use the notion of an optimal symplectic connection and the geometry related to it to construct Kähler metrics with constant scalar curvature and extremal metrics on the total space, in adiabatic classes.

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The moduli space of Fano manifolds with Kähler–Ricci solitons

Cited by 22 publications

References 78 publications

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

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

Lower bound of modified $K$-energy on a Fano manifold with degeneration for Kähler-Ricci solitons

Optimal Symplectic Connections and Deformations of Holomorphic Submersions

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