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Hyperholomorphic connections on coherent sheaves and stability

Misha Verbitsky (2011)

Open Mathematics

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...

Special Lagrangian linear subspaces in product symplectic space

Małgorzata Mikosz (2004)

Banach Center Publications

The notes consist of a study of special Lagrangian linear subspaces. We will give a condition for the graph of a linear symplectomorphism f : ( 2 n , σ = i = 1 n d x i d y i ) ( 2 n , σ ) to be a special Lagrangian linear subspace in ( 2 n × 2 n , ω = π * σ - π * σ ) . This way a special symplectic subset in the symplectic group is introduced. A stratification of special Lagrangian Grassmannian S Λ 2 n S U ( 2 n ) / S O ( 2 n ) is defined.

Special Lagrangian submanifolds in the complex sphere

Henri Anciaux (2007)

Annales de la faculté des sciences de Toulouse Mathématiques

We construct a family of Lagrangian submanifolds in the complex sphere which are foliated by ( n - 1 ) -dimensional spheres. Among them we find those which are special Lagrangian with respect to the Calabi-Yau structure induced by the Stenzel metric.

The even Clifford structure of the fourth Severi variety

Maurizio Parton, Paolo Piccinni (2015)

Complex Manifolds

TheHermitian symmetric spaceM = EIII appears in the classification of complete simply connected Riemannian manifolds carrying a parallel even Clifford structure [19]. This means the existence of a real oriented Euclidean vector bundle E over it together with an algebra bundle morphism φ : Cl0(E) → End(TM) mapping Ʌ2E into skew-symmetric endomorphisms, and the existence of a metric connection on E compatible with φ. We give an explicit description of such a vector bundle E as a sub-bundle of End(TM)....

The restriction theorem for fully nonlinear subequations

F. Reese Harvey, H. Blaine Lawson (2014)

Annales de l’institut Fourier

Let X be a submanifold of a manifold Z . We address the question: When do viscosity subsolutions of a fully nonlinear PDE on Z , restrict to be viscosity subsolutions of the restricted subequation on X ? This is not always true, and conditions are required. We first prove a basic result which, in theory, can be applied to any subequation. Then two definitive results are obtained. The first applies to any “geometrically defined” subequation, and the second to any subequation which can be transformed...

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