×

Finite TYCZ expansions and cscK metrics. (English) Zbl 1431.53078

Summary: Let \((M,g)\) be a Kähler manifold whose associated Kähler form \(\omega\) is integral and let \((L,h)\rightarrow(M,\omega)\) be a quantization Hermitian line bundle. In this paper we study Kähler manifolds \((M,g)\) admitting a finite TYCZ expansion, namely those for which the associated Kempf distortion function \(T_{mg}\) is of the form: \[ T_{mg}(p)=f_s(p)m^s+f_{s-1}(p)m^{s-1}+\cdots+f_r(p)m^r,\quad f_j\in\mathcal{C}^\infty(M),\, s,r\in\mathbb{Z}. \] We show that if the TYCZ expansion is finite then \(T_{mg}\) is indeed a polynomial in \(m\) of degree \(n\), \(n=\dim_{\mathbb{C}}M\), and the log-term of the Szegö kernel of the disc bundle \(D\subset L^\ast\) vanishes (where \(L^\ast\) is the dual bundle of \(L)\). Moreover, we provide a complete classification of the Kähler manifolds admitting finite TYCZ expansion either when \(M\) is a complex curve or when \(M\) is a complex surface with a cscK metric which admits a radial Kähler potential.

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
32Q15 Kähler manifolds
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Arezzo, C.; Loi, A., Quantization of Kähler manifolds and the asymptotic expansion of Tian-Yau-Zelditch, J. Geom. Phys., 47, 87-99 (2003) · Zbl 1027.53081
[2] Arezzo, C.; Loi, A., Moment maps, scalar curvature and quantization of Kähler manifolds, Comm. Math. Phys., 246, 543-549 (2004) · Zbl 1062.32021
[3] Arezzo, C.; Loi, A.; Zuddas, F., Szegö kernel, regular quantizations and spherical CR-structures, Math. Z., 275, 1207-1216 (2013) · Zbl 1290.53070
[4] Boutet de Monvel, L.; Sjöstrand, J., Sur la singularité des noyaux de Bergman et de Szegö, Journes: Equations aux Dérivées Partielles de Rennes (1975). Journes: Equations aux Dérivées Partielles de Rennes (1975), Astèrisque, 3435, 123-164 (1976), Soc. Math. France, Paris · Zbl 0344.32010
[5] Cahen, M.; Gutt, S.; Rawnsley, J. H., Quantization of Kähler manifolds I: geometric interpretation of Berezin’s quantization, J. Geom. Phys., 7, 45-62 (1990) · Zbl 0719.53044
[6] Calabi, E., Isometric imbedding of complex manifolds, Ann. of Math., 58, 1 (1953) · Zbl 0051.13103
[7] Cannas Aghedu, F.; Loi, A., The Simanca metric admits a regular quantization · Zbl 1423.53086
[8] Catlin, D., The Bergman kernel and a theorem of Tian, (Analysis and Geometry in Several Complex Variables. Analysis and Geometry in Several Complex Variables, Katata, 1997. Analysis and Geometry in Several Complex Variables. Analysis and Geometry in Several Complex Variables, Katata, 1997, Trends Math. (1999), Birkhäuser Boston: Birkhäuser Boston Boston, MA), 1-23 · Zbl 0941.32002
[9] Donaldson, S., Scalar curvature and projective embeddings, I, J. Differential Geom., 59, 479-522 (2001) · Zbl 1052.32017
[10] Engliš, M., Berezin quantization and reproducing kernels on complex domains, Trans. Amer. Math. Soc., 348, 411-479 (1996) · Zbl 0842.46053
[11] Engliš, M., A Forelli-Rudin construction and asymptotics of weighted Bergman kernels, J. Funct. Anal., 177, 2, 257-281 (2000) · Zbl 0969.32003
[12] Gramchev, T.; Loi, A., TYZ expansion for the Kepler manifold, Comm. Math. Phys., 289, 825-840 (2009) · Zbl 1175.32013
[13] Ji, S., Inequality for distortion function of invertible sheaves on Abelian varieties, Duke Math. J., 58, 657-667 (1989) · Zbl 0711.14024
[14] Kempf, G. R., Metric on invertible sheaves on abelian varieties, (Topics in Algebraic Geometry (Guanajuato) (1989))
[15] Loi, A., The Tian-Yau-Zelditch asymptotic expansion for real analytic Kähler metrics, Int. J. Geom. Methods Mod. Phys., 1, 253-263 (2004) · Zbl 1084.32015
[16] Loi, A., A Laplace integral, the T-Y-Z expansion and Berezin’s transform on a Kähler manifold, Int. J. Geom. Methods Mod. Phys., 2, 359-371 (2005) · Zbl 1082.53039
[17] Loi, A.; Mossa, R., Berezin quantization of homogeneous bounded domains, Geom. Dedicata, 161, 119-128 (2012) · Zbl 1257.53120
[18] Loi, A.; Mossa, R.; Zuddas, F., The log-term of the disc bundle over a homogeneous Hodge manifold, Ann. Global Anal. Geom., 51, 1, 35-51 (2017) · Zbl 1361.53057
[19] Loi, A.; Salis, F.; Zuddas, F., Two conjectures on Ricci flat metrics, Math. Z., 290, 599-613 (2018) · Zbl 1412.53104
[20] Loi, A.; Salis, F.; Zuddas, F., On the third coefficient of TYZ expansion for radial scalar flat metrics, J. Geom. Phys., 133, 210-218 (2018) · Zbl 1401.53063
[21] Loi, A.; Zedda, M., On the coefficients of TYZ expansion of locally Hermitian symmetric spaces, Manuscripta Math., 148, 303-315 (2015) · Zbl 1327.53096
[22] Loi, A.; Zedda, M., Kähler Immersions of Kähler Manifolds into Complex Space Forms, Lecture Notes of the Unione Matematica Italiana, vol. 23 (2018), Springer · Zbl 1432.32032
[23] Loi, A.; Zedda, M.; Zuddas, F., Some remarks on the Kähler geometry of the Taub-NUT metrics, Ann. Global Anal. Geom., 41, 4, 515-533 (2012) · Zbl 1275.53068
[24] Lu, Z., On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch, Amer. J. Math., 122, 2, 235-273 (2000) · Zbl 0972.53042
[25] Lu, Z.; Tian, G., The log term of Szegö Kernel, Duke Math. J., 125, 351-387 (2004) · Zbl 1072.32014
[26] Ma, X.; Marinescu, G., Holomorphic Morse Inequalities and Bergman Kernels, Progress in Mathematics (2007), Birkhäuser: Birkhäuser Basel · Zbl 1135.32001
[27] Rawnsley, J., Coherent states and Kähler manifolds, Quart. J. Math. Oxford (2), 28, 403-415 (1977) · Zbl 0387.58002
[28] Ruan, W. D., Canonical coordinates and Bergmann metrics, Comm. Anal. Geom., 589-631 (1998) · Zbl 0917.53026
[29] Tian, G., On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom., 32, 99-130 (1990) · Zbl 0706.53036
[30] Zelditch, S., Szegö kernels and a theorem of Tian, Int. Math. Res. Not., 6, 317-331 (1998) · Zbl 0922.58082
[31] Zhang, S., Heights and reductions of semi-stable varieties, Compos. Math., 104, 77-105 (1996) · Zbl 0924.11055
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.