Tian’s Invariant for Flag and Schubert manifolds in the Complex Grassmannian
Palabras clave:
Flag manifold, Schubert variety, Tian’s invariantResumen
We compute Tian’s invariant α(M) for two classes of Kähler manifolds related to the complex Grassmannian Gp,q(C). First, we consider the complete flag variety F(d1, . . . ,dr;n) embedded in a product of Grassmannians. We prove that Tian’s invariant is equal to 1/n, where n is the ambient dimension. Second, we investigate smooth Schubert varieties in Gp,q(C).We establish that for smooth Schubert varieties isomorphic to products of smaller Grassmannians, the invariant is given by the inverse of the minimal dimension of the ambient space of the factors. The methods rely on the transitive action of the unitary group, natural embeddings of products of projective lines, and explicit convergence analysis of integrals involving admissible functions.
Referencias
1. T. Aubin, Réduction du cas positif de l’équation de Monge-Ampère sur les variétés kählériennes compactes à la démonstration d’une inégalité, J. Funct. Anal. 57 (1984), 143–153.
2. W. Fulton, Young Tableaux: With Applications to Representation Theory and Geometry, Cambridge University Press, 1997.
3. P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley & Sons, 1978 (Originally published 1974).
4. J. Grivaux, Tian’s Invariant of the Grassmann Manifold, The Journal of Geometric Analysis, Vol. 16, No. 3, pp. 461–471, 2006.
5. G. Tian, On Kähler-Einstein metrics on certain Kähler manifolds with C1(M) > 0, Invent. Math. 89 (1987), 225–246.
6. S.-T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation I, Comm. Pure Appl. Math. 31 (1978), 339–411.
Descargas
Publicado
Número
Sección
Licencia
Derechos de autor 2026 Matemática
Esta obra está bajo una licencia internacional Creative Commons Atribución-NoComercial 4.0.
