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

Chen, Xiuxiong; Sun, Song

Ann. Math. (2) 180, No. 2, 407-454 (2014).

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.

Reviewer: Carl Tipler (Brest)

MSC:

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

Keywords:

Calabi flow; Kähler metrics; Kähler-Einstein; Kempf-Ness theorem; constant scalar curvature metrics; semi-stability

Citations:

Zbl 0796.32009; Zbl 1052.32017; Zbl 1182.32009; Zbl 1041.58003; Zbl 1204.53050

References:

[1]	S. K. Donaldson, ”Remarks on gauge theory, complex geometry and \(4\)-manifold topology,” in Fields Medallists’ Lectures, River Edge, NJ: World Sci. Publ., 1997, vol. 5, pp. 384-403.
[2]	A. Fujiki, ”Moduli space of polarized algebraic manifolds and Kähler metrics [translation of Sûgaku 42 (1990), no. 3, 231-243; \mr1073369],” Sugaku Expositions, vol. 5, iss. 2, pp. 173-191, 1992. · Zbl 0763.32012
[3]	R. P. Thomas, ”Notes on GIT and symplectic reduction for bundles and varieties,” in Surveys in Differential Geometry. Vol. X, Somerville, MA: Int. Press, 2006, vol. 10, pp. 221-273. · Zbl 1132.14043
[4]	G. Tian, ”Kähler-Einstein metrics with positive scalar curvature,” Invent. Math., vol. 130, iss. 1, pp. 1-37, 1997. · Zbl 0892.53027
[5]	S. K. Donaldson, ”Conjectures in Kähler geometry,” in Strings and Geometry, Providence, RI: Amer. Math. Soc., 2004, vol. 3, pp. 71-78. · Zbl 1161.32010
[6]	J. Ross and R. Thomas, ”A study of the Hilbert-Mumford criterion for the stability of projective varieties,” J. Algebraic Geom., vol. 16, iss. 2, pp. 201-255, 2007. · Zbl 1200.14095
[7]	S. T. Paul, ”Hyperdiscriminant polytopes, Chow polytopes, and Mabuchi energy asymptotics,” Ann. of Math., vol. 175, iss. 1, pp. 255-296, 2012. · Zbl 1243.14038
[8]	S. K. Donaldson, ”Constant scalar curvature metrics on toric surfaces,” Geom. Funct. Anal., vol. 19, iss. 1, pp. 83-136, 2009. · Zbl 1177.53067
[9]	S. K. Donaldson, ”Scalar curvature and projective embeddings. I,” J. Differential Geom., vol. 59, iss. 3, pp. 479-522, 2001. · Zbl 1052.32017
[10]	X. Chen and G. Tian, ”Geometry of Kähler metrics and foliations by holomorphic discs,” Publ. Math. IHES, vol. 107, pp. 1-107, 2008. · Zbl 1182.32009
[11]	S. Bando and T. Mabuchi, ”Uniqueness of Einstein Kähler metrics modulo connected group actions,” in Algebraic Geometry, Sendai, 1985, Amsterdam: North-Holland, 1987, vol. 10, pp. 11-40. · Zbl 0641.53065
[12]	G. Tian and X. Zhu, ”A new holomorphic invariant and uniqueness of Kähler-Ricci solitons,” Comment. Math. Helv., vol. 77, iss. 2, pp. 297-325, 2002. · Zbl 1036.53053
[13]	X. Chen, ”The space of Kähler metrics,” J. Differential Geom., vol. 56, iss. 2, pp. 189-234, 2000. · Zbl 1041.58003
[14]	G. Tian, ”Extremal metrics and geometric stability,” Houston J. Math., vol. 28, iss. 2, pp. 411-432, 2002. · Zbl 1026.53023
[15]	T. Mabuchi, ”Some symplectic geometry on compact Kähler manifolds. I,” Osaka J. Math., vol. 24, iss. 2, pp. 227-252, 1987. · Zbl 0645.53038
[16]	S. Semmes, ”Complex Monge-Ampère and symplectic manifolds,” Amer. J. Math., vol. 114, iss. 3, pp. 495-550, 1992. · Zbl 0790.32017
[17]	S. K. Donaldson, ”Symmetric spaces, Kähler geometry and Hamiltonian dynamics,” in Northern California Symplectic Geometry Seminar, Providence, RI: Amer. Math. Soc., 1999, vol. 196, pp. 13-33. · Zbl 0972.53025
[18]	E. Calabi and X. Chen, ”The space of Kähler metrics II,” J. Differential Geom., vol. 61, iss. 2, pp. 173-193, 2002. · Zbl 1067.58010
[19]	X. Chen, ”Space of Kähler metrics. III. On the lower bound of the Calabi energy and geodesic distance,” Invent. Math., vol. 175, iss. 3, pp. 453-503, 2009. · Zbl 1163.58005
[20]	X. Chen, ”On the lower bound of the Mabuchi energy and its application,” Internat. Math. Res. Notices, iss. 12, pp. 607-623, 2000. · Zbl 0980.58007
[21]	X. Chen and S. Sun, ”Space of Kähler metrics (V) - Kähler quantization,” in Metric and Differential Geometry. The Jeff Cheeger Volume, Boston: Birkhäuer, 2012, vol. 297, pp. 19-41. · Zbl 1252.53084
[22]	B. Berndtsson, Probability measures related to geodesics in the space of Kähler metrics. · Zbl 1408.53001
[23]	S. K. Donaldson and P. B. Kronheimer, The Geometry of Four-Manifolds, New York: The Clarendon Press, Oxford University Press, 1990. · Zbl 0820.57002
[24]	E. Lerman, ”Gradient flow of the norm squared of a moment map,” Enseign. Math., vol. 51, iss. 1-2, pp. 117-127, 2005. · Zbl 1103.53051
[25]	G. R. Kempf, ”Instability in invariant theory,” Ann. of Math., vol. 108, iss. 2, pp. 299-316, 1978. · Zbl 0406.14031
[26]	L. Ness, ”A stratification of the null cone via the moment map,” Amer. J. Math., vol. 106, iss. 6, pp. 1281-1329, 1984. · Zbl 0604.14006
[27]	F. C. Kirwan, Cohomology of Quotients in Symplectic and Algebraic Geometry, Princeton, NJ: Princeton Univ. Press, 1984, vol. 31. · Zbl 0553.14020
[28]	V. Guillemin and S. Sternberg, ”A normal form for the moment map,” in Differential Geometric Methods in Mathematical Physics, Dordrecht: Reidel, 1984, vol. 6, pp. 161-175. · Zbl 0548.58011
[29]	J. Ortega and T. S. Ratiu, Momentum Maps and Hamiltonian Reduction, Boston: Birkhäuser, 2004, vol. 222. · Zbl 1241.53069
[30]	X. Chen and W. He, ”The Calabi flow on Kähler surfaces with bounded Sobolev constant (I),” Math. Ann., vol. 354, iss. 1, pp. 227-261, 2012. · Zbl 1254.53097
[31]	E. Calabi, ”Extremal Kähler metrics,” in Seminar on Differential Geometry, Yau, S. T., Ed., Princeton, N.J.: Princeton Univ. Press, 1982, vol. 102, pp. 259-290.
[32]	E. Calabi, ”Extremal Kähler metrics. II,” in Differential Geometry and Complex Analysis, Chavel, I. and Farkas, H. M., Eds., New York: Springer-Verlag, 1985, pp. 95-114.
[33]	X. Chen and W. Y. He, ”On the Calabi flow,” Amer. J. Math., vol. 130, iss. 2, pp. 539-570, 2008. · Zbl 1204.53050
[34]	X. Chen, H. Li, and B. Wang, ”Kähler-Ricci flow with small initial energy,” Geom. Funct. Anal., vol. 18, iss. 5, pp. 1525-1563, 2009. · Zbl 1169.53051
[35]	G. Tian and X. Zhu, Perelman’s \(\mathcalW\)-functional and stability of Kähler-Ricci flow. · Zbl 1185.53078
[36]	S. Sun and Y-Q. Wang, On the Kähler-Ricci flow near a Kähler-Einstein metric. · Zbl 1314.53123
[37]	L. Simon, ”Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems,” Ann. of Math., vol. 118, iss. 3, pp. 525-571, 1983. · Zbl 0549.35071
[38]	J. Råde, ”On the Yang-Mills heat equation in two and three dimensions,” J. Reine Angew. Math., vol. 431, pp. 123-163, 1992. · Zbl 0760.58041
[39]	S. K. Donaldson, Floer Homology Groups in Yang-Mills Theory, Cambridge: Cambridge Univ. Press, 2002, vol. 147. · Zbl 0998.53057
[40]	A. Fujiki and G. Schumacher, ”The moduli space of extremal compact Kähler manifolds and generalized Weil-Petersson metrics,” Publ. Res. Inst. Math. Sci., vol. 26, iss. 1, pp. 101-183, 1990. · Zbl 0714.32007
[41]	X. Chen, The space of Kähler metrics (IV)-On the lower bound of the K energy.
[42]	G. Székelyhidi, ”The Kähler-Ricci flow and \(K\)-polystability,” Amer. J. Math., vol. 132, iss. 4, pp. 1077-1090, 2010. · Zbl 1206.53075
[43]	V. Tosatti, ”The K-energy on small deformations of constant scalar curvature Kähler manifolds,” in Advances in Geometric Analysis, Somerville, MA: Int. Press, 2012, vol. 21, pp. 139-147. · Zbl 1317.32050
[44]	M. Kuranishi, ”New proof for the existence of locally complete families of complex structures,” in Proc. Conf. Complex Analysis, New York: Springer-Verlag, 1965, pp. 142-154. · Zbl 0144.21102
[45]	A. Fujiki and G. Schumacher, ”The moduli space of Kähler structures on a real compact symplectic manifold,” Publ. Res. Inst. Math. Sci., vol. 24, iss. 1, pp. 141-168, 1988. · Zbl 0655.32020
[46]	J. Fine, ”Calabi flow and projective embeddings,” J. Differential Geom., vol. 84, iss. 3, pp. 489-523, 2010. · Zbl 1202.32018
[47]	H. Ono, Y. Sano, and N. Yotsutani, ”An example of an asymptotically Chow unstable manifold with constant scalar curvature,” Ann. Inst. Fourier \((\)Grenoble\()\), vol. 62, iss. 4, pp. 1265-1287, 2012. · Zbl 1255.53057
[48]	A. Della Vedova and F. Zuddas, ”Scalar curvature and asymptotic Chow stability of projective bundles and blowups,” Trans. Amer. Math. Soc., vol. 364, iss. 12, pp. 6495-6511, 2012. · Zbl 1288.32033
[49]	C. Arezzo, F. Pacard, and M. Singer, ”Extremal metrics on blowups,” Duke Math. J., vol. 157, iss. 1, pp. 1-51, 2011. · Zbl 1221.32008
[50]	R. S. Hamilton, ”The inverse function theorem of Nash and Moser,” Bull. Amer. Math. Soc., vol. 7, iss. 1, pp. 65-222, 1982. · Zbl 0499.58003
[51]	A. Weinstein, ”Symplectic manifolds and their Lagrangian submanifolds,” Advances in Math., vol. 6, pp. 329-346 (1971), 1971. · Zbl 0213.48203
[52]	M. E. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser, 1991, vol. 100. · Zbl 0746.35062