×

Hyperdiscriminant polytopes, Chow polytopes, and Mabuchi energy asymptotics. (English) Zbl 1243.14038

Let us consider a projective complex manifold \(X\) and an ample line bundle \(L\) on \(X\). The Mabuchi energy \(\nu\) is a crucial object to detect the existence of a constant scalar curvature Kähler metric (cscK for short) in \(c_1(L)\). It is an energy functional defined on the Kähler potentials that enjoys some natural geometric properties when there exists a cscK metric in \(c_1(L)\). Given the polarization \(L\), one can also consider the space of holomorphic sections \(H^0(X,L ^k)\) and the associated symmetric space of Bergman metrics \(H_k=\mathrm{GL}(N_k,\mathbb{C})/U(N_k)\) where \(N_k=\dim H^0(X,L^k)\). It is well known from the work of G. Tian than the space of Bergman metrics \(H_k\) is dense in the set of Kähler metrics in the class \(c_1(L)\). Therefore it is natural to understand the behavior of the Mabuchi energy over the space of Bergman metrics and to relate this behavior to the geometry of \((X,L)\).
This paper provides important results in that direction by giving a simple formula for the restriction of the Mabuchi energy \(\nu\) to the space of Bergman metrics. This formula depends on the log norms of the Chow form \(R_X\) and the hyperdiscriminant \(\Delta_X\) of \((X,L^k)\). Let us recall that the Chow form of \(X\) is given by the equation of the divisor \([R_X]=\{ L \in \mathrm{Gr}(N-1-n,N)| L \cap X\neq \emptyset\}\) where \(n=\dim X\). The hyperdiscriminant \(\Delta_X\) of \(X\) is the equation of the dual variety of \(X \times \mathbb{P}^n\) in the corresponding Segre embedding. Namely, if \(\phi_{\sigma}\) is the potential of the Bergman metric induced by \(\sigma\in \mathrm{SL}(N,\mathbb{C})\), then the author proves that \[ \nu(\phi_{\sigma})=\deg(R_X)\log \frac{\| \sigma \cdot \Delta_X \|^2}{\| \Delta_X\|^2}- \deg(\Delta_X)\log \frac{\|\sigma \cdot R_X\|^2}{\| R_X\|^2}. \] There are several nice consequences of this result. For instance, one can see the asymptotic behavior of the Mabuchi energy along any algebraic one parameter subgroup of a maximal algebraic torus of \(\mathrm{SL}(N,\mathbb{C})\). It is completely determined by the Chow polytope and the hyperdiscriminant polytope associated to \(R_X\) and \(\Delta_X\). Furthermore, the boundedness (or properness) of the Mabuchi energy along degenerations in \(\mathrm{SL}(N,\mathbb{C})\) can be rewritten in a geometric way in terms of inclusions of those polytopes (but unfortunately one needs to check the inclusion for all maximal algebraic tori). This leads the author to give a new definition of the notion of K-stability (which is a priori different from the one introduced by G. Tian and S.K. Donaldson) in terms of inclusion of polytopes. Contrarily to the classical notion of K-stability, the deformation of \(X\) does not play the role and so does not use the delicate notion of test-configuration, but requires to check a property on all maximal tori of \(\mathrm{SL}(N,\mathbb{C})\). Then the stability of \((X,L^k)\) as introduced by the author is equivalent to a condition on the positions of the relative polytopes of \(R_X\) and \(\Delta_X\) and hence involves only classical projective complex geometry.

MSC:

14L24 Geometric invariant theory
32Q20 Kähler-Einstein manifolds
51M20 Polyhedra and polytopes; regular figures, division of spaces
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] S. Bando and T. Mabuchi, ”Uniqueness of Einstein Kähler metrics modulo connected group actions,” in Algebraic Geometry, Amsterdam: North-Holland, 1987, vol. 10, pp. 11-40. · Zbl 0641.53065
[2] M. C. Beltrametti, L. M. Fania, and A. J. Sommese, ”On the discriminant variety of a projective manifold,” Forum Math., vol. 4, iss. 6, pp. 529-547, 1992. · Zbl 0780.14023 · doi:10.1515/form.1992.4.529
[3] J. -M. Bismut, H. Gillet, and C. Soulé, ”Analytic torsion and holomorphic determinant bundles. I. Bott-Chern forms and analytic torsion,” Comm. Math. Phys., vol. 115, iss. 1, pp. 49-78, 1988. · Zbl 0658.32025 · doi:10.1007/BF01238853
[4] R. Bott and S. S. Chern, ”Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections,” Acta Math., vol. 114, pp. 71-112, 1965. · Zbl 0148.31906 · doi:10.1007/BF02391818
[5] X. X. Chen and G. Tian, ”Geometry of Kähler metrics and foliations by holomorphic discs,” Publ. Math. Inst. Hautes Études Sci., iss. 107, pp. 1-107, 2008. · Zbl 1182.32009 · doi:10.1007/s10240-008-0013-4
[6] W. Y. Ding and G. Tian, ”Kähler-Einstein metrics and the generalized Futaki invariant,” Invent. Math., vol. 110, iss. 2, pp. 315-335, 1992. · Zbl 0779.53044 · doi:10.1007/BF01231335
[7] S. K. Donaldson, ”Scalar curvature and stability of toric varieties,” J. Differential Geom., vol. 62, iss. 2, pp. 289-349, 2002. · Zbl 1074.53059
[8] S. K. Donaldson, ”Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles,” Proc. London Math. Soc., vol. 50, iss. 1, pp. 1-26, 1985. · Zbl 0529.53018 · doi:10.1112/plms/s3-50.1.1
[9] W. Fulton, Young Tableaux with Applications to Representation Theory and Geometry, Cambridge: Cambridge Univ. Press, 1997, vol. 35. · Zbl 0878.14034
[10] I. M. Gel\('\)fand, M. M. Kapranov, and A. V. Zelevinsky, ”Newton polytopes of the classical resultant and discriminant,” Adv. Math., vol. 84, iss. 2, pp. 237-254, 1990. · Zbl 0721.33002 · doi:10.1016/0001-8708(90)90047-Q
[11] I. M. Gel\('\)fand, M. M. Kapranov, and A. V. Zelevinsky, ”Hyperdeterminants,” Adv. Math., vol. 96, iss. 2, pp. 226-263, 1992. · Zbl 0774.15002 · doi:10.1016/0001-8708(92)90056-Q
[12] I. M. Gel\('\)fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants, Boston, MA: Birkhäuser, 1994. · Zbl 0827.14036 · doi:10.1007/978-0-8176-4771-1
[13] P. Griffiths and J. Harris, ”Algebraic geometry and local differential geometry,” Ann. Sci. École Norm. Sup., vol. 12, iss. 3, pp. 355-452, 1979. · Zbl 0426.14019
[14] P. Griffiths, ”The extension problem in complex analysis. II. Embeddings with positive normal bundle,” Amer. J. Math., vol. 88, pp. 366-446, 1966. · Zbl 0147.07502 · doi:10.2307/2373200
[15] P. Griffiths, ”Hermitian differential geometry, Chern classes, and positive vector bundles,” in Global Analysis (Papers in Honor of K. Kodaira), , 1969, pp. 185-251. · Zbl 0201.24001
[16] M. M. Kapranov, B. Sturmfels, and A. V. Zelevinsky, ”Chow polytopes and general resultants,” Duke Math. J., vol. 67, iss. 1, pp. 189-218, 1992. · Zbl 0780.14027 · doi:10.1215/S0012-7094-92-06707-X
[17] T. Mabuchi, ”\(K\)-energy maps integrating Futaki invariants,” Tohoku Math. J., vol. 38, iss. 4, pp. 575-593, 1986. · Zbl 0619.53040 · doi:10.2748/tmj/1178228410
[18] D. Mumford, ”Stability of projective varieties,” Enseignement Math., vol. 23, iss. 1-2, pp. 39-110, 1977. · Zbl 0363.14003
[19] S. T. Paul, ”Geometric analysis of Chow Mumford stability,” Adv. Math., vol. 182, iss. 2, pp. 333-356, 2004. · Zbl 1050.53061 · doi:10.1016/S0001-8708(03)00081-1
[20] S. T. Paul and G. Tian, ”Analysis of geometric stability,” Int. Math. Res. Not., vol. 2004, iss. 48, pp. 2555-2591, 2004. · Zbl 1076.32018 · doi:10.1155/S1073792804131759
[21] E. A. Tevelev, Projective Duality and Homogeneous Spaces: Invariant Theory and Algebraic Transformation Groups, IV, New York: Springer-Verlag, 2005, vol. 133. · Zbl 1071.14052
[22] G. Tian, ”The \(K\)-energy on hypersurfaces and stability,” Comm. Anal. Geom., vol. 2, iss. 2, pp. 239-265, 1994. · Zbl 0846.32019
[23] G. Tian, ”Kähler-Einstein metrics with positive scalar curvature,” Invent. Math., vol. 130, iss. 1, pp. 1-37, 1997. · Zbl 0892.53027 · doi:10.1007/s002220050176
[24] G. Tian, ”Bott-Chern forms and geometric stability,” Discrete Contin. Dynam. Systems, vol. 6, iss. 1, pp. 211-220, 2000. · Zbl 1022.32009 · doi:10.3934/dcds.2000.6.211
[25] G. Tian, Canonical Metrics in Kähler Geometry, Basel: Birkhäuser, 2000. · Zbl 0978.53002
[26] J. Weyman and A. Zelevinsky, ”Multiplicative properties of projectively dual varieties,” Manuscripta Math., vol. 82, iss. 2, pp. 139-148, 1994. · Zbl 0839.14039 · doi:10.1007/BF02567693
[27] F. L. Zak, Tangents and Secants of Algebraic Varieties, Providence, RI: Amer. Math. Soc., 1993, vol. 127. · Zbl 0795.14018
[28] S. Zhang, ”Heights and reductions of semi-stable varieties,” Compositio Math., vol. 104, iss. 1, pp. 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.