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CHAPTER1 Multilinear algebra
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abelian groups algebra antiholomorphic basis called chart coefﬁcients compact complex manifold complex structure consider continuous map contravariant vector contravariant vector ﬁeld coordinate bundle covariant tensor cross-section deﬁned DEFINITION denote differentiable functions differentiable manifold differentiable structure differential form diﬂerentiable easy to check element equivalent exact sequence exists ﬁber bundle ﬁber space ﬁrst ﬁw ﬁx ﬁxed form of degree forms of type free system function f group G Hermitian metric holomorphic functions induced injective isomorphism lemma Let f Lie group linear map manifold and let morphism n-dimensional open covering open set open subset operator point m e presheaf presheaf homomorphism Proof Let prove remark satisﬁes sequence of sheaves sheaf of germs structure group submanifold subspace surjective tangent space tangent vectors tensor of order THEOREM Let topological group topological space transition functions trivial vector bundle vector space