I had taken course in complex analysis and self-studied differential manifolds and basic topology from Loring Tu Introduction to Manifold. What else do I need (minimally) to prepare myself to read the Principal of Algebraic Geometry by Philip Griffiths? (Do I need Huybrechts Complex Geometry or Rick Miranda's Algebraic curve and Riemann surface?)
What do I need to read Philip Griffths
algebraic-geometrycomplex-geometry
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If you are a complex analyst, you cannot not like holomorphic functions, in many ways they are preferable to meromorphic ones. For instance, your first preference is to solve, say, differential equations so that solutions exist everywhere (in the domain where the equation is defined and is suitably regular), rather than on an open subset. You also may want to have finite dimensionality of the space of solutions (and having estimates on solutions in terms of equations themselves or other data). Your problem, however is that compact complex manifolds lack any nonconstant holomorphic functions. As the result, you compromise and work with sheaves which may come in several different forms. One of these is the sheaf of sections of, say, a line bundle (or, more generally, a vector bundle) given by a divisor, or in form of holomorphic tensors. The latter are no longer functions and thus could exist everywhere on your compact manifold. Of course, in the process you may want to work with meromorphic functions (which are allowed to blow up at the given divisor $D$), but if you do not impose any restrictions along $D$, then you loose finite-dimensionality of the space of solutions, integrability, etc. Another thing which may happen is that solutions of your equations are multi-valued. This is not good for a variety of reasons, so you try to make them single-valued by regarding them as sections of a certain sheaf and then extending holomorphically over the branching divisor. The fact that this is (sometimes) possible is due to the fact that you are working with either holomorphic equations or, at worst, equations which are meromorphic but have controlled singularities along $D$.
The bottom line: Working with holomorphic sheaves is not that different (or, frequently, is the same) than working with meromorphic sheaves where singularities are tightly controlled (where they can occur and what type is allowed). Loosening this control may (and, frequently, does) lead to undesirable results.
I think the best option with little knowledge of differential geometry is definitely Huybrechts' Complex Geometry. His approach is to introduce just the right amount of differential geometry that one needs to understand the proofs of the big theorems. Plus, he mixes in the sheaf theory with the analytic theory in a nice way. It will definitely serve you well if you know at least a bit of de Rham cohomology before starting. If you go with this, don't be afraid to skip chapter 1.2 on first reading and refer back to it when you need it. That chapter has a fantastic treatment of the linear algebra needed for complex geometry, but you might feel more motivated to read it once you know how it will be used.
Voisin's book is a better second course in my opinion. This set of books goes much deeper into both the analysis and the homological algebra of the theory. I've found the proofs in that book to be of greater depth than Huybrechts'. Plus, no analysis punches are pulled, as opposed to Huybrechts' book. So, if you feel like you want to read the proof of the Hodge decomposition, Voisin's book has the detail you'll need.
Griffiths and Harris and Demailly's books are excellent from the analytic viewpoint, but without differential geometry background I would wait on these. They are both almost entirely differential geometric in nature. G&H's chapter 0 does have a ton of great background material for complex geometry that is nice to refer to on occasion, though.
Demailly's book is hard. It is great in that it really starts at the beginning with definitions of smooth manifolds. That being said, if heavy analysis isn't your cup of tea, then that book will be a tough read. I use it more as a reference when I need to dig deep into analytic details.
CMP's book is rather orthogonal to the rest on this list. If you're just trying to learn algebraic geometry, I would suggest reading the first chapter up until Hodge structures are introduced. The way they illustrate the case of elliptic curves and tori is a really nice intro to period domains/maps, but I personally found the next few chapters incomprehensible on first pass. I personally felt that the details were far too sparse in this book.
Best Answer
Also too long for a comment. So I too will put it as an answer.
I was told in a comment to one of my deleted questions that Griffiths' Introduction to Algebraic Curves is a prerequisite to griffiths harris principles of algebraic geometry.
I used Rick Miranda's Algebraic curve and Riemann surface (chapters I - VIII) as an alternative to Griffiths' Introduction to Algebraic Curves.
It appears that griffiths harris principles of algebraic geometry somewhat overlaps with both Rick Miranda's Algebraic curve and Riemann surface (chapters IX onwards) and Huybrechts Complex Geometry.
P.S. I actually have to study Griffiths' Introduction to Algebraic Curves, Huybrechts Complex Geometry and chapters 0 and 1 of griffiths harris principles of algebraic geometry, and I'm hoping it will suffice to study Rick Miranda's Algebraic curve and Riemann surface (chapters I - VIII), chapters 0 and 1 of griffiths harris principles of algebraic geometry and all the complexification I studied in external sources (see my questions eg this one or this one)