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For the preparation of a complex geometry lecture I am looking for a good literature. I already have standard literature like Huybrechts "Complex Geometry. An Introduction" and I am also using it. But this book is very "global", in the sense that it only explains the usual quantities (Hermitian metric, Kähler metric, connections, all kinds of curvature, Complex Laplace-Beltrami operator etc.) from a very abstract and formal point of view and it omits local coordinate computation (like writing the Hermitian metric in local coordinates and deriving all kinds of relations between the coefficients ...).

Question: I would like to ask you, if you know any good reference on complex geometry (book, lecture notes, paper, survey etc.) which explains all these complex geometric quantities as mentioned above (Hermitian metric, ...) but from a "local" point of view (in coordinates, deriving all kinds of relation between the coefficients, ...) ?

Greetings, Raf

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  • $\begingroup$ These notes of Demailly www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf are a step in that direction. $\endgroup$ – Andy Sanders Jan 22 at 17:38
  • $\begingroup$ It doesn't hurt (actually, it does hurt) to start with the local definition of a metric, which is relatively straightforward, and try to derive for yourself the local coordinate formulas for everything else. For complex manifolds, it's a lot more painful than for real geometry, but, if you manage to do it all, you'll know it really really well. That helps streamline your efforts when you run into local coordinate calculations later on. You'll also catch errors, which are quite common, more easily. $\endgroup$ – Deane Yang Jan 29 at 15:55
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A more local perspective, with indices: Thierry Aubin, Nonlinear Analysis on Manifolds: Monge-Ampere Equations, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, 1982. There is a new edition of the same book: Thierry Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer-Verlag, 1998; see chapter 7, p. 251.

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    $\begingroup$ If I remember correctly, for the beginner, Aubin's book doesn't provide much detail as to how, for example, the Riemannian metric (which is of course valued in real numbers) relates to the sesqui-linear tensor he writes out in coordinates as $g_{\mu\bar\nu}dz^{\mu}dz^{\bar\nu}$. He also doesn't really explain (again, if I remember correctly) the relation between the real-valued Levi-Civita connection and the beast he writes in barred and unbarred indices. So while the book is local, I think it takes some effort to expand out some details for yourself. $\endgroup$ – Ben McKay Jan 22 at 8:29
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I think the paragraph "Complex Manifolds"(pp14-34),Chap.0, in Principles of Algebraic Geometry (Ph.Griffiths,J.Harris, 1978), will answer, at least partially,to your question . e.g. Calculus on Complex Manifolds p27.

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