## 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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