Let X be a simply connected compact Kähler manifold with zero first Chern class, and let L be an ample line bundle over X. The pair (X,L) is called a polarized Calabi-Yau manifold. By a theorem of Mumford, the moduli space of the pair (X,L) (CY moduli) exists and is a complex variety. Locally, up to a finite cover, the moduli space is smooth (see [20, 21]). There is a natural Kähler metric, cal...

To every real analytic Riemannian manifold M there is associated a complex structure on a neighborhood of the zero section in the real tangent bundle of M . This structure can be uniquely specified in several ways, and is referred to as a Grauert tube. We say that a Grauert tube is entire if the complex structure can be extended to the entire tangent bundle. We prove here that the complex manif...

Let M be a compact connected special flat affine manifold without boundary equipped with a Gauduchon metric g and a covariant constant volume form. Let G be either a connected reductive complex linear algebraic group or the real locus of a split real form of a complex reductive group. We prove that a flat principal G–bundle EG over M admits a Hermitian–Einstein structure if and only if EG is po...

Our aim in this paper is to study of silver Riemannian structures on manifold and bundle. An integrability condition curvature properties for structure are investigated via the Tachibana operator. Twin metric defined some twin investigated. Examples given tangent cotangent bundles.

Suppose M is an n-dimensional Kähler manifold and L is an ample line bundle over M . Let the Kähler form of M be ωg and the Hermitian metric of L be H. We assume that ωg is the curvature of H, that is, ωg = Ric(H). The Kähler metric of ωg is called a polarized Kähler metric on M . Using H and ωg, for any positive integer m, H 0(M,Lm) becomes a Hermitian inner product space. We use the following...

We show that, in the region where monopoles are well separated, the L-metric on the moduli space of n-monopoles is exponentially close to the Tn-invariant hyperkähler metric proposed by Gibbons and Manton. The proof is based on a description of the Gibbons-Manton metric as a metric on a certain moduli space of solutions to Nahm’s equations, and on twistor methods. In particular, we show how the...

Dirac spinors play crucial role in modern particle physics. However, this crucial application of Dirac spinors is based mostly on the special relativity, where the base manifold M is the flat Minkowski space. Passing to the general relativity, we get a little bit more complicated theory of spinors. LetM be a space-time manifold of the general relativity. It is a four-dimensional orientable mani...

For manifolds M of noncompact type endowed with an affine connection (for example, the Levi-Civita connection) and a closed 2-form (magnetic field) we define a Hilbert algebra structure in the space L 2 (T * M) and construct an irreducible representation of this algebra in L 2 (M). This algebra is automatically extended to polynomial in momenta functions and distributions. Under some natural co...

Proposition 1.2 (Splitting Principle). Let E → M be a complex vector bundle of rank n over a manifold X. Then there is a manifold F (E) and a smooth fibration π : F (E) → M such that • The homomorphism π∗ : H∗(M, Z) → H∗(F (E), Z) is injective. • The bundle π∗E splits into a direct sum of complex line bundles, i.e. π∗E = l1 ⊕ · · ·⊕ ln Proof. Consider the projectivisation p : PC(E) → M and the ...

In the literature, there are several papers establishing a correspondence between deformed kinematics and nontrivial (momentum dependent) metric. this work, we study in detail relationship trajectories given by Hamiltonian geodesic motion obtained from geometry cotangent bundle, finding that both coincide when is identified with squared distance momentum space. Moreover, following natural struc...