Tian's alpha invariant and special Lagrangian sub manifolds from weighted grassmannians

Abstract

This paper presents a comprehensive study of Tian's α-invariant on weighted Grassmannians G_w(p, p+q) and constructs explicit examples of special Lagrangian submanifolds from Calabi-Yau hypersurfaces in weighted Grassmannians. We develop a systematic framework using group actions to reduce the analysis to diagonal matrices, construct an explicit extremal function Ψ_w, and compute the α-invariant for G_w(2,4) with weights (1,1,1,1,1,m), obtaining the explicit formula α(G_w(2,4)) = min{1/2, 1/m}. Furthermore, we generalize the construction of special Lagrangian submanifolds from G_2,4ℂ to G_2,5ℂ and G_3,6ℂ. We prove that hypersurfaces X_m ⊂ G_2,5ℂ and Y_m ⊂ G_3,6ℂ defined by weighted homogeneous polynomials have vanishing first Chern class and thus are Calabi-Yau manifolds. The real loci L_m = X_m^ℝ and L_m^(3,6) = Y_m^ℝ are shown to be special Lagrangian submanifolds diffeomorphic to circle bundles over ℝℙ⁴ and ℝℙ⁵, respectively. For generic m, these are spherical space forms S⁵/ℤ_5m and S⁸/ℤ_6m (with parity-dependent modifications). A graphical illustration of the α-invariant and tables of topological invariants are provided.

1. Introduction

The study of Kähler-Einstein metrics on Fano manifolds has been a central theme in complex geometry since the resolution of the Calabi conjecture by Yau [7]. A fundamental obstruction to the existence of such metrics is encoded in Tian's α-invariant [6], which measures the singularity of plurisubharmonic functions. For toric varieties, this invariant can be computed combinatorially. For homogeneous spaces like Grassmannians, symmetry arguments provide powerful tools.

This page shows an excerpt of the introduction. The full text, including the theorems, proofs and examples, is available in the PDF.

References

1. A. Ben Abdesselem and R. Jelloul, Enveloppe inférieure de fonctions admissibles sur la Grassmannienne G_2,4ℂ en présence de symétries, Bull. Sci. Math. 138 (2014), 139-146.Google Scholar
2. A. Ben Abdesselem, Enveloppes inférieures de fonctions admissibles sur l'espace projectif complexe. Cas symétrique, Bull. Sci. Math. 130 (2006), 341-353.Google Scholar
3. A. Corti and M. Reid, Weighted Grassmannians, in: Higher Dimensional Varieties and Rational Points, Bolyai Soc. Math. Stud. 12 (2003), 141-163.Google Scholar
4. S. K. Donaldson, Scalar curvature and stability of toric varieties, J. Differential Geom. 62 (2002), 289-349.Google Scholar
5. G. Tian, Kähler-Einstein metrics with positive scalar curvature, Invent. Math. 130 (1997), 1-37.Google Scholar
6. G. Tian, On Kähler-Einstein metrics on certain Kähler manifolds with c₁(M) > 0, Invent. Math. 89 (1987), 225-246.Google Scholar
7. 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.Google Scholar
8. R. Harvey and H. Lawson, Calibrated Geometries, Acta Math. 148 (1982), 47-157.Google Scholar
9. A. Strominger, S.T. Yau, and E. Zaslow, Mirror symmetry is T-duality, Nucl. Phys. B479 (1996), 243-259.Google Scholar
10. P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley & Sons, New York, 1978.Google Scholar
11. P. Griffiths, On the periods of certain rational integrals I, II, Ann. of Math. 90 (1969), 460-541.Google Scholar
12. J. Milnor and J. Stasheff, Characteristic Classes, Princeton University Press, 1974.Google Scholar
13. H. Whitney, Differentiable manifolds, Ann. of Math. 37 (1936), 645-680.Google Scholar
14. J. Wolf, Spaces of Constant Curvature, Publish or Perish, Houston, 1984.Google Scholar
15. A. Ben Abdesselem and R. Jelloul, Special Lagrangian submanifolds from Grassmannians, J. Geom. Phys. 86 (2014), 1-12.Google Scholar
16. A. Corti and M. Reid, Weighted Grassmannians, in: Higher Dimensional Varieties and Rational Points, Bolyai Soc. Math. Stud. 12 (2003), 141-163.Google Scholar
17. I. Dolgachev, Weighted projective varieties, in: Group Actions and Vector Fields, Lecture Notes in Math. 956, Springer (1982), 34-71.Google Scholar
18. E. Arbarello, M. Cornalba, P. Griffiths, and J. Harris, Geometry of Algebraic Curves, Springer-Verlag, 1985.Google Scholar
19. A. Beauville, Vanishing theorems and the Hodge theorem, in: Complex Geometry and Lie Theory, Springer, 1983, pp. 1-28.Google Scholar
20. A. Borel, Linear Algebraic Groups, Benjamin, 1973.Google Scholar
21. E. Calabi, The space of Kähler metrics, Proc. Int. Congress Math. Amsterdam 2 (1954), 206-207.Google Scholar
22. J.-P. Demailly, Complex Analytic and Differential Geometry, available at https://www-fourier.ujf-grenoble.fr/~demailly/, 2012.Google Scholar
23. J.-P. Demailly, A numerical criterion for the existence of Kähler-Einstein metrics on Fano manifolds, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), 597-602.Google Scholar
24. W. Fulton, Intersection Theory, Springer, 1990.Google Scholar
25. P. Griffiths, Periods of integrals on algebraic manifolds, Bull. Amer. Math. Soc. 76 (1970), 535-541.Google Scholar
26. P. Griffiths, Some remarks on the Hodge conjecture, Ann. of Math. 95 (1972), 460-541.Google Scholar
27. P. Griffiths and W. Schmid, Recent developments in Hodge theory: a discussion of techniques and results, in: Discrete Subgroups of Lie Groups and Applications to Moduli, Oxford Univ. Press, 1973, pp. 31-127.Google Scholar
28. F. Hirzebruch, Topological Methods in Algebraic Geometry, Springer, 1964.Google Scholar
29. D. Huybrechts, Complex Geometry: An Introduction, Springer, 2005.Google Scholar
30. R. Jelloul, On the α-invariant of weighted Grassmannians, Preprint, 2020.Google Scholar
31. S. Kobayashi, Differential Geometry of Complex Vector Bundles, Princeton Univ. Press, 1987.Google Scholar
32. N. Koike, On the α-invariant of Fano manifolds, Tokyo J. Math. 13 (1990), 1-16.Google Scholar
33. C. LeBrun, On the scalar curvature of complex surfaces, Geom. Funct. Anal. 5 (1995), 619-628.Google Scholar
34. D. Mumford and J. Shah, The α-invariant and Kähler-Einstein metrics on Fano surfaces, Invent. Math. 83 (1986), 1-28.Google Scholar
35. I. Nakamura, On the α-invariant of Fano 3-folds, Math. Ann. 280 (1988), 1-18.Google Scholar
36. M. Reid, Canonical 3-folds, in: Journées de Géométrie Algébrique d'Angers, 1979, pp. 273-310.Google Scholar
37. R. Robinson, Special Lagrangian submanifolds in Calabi-Yau manifolds, PhD thesis, Oxford Univ., 1977.Google Scholar
38. Y.-T. Siu, The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds, Ann. of Math. 112 (1980), 73-111.Google Scholar
39. A. Strominger and S.-T. Yau, The mirror formula for CY manifolds, Nucl. Phys. B 479 (1996), 243-259.Google Scholar
40. G. Tian, On Calabi's conjecture for complex surfaces with positive first Chern class, Invent. Math. 101 (1990), 101-172.Google Scholar
41. G. Tian, The α-invariant of Fano manifolds, J. Differential Geom. 35 (1992), 507-535.Google Scholar
42. G. Tian, Kähler-Einstein metrics on algebraic manifolds, in: International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), pp. 586-597, Birkhäuser, 1996.Google Scholar
43. G. Tian, The α-invariant of some Fano manifolds, Asian J. Math. 1 (1997), 1-20.Google Scholar
44. A. Weil, Foundations of Algebraic Geometry, Amer. Math. Soc. Colloq. Publ. 29, 1962.Google Scholar
45. S.-T. Yau, Calabi's conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), 1798-1799.Google Scholar
46. S. Zelditch, The α-invariant and the existence of Kähler-Einstein metrics, J. Differential Geom. 48 (1998), 1-28.Google Scholar
47. X. Zhu, On the α-invariant of Fano manifolds, Ph.D. thesis, MIT, 2000.Google Scholar

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Jelloul, R. (2026). Tian's alpha invariant and special Lagrangian sub manifolds from weighted grassmannians. Asia Mathematika, 10(2), 99–123. https://doi.org/10.5281/zenodo.23122661

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