Spin(9) and almost complex structures on 16-dimensional manifolds

For a Spin(9)-structure on a Riemannian manifold M ¹⁶ we write explicitly the matrix ψ of its Kähler 2-forms and the canonical 8-form Φ_Spin(9). We then prove that Φ_Spin(9) coincides up to a constant with the fourth coefficient of the characteristic polynomial of ψ. This is inspired by lower dimensional situations, related to Hopf fibrations and to Spin(7). As applications, formulas are deduced for Pontrjagin classes and integrals of Φ_Spin(9) and \({\Phi_{\rm Spin(9)}^2}\) in the special case of holonomy Spin(9).     

Compatible complex structures on almost quaternionic manifolds

On an almost quaternionic manifold we study the integrability of almost complex structures which are compatible with the almost quaternionic structure . If , we prove that the existence of two compatible complex structures forces to be quaternionic. If , that is is an oriented conformal 4-manifold, we prove a maximum principle for the angle function of two compatible complex structures and deduce an application to anti-self-dual manifolds. By considering the special class of Oproiu connections we prove the existence of a well defined almost complex structure on the twistor space of an almost quaternionic manifold and show that is a complex structure if and only if is quaternionic. This is a natural generalization of the Penrose twistor constructions.

     

Nonpositively curved almost Hermitian metrics on product of compact almost complex manifolds

In this paper, we give a classification of almost Hermitian metrics with nonpositive holomorphic bisectional curvature on a product of compact almost complex manifolds. This generalizes previous results of Zheng [Ann. of Math.(2), 137(3), 671–673(1993)] and the author [Proc. Amer.Math. Soc., 139(4), 1469–1472(2011)].     

The Albanese mapping for compact almost complex manifolds

Partially supported by the Forschungsinstitut für Mathematik at the ETH Zuerich     

Note on the existence of almost omplex structures on compact manifolds

In the paper we define a multiplicative genus of a compact orientable manifold. We use this genus for the study of the existence of almost complex structures on manifolds. A few applications are given, namely, we prove the nonexistence of an almost complex structure on quaternionic flag manifolds and give a theorem on the existence of an almost complex structure on the product of manifolds.     

Stably and almost complex structures on bounded flag manifolds

We study the enumeration problem of stably complex structures on bounded flag manifolds arising from omniorientations, and determine those induced by almost complex structures. We also enumerate the stably complex structures on these manifolds which bound, therefore representing zero in the complex cobordism ring .

     

Almost complex structures on connected sums of almost complex manifolds and complex projective spaces

Archiv der Mathematik - For a smooth curve C in $${{mathbb {P}}}^{r_0}$$ lying on a rational surface scroll, we try to identify those complete and base point free linear series of small degree...     

Lelong functional on almost complex manifolds

We establish plurisubharmonicity of the envelope of Lelong functional on almost complex manifolds of real dimension four, thereby we generalize the corresponding result for complex manifolds.     

Extension formulae on almost complex manifolds

We give the extension formulae on almost complex manifolds and give decompositions of the extension formulae.As applications,we study(n,0)-forms,the(n,0)-Dolbeault cohomology group and(n,q)-forms on almost complex manifolds.     

Invariant almost Hermitian structures on flag manifolds

Let G be a complex semi-simple Lie group and form its maximal flag manifold where P is a minimal parabolic (Borel) subgroup, U a compact real form and T=U∩P a maximal torus of U. We study U-invariant almost Hermitian structures on . The (1,2)-symplectic (or quasi-Kähler) structures are naturally related to the affine Weyl groups. A special form for them, involving abelian ideals of a Borel subalgebra, is derived. From the (1,2)-symplectic structures a classification of the whole set of invariant structures is provided showing, in particular, that nearly Kähler invariant structures are Kähler, except in the A₂ case.     

J-pluripolar subsets and currents on almost complex manifolds

We prove that every almost complex submanifold of an almost complex manifold is locally J-pluripolar. This generalizes a result of Rosay for J-holomorphic submanifolds. Our second main result is an almost complex version of El Mir’s theorem for the extension of positive currents across locally complete pluripolar sets. As a consequence, we extend some results proved by Dabbek–Elkhadhra–El Mir and Dinh–Sibony in the standard complex case. We also obtain a version of the well-known results of Federer and Bassanelli for flat and mathbb C{mathbb {C}}-flat currents in the almost complex setting.     

Chern numbers of almost complex manifolds

It is shown that any system of numbers that can be realised as the system of Chern numbers of an almost complex manifold of dimension , , can also be realised in this way by a connected almost complex manifold. This answers an old question posed by Hirzebruch.

     

Special metrics on compact complex manifolds

We study the existence of special metrics on compact complex manifolds. We show that every considered metric can be caracterized using conditions on the space of positive currents. We investigate what happens under holomorphic submersions or modifications. We show which metrics exist on some classical examples.