Moduli of Vector Bundles on a Compact Riemann Surface

Narasimhan, M. S.; Ramanan, S.

doi:10.2307/1970807

Cited by 261 publications

(229 citation statements)

References 11 publications

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“…Our approach also yields a proof of the following result, established by Narasimhan and Ramanan by other methods [17], cf. also [20].…”

Section: Introductionsupporting

confidence: 53%

“…In particular, for genus ℓ ≥ 2, the space K = N G ∪ N (T ) is known to be the Kummer variety of Σ associated with its Jacobien J and the canonical involution thereupon. Theorem 1 above implies in particular that, for genus ≥ 3, the Kummer variety K is precisely the singular locus of N , a result due to Narasimhan-Ramanan [17]. Theorem 1 above has the following consequence:…”

Section: Introductionmentioning

confidence: 89%

“…This space and related ones have been studied extensively in the literature [17], [18], [19]. In particular, for genus ℓ ≥ 2, the space K = N G ∪ N (T ) is known to be the Kummer variety of Σ associated with its Jacobien J and the canonical involution thereupon.…”

Section: Introductionmentioning

confidence: 99%

“…The space N then equals complex projective 3-space and K is the Kummer surface associated with the Jacobien of Σ, cf. Narasimhan-Ramanan [17]. In the literature, this case has been considered somewhat special since as a space N is then actually smooth.…”

Section: Introductionmentioning

confidence: 99%

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Poisson geometry of flat connections for SU(2)-bundles on surfaces

Huebschmann

1996

Math Z

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Abstract. In earlier work we have shown that the moduli space N of flat connections for the (trivial) SU(2)-bundle on a closed surface of genus ℓ ≥ 2 is a stratified symplectic space so that, in particular, the data give rise to a Poisson algebra (C ∞ N, {·, ·}) of continuous functions on N; furthermore, the strata are Kähler manifolds, and the stratification consists of an open connected and dense submanifold N Z of real dimension 6(ℓ − 1), a connected stratum N (T ) of real dimension 2ℓ, and 2 2ℓ isolated points. In this paper we show that, close to each point of N (T ) , the space N and Poisson algebra (C ∞ N, {·, ·}) look like a product of C ℓ endowed with the standard symplectic Poisson structure with the reduced space and Poisson algebra of the system of (ℓ − 1) particles in the plane with total angular momentum zero, while close to one of the isolated points, the Poisson algebra (C ∞ N, {·, ·}) looks like that of the reduced system of ℓ particles in R 3 with total angular momentum zero. Moreover, in the genus two case where the space N is known to be smooth we locally describe the Poisson algebra (C ∞ N, {·, ·}) and the various underlying symplectic structures on the strata and their mutual positions explicitly in terms of the Poisson structure.1991 Mathematics Subject Classification. 32G13, 32G15, 32S60, 58C27, 58D27, 58E15, 81T13.

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“…Our approach also yields a proof of the following result, established by Narasimhan and Ramanan by other methods [17], cf. also [20].…”

Section: Introductionsupporting

confidence: 53%

Section: Introductionmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Poisson geometry of flat connections for SU(2)-bundles on surfaces

Huebschmann

1996

Math Z

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“…3.2] for the finiteness result. Now, in our situation of rank 2 bundles on a curve of genus 2, it is a theorem of Narasimhan and Ramanan that M 2 ∼ = P 3 ; see [24,Thm. 2,§7], and note that despite the Riemann surface language, the argument goes through unmodified in arbitrary odd characteristic.…”

Section: Preliminaries On Degreesmentioning

confidence: 99%

The generalized Verschiebung map for curves of genus 2

Osserman

2006

Math. Ann.

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Abstract. Let C be a smooth curve, and Mr(C) the coarse moduli space of vector bundles of rank r and trivial determinant on C. We examine the generalized Verschiebung map Vr : Mr(C (p) ) Mr(C) induced by pulling back under Frobenius. Our main result is a computation of the degree of V 2 for a general C of genus 2, in characteristic p > 2. We also give several general background results on the Verschiebung in an appendix.

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The Quadric Line Complex

Griffiths¹,

Harris²

1994

Principles of Algebraic Geometry

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Moduli of Vector Bundles on a Compact Riemann Surface

Cited by 261 publications

References 11 publications

Poisson geometry of flat connections for SU(2)-bundles on surfaces

Poisson geometry of flat connections for SU(2)-bundles on surfaces

The generalized Verschiebung map for curves of genus 2

The Quadric Line Complex

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