Sean Timothy Paul, Gang Tian
The Mabuchi K-energy map is exhibited as a singular metric on the refined CM polarization of any equivariant family $\mathbf{X}\overset{p}{\to} S$. Consequently we show that the generalized Futaki invariant is the leading term in the asymptotics of the reduced K-energy of the generic fiber of the map $p$. Properness of the K-energy implies that the generalized Futaki invariant is strictly negative.
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The interpretation, due to T. Mabuchi, of the classical Futaki invariant of Fano toric manifolds is extended to the case of the Generalized Futaki invariant, introduced by W. Ding and G. Tian, of almost Fano toric varieties. As an application it is shown that the real part of the Generalized Futaki invariant is positive for all degenerations of the Fano manifold V_{38}, obtained by intersection of the Veronese embedding of ${\bf P}^3\times{\bf P}^2 \subset {\bf P}^{11}$ with codimension-two hyperplanes.
We study the algebraic properties of the generalized Futaki invariant of an almost Fano variety and prove that it is in fact a pushforward to a point of an appropriate equivariant Chow cohomology class of the variety. This allows us to use Bott-type formulae for calculating the invariant. We show this use on some examples.
Let X be a Fano manifold. G.Tian proves that if X admits a Kaehler-Einstein metric, then it satisfies two different stability conditions: one involving the Futaki invariant of a special degeneration of X, the other Hilbert-Mumford-stability of X w.r.t. a certain polarization. He conjectures that each of these conditions is also sufficient for the existence of such a metric. If this is true, then in particular the two stability conditions would be equivalent. We show that for Fano hypersurfaces in projective space, where due to the work of Lu and Yotov an explicit formula for the Futaki invaria…
In this paper we compute the Futaki invariant of adiabatic Kaehler classes on resolutions of Kaehler orbifolds with isolated singularities. Combined with previous existence results of extremal metrics by Arezzo-Lena-Mazzieri, this gives a number of new existence and non-existence results for cscK metrics.
K-stability and large complex structure limits
研究 Kähler 测试配置的镜像 Landau-Ginzburg 模型，探讨 Kähler 形式大尺度缩放下 Donaldson-Futaki 不变量镜像公式的行为，并在大复结构极限下导出基于 theta 函数的新公式，且在极化曲面非平凡示例中验证该极限行为。
We study the convergence of the Kähler-Ricci flow on a compact Kähler manifold $(M,J)$ with positive first Chern class $c_1(M;J)$ and vanished Futaki invariant on $πc_1(M;J)$. As the application we establish a criterion for the stability of the Kähler-Ricci flow (with perturbed complex structure) around a Kähler-Einstein metric with positive scalar curvature, under certain local stable condition on the dimension of holomorphic vector fields. In particular this gives a stability theorem for the existence of Kähler-Einstein metrics on a Kähler manifold with possibly nontrivial holomorphic vector…
Futaki invariant on Hopf manifolds
本文证明了Futaki不变量在Hopf流形上为零，该不变量是Kähler几何中度量存在性的障碍，此前已推广至紧致复流形。
We consider the problem of existence of constant scalar curvature Kaehler metrics on complete intersections of sections of vector bundles. In particular we give general formulas relating the Futaki invariant of such a manifold to the weight of sections defining it and to the Futaki invariant of the ambient manifold. As applications we give a new Mukai-Umemura-Tian like example of Fano 5-fold admitting no Kaehler-Einstein metric and a strong evidence of K-stability of complete intersections on Grassmannians.
In this note, given a polarized algebraic manifold $(X,L)$, we define the Donaldson-Futaki invariant for a sequence of test configurations for $(X,L)$ with exponents tending to infinity. This then allows us to define a strong version of K-stability or K-semistability for $(X,L)$. In particular, $(X,L)$ will be shown to be K-semistable in this strong sense if the polarization class $c_1(L)$ admits a constant scalar curvature Kaehler metric.
It is known that a necessary condition for the existence of Kähler-Ricci solitons is the vanishing of the modified Futaki invariant introduced by Tian-Zhu. In a recent work of Berman-Nyström, it was generalized for (singular) Fano varieties and the notion of algebro-geometric stability of the pair $(M,V)$ of a Fano manifold $M$ and a holomorphic vector field $V$ was introduced. In this paper, we propose a method of computing the modified Futaki invariant for Fano complete intersections in projective spaces.
We give a formula of the Donaldson-Futaki invariants for certain type of semi test configurations, which essentially generalizes Ross-Thomas' slope theory. The positivity (resp. non-negativity) of those "a priori special" Donaldson-Futaki invariants implies K-stability (resp. K-semistability). We show its applicability by proving K-(semi)stability of certain polarized varieties with semi-log-canonical singularities, generalizing some results by Ross-Thomas.
We find a new obstruction to the existence of solutions of the Hull-Strominger system, which goes beyond the balanced property of the Calabi-Yau manifold $(X,Ω)$ and the Mumford-Takemoto slope stability of the bundle over it. The basic principle is the construction of a (possibly indefinite) Hermitian-Einstein metric on the holomorphic string algebroid associated to a solution of the system, provided that the connection $\nabla$ on the tangent bundle is Hermitian-Yang Mills. Using this, we define a family of Futaki invariants obstructing the existence of solutions in a given balanced class. Ou…
We study logarithmic K-stability for pairs by extending the formula for Donaldson-Futaki invariants to log setting. We also provide algebro-geometric counterparts of recent results of existence of Kahler-Einstein metrics with cone singularities.
Weighted K-stability and coercivity with applications to extremal Kahler and Sasaki metrics
本文证明紧加权极值Kähler流形具有强制的加权Mabuchi能量，统一了特殊曲率条件下的Kähler度量结果，并据此刻画了极值Kähler度量和Sasaki-Einstein度量的存在性。
In this paper, we discuss the relative $K$-stability and the modified $K$-energy associated to the Calabi's extremal metric on toric manifolds. We give a sufficient condition in the sense of convex polytopes associated to toric manifolds for both the relative $K$-stability and the properness of modified $K$-energy. In particular, our results hold for toric Fano manifolds with vanishing Futaki-invariant. We also verify our results on the toric Fano surfaces.
In this paper, we study the limiting properties of the $K$ energy for smooth hypersurfaces in the projective spaces. Our result generalizes the result of Ding-Tian (W. Ding and G. Tian. Kähler-Einstein metrics and the generalized Futaki invariant. {\em Invent Math}, 110:315-335, 1992.) in the case of hypersurfaces. In particular, we allow the center fiber of a special degeneration (a degeneration by a one-parameter group) to have multiplicity great than 1.
Toric mirrors and test configurations
研究toric极化 manifold与其镜像Landau-Ginzburg模型的K稳定性关系，揭示Z稳定性在镜像对称下的自然出现。
The holomorphic invariants introduced by Futaki as obstruction to the asymptotic Chow semistability are studied by an algebraic-geometric point of view and are shown to be the Mumford weights of suitable line bundles on the Hilbert scheme. These invariants are calculated in two special cases. The first is a projective bundle over a curve of genus at least two, and it is shown that it is asymptotically Chow polystable (with every polarization) if and only the underlying vector bundle is slope polystable. This proves a conjecture of Morrison with the extra assumption that the involved polarizati…
We extend the framework of K-stability (Tian, Donaldson) to more general algebro-geometric setting, such as partial desingularisations of (fixed) singularities, (not necessarily flat) families over higher dimensional base and the classical birational geometry of surfaces. We also observe that "concavity" of the volume function implies decrease of the (generalised) Donaldson-Futaki invariants along the Minimal Model Program, in our generalised settings. Several related results on the connection with the MMP theory, some of which are new even in the original setting of families over curves, are …
In this paper, we prove that on a Fano manifold $M$ which admits a Kähler-Ricci soliton $(\om,X)$, if the initial Kähler metric $\om_{\vphi_0}$ is close to $\om$ in some weak sense, then the weak Kähler-Ricci flow exists globally and converges in Cheeger-Gromov sense. Moreover, if $\vphi_0$ is also $K_X$-invariant, then the weak modified Kähler-Ricci flow converges exponentially to a unique Kähler-Ricci soliton nearby. Especially, if the Futaki invariant vanishes, we may delete the $K_X$-invariant assumption. The methods based on the metric geometry of the space of the Kähler metrics are poten…
As a generalization of Kahler-Einstein metrics for Fano manifolds with nonvanishing Futaki invariant, Mabuchi solitons are critical points of a Calabi-type energy functional. We study their existence on toric Fano varieties and the underlying algebraic stability notion: relative Ding stability. As a toy model for a YTD type correspondence, a new feature is the emergence of a non-uniformly stable case. We show a partial coercivity for the modified Ding functionals in this case, and obtain singular Mabuchi solitons via a variational approach. In the unstable case, we determine the maximal destab…
On a compact complex manifold $(M, J)$ endowed with a holomorphic Poisson tensor $π_J$ and a deRham class $α\in H^2(M, \mathbb R)$, we study the space of generalized Kähler (GK) structures defined by a symplectic form $F\in α$ and whose holomorphic Poisson tensor is $π_J$. We define a notion of generalized Kähler class of such structures, and use the moment map framework of Boulanger and Goto to extend the Calabi program to GK geometry. We obtain generalizations of the Futaki--Mabuchi extremal vector field and Calabi--Lichnerowicz--Matsushima result for the Lie algebra of the group of automorp…
Sharp parameter gaps for quantized Mabuchi energies and cscK metrics
该研究针对极化流形，建立了量子化Mabuchi能量半稳定性与一致K-稳定性、cscK度量存在性的等价关系，得到了相关斜率的尖锐比较结果，并构造了具有无界参数的cscK极化，为Kähler几何中的相关问题提供了关键理论依据。
We give a complete criterion for the existence of generalized Kähler Einstein metrics on toric Fano manifolds from view points of a uniform stability in a sense of GIT and the properness of a functional on the space of Kähler metrics.
The Stability of Generalized Ricci Solitons
本文研究了广义里奇溶子的稳定性，证明了在特定曲率条件下，偏微分平坦爱因斯坦流形的线性稳定性，并展示了稳态梯度广义里奇溶子的动力稳定性与线性稳定性等价性。
Dynamical stability of Pluriclosed and Generalized Ricci solitons
本研究探讨 pluriclosed 流与广义 Ricci 流的稳定性，证明了广义爱因斯坦 - 希尔伯特泛函二阶变分非正且无穷小形变可积时流动力学稳定，以及第一陈类消失时 pluriclosed 稳态孤子动力学稳定的结论。
In this paper we obtain a stability theorem of generalized Kahler structures with one pure spinor under small deformations of generalized complex structures. (This is analogous to the stability theorem of Kahler manifolds by Kodaira-Spencer.) We apply the stability theorem to a class of compact Kahler manifolds which admits deformations to generalized complex manifolds and obtain non-trivial generalized Kahler structures on Fano surfaces and toric Kahler manifolds. In particular, we show that holomorphic Poisson structures on a Kahler manifold induce deformations of generalized Kahler structur…
We consider principal bundles as generalized morphisms between topological groupoids. In the category of these generalized morphisms two topological groupoids are isomorphic if and only if they are Morita equivalent. We show that the fibers of a generalized morphism from H to G induce a singular foliation of the topological groupoid H, and we prove a Reeb-Thurston stability theorem for such foliations. Next, we use generalized morphisms to study some Morita invariants of topological groupoids, in particular the homotopy groups of a topological groupoid and the Connes convolution algebra of an …
Let (M,g) be an analytic, compact, Riemannian manifold with boundary, of dimension n >= 2. We study a class of generalized Radon transforms, integrating over a family of hypersurfaces embedded in M, satisfying the Bolker condition [23]. Using analytic microlocal analysis, we prove a microlocal regularity theorem for generalized Radon transforms on analytic manifolds defined on an analytic family of hypersurfaces. We then show injectivity and stability for an open, dense subset of smooth generalized Radon transforms satisfying the Bolker condition, including the analytic ones.
Goto's generalized Kaehler stability theorem
本文给出了Goto的广义Kaehler稳定性定理的简短且更直接的证明，表明在J_2由无处为零的闭形式确定时，J_1与J_2的小变形可耦合以保持广义Kaehler结构。
In this paper we study the (asymptotic and exponential) stability of the $m$-fold circle as a solution of the $p$-curve shortening flow ($p\geq 1$ an integer).
Stability of the generalized Lagrangian mean curvature flow in cotangent bundle
本文研究余切丛中图情形的广义拉格朗日平均曲率流稳定性，通过新的导数估计弱化初始条件并去除正曲率条件，证明特殊拉格朗日子流形附近该流是稳定的。
Generalized $S^1$-stability theorem
本文利用等变 $\mu$-气泡技术，将 Rosenberg 的 $S^1$-稳定性猜想推广到带边界的紧流形，证明了在维数 $n\in\{3,5,6\}$ 时，$M^n$ 与 $M^n\times S^1$ 的 Yamabe 不变量正性等价。
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