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Abstract Let 𝑋 be a Kähler manifold with semiample canonical bundle K_{X}.It is proved in [W. Jian, Y. Shi and J. Song, A remark on constant scalar curvature Kähler metrics on minimal models,Proc. Amer. Math. Soc.147(2019), 8, 3507–3513] that, for any Kähler class 𝛾, there exists \delta>0such that, for all t\in(0,\delta), there exists a unique cscK metric g_{t}in K_{X}+t\gamma.In this paper, we prove that \{(X,g_{t})\}_{t\in(0,\delta)}have uniformly bounded Kähler potentials, volume forms and diameters.As a consequence, these metric spaces are pre-compact in the Gromov–Hausdorff sense.
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This content will become publicly available on February 11, 2026
The Riemannian and symplectic geometry of the space of generalized Kähler structures
On a compact complex manifold(M, J)endowed with a holomorphic Poisson tensor \pi_{J}and a de Rham class\alpha\in H^{2}(M, \mathbb{R}), we study the space of generalized Kähler (GK) structures defined by a symplectic formF\in \alphaand whose holomorphic Poisson tensor is\pi_{J}. We define a notion of generalized Kähler class of such structures, and use the moment map framework of Boulanger (2019) and Goto (2020) to extend the Calabi program to GK geometry. We obtain generalizations of the Futaki–Mabuchi extremal vector field (1995) and the Calabi–Lichnerowicz–Matsushima result (1982, 1958, 1957) for the Lie algebra of the group of automorphisms of(M, J, \pi_{J}). We define a closed1-form on a GK class, which yields a generalization of the Mabuchi energy and thus a variational characterization of GK structures of constant scalar curvature. Next we introduce a formal Riemannian metric on a given GK class, generalizing the fundamental construction of Mabuchi–Semmes–Donaldson (1987, 1992, 1997) We show that this metric has nonpositive sectional curvature, and that the Mabuchi energy is convex along geodesics, leading to a conditional uniqueness result for constant scalar curvature GK structures. We finally examine the toric case, proving the uniqueness of extremal generalized Kähler structures and showing that their existence is obstructed by the uniform relative K-stability of the corresponding Delzant polytope. Using the resolution of the Yau–Tian–Donaldson conjecture in the toric case by Chen–Cheng (2021) and He (2019), we show in some settings that this condition suffices for existence and thus construct new examples.
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- Award ID(s):
- 2203536
- PAR ID:
- 10625357
- Publisher / Repository:
- EMS Press
- Date Published:
- Journal Name:
- Commentarii Mathematici Helvetici
- Volume:
- 100
- Issue:
- 1
- ISSN:
- 0010-2571
- Page Range / eLocation ID:
- 147 to 223
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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