Toric Kempf--Ness Sets

T. E. Panov

Informatics and Automation, Geometry, topology, and mathematical physics. I, Tome 263 (2008), pp. 159-172.

Résumé

In the theory of algebraic group actions on affine varieties, the concept of a Kempf–Ness set is used to replace the categorical quotient by the quotient with respect to a maximal compact subgroup. Using recent achievements of “toric topology”, we show that an appropriate notion of a Kempf–Ness set exists for a class of algebraic torus actions on quasiaffine varieties (coordinate subspace arrangement complements) arising in the Batyrev–Cox “geometric invariant theory” approach to toric varieties. We proceed by studying the cohomology of these “toric” Kempf–Ness sets. In the case of projective nonsingular toric varieties the Kempf–Ness sets can be described as complete intersections of real quadrics in a complex space.

Bibliographie

[1] Baskakov I. V., “Troinye proizvedeniya Massi v kogomologiyakh moment-ugol kompleksov”, UMN, 58:5 (2003), 199–200 | MR | Zbl

[2] Bukhshtaber V. M., Panov T. E., Toricheskie deistviya v topologii i kombinatorike, MTsNMO, M., 2004 | MR

[3] Vinberg E. B., Popov V. L., “Teoriya invariantov”, Algebraicheskaya geometriya – 4, Itogi nauki i tekhniki. Sovr. probl. matematiki. Fund. napr., 55, VINITI, M., 1989, 137–309 | MR | Zbl

[4] Grbich E., Terio S., “Gomotopicheskii tip dopolneniya konfiguratsii koordinatnykh podprostranstv korazmernosti dva”, UMN, 59:6 (2004), 203–204 | MR | Zbl

[5] Danilov V. I., “Geometriya toricheskikh mnogoobrazii”, UMN, 33:2 (1978), 85–134 | MR | Zbl

[6] Batyrev V. V., “Quantum cohomology rings of toric manifolds”, Astérisque, 218, 1993, 9–34 ; arxiv: alg-geom/9310004v2 | MR | Zbl

[7] Bosio F., Meersseman L., “Real quadrics in $\mathbb C^n$, complex manifolds and convex polytopes”, Acta math., 197:1 (2006), 53–127 | DOI | MR | Zbl

[8] Buchstaber V. M., Panov T. E., Torus actions and their applications in topology and combinatorics, Univ. Lect. Ser., 24, Amer. Math. Soc., Providence, RI, 2002 | MR | Zbl

[9] Buchstaber V. M., Panov T. E., Ray N., “Spaces of polytopes and cobordism of quasitoric manifolds”, Moscow Math. J., 7:2 (2007), 219–242 ; arxiv: math/0609346v2[math.AT] | MR | Zbl

[10] Cox D. A., “The homogeneous coordinate ring of a toric variety”, J. Alg. Geom., 4:1 (1995), 17–50 ; arxiv: alg-geom/9210008v2 | MR | Zbl

[11] Cox D. A., “Recent developments in toric geometry”, Algebraic geometry, Proc. Summer Res. Inst. (Santa Cruz, 1995), Proc. Symp. Pure Math., 62, Pt. 2, Amer. Math. Soc., Providence, RI, 1997, 389–436 ; arxiv: alg-geom/9606016v1 | MR | Zbl

[12] Davis M. W., Januszkiewicz T., “Convex polytopes, Coxeter orbifolds and torus actions”, Duke Math. J., 62:2 (1991), 417–451 | DOI | MR | Zbl

[13] Denham G., SuciuA. I., “Moment-angle complexes, monomial ideals, and Massey products”, Pure and Appl. Math. Quart., 3:1 (2007), 25–60 ; arxiv: math/0512497v4[math.AT] | MR | Zbl

[14] Hochster M., “Cohen–Macaulay rings, combinatorics, and simplicial complexes”, Ring theory II, Proc. 2nd Oklahoma Conf., eds. B. R. McDonald, R. A. Morris, M. Dekker, New York, 1977, 171–223 | MR

[15] Masuda M., Panov T., “On the cohomology of torus manifolds”, Osaka J. Math., 43 (2006), 711–746 ; arxiv: math/0306100v2[math.AT] | MR | Zbl

[16] Panov T., “Cohomology of face rings, and torus actions”, Surveys in contemporary mathematics, LMS Lect. Note Ser., 347, Cambridge Univ. Press, Cambridge, 2008, 165–201 ; arxiv: math/0506526v3[math.AT] | MR | Zbl

[17] Schwarz G. W., “The topology of algebraic quotients”, Topological methods in algebraic transformation groups, Progr. Math., 80, Birkhäuser, Boston, 1989, 135–151 | MR

[18] Stanley R. P., Combinatorics and commutative algebra, 2nd ed., Progr. Math., 41, Birkhäuser, Boston, 1996 | MR | Zbl