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Extra resources for Communications in Mathematical Physics - Volume 203

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35. 36) = 4| | . 2 Then he computes ρ and uses a maximum principle to argue that if M is complete, then ρ = 0, from which and then R vanish. Next, we discuss special coordinates. 37. Let (M, ω, ∇) be a special Kähler manifold. (a) A holomorphic coordinate system {zi } is special if ∇ Re(dzi ) = 0. (b) We say that special coordinates {zi } and flat Darboux coordinates {x i , yj } are adapted if Re(zi ) = x i . (c) Special coordinate systems {zi }, {wj } are said to be conjugate if there exists a flat Darboux coordinate system {x i , yj } such that Re(zi ) = x i and Re(wj ) = −yj .

24 below. 18. 37. 36 D. S. 21. Nowhere do we use the positive definiteness of ω. Hence our discussion applies also to pseudo-Kähler manifolds. ) We have the following easy result. 22. (a) Let (M, ω, ∇) be a special Kähler manifold. The connection ∇ determines a horizontal distribution H in the real cotangent bundle T ∗ M. Then H is invariant under the complex structure of T ∗ M. (b) The (0, 1) part of the connection ∇ on the complex tangent bundle T M equals the ∂¯ operator. Proof. (a) Choose a flat Darboux coordinate system {x i , yj }.

26) in special coordinates {zi } introduced above. 23) and the fact that ω has type (1, 1), we have ∂ ∂ , ∇(dzj ⊗ j ) ∂zi ∂z 1 ∂ 1 ∂τj ∂ ∂ ), − dzk ⊗ = −dzi ⊗ dzj ω ( i − τim k 2 ∂x ∂ym 2 ∂z ∂y 1 ∂τj i dzi ⊗ dzj ⊗ dzk = 4 ∂zk 1 ∂ 3F = dzi ⊗ dzj ⊗ dzk . 14) as well. The cubic form can also be used7 to relate the special Kähler connection ∇ to the Levi–Civita connection D. Write ∇ = D + AR , 7 I learned this from the account in [BCOV], though it also appears in many other works. 29) 38 D. S. Freed where AR ∈ 1 (M, End R T M).

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