Calabi flow, geodesic rays, and uniqueness of constant scalar curvature Kähler metrics. (English) Zbl 1307.53058

In this article, the authors prove a uniqueness theorem for constant scalar curvature Kähler metrics (cscK for short) adjacent to a fixed Kähler class. Consider a compact Kähler manifold \((M,\omega,J)\). Denote by \(\mathcal{G}\) the group of Hamiltonian diffeomorphisms of \((M,\omega)\), which acts on \(\mathcal{J}\), the space of almost-complex structures on \(M\) that are \(\omega\)-compatible. Following work of S. Donaldson [“Remarks on gauge theory, complex geometry and 4-manifold topology”, World Sci. Ser. 20th Century Math. 5, 384–403 (1997)] and A. Fujiki [Sugaku Expo. 5, No. 2, 173–191 (1992); translation from Sugaku 42, No. 3, 231–243 (1990; Zbl 0796.32009)], the scalar curvature appears as a moment map for the \(\mathcal{G}\)-action on \(\mathcal{J}\) endowed with a natural Kähler structure. It is possible to make sense of complexified \(\mathcal{G}\)-orbits by complexifying the infinitesimal action of \(\mathcal{G}\). If \(\mathcal{J}\) and \(\mathcal{G}\) were finite-dimensional and compact, by the Kempf-Ness theorem, a \(\mathcal{G}^\mathbb{C}\)-orbit would admit a metric of constant scalar curvature if and only if it were poly-stable. This is the content of the Yau-Tian-Donaldson conjecture.
Moreover, such cscK metrics should form a unique \(\mathcal{G}\)-orbit. This uniqueness result was proved in [S. K. Donaldson, J. Differ. Geom. 59, No. 3, 479–522 (2001; Zbl 1052.32017) ] under some restrictions, and in [X. Chen and G. Tian, Publ. Math., Inst. Hautes Étud. Sci. 107, 1–107 (2008; Zbl 1182.32009)] and [B. Berndtsson and R. Berman, “Convexity of the K-energy on the space of Kähler metrics and uniqueness of extremal metrics, Preprint, arXiv:1405.0401] in full generality. Last, a \(\mathcal{G}^\mathbb{C}\)-orbit should be semi-stable if and only if its closure contains a constant scalar curvature Kähler metric. Again, such a poly-stable orbit should be unique, and this is the main result of this article.
More precisely, if there are two cscK structures \(J_1\) and \(J_2\) in the closure of the \(\mathcal{G}^\mathbb{C}\)-orbit of \(J\), then there exists a symplectic diffeomorphism \(\phi\) such that \(\phi^*J_1=J_2\).
In more algebraic terms, this result implies uniqueness for smooth cscK central fibers for test configurations of a given polarized compact complex manifold. The proof relies on several new ideas and the main technical tool is the Calabi flow. Initially defined on the space of Kähler potentials in a given Kähler class, the authors consider this flow on the whole space \(\mathcal{J}\). A Łojasiewicz type inequality is proved for the scalar curvature fonctional on \(\mathcal{J}\) which is used to obtain a stability result for the Calabi flow, and to detect adjacent cscK structures. The lack of compactness is then bypassed by mean of geodesic rays. Performing a local study of the \(\mathcal{G}\)-action and of the moment map setting, the authors show that in their situation, the Calabi flow is asymptotic to a smooth geodesic ray. Then, results of X. Chen [J. Differ. Geom. 56, No. 2, 189–234 (2000; Zbl 1041.58003)] and X. X. Chen and W. Y. He [Am. J. Math. 130, No. 2, 539–570 (2008; Zbl 1204.53050)] on the space of Kähler potentials and its geodesics enable to conclude \(\mathcal{C}^0\)-estimates and the proof of the main result.


53C55 Global differential geometry of Hermitian and Kählerian manifolds
53D25 Geodesic flows in symplectic geometry and contact geometry
53D20 Momentum maps; symplectic reduction
32G05 Deformations of complex structures
53C22 Geodesics in global differential geometry
32Q26 Notions of stability for complex manifolds
32Q60 Almost complex manifolds
58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable)
35K55 Nonlinear parabolic equations
Full Text: DOI arXiv


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