Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



A study of the Hilbert-Mumford criterion for the stability of projective varieties

Authors: Julius Ross and Richard Thomas
Journal: J. Algebraic Geom. 16 (2007), 201-255
Published electronically: November 28, 2006
MathSciNet review: 2274514
Full-text PDF

Abstract | References | Additional Information

Abstract: We make a systematic study of the Hilbert-Mumford criterion for different notions of stability for polarised algebraic varieties $ (X,L)$; in particular for K- and Chow stability. For each type of stability this leads to a concept of slope $ \mu$ for varieties and their subschemes; if $ (X,L)$ is semistable, then $ \mu(Z)\le\mu(X)$ for all $ Z\subset X$. We give examples such as curves, canonical models and Calabi-Yaus. We prove various foundational technical results towards understanding the converse, leading to partial results; in particular this gives a geometric (rather than combinatorial) proof of the stability of smooth curves.

References [Enhancements On Off] (What's this?)

  • [Au] Aubin, T. (1976). Équations du type Monge-Ampère sur les variétés kähleriennes compactes. C. R. Acad. Sci. Paris Sr. A-B 283, A119-A121. MR 0433520 (55:6496)
  • [De] Demailly, J.-P (1994). $ L\sp 2$ vanishing theorems for positive line bundles and adjunction theory, Transcendental methods in algebraic geometry (Cetraro), 1-97. MR 1603616 (99k:32051)
  • [Do1] Donaldson, S. K. (2001). Scalar curvature and projective embeddings, I. Jour. Diff. Geom. 59, 479-522. MR 1916953 (2003j:32030)
  • [Do2] Donaldson, S. K. (2002). Scalar curvature and stability of toric varieties. Jour. Diff. Geom. 62, 289-349. MR 1988506 (2005c:32028)
  • [FR] Fine, J. and Ross, J. (2006) A note on positivity of the CM line bundle. To appear in Int. Math. Res. Notices. math.AG/0605302.
  • [Fu] Fulton, W. (1984). Intersection theory. Springer-Verlag, Berlin. MR 0732620 (85k:14004)
  • [Gi] Gieseker, D. (1977). Global moduli for surfaces of general type. Invent. Math. 43, 233-282. MR 0498596 (58:16687)
  • [Gr] Grothendieck, A. (1960/61). Techniques de construction et théorèmes d'existence en géométrie algébrique. IV. Les schémas de Hilbert. Séminaire Bourbaki, No. 221.
  • [Ha] Hartshorne, R. (1977). Algebraic Geometry. Graduate Texts in Mathematics 52, Springer-Verlag. MR 0463157 (57:3116)
  • [Hi] Hildebrand, F. B. (1974). Introduction to numerical analysis, 2nd Ed., McGraw-Hill, New York. MR 0347033 (49:11753)
  • [HL] Huybrechts, D. and Lehn, M. (1997). Geometry of moduli spaces of shaves. Aspects in Mathematics Vol. E31, Vieweg. MR 1450870 (98g:14012)
  • [Ka] Karu, K. (2000). Minimal models and boundedness of stable varieties. Jour. Alg. Geom. 9, 93-109. MR 1713521 (2001g:14059)
  • [Ke] Kempf, G. (1978). Instability in invariant theory. Ann. of Math. 108, 299-316. MR 0506989 (80c:20057)
  • [KM] Kirby, D. and Mehran, A. (1982) A note on the coefficients of the Hilbert-Samuel polynomial for a Cohen-Macaulay module. Jour. London Math. Soc. 25, 449-457. MR 0657501 (84a:13022)
  • [Kl] Kleiman, S. L. (1966). Toward a numerical theory of ampleness. Ann. of Math. 84, 293-344. MR 0206009 (34:5834)
  • [Ko] Kollár, J. (1994). Projectivity of complete moduli. Jour. Diff. Geom 32, 235-268. MR 1064874 (92e:14008)
  • [La] Lazarsfeld, R. (2004). Positivity in algebraic geometry. I. Classical setting: line bundles and linear series. Ergeb. Math. Grenzgeb. (3), Springer-Verlag. MR 2095471 (2005k:14001a)
  • [Li] Li, J. (1993). Algebraic geometric interpretation of Donaldson's polynomial invariants. Jour. Diff. Geom. 37, 417-466.
  • [Ma] Matlis, E. (1973). The multiplicity and reduction number of a one-dimensional local ring. Proc. London Math. Soc. 26, 273-288. MR 0313247 (47:1802)
  • [Mo] Morrison, I. (1980). Projective stability of ruled surfaces. Invent. Math. 56, 269-304. MR 0561975 (81c:14007)
  • [Mor] Mori, S. (1982). Threefolds whose canonical bundles are not numerically effective. Ann. of Math. 116, 133-176. MR 0662120 (84e:14032)
  • [Mu] Mumford, D. (1977). Stability of projective varieties. Enseignement Math. (2) 23, 39-110. MR 0450272 (56:8568)
  • [GIT] Mumford, D., Fogarty, J. and Kirwan, F. (1994). Geometric Invariant Theory. Third edition, Erg. Math. 34, Springer-Verlag, Berlin. MR 1304906 (95m:14012)
  • [No] Northcott, D. (1960). A note on the coefficients of the abstract Hilbert function. J. London Math. Soc. 35, 209-214. MR 0110731 (22:1599)
  • [PT] Paul, S. and Tian, G. (2004). Algebraic and Analytic K-Stability. Preprint math.DG/0405530.
  • [Ro] Ross, J. (2003). Instability of polarised algebraic varieties. Ph.D. thesis, Imperial College.
  • [RT] Ross, J. and Thomas, R. P. (2004). An obstruction to the existence of constant scalar curvature Kähler metrics. Jour. Diff. Geom. 72, 429-466.
  • [Sz] Székelyhidi, G. (2004). Extremal metrics and K-stability. To appear in Bull. LMS. math.AG/0410401.
  • [Ti1] Tian, G. (1994). The $ K$-energy on hypersurfaces and stability. Comm. Anal. Geom. 2, 239-265. MR 1312688 (95m:32030)
  • [Ti2] Tian, G. (1997). Kähler-Einstein metrics with positive scalar curvature. Invent. Math. 130, 1-37. MR 1471884 (99e:53065)
  • [V] Viehweg, E. (1995). Quasi-projective moduli for polarized manifolds. Erg. Math. (3) 30. Springer-Verlag, Berlin. MR 1368632 (97j:14001)
  • [Y] Yau, S.-T. (1978). On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I. Comm. Pure Appl. Math. 31, 339-411. MR 0480350 (81d:53045)
  • [Zh] Zhang, S. (1996). Heights and reductions of semi-stable varieties. Compositio Math. 104, 77-105. MR 1420712 (97m:14027)

Additional Information

Julius Ross
Affiliation: Department of Mathematics, Columbia University, New York, New York 10027

Richard Thomas
Affiliation: Department of Mathematics, Imperial College, London SW7 2AZ, United Kingdom

Received by editor(s): April 1, 2005
Received by editor(s) in revised form: May 1, 2006
Published electronically: November 28, 2006
Additional Notes: The first author was supported by an EPSRC Ph.D. studentship. The second author was partially supported by the Royal Society and the Leverhulme Trust.

American Mathematical Society