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] Thierry Aubin, Équations du type Monge-Ampère sur les variétés kähleriennes compactes, C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 3, Aiii, A119–A121. MR 0433520
  • [De] Jean-Pierre Demailly, 𝐿² vanishing theorems for positive line bundles and adjunction theory, Transcendental methods in algebraic geometry (Cetraro, 1994) Lecture Notes in Math., vol. 1646, Springer, Berlin, 1996, pp. 1–97. MR 1603616,
  • [Do1] S. K. Donaldson, Scalar curvature and projective embeddings. I, J. Differential Geom. 59 (2001), no. 3, 479–522. MR 1916953
  • [Do2] S. K. Donaldson, Scalar curvature and stability of toric varieties, J. Differential Geom. 62 (2002), no. 2, 289–349. MR 1988506
  • [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] D. Gieseker, Global moduli for surfaces of general type, Invent. Math. 43 (1977), no. 3, 233–282. MR 0498596,
  • [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] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
  • [Hi] F. B. Hildebrand, Introduction to numerical analysis, 2nd ed., McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. International Series in Pure and Applied Mathematics. MR 0347033
  • [HL] Daniel Huybrechts and Manfred Lehn, The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig, 1997. MR 1450870
  • [Ka] Kalle Karu, Minimal models and boundedness of stable varieties, J. Algebraic Geom. 9 (2000), no. 1, 93–109. MR 1713521
  • [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] Steven L. Kleiman, Toward a numerical theory of ampleness, Ann. of Math. (2) 84 (1966), 293–344. MR 0206009,
  • [Ko] János Kollár, Projectivity of complete moduli, J. Differential Geom. 32 (1990), no. 1, 235–268. MR 1064874
  • [La] Robert Lazarsfeld, Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. MR 2095471
  • [Li] Li, J. (1993). Algebraic geometric interpretation of Donaldson's polynomial invariants. Jour. Diff. Geom. 37, 417-466.
  • [Ma] Eben Matlis, The multiplicity and reduction number of a one-dimensional local ring, Proc. London Math. Soc. (3) 26 (1973), 273–288. MR 0313247,
  • [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] David Mumford, Stability of projective varieties, Enseignement Math. (2) 23 (1977), no. 1-2, 39–110. MR 0450272
  • [GIT] D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906
  • [No] D. G. Northcott, A note on the coefficients of the abstract Hilbet function, J. London Math. Soc. 35 (1960), 209–214. MR 0110731,
  • [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] Gang Tian, The 𝐾-energy on hypersurfaces and stability, Comm. Anal. Geom. 2 (1994), no. 2, 239–265. MR 1312688,
  • [Ti2] Gang Tian, Kähler-Einstein metrics with positive scalar curvature, Invent. Math. 130 (1997), no. 1, 1–37. MR 1471884,
  • [V] Eckart Viehweg, Quasi-projective moduli for polarized manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 30, Springer-Verlag, Berlin, 1995. MR 1368632
  • [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] Shouwu Zhang, Heights and reductions of semi-stable varieties, Compositio Math. 104 (1996), no. 1, 77–105. MR 1420712

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