×

Compact Kähler quotients of algebraic varieties and geometric invariant theory. (English) Zbl 1216.14044

This is a continuation of the previous work of the author [Trans. Am. Math. Soc. 362, No. 6, 3243–3271 (2010; Zbl 1216.14045)]. Let \(G\) be a complex reductive group with a linearized action on a smooth polarized complex manifold \(X\). Let \(K\) be the maximal compact subgroup of \(G\). The action of \(K\) is Hamiltonian. If one considers the induced moment map \(\mu: X\rightarrow {Lie}(K)^*\), then it is well known from geometric invariant theory (GIT) that the two quotients \(X/\!\!/G\) and \(\mu^{-1}(0)/K\) are symplectomorphic.
In the paper under review, the author studies the more general case of a complex algebraic irreducible \(G\)-variety \(X\) equipped with a symplectic form (with respect to which the action of \(K\) is still Hamiltonian) which is no longer necessarily the curvature form of a connection on an ample line bundle. Let us give some details. Recall that \(X\) is a \(G\)-variety if the action map \(G\times X\rightarrow X\) is regular. Furthermore, it is irreducible if \(G\) acts transitively on the set of irreducible components of \(X\). In this setting, one can still define the set \(X(\mu)\) of semistable points. They are the points whose closure of the \(G\)-orbit meets the \(0\)-level set of the moment map. Now, one says that \(X\) has only 1-rational singularities if for any resolution of singularities \(f:\tilde{X}\rightarrow X\), the sheaf \(R^1 f_* \mathcal{O}_{\tilde{X}}\) vanishes.
The main result of this paper says that if the level set of the moment map is compact and non empty, and \(X\) has at worst 1-rational singularities, then one has the following nice consequences, the first two of which were proved in the previous work of the author [loc. cit.]: The set \(X(\mu)\subset X\) is Zariski open, the analytic Hilbert quotient \(X(\mu)/\!\!/G\) is a projective algebraic variety, the map \(X(\mu)\rightarrow X(\mu)/\!\!/G\) is a good quotient, and finally, \(X(\mu)\) coincides with the GIT-semistable set for the linearization of the \(G\)-action with respect to a certain Weil divisor. The generalization of the geometric invariant theory for linearizations induced by Weil divisor has been studied by J. Hausen [Compos. Math. 140, No. 6, 1518–1536 (2004; Zbl 1072.14057)].
The finiteness of the momentum map quotients for a given \(G\)-variety is also discussed. The author shows that there exist only finitely many subsets of \(X\) that can be realized as the set of semistable points with respect to some \(K\)-invariant Kähler structure and some momentum map \(\mu\) with compact zero fibre \(\mu^{-1}(0)\). A refinement of the main result is given in the case the group \(G\) is semi-simple.
Finally, to sum up, this nice work shows that momentum map quotients of algebraic Hamiltonian \(G\)-varieties have very strong algebraic properties.

MSC:

14L30 Group actions on varieties or schemes (quotients)
14L24 Geometric invariant theory
32M05 Complex Lie groups, group actions on complex spaces
53D20 Momentum maps; symplectic reduction
53C55 Global differential geometry of Hermitian and Kählerian manifolds
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Berchtold, F.; Hausen, J., GIT equivalence beyond the ample cone, Michigan Math. J., 54, 3, 483-515 (2006) · Zbl 1171.14028
[2] Białynicki-Birula, A., Finiteness of the number of maximal open subsets with good quotients, Transform. Groups, 3, 4, 301-319 (1998) · Zbl 0940.14034
[3] Białynicki-Birula, A., Quotients by actions of groups, (Algebraic Quotients. Torus Actions and Cohomology. The Adjoint Representation and the Adjoint Action. Algebraic Quotients. Torus Actions and Cohomology. The Adjoint Representation and the Adjoint Action, Encyclopaedia Math. Sci., vol. 131 (2002), Springer: Springer Berlin), 1-82 · Zbl 1061.14046
[4] Białynicki-Birula, A.; Święcicka, J., On exotic orbit spaces of tori acting on projective varieties, (Group Actions and Invariant Theory. Group Actions and Invariant Theory, Montreal, PQ, 1988. Group Actions and Invariant Theory. Group Actions and Invariant Theory, Montreal, PQ, 1988, CMS Conf. Proc., vol. 10 (1989), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 25-30 · Zbl 0705.14043
[5] Dolgachev, I. V.; Hu, Y., Variation of geometric invariant theory quotients, Inst. Hautes Études Sci. Publ. Math., 87, 5-56 (1998), with an appendix by Nicolas Ressayre · Zbl 1001.14018
[6] Fischer, G., Ein relativer Satz von Chow und die Elimination der Unbestimmtheitsstellen meromorpher Funktionen, Math. Ann., 217, 2, 145-152 (1975) · Zbl 0313.32035
[7] Fischer, G., Complex Analytic Geometry, Lecture Notes in Math., vol. 538 (1976), Springer-Verlag: Springer-Verlag Berlin · Zbl 0343.32002
[8] Fujiki, A., Kählerian normal complex surfaces, Tohoku Math. J. (2), 35, 1, 101-117 (1983) · Zbl 0562.32015
[9] Fujiki, A., Kähler quotient and equivariant cohomology, (Moduli of Vector Bundles. Moduli of Vector Bundles, Sanda, 1994; Kyoto, 1994. Moduli of Vector Bundles. Moduli of Vector Bundles, Sanda, 1994; Kyoto, 1994, Lect. Notes Pure Appl. Math., vol. 179 (1996), Dekker: Dekker New York), 39-53 · Zbl 0883.14028
[10] Grauert, H., Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann., 146, 331-368 (1962) · Zbl 0173.33004
[12] Greb, D., Projectivity of analytic Hilbert and Kaehler quotients (2008), Trans. Amer. Math. Soc., in press · Zbl 1294.14003
[13] Greb, D., 1-Rational singularities and quotients by reductive groups (2009)
[14] Greb, D., Rational singularities and quotients by holomorphic group actions (2009)
[15] Greb, D.; Heinzner, P., Kaehlerian reduction in steps, (Campbell, E.; Helminck, A. G.; Kraft, H.; Wehlau, D., (In Honour of Gerry Schwarz). (In Honour of Gerry Schwarz), Progr. Math., vol. 278 (2010), Birkhäuser: Birkhäuser Boston), pp. 63-82 · Zbl 1203.53083
[16] Grothendieck, A., Revêtements étales et groupe fondamental (SGA 1), Lecture Notes in Math., vol. 224 (1971), Springer-Verlag: Springer-Verlag Berlin
[17] Guillemin, V.; Sternberg, S., Geometric quantization and multiplicities of group representations, Invent. Math., 67, 3, 515-538 (1982) · Zbl 0503.58018
[18] Hartshorne, R., Algebraic Geometry, Grad. Texts in Math., vol. 52 (1977), Springer-Verlag: Springer-Verlag New York · Zbl 0367.14001
[19] Hashimoto, M., “Geometric quotients are algebraic schemes” based on Fogarty’s idea, J. Math. Kyoto Univ., 43, 4, 807-814 (2004) · Zbl 1129.14066
[20] Hausen, J., Geometric invariant theory based on Weil divisors, Compos. Math., 140, 6, 1518-1536 (2004) · Zbl 1072.14057
[21] Hausen, J.; Heinzner, P., Actions of compact groups on coherent sheaves, Transform. Groups, 4, 1, 25-34 (1999) · Zbl 0933.32033
[22] Heinzner, P., Geometric invariant theory on Stein spaces, Math. Ann., 289, 4, 631-662 (1991) · Zbl 0728.32010
[23] Heinzner, P.; Loose, F., Reduction of complex Hamiltonian \(G\)-spaces, Geom. Funct. Anal., 4, 3, 288-297 (1994) · Zbl 0816.53018
[24] Heinzner, P.; Migliorini, L., Projectivity of moment map quotients, Osaka J. Math., 38, 1, 167-184 (2001) · Zbl 0982.32020
[25] Heinzner, P.; Huckleberry, A.; Loose, F., Kählerian extensions of the symplectic reduction, J. Reine Angew. Math., 455, 123-140 (1994) · Zbl 0803.53042
[26] Heinzner, P.; Migliorini, L.; Polito, M., Semistable quotients, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 26, 2, 233-248 (1998) · Zbl 0922.32017
[27] Kapranov, M., Chow quotients of Grassmannians I, (I.M. Gel’fand Seminar. I.M. Gel’fand Seminar, Adv. Soviet Math., vol. 16(2) (1993), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 29-110 · Zbl 0811.14043
[28] Kirwan, F. C., Cohomology of Quotients in Symplectic and Algebraic Geometry, Math. Notes, vol. 31 (1984), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0553.14020
[29] Kollár, J., Rational Curves on Algebraic Varieties, Ergeb. Math. Grenzgeb. (3), vol. 32 (1996), Springer-Verlag: Springer-Verlag Berlin
[30] Lieberman, D. I., Compactness of the Chow scheme: applications to automorphisms and deformations of Kähler manifolds, (Fonctions de Plusieurs Variables Complexes, III, Sém. François Norguet, 1975-1977. Fonctions de Plusieurs Variables Complexes, III, Sém. François Norguet, 1975-1977, Lecture Notes in Math., vol. 670 (1978), Springer: Springer Berlin), 140-186
[31] Luna, D., Slices étales, (Bull. Soc. Math. France Mem., vol. 33 (1973), Soc. Math. France: Soc. Math. France Paris), 81-105 · Zbl 0286.14014
[32] Luna, D., Fonctions différentiables invariantes sous l’opération d’un groupe réductif, Ann. Inst. Fourier (Grenoble), 26, 1, 33-49 (1976) · Zbl 0315.20039
[33] Matsushima, Y., Espaces homogènes de Stein des groupes de Lie complexes, Nagoya Math. J., 16, 205-218 (1960) · Zbl 0094.28201
[34] Morosawa, S.; Nishimura, Y.; Taniguchi, M.; Ueda, T., Holomorphic Dynamics, Cambridge Stud. Adv. Math., vol. 66 (2000), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0979.37001
[35] Mumford, D.; Fogarty, J.; Kirwan, F. C., Geometric Invariant Theory, Ergeb. Math. Grenzgeb. (2), vol. 34 (1994), Springer-Verlag: Springer-Verlag Berlin · Zbl 0797.14004
[36] Namikawa, Y., Projectivity criterion of Moishezon spaces and density of projective symplectic varieties, Internat. J. Math., 13, 2, 125-135 (2002) · Zbl 1055.32015
[37] Neeman, A., A weak GAGA statement for arbitrary morphisms, Proc. Amer. Math. Soc., 100, 3, 429-432 (1987) · Zbl 0622.14013
[38] Neeman, A., Analytic questions in geometric invariant theory, (Invariant Theory. Invariant Theory, Denton, TX, 1986. Invariant Theory. Invariant Theory, Denton, TX, 1986, Contemp. Math., vol. 88 (1989), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 11-23
[39] Rosenlicht, M., Some basic theorems on algebraic groups, Amer. J. Math., 78, 401-443 (1956) · Zbl 0073.37601
[40] Schmitt, A., Walls for Gieseker semistability and the Mumford-Thaddeus principle for moduli spaces of sheaves over higher dimensional bases, Comment. Math. Helv., 75, 2, 216-231 (2000) · Zbl 0981.14008
[41] Schröer, S., On non-projective normal surfaces, Manuscripta Math., 100, 3, 317-321 (1999) · Zbl 0987.14031
[42] Serre, J.-P., Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier (Grenoble), 6, 1-42 (1955-1956) · Zbl 0075.30401
[43] Shafarevich, I. R., Basic Algebraic Geometry. 2 (1994), Springer-Verlag: Springer-Verlag Berlin · Zbl 0675.14001
[44] Sjamaar, R., Holomorphic slices, symplectic reduction and multiplicities of representations, Ann. of Math. (2), 141, 1, 87-129 (1995) · Zbl 0827.32030
[45] Snow, D. M., Reductive group actions on Stein spaces, Math. Ann., 259, 1, 79-97 (1982) · Zbl 0509.32021
[46] Sumihiro, H., Equivariant completion, J. Math. Kyoto Univ., 14, 1-28 (1974) · Zbl 0277.14008
[47] Thaddeus, M., Geometric invariant theory and flips, J. Amer. Math. Soc., 9, 3, 691-723 (1996) · Zbl 0874.14042
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.