Tian's invariant on \(G_{p,p+q}(\mathbb{C})\) and \(\Sigma_\lambda\)-Schubert manifold

Abstract

In a preceding work [1], prove the existence of a lower bound of all admissible functions with sup equal to zero on some toric algebraic manifolds. The considered functions are invariant under the action of automorphisms group, obtained from the one on \(G_{2,4}(\mathbb{C})\) and prove an analogous result on the Grassmannian \(G_{p,p+q}(\mathbb{C})\) by considering an automorphisms group. This gives a new method for computing Tian constant relative to this class of functions.

1. Introduction

The Tian invariant, denoted \(\alpha(M)\), is an important concept in Kahler geometry and is used to study the existence of Einstein-Kahler metrics on compact complex manifolds. It was introduced by Gang Tian and plays a key role in Cheeger-Colding-Tian theory. For a compact complex manifold M and an ample line bundle L over M, the Tian invariant \(\alpha(M)\) is a positive real number that is related to the existence of Einstein-Kahler metrics on M. More precisely, if \(\alpha(M)\) is greater than a certain critical value (which depends on the dimension of the manifold), then there exists an Einstein-Kahler metric on M.

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References

1. Ben Abdesselem, A., Jelloul, R., Enveloppe inférieure de fonctions admissibles sur la Grassmannienne G_2,4(ℂ) en présence de symétries, Bulletin des Sciences Mathématiques, (2013), 139-146.Google Scholar
2. Grivaux, J., Tian constant on Grassmann manifolds. Kähler-Einstein metrics, Journal of Geometric Analysis, 16 (2006), 523-533.Google Scholar
3. Tian, G., On Kähler-Einstein metrics on certain Kähler manifolds with c₁(M) > 0, Inventiones Mathematicae, 89 (1987), 225-246.Google Scholar

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Jelloul, R. (2026). Tian's invariant on \(G_{p,p+q}(\mathbb{C})\) and \(\Sigma_\lambda\)-Schubert manifold. Asia Mathematika, 10(1), 96–115. https://doi.org/10.5281/zenodo.21039614

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