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Fano manifolds of degree ten and EPW sextics

Atanas Iliev, Laurent Manivel (2011)

Annales scientifiques de l'École Normale Supérieure

O’Grady showed that certain special sextics in ℙ 5 called EPW sextics admit smooth double covers with a holomorphic symplectic structure. We propose another perspective on these symplectic manifolds, by showing that they can be constructed from the Hilbert schemes of conics on Fano fourfolds of degree ten. As applications, we construct families of Lagrangian surfaces in these symplectic fourfolds, and related integrable systems whose fibers are intermediate Jacobians.

From Sasakian 3-structures to quaternionic geometry

Yoshiyuki Watanabe, Hiroshi Mori (1998)

Archivum Mathematicum

We construct a family of almost quaternionic Hermitian structures from an almost contact metric 3-structure and also do three kinds of quaternionic Kähler structures from a Sasakian 3-structure. In particular we have a generalization of the second main result of Boyer-Galicki-Mann [5].

H-anti-invariant submersions from almost quaternionic Hermitian manifolds

Kwang-Soon Park (2017)

Czechoslovak Mathematical Journal

As a generalization of anti-invariant Riemannian submersions and Lagrangian Riemannian submersions, we introduce the notions of h-anti-invariant submersions and h-Lagrangian submersions from almost quaternionic Hermitian manifolds onto Riemannian manifolds. We obtain characterizations and investigate some properties: the integrability of distributions, the geometry of foliations, and the harmonicity of such maps. We also find a condition for such maps to be totally geodesic and give some examples...

H-conformal anti-invariant submersions from almost quaternionic Hermitian manifolds

Kwang Soon Park (2020)

Czechoslovak Mathematical Journal

We introduce the notions of h-conformal anti-invariant submersions and h-conformal Lagrangian submersions from almost quaternionic Hermitian manifolds onto Riemannian manifolds as a generalization of Riemannian submersions, horizontally conformal submersions, anti-invariant submersions, h-anti-invariant submersions, h-Lagrangian submersion, conformal anti-invariant submersions. We investigate their properties: the integrability of distributions, the geometry of foliations, the conditions for such...

Homogeneous quaternionic Kähler structures on Alekseevskian 𝒲-spaces

Wafaa Batat, P. M. Gadea, Jaime Muñoz Masqué (2012)

Annales Polonici Mathematici

The homogeneous quaternionic Kähler structures on the Alekseevskian 𝒲-spaces with their natural quaternionic structures, each of these spaces described as a solvable Lie group, and the type of such structures in Fino's classification, are found.

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...

Hypersurfaces with almost complex structures in the real affine space

Mayuko Kon (2007)

Colloquium Mathematicae

We study affine hypersurface immersions f : M → ℝ 2 n + 1 , where M is an almost complex n-dimensional manifold. The main purpose is to give a condition for (M,J) to be a special Kähler manifold with respect to the Levi-Civita connection of an affine fundamental form.

Lagrangian fibrations on hyperkähler manifolds – On a question of Beauville

Daniel Greb, Christian Lehn, Sönke Rollenske (2013)

Annales scientifiques de l'École Normale Supérieure

Let  X be a compact hyperkähler manifold containing a complex torus L as a Lagrangian subvariety. Beauville posed the question whether X admits a Lagrangian fibration with fibre L . We show that this is indeed the case if X is not projective. If X is projective we find an almost holomorphic Lagrangian fibration with fibre L under additional assumptions on the pair ( X , L ) , which can be formulated in topological or deformation-theoretic terms. Moreover, we show that for any such almost holomorphic Lagrangian...

New hyper-Käahler structures on tangent bundles

Xuerong Qi, Linfen Cao, Xingxiao Li (2014)

Communications in Mathematics

Let ( M , g , J ) be an almost Hermitian manifold, then the tangent bundle T M carries a class of naturally defined almost hyper-Hermitian structures ( G , J 1 , J 2 , J 3 ) . In this paper we give conditions under which these almost hyper-Hermitian structures ( G , J 1 , J 2 , J 3 ) are locally conformal hyper-Kähler. As an application, a family of new hyper-structures is obtained on the tangent bundle of a complex space form. Furthermore, by restricting these almost hyper-Hermitian structures on the unit tangent sphere bundle T 1 M , we obtain a class of almost...

On almost hyperHermitian structures on Riemannian manifolds and tangent bundles

Serge Bogdanovich, Alexander Ermolitski (2004)

Open Mathematics

Some results concerning almost hyperHermitian structures are considered, using the notions of the canonical connection and the second fundamental tensor field h of a structure on a Riemannian manifold which were introduced by the second author. With the help of any metric connection ∇ ˜ on an almost Hermitian manifold M an almost hyperHermitian structure can be constructed in the defined way on the tangent bundle TM. A similar construction was considered in [6], [7]. This structure includes two basic...

On holomorphically projective mappings of e -Kähler manifolds

Irena Hinterleitner (2012)

Archivum Mathematicum

In this paper we study fundamental equations of holomorphically projective mappings of e -Kähler spaces (i.e. classical, pseudo- and hyperbolic Kähler spaces) with respect to the smoothness class of metrics. We show that holomorphically projective mappings preserve the smoothness class of metrics.

On the asymptotic geometry of gravitational instantons

Vincent Minerbe (2010)

Annales scientifiques de l'École Normale Supérieure

We investigate the geometry at infinity of the so-called “gravitational instantons”, i.e. asymptotically flat hyperkähler four-manifolds, in relation with their volume growth. In particular, we prove that gravitational instantons with cubic volume growth are ALF, namely asymptotic to a circle fibration over a Euclidean three-space, with fibers of asymptotically constant length.

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