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.
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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 = 2 on the projective plane is studied in more detail. Proofs are only indicated and will appear in separate papers.
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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.
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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 torsion. In the language of Hitchin’s and Gualtieri’s generalized complex geometry, (4,4)-manifolds are called “generalized hyperkähler manifolds”. We show that the moduli space of anti-self-dual connections on M is a (4,4)-manifold if M is equipped with a (4,4)-structure.
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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.
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Let M be a hyperkähler manifold, and F a reflexive sheaf on M. Assume that F (away from its singularities) admits a connection ▿ with a curvature Θ which is invariant under the standard SU(2)-action on 2-forms. If Θ is square-integrable, such sheaf is called hyperholomorphic. Hyperholomorphic sheaves were studied at great length in [21]. Such sheaves are stable and their singular sets are hyperkähler subvarieties in M. In the present paper, we study sheaves admitting a connection with SU(2)-invariant curvature which is not necessary L 2-integrable. We show that such sheaves are polystable.
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