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Bubble tree compactification of moduli spaces of vector bundles on surfaces

Dimitri Markushevich, Alexander Tikhomirov, Günther Trautmann (2012)

Open Mathematics

We announce some results on compactifying moduli spaces of rank 2 vector bundles on surfaces by spaces of vector bundles on trees of surfaces. This is thought as an algebraic counterpart of the so-called bubbling of vector bundles and connections in differential geometry. The new moduli spaces are algebraic spaces arising as quotients by group actions according to a result of Kollár. As an example, the compactification of the space of stable rank 2 vector bundles with Chern classes c 1 = 0, c 1...

Decomposability criterion for linear sheaves

Marcos Jardim, Vitor Silva (2012)

Open Mathematics

We establish a decomposability criterion for linear sheaves on ℙn. Applying it to instanton bundles, we show, in particular, that every rank 2n instanton bundle of charge 1 on ℙn is decomposable. Moreover, we provide an example of an indecomposable instanton bundle of rank 2n − 1 and charge 1, thus showing that our criterion is sharp.

General theory of Lie derivatives for Lorentz tensors

Lorenzo Fatibene, Mauro Francaviglia (2011)

Communications in Mathematics

We show how the ad hoc prescriptions appearing in 2001 for the Lie derivative of Lorentz tensors are a direct consequence of the Kosmann lift defined earlier, in a much more general setting encompassing older results of Y. Kosmann about Lie derivatives of spinors.

On the graph labellings arising from phylogenetics

Weronika Buczyńska, Jarosław Buczyński, Kaie Kubjas, Mateusz Michałek (2013)

Open Mathematics

We study semigroups of labellings associated to a graph. These generalise the Jukes-Cantor model and phylogenetic toric varieties defined in [Buczynska W., Phylogenetic toric varieties on graphs, J. Algebraic Combin., 2012, 35(3), 421–460]. Our main theorem bounds the degree of the generators of the semigroup by g + 1 when the graph has first Betti number g. Also, we provide a series of examples where the bound is sharp.

Representations of the Kauffman bracket skein algebra of the punctured torus

Jea-Pil Cho, Răzvan Gelca (2014)

Fundamenta Mathematicae

We describe the action of the Kauffman bracket skein algebra on some vector spaces that arise as relative Kauffman bracket skein modules of tangles in the punctured torus. We show how this action determines the Reshetikhin-Turaev representation of the punctured torus. We rephrase our results to describe the quantum group quantization of the moduli space of flat SU(2)-connections on the punctured torus with fixed trace of the holonomy around the boundary.

Stable bundles on hypercomplex surfaces

Ruxandra Moraru, Misha Verbitsky (2010)

Open Mathematics

A hypercomplex manifold is a manifold equipped with three complex structures I, J, K satisfying the quaternionic relations. Let M be a 4-dimensional compact smooth manifold equipped with a hypercomplex structure, and E be a vector bundle on M. We show that the moduli space of anti-self-dual connections on E is also hypercomplex, and admits a strong HKT metric. We also study manifolds with (4,4)-supersymmetry, that is, Riemannian manifolds equipped with a pair of strong HKT-structures that have opposite...

Surfaces kählériennes de volume fini et équations de Seiberg-Witten

Yann Rollin (2002)

Bulletin de la Société Mathématique de France

Soit M = ℙ ( ℰ ) une surface complexe réglée. Nous introduisons des métriques de volume fini sur M dons les singularités sont paramétrisées par une structure parabolique sur le fibré ℰ . Nous généralisons alors un résultat de Burns-deBartolomeis et Le Brun, en montrant que l’existence de métriques kählériennes singulières, de volume fini, à courbure scalaire constante négative ou nulle sur M est équivalente à une condition de polystabilité parabolique sur ℰ  ; de plus ces métriques proviennent toutes de quotients...

Symplectic structures on moduli spaces of framed sheaves on surfaces

Francesco Sala (2012)

Open Mathematics

We provide generalizations of the notions of Atiyah class and Kodaira-Spencer map to the case of framed sheaves. Moreover, we construct closed two-forms on the moduli spaces of framed sheaves on surfaces. As an application, we define a symplectic structure on the moduli spaces of framed sheaves on some birationally ruled surfaces.

The geometry of Calogero-Moser systems

Jacques Hurtubise, Thomas Nevins (2005)

Annales de l’institut Fourier

We give a geometric construction of the phase space of the elliptic Calogero-Moser system for arbitrary root systems, as a space of Weyl invariant pairs (bundles, Higgs fields) on the r -th power of the elliptic curve, where r is the rank of the root system. The Poisson structure and the Hamiltonians of the integrable system are given natural constructions. We also exhibit a curious duality between the spectral varieties for the system associated to a root system, and the Lagrangian varieties for...

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