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Symplectic connections on a Riemann surface and holomorphic immersions in the Lagrangian homogeneous space. (English) Zbl 1159.14010

Let \(X\) be a compact connected Riemann surface and fix a universal cover \(\gamma: \tilde{X}\rightarrow X\). Let \(V\) be a finite dimensional complex vector space equipped with a symplectic form \(\omega\) and \(\dim(V)=2n\). Let \(Gr_L\) be the subvariety of \(Gr(n,V)\) parametrizing all Lagragian subspaces. Consider \(A_X\) the space of all equivalence classes of holomorphic maps (actually they are immersions) from \(\tilde{X} \) to \(Gr_L\) such that
1) the differential \(df\), seen as a symmetric bilinear form on the pullback to \(\tilde{X}\) of the tautological bundle over \(Gr_L\), is fibrewise nondegenerate.
2) there is a morphism \(\rho_f: Gal(\gamma) \rightarrow Sp(V)\) from the Galois group for the covering \(\gamma\) to the group of automorphisms of \(V\) preserving \(\omega\), and this morphism satisfies: \[ \rho_f(g)(f(z))= f(g(z)) \] for all \(z \in \tilde{X}\) and all \(g \in Gal(\gamma)\), i.e \(f\) is equivariant with respect to \(\rho_f\). Note that two elements \(f_1,f_2:\tilde{X}\rightarrow Gr_L\) are equivalent if they differ by the action of some fixed element in \(Sp(V)\).
A flat \(O(n,\mathbb{C})\)-bundle over \(X\) is parametrized by a triple \((F,B,\nabla)\) where \(F\) is a holomorphic vector bundle of rank \(n\) over \(X\), \(B\) is a holomorphic symmetric bilinear form on \(F\) which is fibrewise non degenerate and \(\nabla\) is a holomorphic connection on \(F\) preserving \(B\). On another hand, for a flat \(O(n,\mathbb{C})\)-bundle over \(X\) we can define the quotient group \(PO(n,\mathbb{C})=O(n,\mathbb{C})/\{\pm Id\}\) and two flat \(O(n,\mathbb{C})\)-bundles are said to be equivalent if the corresponding \(PO(n,\mathbb{C})\)-bundles over \(X\) are isomorphic.
The main result of the authors is the following. The set \(A_X\) is in bijective correspondence with the pairs of the form \((P,(F,\nabla))\) where \(P\) is a projective structure on \(X\) and \((F,\nabla)\) is an equivalence class of flat \(O(n,\mathbb{C})\)-connection on \(X\). The proof uses the fact that an element of \(A_X\) gives a holomorphic differential operator of order 2 on \(X\) and that all differential operators on \(X\) arise this way.

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
32C38 Sheaves of differential operators and their modules, \(D\)-modules
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References:

[1] Atiyah, M. F., Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc., 85, 181-207 (1957) · Zbl 0078.16002
[2] Biswas, I., Orbifold projective structures, differential operators, and logarithmic connections on a pointed Riemann surface, J. Geom. Phys., 56, 2345-2378 (2006) · Zbl 1106.32009
[3] Biswas, I.; Raina, A. K., Projective structures on a Riemann surface, II, Int. Math. Res. Not., 13, 685-716 (1999) · Zbl 0939.14014
[4] Deligne, P., (Equations Différentielles à Points Singuliers Réguliers. Equations Différentielles à Points Singuliers Réguliers, Lecture Notes in Math., no. 163 (1970), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0244.14004
[5] Griffiths, P.; Harris, J., Principles of Algebraic Geometry (1978), John Wiley & Sons Inc.: John Wiley & Sons Inc. New York · Zbl 0408.14001
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