Udied in [13]. In particular, Ref. [13] offers quite a few examples for K ler hyperbolic manifolds, for example symmetric spaces, bounded symmetric domains in Cn , hyperconvex bounded domains, and so on. Definitely, Theorem 1 is valid on these manifolds. Remark 4. Each of the outcomes are nonetheless valid if L is twisted by a Nakano semi-positive [16] vector bundle E. The proof requires absolutely nothing new therefore we omit it. The plan of this paper is as follows: we will first recall the background materials in Section two. The K ler hyperbolicity is discussed in Section three. Then, we go over the Hodge decomposition on a non-compact manifold in Section 4. In Section five, we prove the injectivity theorem along with the extension theorem.1.Then, the natural morphismSymmetry 2021, 13,3 of2. Preliminarily 2.1. Singular Metric Recall that a smooth Hermitian metric h on a line bundle L is provided in any trivialization : L|U U C by 2 = | |two e-2( x) , x U, L x h exactly where C (U ) is an arbitrary function, named the weight of the metric with respect towards the trivialization . Then, the singular Hermitian metric is defined in [16] as follows: Definition 1 (Singular metric). A singular Hermitian metric h on a line bundle L is provided in any trivialization : L|U U C by2 h= | |2 e-2(x) , x U, L xwhere L1 (U ) is an arbitrary function, called the weight of your metric with respect to the trivialization . Often, we are going to directly say that is actually a (singular) metric on L if nothing at all is confused. 2.two. Multiplier Ideal Sheaf The multiplier best sheaf is definitely an crucial tool in modern day complex geometry, which was originally introduced in [16,17]. Definition 2 (Multiplier ideal sheaf). Let L be a line bundle. Let be a singular metric on L such that i L, to get a smooth real (1, 1)-form on X. Then, the multiplier perfect sheaf is defined as I x := f .Note that X is non-compact, and f ( X, I ) generally won’t imply thatX| f |two e-2 .Nevertheless, when X is moreover assumed to be weakly pseudoconvex, we could substitute for . Right here, is a convex rising function of arbitrary rapidly development at infinity and will be the smooth plurisubharmonic exhaustion function provided by the weak pseudoconvexity of X. This element is usually utilised to ensure the convergence of integrals at infinity. Moreover, we have I ( ) = I and i L, . Therefore, we can normally assume without loss of generality that, for each f ( X, I ),X| f |two e-2 .3. The K ler Manifold with Negative Betamethasone disodium Purity curvature 3.1. Negative Curvature Firstly, let us recall the definition for a manifold with negative sectional curvature. Definition three. Let ( X, ) be a K ler manifold. Let (i j )1 i,j,, n be the curvature associated with . Then, X is mentioned to possess unfavorable sectional curvature, if there exists a positive continuous K such that, for any non-zero complex vector = ( 1 , …, n ), ii j i -K 2 .Symmetry 2021, 13,4 ofIt is denoted by sec-K.A full K ler manifold with negative sectional curvature will probably be K ler hyperbolic (see Proposition 1). The K ler hyperbolicity was initially introduced in [13] for any compact K ler manifold. Even so, there is certainly no obstacle to extend it for the non-compact case. Firstly, let us recall the C6 Ceramide In Vitro d-boundedness of a differential type. Definition four. Let be a differential type on X. Let : X X be the universal covering of X. Then, (i) is named bounded (with respect to ) if the L -norm of is finite,L:= sup || .xXHere, || may be the pointwise norm induced.
