A Poincaré type Kähler metric on the complement of a simple normal crossing divisor , in a compact Kähler manifold , is a Kähler metric on with cusp singularity along . We relate the Futaki character for holomorphic vector fields parallel to the divisor, defined for any fixed Poincaré type Kähler class, to the classical Futaki character for the relative smooth class. As an application we express a numerical obstruction to the existence of extremal Poincaré type Kähler metrics, in terms of mean scalar curvatures and Futaki characters.
On appelle métrique kählérienne de type Poincaré, sur le complémentaire d’un diviseur à croisements normaux simples dans une variété kählérienne compacte , une métrique kählérienne sur à singularités cusp le long de . On relie le caractère de Futaki des champs de vecteurs holomorphes parallèles au diviseur, défini pour toute classe de Kähler de métriques de type Poincaré fixée, au caractère de Futaki classique de la classe lisse sous-jacente. On donne en application une obstruction numérique à l’existence de métriques extrémales de type Poincaré, exprimée en termes de courbures scalaires moyennes et de caractères de Futaki.
Revised:
Accepted:
Published online:
Keywords: Extremal Kähler metrics, Poincaré type Kähler metrics, Futaki character/invariant, Yau–Tian–Donaldson conjecture.
Mot clés : Métriques kählériennes extrémales, métriques kählériennes de type Poincaré, caractère/invariant de Futaki, conjecture de Yau–Tian–Donaldson.
@article{AIF_2018__68_1_319_0, author = {Auvray, Hugues}, title = {Note on {Poincar\'e} type {Futaki} characters}, journal = {Annales de l'Institut Fourier}, pages = {319--344}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {68}, number = {1}, year = {2018}, doi = {10.5802/aif.3162}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.3162/} }
TY - JOUR AU - Auvray, Hugues TI - Note on Poincaré type Futaki characters JO - Annales de l'Institut Fourier PY - 2018 SP - 319 EP - 344 VL - 68 IS - 1 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.3162/ DO - 10.5802/aif.3162 LA - en ID - AIF_2018__68_1_319_0 ER -
Auvray, Hugues. Note on Poincaré type Futaki characters. Annales de l'Institut Fourier, Volume 68 (2018) no. 1, pp. 319-344. doi : 10.5802/aif.3162. http://archive.numdam.org/articles/10.5802/aif.3162/
[1] Extremal metrics of Poincaré type on toric varieties (2017) (https://arxiv.org/abs/1711.08424)
[2] Metrics of Poincaré type with constant scalar curvature: a topological constraint, J. Lond. Math. Soc., Volume 87 (2013) no. 2, pp. 607-621 | DOI | MR | Zbl
[3] Asymptotic properties of extremal Kähler metrics of Poincaré type, Proc. Lond. Math. Soc., Volume 115 (2017) no. 4, pp. 813-853 | DOI | Zbl
[4] The space of Poincaré type Kähler metrics on the complement of a divisor, J. Reine Angew. Math., Volume 722 (2017), pp. 1-64 | DOI | MR | Zbl
[5] Einstein manifolds, Classics in Mathematics, Springer, 2008, xii+516 pages (Reprint of the 1987 edition) | MR | Zbl
[6] Kähler-Einstein metrics and stability, Int. Math. Res. Not. (2014) no. 8, pp. 2119-2125 | DOI | MR | Zbl
[7] Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities, J. Am. Math. Soc., Volume 28 (2015) no. 1, pp. 183-197 | DOI | Zbl
[8] Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than , J. Am. Math. Soc., Volume 28 (2015) no. 1, pp. 199-234 | DOI | Zbl
[9] Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches and completion of the main proof, J. Am. Math. Soc., Volume 28 (2015) no. 1, pp. 235-278 | DOI | Zbl
[10] Scalar curvature and stability of toric varieties, J. Differ. Geom., Volume 62 (2002) no. 2, pp. 289-349 http://projecteuclid.org/getRecord?id=euclid.jdg/1090950195 | DOI | MR | Zbl
[11] Kähler-Einstein metrics and integral invariants, Lecture Notes in Mathematics, 1314, Springer, 1988, iv+140 pages | MR | Zbl
[12] A special Stokes’s theorem for complete Riemannian manifolds, Ann. Math., Volume 60 (1954), pp. 140-145 | DOI | MR | Zbl
[13] Calabi’s extremal metrics: an elementary introduction, Lecture notes
[14] Principles of algebraic geometry, Wiley Classics Library, John Wiley & Sons, 1994, xiv+813 pages (Reprint of the 1978 original) | DOI | Zbl
[15] Scalar curvature and Futaki invariant of Kähler metrics with cone singularities along a divisor (2017) (to appear in Ann. Inst. Fourier, https://arxiv.org/abs/1508.02640v2)
[16] Stability of extremal Kähler manifolds, Osaka J. Math., Volume 41 (2004) no. 3, pp. 563-582 http://projecteuclid.org/getRecord?id=euclid.ojm/1153494139 | MR | Zbl
[17] K-stability of constant scalar curvature Kähler manifolds, Adv. Math., Volume 221 (2009) no. 4, pp. 1397-1408 | DOI | MR | Zbl
[18] Extremal metrics and K-stability, Imperial College, University of London (UK) (2006) (Ph. D. Thesis https://arxiv.org/abs/math/0611002)
[19] Kähler-Einstein metrics with positive scalar curvature, Invent. Math., Volume 130 (1997) no. 1, pp. 1-37 | DOI | MR | Zbl
[20] K-stability and Kähler-Einstein metrics, Commun. Pure Appl. Math., Volume 68 (2015) no. 7, pp. 1085-1156 Corrigendum in ibid. 68 (2015), no. 11, p. 2082-2083 | DOI | MR | Zbl
[21] Existence of Kähler-Einstein metrics on complete Kähler manifolds and their applications to algebraic geometry, Mathematical aspects of string theory (San Diego, 1986) (Advanced Series in Mathematical Physics), Volume 1, World Science Publishing, 1987, pp. 574-628 | MR | Zbl
[22] Kähler-Einstein metrics of negative Ricci curvature on general quasi-projective manifolds, Commun. Anal. Geom., Volume 16 (2008) no. 2, pp. 395-435 http://projecteuclid.org/getRecord?id=euclid.cag/1216396331 | DOI | MR | Zbl
[23] Open problems in geometry, Differential geometry: partial differential equations on manifolds (Los Angeles, 1990) (Proceedings of Symposia in Pure Mathematics), Volume 54, American Mathematical Society, 1993, pp. 1-28 | MR | Zbl
Cited by Sources: