Trivial Tangent Bundle Research Articles

The Lipman–Zariski Conjecture in Low Genus

Abstract We prove the Lipman–Zariski conjecture for complex surface singularities of genus 1 and also for those of genus 2 whose link is not a rational homology sphere. As an application, we characterize complex $2$-tori as the only normal compact complex surfaces whose smooth locus has trivial tangent bundle. We also deduce that all complex-projective surfaces with locally free and generically nef tangent sheaf are smooth, and we classify them.

ON VARIETIES WITH TRIVIAL TANGENT BUNDLE IN CHARACTERISTIC

In this article, I give a crystalline characterization of abelian varieties amongst the class of smooth projective varieties with trivial tangent bundles in characteristic $p>0$. Using my characterization, I show that a smooth, projective, ordinary variety with trivial tangent bundle is an abelian variety if and only if its second crystalline cohomology is torsion-free. I also show that a conjecture of KeZheng Li about smooth projective varieties with trivial tangent bundles in characteristic $p>0$ is true for smooth projective surfaces. I give a new proof of a result by Li and prove a refinement of it. Based on my characterization of abelian varieties, I propose modifications of Li’s conjecture, which I expect to be true.

Hörmander index in finite-dimensional case

We calculate the Hormander index in the finite-dimensional case. Then we use the result to give some iteration inequalities, and prove almost existence of mean indices for given complete autonomous Hamiltonian system on compact symplectic manifold with symplectic trivial tangent bundle and given autonomous Hamiltonian system on regular compact energy hypersurface of symplectic manifold with symplectic trivial tangent bundle.

Weakly complex homogeneous spaces

Abstract. We complete our recent classification (2011) of compact inner symmetric spaces with weakly complex tangent bundle by filling up a case which was left open, and extend this classification to the larger category of compact homogeneous spaces with positive Euler characteristic. We show that a simply connected compact equal rank homogeneous space has weakly complex tangent bundle if and only if it is a product of compact equal rank homogeneous spaces which either carry an invariant almost complex structure (and are classified by Hermann (1955)), or have stably trivial tangent bundle (and are classified by Singhof and Wemmer (1986)), or belong to an explicit list of weakly complex spaces which have neither stably trivial tangent bundle, nor carry invariant almost complex structures.

Principal bundles on compact complex manifolds with trivial tangent bundle

Let G be a connected complex Lie group and $${\Gamma \subset G}$$ a cocompact lattice. Let H be a complex Lie group. We prove that a holomorphic principal H-bundle E H over G/Γ admits a holomorphic connection if and only if E H is invariant. If G is simply connected, we show that a holomorphic principal H-bundle E H over G/Γ admits a flat holomorphic connection if and only if E H is homogeneous.

Essentially finite vector bundles on varieties with trivial tangent bundle

Let X X be a smooth projective variety, defined over an algebraically closed field of positive characteristic, such that the tangent bundle T X TX is trivial. Let F X : X ⟶ X F_X\, :\, X\,\longrightarrow \, X be the absolute Frobenius morphism of X X . We prove that for any n ≥ 1 n\, \geq \,1 , the n n –fold composition F X n F^n_X is a torsor over X X for a finite group–scheme that depends on n n . For any vector bundle E ⟶ X E\,\longrightarrow \, X , we show that the direct image ( F X n ) ∗ E (F^n_X)_*E is essentially finite (respectively, F F –trivial) if and only if E E is essentially finite (respectively, F F –trivial).

On Manifolds with Trivial Logarithmic Tangent Bundle: The Non-Kähler Case

We study non-Kahler manifolds with trivial logarithmic tangent bundle. We show that each such manifold arises as a fibre bundle with a compact complex parallelizable manifold as basis and a compactficiation of a semi-torus as fibre.

Homotopy types of locally linear representation forms

The authors of [6] investigated certain locally linear actions of a cyclic groupG of odd order on homotopy spheres, the so-calledG-representation forms [16]. In particular, several conditions on a dimension function were described that made sure that it can be realized as the dimension function of aG-representation form. It remained unclear, whether all homotopy types with those dimension functions would support a locally linear structure. It is the aim of this note to show that this is not the case, i.e., to give examples of homotopy representations [17] with the same dimension functions some of which support a locally linear structure with stably trivial tangent bundle and others do not. The main tools are formulated as general splitting principles for fixed point and restriction functors that may have some interest in their own right, too.