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Gromov invariants for holomorphic maps from Riemann surfaces to Grassmannians. (English) Zbl 0865.14017
The space \({\mathcal M}\) of holomorphic mappings of fixed degree \(d\) from a compact Riemannian surface of genus \(g\) to the Grassmannian of complex \(r\)-planes in \(\mathbb{C}^k\) are considered under the assumptions \(g\geq 1\) and \(d>2r(g-1)\). The Uhlenbeck compactification [see J. Sacks and K. Uhlenbeck, Ann. Math., II. Ser. 113, 1-24 (1981; Zbl 0462.58014) and J. E. Wolfson, J. Differ. Geom. 28, No. 3, 383-405 (1988; Zbl 0661.53024)] of \({\mathcal M}\) is shown to have the structure of a projective variety. Also an algebraic compactification of \({\mathcal M}\) is considered, which is obtained as particular case of the Grothendieck Quot Scheme [A. Grothendieck, “Techniques de construction et théorèmes d’existence en géometrie algébrique. IV: Les schémas de Hilbert”, Sémin. Bourbaki 13 (1960/61), No. 221 (1961; Zbl 0236.14003)]. It is shown that there is an algebraic surjection from the latter compactification to the first, which is an isomorphism in case of projective space (i.e. \(r=1)\). This latter compactification is shown to embed into the moduli space of solutions of a certain nonlinear partial differential equation, i.e. a generalized version of the vortex equations [see S. B. Bradlow and G. D. Daskalopoulos, Int. J. Math. 2, No. 5, 477-513 (1991; Zbl 0759.32013)]. That this can be used to calculate the Gromov invariants [certain intersection numbers, see M. Gromov in: Proc. Int. Congr. Math., Berkeley 1986, Vol. 1, 81-98 (1987; Zbl 0664.53016)] is demonstrated for a Riemannian surface of genus one.
Reviewer: A.Kriegl (Wien)

MSC:
14H55 Riemann surfaces; Weierstrass points; gap sequences
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14M15 Grassmannians, Schubert varieties, flag manifolds
14D20 Algebraic moduli problems, moduli of vector bundles
32G13 Complex-analytic moduli problems
58D27 Moduli problems for differential geometric structures
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References:
[1] M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523 – 615. · Zbl 0509.14014 · doi:10.1098/rsta.1983.0017 · doi.org
[2] E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. · Zbl 0559.14017
[3] Steven B. Bradlow, Special metrics and stability for holomorphic bundles with global sections, J. Differential Geom. 33 (1991), no. 1, 169 – 213. · Zbl 0697.32014
[4] Steven B. Bradlow and Georgios D. Daskalopoulos, Moduli of stable pairs for holomorphic bundles over Riemann surfaces, Internat. J. Math. 2 (1991), no. 5, 477 – 513. · Zbl 0759.32013 · doi:10.1142/S0129167X91000272 · doi.org
[5] Bradlow, S. B. and G. D. Daskalopoulos, Moduli of stable pairs for holomorphic bundles over Riemann surfaces II, Int. J. Math. 4 (No. 6) (1993), 903–925. CMP 94:05 · Zbl 0798.32020
[6] Bradlow, S., G. Daskalopoulos, and R. Wentworth, Birational equivalences of vortex moduli, Topology (to appear). · Zbl 0856.32019
[7] Raoul Bott and Loring W. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics, vol. 82, Springer-Verlag, New York-Berlin, 1982. · Zbl 0496.55001
[8] Aaron Bertram, Moduli of rank-2 vector bundles, theta divisors, and the geometry of curves in projective space, J. Differential Geom. 35 (1992), no. 2, 429 – 469. · Zbl 0787.14014
[9] A. Bruguières, The scheme of morphisms from an elliptic curve to a Grassmannian, Compositio Math. 63 (1987), no. 1, 15 – 40. · Zbl 0664.14005
[10] Daskalopoulos, G. and K. Uhlenbeck, An application of transversality to the topology of the moduli space of stable bundles, Topology 34 (1994), 203–215. · Zbl 0835.58005
[11] Andreas Floer, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys. 120 (1989), no. 4, 575 – 611. · Zbl 0755.58022
[12] William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. · Zbl 0541.14005
[13] M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307 – 347. · Zbl 0592.53025 · doi:10.1007/BF01388806 · doi.org
[14] Mikhael Gromov, Soft and hard symplectic geometry, ICM Series, American Mathematical Society, Providence, RI, 1988. A plenary address presented at the International Congress of Mathematicians held in Berkeley, California, August 1986; Introduced by Jeff Cheeger. · Zbl 0917.53012
[15] Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. · Zbl 0408.14001
[16] V. Guillemin and S. Sternberg, Birational equivalence in the symplectic category, Invent. Math. 97 (1989), no. 3, 485 – 522. · Zbl 0683.53033 · doi:10.1007/BF01388888 · doi.org
[17] Séminaire Bourbaki, 13ième année: 1960/61. Textes des conférences. Exposés 205 à 222, Secrétariat mathématique, Paris, 1961 (French). Deuxième édition, corrigée. 3 fascicules.
[18] Kenneth Intriligator, Fusion residues, Modern Phys. Lett. A 6 (1991), no. 38, 3543 – 3556. · Zbl 1020.81847 · doi:10.1142/S0217732391004097 · doi.org
[19] Frances Kirwan, On the homology of compactifications of moduli spaces of vector bundles over a Riemann surface, Proc. London Math. Soc. (3) 53 (1986), no. 2, 237 – 266. · Zbl 0607.14017 · doi:10.1112/plms/s3-53.2.237 · doi.org
[20] Shoshichi Kobayashi, Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, vol. 15, Princeton University Press, Princeton, NJ; Princeton University Press, Princeton, NJ, 1987. Kanô Memorial Lectures, 5. · Zbl 0708.53002
[21] P. E. Newstead, Introduction to moduli problems and orbit spaces, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 51, Tata Institute of Fundamental Research, Bombay; by the Narosa Publishing House, New Delhi, 1978. · Zbl 0411.14003
[22] Ruan, Y., Toplogical sigma model and Donaldson type invariants in Gromov theory, preprint, 1993.
[23] J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. of Math. (2) 113 (1981), no. 1, 1 – 24. · Zbl 0462.58014 · doi:10.2307/1971131 · doi.org
[24] Stein Arild Strømme, On parametrized rational curves in Grassmann varieties, Space curves (Rocca di Papa, 1985) Lecture Notes in Math., vol. 1266, Springer, Berlin, 1987, pp. 251 – 272. · Zbl 0625.14027 · doi:10.1007/BFb0078187 · doi.org
[25] Michael Thaddeus, Conformal field theory and the cohomology of the moduli space of stable bundles, J. Differential Geom. 35 (1992), no. 1, 131 – 149. · Zbl 0772.53013
[26] Michael Thaddeus, Stable pairs, linear systems and the Verlinde formula, Invent. Math. 117 (1994), no. 2, 317 – 353. · Zbl 0882.14003 · doi:10.1007/BF01232244 · doi.org
[27] Tiwari, S., preprint.
[28] Cumrun Vafa, Topological mirrors and quantum rings, Essays on mirror manifolds, Int. Press, Hong Kong, 1992, pp. 96 – 119. · Zbl 0827.58073
[29] J. G. Wolfson, Gromov’s compactness of pseudo-holomorphic curves and symplectic geometry, J. Differential Geom. 28 (1988), no. 3, 383 – 405. · Zbl 0661.53024
[30] Edward Witten, Topological sigma models, Comm. Math. Phys. 118 (1988), no. 3, 411 – 449. · Zbl 0674.58047
[31] Zagier, D., unpublished.
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