Relative Chow stability and optimal weights

Tipler, Carl

doi:10.1090/proc/16426

Relative Chow stability and optimal weights
HTML articles powered by AMS MathViewer

by Carl Tipler

Proc. Amer. Math. Soc. 151 (2023), 4015-4026

DOI: https://doi.org/10.1090/proc/16426

Published electronically: May 19, 2023

HTML | PDF | Request permission

Abstract:

For a polarized Kähler manifold $(X,L)$, we show the equivalence between relative balanced embeddings introduced by Mabuchi and $\sigma$-balanced embeddings introduced by Sano, answering a question of Hashimoto. We give a GIT characterization of the existence of a $\sigma$-balanced embedding, and relate the optimal weight $\sigma$ to the action of $\mathrm {Aut}_0(X,L)$ on the Chow line of $(X,L)$.

References

Vestislav Apostolov and Hongnian Huang, A splitting theorem for extremal Kähler metrics, J. Geom. Anal. 25 (2015), no. 1, 149–170. MR 3299273, DOI 10.1007/s12220-013-9417-6
Eugenio Calabi, Extremal Kähler metrics, Seminar on Differential Geometry, Ann. of Math. Stud., No. 102, Princeton Univ. Press, Princeton, NJ, 1982, pp. 259–290. MR 645743
Alberto Della Vedova and Fabio Zuddas, Scalar curvature and asymptotic Chow stability of projective bundles and blowups, Trans. Amer. Math. Soc. 364 (2012), no. 12, 6495–6511. MR 2958945, DOI 10.1090/S0002-9947-2012-05587-5
S. K. Donaldson, Scalar curvature and projective embeddings. I, J. Differential Geom. 59 (2001), no. 3, 479–522. MR 1916953, DOI 10.4310/jdg/1090349449
S. K. Donaldson, Scalar curvature and projective embeddings. II, Q. J. Math. 56 (2005), no. 3, 345–356. MR 2161248, DOI 10.1093/qmath/hah044
Akito Futaki, Asymptotic Chow semi-stability and integral invariants, Internat. J. Math. 15 (2004), no. 9, 967–979. MR 2106156, DOI 10.1142/S0129167X04002612
Yoshinori Hashimoto, Quantisation of extremal Kähler metrics, J. Geom. Anal. 31 (2021), no. 3, 2970–3028. MR 4225832, DOI 10.1007/s12220-020-00381-7
Yoshinori Hashimoto, Relative stability associated to quantised extremal Kähler metrics, J. Math. Soc. Japan 71 (2019), no. 3, 861–880. MR 3984245, DOI 10.2969/jmsj/79947994
Shoshichi Kobayashi, Transformation groups in differential geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], Band 70, Springer-Verlag, New York-Heidelberg, 1972. MR 355886, DOI 10.1007/978-3-642-61981-6
Mehdi Lejmi, Extremal almost-Kähler metrics, Internat. J. Math. 21 (2010), no. 12, 1639–1662. MR 2747965, DOI 10.1142/S0129167X10006690
Chi Li, Constant scalar curvature Kähler metric obtains the minimum of K-energy, Int. Math. Res. Not. IMRN 9 (2011), 2161–2175. MR 2806561, DOI 10.1093/imrn/rnq152
Huazhang Luo, Geometric criterion for Gieseker-Mumford stability of polarized manifolds, J. Differential Geom. 49 (1998), no. 3, 577–599. MR 1669716
Toshiki Mabuchi, Stability of extremal Kähler manifolds, Osaka J. Math. 41 (2004), no. 3, 563–582. MR 2107663
Toshiki Mabuchi, An obstruction to asymptotic semistability and approximate critical metrics, Osaka J. Math. 41 (2004), no. 2, 463–472. MR 2069096
Toshiki Mabuchi, Uniqueness of extremal Kähler metrics for an integral Kähler class, Internat. J. Math. 15 (2004), no. 6, 531–546. MR 2078878, DOI 10.1142/S0129167X04002429
Toshiki Mabuchi, An energy-theoretic approach to the Hitchin-Kobayashi correspondence for manifolds. I, Invent. Math. 159 (2005), no. 2, 225–243. MR 2116275, DOI 10.1007/s00222-004-0387-y
Hajime Ono, Yuji Sano, and Naoto Yotsutani, An example of an asymptotically Chow unstable manifold with constant scalar curvature, Ann. Inst. Fourier (Grenoble) 62 (2012), no. 4, 1265–1287 (English, with English and French summaries). MR 3025743, DOI 10.5802/aif.2722
Sean Timothy Paul, Geometric analysis of Chow Mumford stability, Adv. Math. 182 (2004), no. 2, 333–356. MR 2032032, DOI 10.1016/S0001-8708(03)00081-1
D. H. Phong and Jacob Sturm, Stability, energy functionals, and Kähler-Einstein metrics, Comm. Anal. Geom. 11 (2003), no. 3, 565–597. MR 2015757, DOI 10.4310/CAG.2003.v11.n3.a6
D. H. Phong and Jacob Sturm, Scalar curvature, moment maps, and the Deligne pairing, Amer. J. Math. 126 (2004), no. 3, 693–712. MR 2058389, DOI 10.1353/ajm.2004.0019
Y. Sano, Lecture on $\sigma$-balanced metrics, 2011 Complex Geometry and Symplectic Geometry Conference, University of Science and Technology in China.
Yuji Sano and Carl Tipler, Extremal metrics and lower bound of the modified K-energy, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 9, 2289–2310. MR 3420508, DOI 10.4171/JEMS/557
Yuji Sano and Carl Tipler, A moment map picture of relative balanced metrics on extremal Kähler manifolds, J. Geom. Anal. 31 (2021), no. 6, 5941–5973. MR 4267632, DOI 10.1007/s12220-020-00510-2
Reza Seyyedali, Relative Chow stability and extremal metrics, Adv. Math. 316 (2017), 770–805. MR 3672920, DOI 10.1016/j.aim.2017.06.031
Gábor Székelyhidi, Extremal metrics and $K$-stability, Bull. Lond. Math. Soc. 39 (2007), no. 1, 76–84. MR 2303522, DOI 10.1112/blms/bdl015
Xiaowei Wang, Moment map, Futaki invariant and stability of projective manifolds, Comm. Anal. Geom. 12 (2004), no. 5, 1009–1037. MR 2103309, DOI 10.4310/CAG.2004.v12.n5.a2
Shouwu Zhang, Heights and reductions of semi-stable varieties, Compositio Math. 104 (1996), no. 1, 77–105. MR 1420712