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I am interested to pursue my graduate studies in complex geometry but sadly I did not find a lot of references regarding the learning roadmap for complex geometry on the website. (most of them are algebraic geometry) I wish to study myself a bit as preparation before going into the graduate program. What prerequisite do I actually need so that in future I can read books by Philip Griffith? Currently, I have studied/taken courses in

1) Modern Algebra (up to field extension)

2) Differential Manifold, Riemannian Geometry

3) Complex, Real Functional Analysis.

4) Basic PDE.

5) Topology, Algebraic Topology (First 2 chapters of Hatcher)

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    $\begingroup$ I would recommend to start with Griffiths-Harris, which contains a lot of foundational material. If you happen to be blocked at some point you can always go back and find some appropriate reference. $\endgroup$
    – abx
    Commented Dec 6, 2019 at 13:43
  • $\begingroup$ I agree with @abx. It looks like you have the foundations needed to start trying to read Griffiths and Harris. Ben McKay's recommendations below also look good. Look for one that heads in the direction you want and that you find easiest to read. $\endgroup$
    – Deane Yang
    Commented Dec 6, 2019 at 16:10
  • $\begingroup$ @DeaneYang, I thought I will need several complex variables, commutative algebra etc to start reading it? $\endgroup$
    – Nothing
    Commented Dec 6, 2019 at 17:06
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    $\begingroup$ Since Griffiths-Harris takes an analytic approach, commutative algebra is not needed. There is some several complex variables needed, but you can wait until it's actually being used. And you can temporarily just accept the needed theorems as given. $\endgroup$
    – Deane Yang
    Commented Dec 6, 2019 at 19:38
  • $\begingroup$ @DeaneYang Thank you very much for the suggestion. $\endgroup$
    – Nothing
    Commented Dec 7, 2019 at 6:58

4 Answers 4

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I can’t recommend Voisin’s book “Hodge Theory and Complex Algebraic Geometry” enough. It’s very lucid and precise, and it covers quite a lot of material.

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Demailly, Complex Analytic and Differential Geometry

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Huybrechts, Complex Geometry: An Introduction

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  • $\begingroup$ A very good recommendation. It has these very nice appendices covering eg. sheaf cohomology $\endgroup$
    – Piotr Achinger
    Commented Dec 6, 2019 at 19:16
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S. S. Chern, Complex Manifolds Without Potential Theory

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