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Unformatted text preview: SYMPLECTIC GEOMETRY, LECTURE 12 Prof. Denis Auroux 1. Existence of Almost-Complex Structures Let ( M, ) be a symplectic manifold. If J is a compatible almost-complex structure, we obtain invariants c j ( T M, J ) H 2 j ( M, Z ) of the deformation equivalence class of ( M, ). Remark. There exist 4-manifolds ( M 4 , 1 ), ( M 4 , 2 ) s.t. c 1 ( T M, 1 ) = c 1 ( T M, 2 ). We can use this to obtain an obstruction to the existence of an almost-complex structure on a 4-manifold: note that we have two Chern classes c 1 ( T M, J ) H 2 ( M, Z ) and c 2 ( T M, J ) = e ( T M ) H 4 ( M, Z ) = Z if M 4 is closed, compact. Then the class (1) (1 + c 1 + c 2 )(1 c 1 + c 2 ) 1 = c 2 + 2 c 2 = c 2 ( T M T M, J J ) = c 2 ( T M R C , i ) 1 is independent of J . More generally, for E a real vector space with complex structure J , we have an equivalence ( E R C , i ) = E E = ( E, J ) ( E, J ). Indeed, J extends C-linearly to an almost complex structure J C which is diagonalizable with eigenvalues i . Applying this to vector bundles, we obtain the Pontrjagin classes (2) p 1 ( T M ) = c 2 ( T M R C ) H 4 ( M, Z ) = Z for a 4-manifold M ....
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