Canonical L2 -extension theorem [21], l extends to a holomorphic L-valued (n – q, 0)-form on X, which can be denoted by l . Repair l0 . Then, for l l0 ,l2 l0 ,l2 l ,2 , ,hence l is uniformly bounded in L2 -norm l , . Consequently, it converges to a 0 holomorphic L-valued (n – q, 0)-form, say . Furthermore, as l0 tends to , we obtain that 2 two . Now, it is actually quick to confirm that , ,[ q ] H q ( X, KX L I ).We denote this morphism by i = [ q ]. Conversely, let [] H q ( X, KX L I ). Let Lq be the sheaf of germs of (n, q)types on X with values in L and with measurable coefficients, such that each | |2 and , ||2 are locally integrable. The operator defines a SC-19220 In stock complex of sheaves (L, ), and it is , effortless to confirm that (L, ) is often a resolution of KX L I . Every sheaf Lq is a C -module, is a resolution by acyclic sheaves. so LSymmetry 2021, 13,12 ofThen, we can find a representative ( X, Lq ) of[] H q ( X, KX L I )through this resolution by acyclic sheaves. In other words, is actually a -closed L-valued (n, q)two |2 are locally integrable. Furthermore, through the kind on X such that ||, and | , discussions in Section 2.two, we could arrange the factors so thatn,q 2 , and2 , .In unique, |Y L(2) (Y, L). Now let l be the harmonic representative of |Y in n,q L (Y, L). Equivalently, l = l = 0. Applying the exact same argument of the very first part, we(2)lwill eventually acquire a sequence of holomorphic L-valued (n – q, 0)-forms l and its limit on X. Alternatively, l2 l ,|Y2 l , q2 , ,^ the sequence l is convergent to, say . Due to the fact l l = l , ^ = lim l = lim (l l ) = q .l l q^ ^ For that reason, Hn,q ( L, ) by definition. We denote this morphism by j([]) = . It truly is uncomplicated to verify that i j = Id and j i = Id. The proof is finished. Now, we’re prepared to prove the injectivity theorem on a non-compact manifold. A single could consult [3,5,7,8] to get a sophisticated comprehension for the injectivity theorem on a compact manifold. Theorem two (=Theorem 1). Let ( X, ) be a weakly pseudoconvex K ler manifold such that sec-Kfor some good continual K. Let ( L, L ) and ( H, H ) be two (singular) Hermitian line bundles on X. Assume the following situations: 1. two. three. There exists a closed subvariety Z on X such that L and H are both smooth on X \ Z; i L, L 0 and i H, H 0 on X; i L, L i H, H for some optimistic quantity .To get a (non-zero) section s of H with supX |s|two e- H , the SB 271046 In stock multiplication map induced by the tensor product 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.Proof. By Proposition 5, it’s adequate to prove thats : Hn,q ( L, L ) Hn,q ( L H, L H )is well-defined, hence injective. In other words, let Hn,q ( L, L ), and we should prove that s Hn,q ( L H, L H ). n,q In reality, considering that Hn,q ( L, L ), there exists l Hl ( L) and l L (two)n,q-(Y, L)Symmetry 2021, 13,13 ofwith = l l . Applying Proposition 2, we receive that 0 = ( L l , L l )l , L ([i L, L , ]l , l )l , L . Notice that i L, L 0, ([i L, L , ]l , l )l , L 0. Hence,( L l , L l )l , L = ([i L, L , ]l , l )l , L = 0.In particular, L l = 0. Now, apply Proposition two once more on sl and observe that (sl ) = 0, we acquire that 0 ( (sl ), (sl ))l , L H l l =( L H (sl ), L H (sl ))l , L H ([i L H, L H , ](sl ), sl )l , L H .Due to the fact L H (sl ) = s L l = 0, and([i L H, L H , ](sl ), sl )l , L Hsup |s|2 e- H ([i L H, L H , ]l , l )l , LX1 (1 ) sup |s|2 e- H ([i L, L , ]l , l )l , L X=0,it can be effortless to se.