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By Khinchin A.

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6. For each point p0 in M , there is a smooth real function f in a neighborhood of p0 such that ω = i∂∂f . 7. For each point p0 in M , there are holomorphic coordinates z centered at p0 such that g(z) = 1 + O(|z|2 ). A function f as in (6) will be called a K¨ ahler potential, holomorphic coordinates as in (7) will be called normal coordinates at p0 . 10) up to terms of order two and higher. 17. By definition, (1) and (2) are equivalent. 16. The equivalence of the two assertions in (4) follows from barring the respective equation.

With our conventions, we have Zj♭ := Zj , · = 1 ∗ Z 2 j and Zj♭ := Zj , · = 1 ∗ Z . 31) 36 ¨ hler Manifolds Lectures on Ka Since Xj = Zj + Zj and Yj = i(Zj − Zj ), we get Xj∗ = 1 ∗ i (Z + Zj∗ ) and Yj∗ = − (Zj∗ − Zj∗ ) 2 j 2 ∗ for the dual frame (X1∗ , Y1∗ , . . , Xm , Ym∗ ) of (X1 , Y1 , . . , Xm , Ym ). 32) + ∂, meaning that the sum of the first terms is equal to ∂ D and the sum of the second to ∂. 26 also hold for complex tangent vectors. 26 for the first equality and comparison of types for the second.

It follows that π ′ ◦ f maps Si to a point in T . In particular, there is a holomorphic map g : T → T such that π ′ ◦ f = g ◦ π. Suppose now that f is biholomorphic. Then f is not constant on Si (we assume m ≥ 2). By what we said above, it follows that f maps each Si biholomorphically to an Sj′ . Thus k = l and, up to renumeration, f (Si ) = Si′ for all i. Moreover, the induced map g : T → T is biholomorphic with g(pi ) = p′i for all i. Let now k = l = 2, p1 = p′1 = 0, p2 = p, p′2 = p′ . 6 above we showed that any biholomorphic map of T is of the form h(z) = Az + b with b ∈ T and A ∈ Gl(m, C) such that A(Γ) = Γ.

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A course of mathematical analysis by Khinchin A.


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