By Almeida L., Damascelli L., Ge Y.
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Extra info for A few symmetry results for nonlinear elliptic PDE on noncompact manifolds
We saw that E is a subbundle of the trivial bundle C2 × CP1 . Let p stand for the orthogonal projection from the trivial bundle to E. Over the point [z0 : z1 ] ∈ CP1 this projection is given by w0 z¯0 + w1 z¯1 p(w0 , w1 ) = (z0 , z1 ). |z0 |2 + |z1 |2 This gives a connection ∇ on E if we take the exterior derivative d as a connection on the trivial bundle followed by p. Let us take a section s of E ((z0 , z1 ) at a point [z0 : z1 ]), then the one-form ∇s with values in E is given by ∇s = p(dz0 , dz1 ) = z¯0 dz0 + z¯1 dz1 (z0 , z1 ).
Because the matrices Mij are uppertriangular we have an inclusion of the vector bundle F into E (corresponding to the upper left corner of the transition matrices Mij . The presence 1 in the lower right corner indicates that the quotient line bundle is trivial: 0 → F → E → 1 → 0. The corresponding cohomology ˇ 1 (X, F ) is the class of the cocycle hij and when it is not a class in H coboundary, the extension is not split. ˇ 1 (CP1 , T ∗ CP1 ) is one-dimensional Example. T ∗ CP1 = L⊗−2 and hence H and there is a distinguished non-zero element κ in this group with the corresponding non-trivial bundle extension 0 → T ∗ CP1 → P → 1 → 0.
Next we look at the zeros of s and they define a divisor D = p vp (s)[p]. We observe that furthermore s defines a nowhere vanishing holomorphic section of L−D ⊗ V . Thus we have a subbundle F ⊂ L−D ⊗ V spanned by s and hence an exact sequence 0 → F → L−D ⊗ V → Q → 0. 3 we have established the following: Fact. If V is holomorphic vector bundle over CP1 then there exists a divisor D such that Γ(V ⊗ LD ) = 0. Hence there exists an integer N such that for any n ≥ N one has Γ(V ⊗ L⊗n ) = 0 simply because one may take N = deg(D).
A few symmetry results for nonlinear elliptic PDE on noncompact manifolds by Almeida L., Damascelli L., Ge Y.