And bigger compact subsets of X). We prove that Theorem 1. Let ( X,

And bigger compact subsets of X). We prove that Theorem 1. Let ( X, ) be a weakly pseudoconvex K ler manifold such that the sectional curvature secCitation: Wu, J. The Injectivity Theorem on a Non-Compact K ler Manifold. Symmetry 2021, 13, 2222. https://doi.org/10.3390/Goralatide Technical Information sym13112222 Academic Editor: Roman Ger Received: 20 October 2021 Accepted: 9 November 2021 Published: 20 November-K (see Definition 3)Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.for some optimistic constant K. Let ( L, L ) and ( H, H ) be two (singular) Hermitian line bundles on X. Assume the following circumstances: 1. two. three. There exists a closed subvariety Z on X such that L and H are smooth on X \ Z; i L, L 0 and i H, H 0 on X; i L, L i H, H for some optimistic number .Copyright: 2021 by the author. Licensee MDPI, Basel, Switzerland. This short article is an open access short article distributed beneath the terms and circumstances of the Inventive Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ four.0/).For any (non-zero) section s of H with supX |s|2 e- H , the multiplication map induced by the tensor solution with s : H q ( X, KX L I ( L )) H q ( X, KX L H I ( L H )) is (well-defined and) injective for any q 0.Remark 1. The assumption (1) is often quickly removed if Demailly’s approximation method [12] is valid within this situation. Nonetheless, it appears to me that the compactness in the baseSymmetry 2021, 13, 2222. https://doi.org/10.3390/symhttps://www.mdpi.com/journal/symmetrySymmetry 2021, 13,two ofmanifold is of vital significance in his original proof. Thus, it can be challenging to straight apply his argument right here. We’re interested to know no matter if such an approximation exists on a non-compact manifold. We will recall the definition of singular metric and multiplier perfect sheaf I ( L ) in Section 2, as well as the elementary properties of manifolds with damaging sectional curvature in Section three. Theorem 1 implies the following L2 -extension theorem concerning the subvariety that’s not necessary to be lowered. Such sort of extension problem was studied in [10] prior to. Corollary 1. Let ( X, ) be a weakly pseudoconvex K ler manifold such that sec-Kfor some constructive constant K. Let ( L, L ) be a (singular) Hermitian line bundle on X, and let be a quasi-plurisubharmonic function on X. Assume the following circumstances: 1. 2. 3. There exists a closed subvariety Z on X such that L is smooth on X \ Z; i L, L 0; i L, L (1 )i 0 for all non-negative quantity [0, ) with 0 H 0 ( X, KX L I ( L )) H 0 ( X, KX L I ( L )/I ( L )) is surjective. Remark 2. If L is smooth, we have I ( L ) = O X and I ( L )/I ( L ) = O X /I =: OY , where Y is the subvariety defined by the excellent sheaf I . In distinct, Y will not be necessary to be decreased. Then, the surjectivity statement can interpret an extension theorem for holomorphic sections, with respect towards the restriction morphism H 0 ( X, KX L) H 0 (Y, (KX L)|Y ). So that you can prove Theorem 1, we improve the L2 -Hodge theory introduced in [13], such that it is appropriate for the forms taking worth inside a line bundle. The essential issue is definitely the Hodge decomposition [14,15] on a non-compact manifold. Since the base manifold has adverse sectional curvature, it is actually K ler hyperbolic by [13]. We then apply the K ler hyperbolicity to establish the Hodge decomposition. We leave all the specifics inside the text. Remark 3. The K ler hyperbolic manifold was GYY4137 Purity & Documentation deeply st.