International Journal of Mathematics

Following [T.-J. Li and W. Zhang, Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Comm. Anal. Geom.17(4) (2009) 651–683], we continue to study the link between the cohomology of an almost-complex manifold and its almost-complex structure. In particular, we apply the same argument in [T.-J. Li and W. Zhang, Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Comm. Anal. Geom.17(4) (2009) 651–683] and the results obtained by [D. Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math.36(1) (1976) 225–255] to study the cone of semi-Kähler structures on a compact semi-Kähler manifold.

In this paper we investigate the Cauchy problem for the three-dimensional incompressible magnetohydrodynamic equations. A logarithmal improved blow-up criterion of smooth solutions is obtained.

We show that the usual sufficient criterion for a very general hypersurface in a smooth projective manifold to have the same Picard number as the ambient variety can be generalized to quasi-smooth hypersurfaces in complete simplicial toric varieties. This sufficient condition always holds for very general K3 surfaces embedded in Fano toric 3-folds.

We will give an explicit ring structure of the graded ring of Siegel modular forms of degree two with respect to a certain discrete subgroup of Sp(2;ℝ) contained in a non-split ℚ-form. We will also give the fundamental relations among the generators.

By using exponential functions, a fundamental set of power series solutions for singular linear systems of the first kind is explicitly computed.

Positive Quaternion Kähler Manifolds are Riemannian manifolds with holonomy contained in Sp(n)Sp(1) and with positive scalar curvature. Conjecturally, they are symmetric spaces. We prove this conjecture in dimension 20 under additional assumptions and we provide recognition theorems for the real Grassmannian in almost all dimensions.

A point q in a contact manifold is called a translated point for a contactomorphism ϕ with respect to some fixed contact form if ϕ(q) and q belong to the same Reeb orbit and the contact form is preserved at q. The problem of existence of translated points has an interpretation in terms of Reeb chords between Legendrian submanifolds, and can be seen as a special case of the problem of leafwise coisotropic intersections. For a compactly supported contactomorphism ϕ of ℝ²ⁿ⁺¹ or ℝ²ⁿ × S¹ contact isotopic to the identity, existence of translated points follows immediately from Chekanov's theorem on critical points of quasi-functions and Bhupal's graph construction. In this article we prove that if ϕ is positive then there are infinitely many nontrivial geometrically distinct iterated translated points, i.e. translated points of some iteration ϕ^k. This result can be seen as a (partial) contact analog of the result of Viterbo on existence of infinitely many iterated fixed points for compactly supported Hamiltonian symplectomorphisms of ℝ²ⁿ, and is obtained with generating functions techniques.

In this article, we obtain the following generalization of isometric C¹-immersion theorem of Nash and Kuiper. Let M be a smooth manifold of dimension m and H a rank k subbundle of the tangent bundle TM with a Riemannian metric g_H. Then the pair (H, g_H) defines a sub-Riemannian structure on M. We call a C¹-map f : (M, H, g_H) → (N, h) into a Riemannian manifold (N, h) a partial isometry if the derivative map df restricted to H is isometric, that is if f*h|_H = g_H. We prove that if f₀ : M → N is a smooth map such that df₀|_H is a bundle monomorphism and , then f₀ can be homotoped to a C¹-map f : M → N which is a partial isometry, provided dim N > k. As a consequence of this result, we obtain that every sub-Riemannian manifold (M, H, g_H) admits a partial isometry in ℝⁿ, provided n ≥ m + k.

We present a novel notion of stable objects in the derived category of coherent sheaves on a smooth projective variety. As one application we compactify a moduli space of stable bundles using genuine complexes.