Understanding how to utilize the properties of Kähler manifolds Part2(Differential Geometry)
- On harmonic symmetries for locally conformally Kähler manifolds(arXiv)
Author : Teng Huang
Abstract : In this article, we study harmonic symmetries on the compact locally conformally Kähler manifold M of dimC=n. The space of harmonic symmetries is a subspace of harmonic differential forms which defined by the kernel of a certain Laplacian-type operator □. We observe that the spaces ker(□)∩Ωl={0} for any |l−n|≥2 and kerΔ∂¯∩Pk,n−1−k∩ker(iθ♯)≅ker(□k,n−1−k), kerΔ∂¯∩Pk,n−k≅ker(□k,n−k). Furthermore, suppose that M is a Vaisman manifold, we prove that (i) α is (n−1)-form in ker(□) if only if α is a transversally harmonic and transversally effective V-foliate form; (ii) α is a (p,n−p)-form in ker(□p,n−p) if only if there are two forms β1∈Sp−1,n−p and β2∈Sp,n−p−1 such that α=θ1,0∧β1+θ0,1∧β2
2.Degenerate J-flow on compact Kähler manifolds (arXiv)
Author : Tat Dat Tô
Abstract : In this note, we study a degenerate twisted J-flow on compact Kähler manifolds. We show that it exists for all time, it is unique and converges to a weak solution of a degenerate twisted J-equation. In particular, this confirms an expectation formulated by Song-Weinkove for the J-flow. As a consequence, we establish the properness of the Mabuchi K-energy twisted by a certain semi-positive closed (1,1)-form for Kähler classes in a certain subcone
3.On fibrations and measures of irrationality of hyper-Kähler manifolds (arXiv)
Author : Claire Voisin
Abstract : We prove some results on the fibers and images of rational maps from a hyper-Kähler manifold. We study in particular the minimal genus of fibers of a fibration into curves. The last section of this paper is devoted to the study of the rational map defined by a linear system on a hyper-Kähler fourfold satisfying numerical conditions similar to those considered by O’Grady in his study of fourfolds numerically equivalent to K3[2]. We extend his results to this more general context.
4.Stein complements in compact Kähler manifolds (arXiv)
Author : Andreas Höring, Thomas Peternell
Abstract : Given a projective or compact Kähler manifold X and a (smooth) hypersurface Y, we study conditions under which X∖Y could be Stein. We apply this in particular to the case when X is the projectivization of the so-called canonical extension of the tangent bundle TM of a projective manifold M with Y being the projectivization of TM itself.
