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The even Clifford structure of the fourth Severi variety

100%

Parton M., Piccinni P.

Complex Manifolds

2015

tom 2

nr 1

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). From this we construct a canonical differential 8-form on EIII, associated with its holonomy Spin(10) · U(1) ⊂ U(16), that represents a generator of its cohomology ring. We relate it with a Schubert cycle structure by looking at EIII as the smooth projective variety V(4) ⊂ CP26 known as the fourth Severi variety.

Hyperholomorphic connections on coherent sheaves and stability

68%

Verbitsky M.

Open Mathematics

2011

tom 9

nr 3

535-557

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.

Ograniczanie wyników

1 Complex Manifolds

1 Open Mathematics

1 Parton M.

1 Piccinni P.

1 Verbitsky M.

1 2015

1 2011

Wyniki wyszukiwania

The even Clifford structure of the fourth Severi variety

Hyperholomorphic connections on coherent sheaves and stability