This chapter examines complex manifolds. It discusses some basics concerning holomorphic functions on ℂn; the definition of a complex manifold; first examples of complex manifolds; the Newlander-Nirenberg theorem; metrics and almost complex structures on TM; the almost Kähler 2-form; symplectic forms; Kähler manifolds; complex manifolds with closed almost Kähler form; and examples of Kähler manifolds.
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