Compact complex surfaces and constant scalar curvature Kähler metrics

Let X be a compact Kähler manifold and γ Kähler class. For a Kàhler metric g on X we denote by Rg the scalar curvature on X According to Calabi [3][4], consider the functional defined on the set of all the Kähler metrics g whose Kähler forms belong to γ, where dvg is the volume form associated to g. Such a Kähler metric is called extremal if it gives a critical point of Ф. In particular, if Rg is constant, g is extremal. The converse is also true if dim L(X) = 0, where L(X) is the maximal connected linear algebraic subgroup of AutoX (cf. [5]). Note also that any Kähler-Einstein metric is of constant scalar curvature.

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KÄHLER GEOMETRY OF TORIC VARIETIES AND EXTREMAL METRICS

International Journal of Mathematics ◽

10.1142/s0129167x98000282 ◽

1998 ◽

Vol 09 (06) ◽

pp. 641-651 ◽

Cited By ~ 71

Author(s):

MIGUEL ABREU

Keyword(s):

Scalar Curvature ◽

Convex Polytopes ◽

Constant Scalar Curvature ◽

Affine Function ◽

Combinatorial Identity ◽

Kähler Metrics ◽

Kahler Metrics ◽

Total Integral ◽

First Chern Class ◽

Moment Polytope

A (symplectic) toric variety X, of real dimension 2n, is completely determined by its moment polytope Δ ⊂ ℝn. Recently Guillemin gave an explicit combinatorial way of constructing "toric" Kähler metrics on X, using only data on Δ. In this paper, differential geometric properties of these metrics are investigated using Guillemin's construction. In particular, a nice combinatorial formula for the scalar curvature R is given, and the Euler–Lagrange condition for such "toric" metrics being extremal (in the sense of Calabi) is proven to be R being an affine function on Δ ⊂ ℝn. A construction, due to Calabi, of a 1-parameter family of extremal Kähler metrics of non-constant scalar curvature on [Formula: see text] is recast very simply and explicitly using Guillemin's approach. Finally, we present a curious combinatorial identity for convex polytopes Δ ⊂ ℝn that follows from the well-known relation between the total integral of the scalar curvature of a Kähler metric and the wedge product of the first Chern class of the underlying complex manifold with a suitable power of the Kähler class.

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An ℓ-th root of a test configuration of exponent ℓ

Complex Manifolds ◽

10.1515/coma-2016-0005 ◽

2016 ◽

Vol 3 (1) ◽

Author(s):

Toshiki Mabuchi

Keyword(s):

Scalar Curvature ◽

Moduli Space ◽

Constant Scalar Curvature ◽

Algebraic Manifold ◽

Positive Real ◽

Kähler Metrics ◽

Test Configuration ◽

Kahler Metrics ◽

Integral Divisor

AbstractLet (X, L) be a polarized algebraic manifold. Then for every test configuration μ = (X, L,Ψ) for (X, L) of exponent ℓ, we obtain an ℓ-th root (κ, D) of μ and Gm-equivariant desingularizations ι : ^X → X and η : ^X → Y, both isomorphic on^X \^X 0, such thatwhereκ= (Y, Q, η) is a test configuration for (X, L) of exponent 1, and D is an effective Q-divisor on^X such that ℓD is an integral divisor with support in the fiber X0. Then (κ, D) can be chosen in such a way thatwhere C1 and C2 are positive real constants independent of the choice of μ and ℓ. This plays an important role in our forthcoming papers on the existence of constant scalar curvature Kähler metrics (cf. [6]) and also on the compactified moduli space of test configurations (cf. [5],[7]).

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