Kirwan surjectivity for the equivariant Dolbeault cohomology. (English) Zbl 1423.53089
Summary: Consider the holomorphic Hamiltonian action of a compact Lie group \(K\) on a compact Kähler manifold \(M\) with a moment map \(\varPhi : M \rightarrow \mathfrak{k}^\ast\). Assume that 0 is a regular value of the moment map. Weitsman raised the question of what we can say about the cohomology of the Kähler quotient \(M_0 := \varPhi^{- 1}(0) / K\) if all the ordinary cohomology of \(M\) is of type \((p, p)\).
In this paper, using the Cartan-Chern-Weil theory we show that in the above context there is a natural surjective Kirwan map from an equivariant version of the Dolbeault cohomology of \(M\) onto the Dolbeault cohomology of the Kähler quotient \(M_0\). As an immediate consequence, this result provides an answer to the question posed by Weitsman.
In this paper, using the Cartan-Chern-Weil theory we show that in the above context there is a natural surjective Kirwan map from an equivariant version of the Dolbeault cohomology of \(M\) onto the Dolbeault cohomology of the Kähler quotient \(M_0\). As an immediate consequence, this result provides an answer to the question posed by Weitsman.
MSC:
| 53C55 | Global differential geometry of Hermitian and Kählerian manifolds |
| 53D20 | Momentum maps; symplectic reduction |
| 32Q20 | Kähler-Einstein manifolds |
References:
| [1] | Battaglia, F.; Zaffran, D., Foliations modeling nonrational simplicial toric varieties, Int. Math. Res. Not. IMRN, 22, 11785-11815 (2015) · Zbl 1351.14032 |
| [2] | G. Bazzoni, I. Biswas, M. Fernández, V. Munoz, Aleksy Tralle, Homotopic Properties of Kähler Orbifolds, Special Metrics and Group Actions in Geometry, pp. 23-57.; G. Bazzoni, I. Biswas, M. Fernández, V. Munoz, Aleksy Tralle, Homotopic Properties of Kähler Orbifolds, Special Metrics and Group Actions in Geometry, pp. 23-57. · Zbl 1411.53053 |
| [3] | Cox, David A.; Little, John B.; Schenck, Henry K., (Toric Varieties. Toric Varieties, Graduate Studies in Mathematics, vol. 124 (2011), American Mathematical Society: American Mathematical Society Providence, Rhode Island) · Zbl 1223.14001 |
| [4] | Deligne, P.; Griffiths, P.; Morgan, J.; Sullivan, D., Real homotopy theory of Kähler manifolds, Invent. Math., 29, 245-274 (1975) · Zbl 0312.55011 |
| [5] | El Kacimi-Alaoui, A., Opérateurs Transversalement elliptique sur un feuilletage riemannien et applications, Compos. Math., 73, 57-106 (1990) · Zbl 0697.57014 |
| [6] | Ginzburg, V. A., Equivariant cohomology and Kähler geometry, Funktsional. Anal. i Prilozhen., 21, 4, 19-34 (1987), 96. English translation in Functional Anal. Appl. 21 (1987) (4) 271-283 · Zbl 0656.53062 |
| [7] | Guillemin, V.; Sternberg, S., Geometric quantization and multiplicities of group representations, Invent. Math., 67, 3, 515-538 (1982) · Zbl 0503.58018 |
| [8] | Guillemin, V.; Sternberg, S., (Supersymmetry and Equivariant de Rham Theory. Supersymmetry and Equivariant de Rham Theory, Mathematics Past and Present (1999), springer-Verlag: springer-Verlag Berlin) · Zbl 0934.55007 |
| [9] | M. Harada, AIM Workshop on Moment Maps and Surjectivity in Various Geometries (August 9-13, 2004): Conjectures and Further Questions, available online at https://aimath.org/WWN/momentmaps/momentmaps.pdf; M. Harada, AIM Workshop on Moment Maps and Surjectivity in Various Geometries (August 9-13, 2004): Conjectures and Further Questions, available online at https://aimath.org/WWN/momentmaps/momentmaps.pdf |
| [10] | Kempf, G.; Ness, L., The length of vectors in representation spaces, (Algebraic Geometry. Algebraic Geometry, Lecture Notes in Mathematics, vol. 732 (1979), Springer Berlin / Heidelberg), 233-243 · Zbl 0407.22012 |
| [11] | Kirwan, Frances, (Cohomology of Quotients in Symplectic and Algebraic Geometry. Cohomology of Quotients in Symplectic and Algebraic Geometry, Mathematical Notes, vol. 31 (1984), Princeton University Press: Princeton University Press Princeton, NJ) · Zbl 0553.14020 |
| [12] | Lerman, E.; Tolman, S., Hamiltonian torus actions on symplectic orbifolds and toric varieties, Trans. AMS, 349, 10, 4201-4230 (1997) · Zbl 0897.58016 |
| [13] | Lillywhite, S., The Topology of Symplectic Quotients of Loop Spaces (1998), The University of Maryland: The University of Maryland College Park, MD, Ph.D. thesis |
| [14] | Lillywhite, S., Formality in an equivariant setting, Trans. Amer. Math. Soc., 335, 7, 2771-2793 (2003) · Zbl 1021.55006 |
| [15] | Y. Lin, R. Sjamaar, Cohomological localization for transverse Lie algebra actions on Riemannian foliations, Preprint, arXiv:1806.01908v1; Y. Lin, R. Sjamaar, Cohomological localization for transverse Lie algebra actions on Riemannian foliations, Preprint, arXiv:1806.01908v1 |
| [16] | Y. Lin, X. Yang, Basic Kirwan injectivity and its applications, Preprint, arXiv:1902.06187v2; Y. Lin, X. Yang, Basic Kirwan injectivity and its applications, Preprint, arXiv:1902.06187v2 |
| [17] | Molino, P., Riemannian Foliations, with Appendices By G. Cairns, Y. Carrière, É. Ghys, E. Salem and V. Sergiescu (1988), Birkhäuser: Birkhäuser Boston · Zbl 0633.53001 |
| [18] | Prato, E., Simple non-rational convex polytopes via symplectic geometry, Topology, 40, 961-975 (2001) · Zbl 1013.53054 |
| [19] | Teleman, C., The quantization conjecture revistied, Ann. of Math. (2), 152, 1, 1-43 (2000) · Zbl 0980.53102 |
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.
