Universal property of chern character forms of the canonical connection

Abstract

Let γ q,n denote the complex Stiefel bundle over the complex Grassmannian Gr n (C q ) and let ω 0 be the universal connection on this bundle. Consider the Chern character form of ω 0 defined by the formulawhere Ω 0 is the curvature form of the connection ω 0 . Let M be a manifold of dimension ≤ m and σ a closed 2k-form on M . Suppose, there exists a continuous map f 0 : M → Gr n (C q ) which pulls back the cohomology class of ch k (ω 0 ) onto the cohomology class of σ. We prove that if q and n are greater than certain numbers (which we determine in this paper) then there exists a smooth map f : M → Gr n (C q ) such that f * ch k (ω 0 ) = σ.

Smooth maps into quaternionic Grassmannians inducing a prescribed 4-form

“…The results of this section were proved earlier in [1] and [2]. For the sake of completeness we present the relevant part from there.…”

“…We shall show that p 1 (α Q ) = p 1 (α P ). We recall that the Symplectic Pontrjagin form p 1 (α Q ) is uniquely determined by the equation (2) π * Q p 1 (α Q ) = trace (Dα Q ) 2 , where D stands for the covariant differentiation and π Q denotes the projection map Q −→ M . Similarly, π * P p 1 (α P ) = trace (Dα P ) 2 ([5]).…”

“…In [2] we proved that the complex Grassmannians Gr k (C n ) admit some even degree differential forms σ i of degree 2i which are universal. This means that any closed differential 2i-form ω on a manifold M can be obtained as the pullback of σ i by an immersion f : M → Gr k (C n ) (for sufficiently large n) provided there is a continuous map f 0 : M → Gr k (C n ) which pulls back the deRham cohomology class of σ i onto that of ω.…”

Hodge classes of Chern character forms on compact Kähler manifolds

“…Since every d-closed form on U α is d-exact, hence logarithmically d-exact, every element ζ of Č0 (U; Z 1 ) is of the form ζ α = d log f α for some invertible functions f α ∈ C ∞ (U α ). So every coboundary δζ ∈ Ž1 (U; Z 1 ) is of the form (δζ) αβ = d log f α − d log f β , and hence every d-exact 2-form can be expressed as Datta1,Thm. 4.1], [PiTa1, Prop.…”

A priori estimates for a generalized Monge–Ampère PDE on some compact Kähler manifolds

“…It is not even obvious as to whether there is any connection satisfying this requirement, leave aside a Chern connection. Work along these lines was done by Datta in [9] using the h-principle. Therefore, it is more reasonable to ask whether equality can be realised for the top Chern character form.…”