arXiv research

Formula proves invariant matches for smooth and orbifold test configurations.

Only products of projective lines have vanishing Futaki invariants for all Kähler classes.

Futaki invariant vanishes on Hopf manifolds.

Study shows Futaki invariant vanishes on most Fano threefolds.

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 positiv…

In this paper, the Bando-Futaki invariants on hypersurfaces are derived in terms of the degree of the defining polynomials, the dimension of the underlying projective space, and the given holomorphic vector field. In addition, the holomorphic invariant introduced by Tian and Chen (Ricci Flow on Kähler-Einstein surfaces…

We study the Futaki invariant and the Mabuchi K-energy of a Kähler manifold $M$ using the Deligne pairing technique developed in earlier papers. We first prove a rather simple characterization of the Futaki character: The Futaki character on a Q-Fano variety is the eigenvalue of the action of $Aut(M)$ on $Chow(M)$, the…

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…

In this note we study the localization of Futaki-Morita integrals at isolated degenerate zeros by giving a streamlined exposition in the spirit of Bott and implement the localization procedure for a holomorphic vector field on $CP^n$ with a maximally degenerate zero, giving an essentially unique formula for the Futaki-…

Formula for α-Futaki character on toric manifolds.

Uniformly K-stable toric varieties are asymptotically Chow stable if their Futaki-Ono invariant vanishes.

In this paper, we give an explicit formula for the Futaki invariants of complete intersections. The result is new in the case where the variety is smooth or has orbifold singularities.

In this work we prove a residue formula for Morita-Futaki-Bott invariant with respect any holomorphic vector fields with isolated (possibly degenerated) singularities in terms of Grothendieck's residues.

Odaka and Wang proved the intersection formula for the Donaldson-Futaki invariant. In this paper, we generalize this result for the higher Futaki invariants which are obstructions to asymptotic Chow semistability.

We prove the expansion formula for the classical Futaki invariants on the blowup of Kähler surfaces, which explains the balancing condition of Arezzo-Pacard. The relation with Stoppa's result is also discussed.

We establish a lower bound for the Donaldson-Futaki invariant of optimal degenerations produced by the Kähler-Ricci flow in terms of the greatest Ricci lower bound on arbitrary Fano manifolds. As an application, we can generalize the finiteness of the Futaki invariants on Kähler-Ricci solitons obtained by Guo-Phong-Son…

Smooth approximations of Kähler-Ricci solitons found using quantized metrics and Futaki invariants.

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 man…

We show that the Einstein-Hilbert functional, as a functional on the space of Reeb vector fields, detects the vanishing Sasaki-Futaki invariant. In particular, this provides an obstruction to the existence of a constant scalar curvature Sasakian metric. As an application we prove that K-semistable polarized Sasaki mani…

The study connects K-stability and large complex structure limits in mirror symmetry.

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 examp…

In this note, we consider a sequence of test configurations compatible with a Kaehler metric in $c_1(L)$ on a polarized algebraic manifold $(X,L)$. Then an explicit formula for the Donaldson-Futaki invariant $F_1$ for the sequence will be given.

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.

We present an analytic proof of the relationship between the Calabi-Futaki invariant for a Kähler manifold relative to a holomorphic vector field with a nondegenerate zero and the corresponding invariant of its blowup at that zero, restricting to the case that zeros on the exceptional divisor are isolated. This extends…

This note discusses the higher K-energy functionals which were defined by Bando and Mabuchi, and integrate higher Futaki invariants. Two new formulas for the higher K-energy functionals are given, and the second K-energy is shown to be related to Donaldson's Lagrangian applied to metrics on the tangent bundle.

In this paper, we give a new version of the modified Futaki invariant for a test configuration associated to the soliton action on a Fano manifold. Our version will naturally come from toric test configurations defined by Donaldson for toric manifolds. As an application, we show that the modified $K$-energy is proper f…

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 condition…

Paper computes stability of Q-Fano spherical varieties using test configurations and Futaki invariants.

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…

New obstruction found for Hull-Strominger system solutions.

We study the scalar curvature of Kähler metrics that have cone singularities along a divisor, with a particular focus on certain specific classes of such metrics that enjoy some curvature estimates. Our main result is that, on the projective completion of a pluricanonical bundle over a product of Kähler--Einstein Fano …

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…

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…

Analytic K-semistability connects curvature to metric existence.

We derive an explicit formula for the asymptotic slope of the Aubin-Yau functional along a Bergman geodesic on a surface of complex dimension 2, extending the work of Phong-Sturm on Riemann surfaces. This is equivalent to an explicit calculation of the Donaldson-Futaki invariant of a test configuration. The slope is gi…

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-Einste…

Introduces new equations linking Kähler-Einstein and Hermitian-Yang-Mills theories.

In this note we revisit and extend few classical and recent results on the definition and use of the Futaki invariant in connection with the existence problem for Kaehler constant scalar curvature metrics on polarized algebraic manifolds, especially in the case of resolution of singularities. The general inspiration be…

We study obstructions to the existence of closed Fedosov's star products on a given Kähler manifold. In our previous paper, we proved that the Levi-Civita connection of a Kähler manifold will produce a closed (in the sense of Connes-Flato-Sternheimer) Fedosov's star product only if it is a zero of the Cahen-Gutt moment…

The paper studies K-stability of spherical varieties and their degenerations.

The paper classifies test configurations and derives a criterion for uniform K-stability of certain algebraic varieties.

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.

For smooth test configurations, there always exist C^{1,1} geodesic rays in Kahler metric space parallel to the algebraic ray. The $\yen$ invariant agrees with Futaki invariant, at least under nice assumptions. Explicit examples in Toric cases are calculated. On simple test configurations, Donaldson's correspondence be…

Let (X,Ω) be a closed polarized complex manifold, g be an extremal metric on X that represents the Kähler class Ω, and G be a compact connected subgroup of the isometry group Isom(X,g). Assume that the Futaki invariant relative to G is nondegenerate at g. Consider a smooth family $(M \to B)$ of polarized complex deform…

Derives scalar curvature formula in generalized Kähler geometry.

We give a new formula for the energy functionals E_k defined by Chen-Tian, and discuss the relations between these functionals. We also apply our formula to give a new proof of the fact that the holomorphic invariants corresponding to the E_k functionals are equal to the Futaki invariant.

In this paper, we establish several sufficient and necessary conditions for the convergence of a Kähler-Ricci flow, on a Kähler manifold with positive first Chern class, to a Kähler-Einstein metric (or a shrinking Kähler-Ricci soliton).

Study complex structures and curvature equations on compact manifolds.