# Algebraic geometry reference for differential geometer

I am a graduate student in differential geometry and would like to learn more about algebraic geometry recently. Are there any recommended textbook/reference/lecture notes which is easier for a differential geometer to approach? Or are there any standard textbooks that can build my way up to algebraic geometry instead of some AG books for DG students?

My background for algebra is only an honor course I took in undergraduate which includes group, ring, field, and module.

Your help is very much appreciated!!

• Among the classic references, Griffiths and Harris's Principles of algebraic geometry is one of the more accessible ones to more (complex) analytically minded geometers. But maybe you're looking for something even more specifically aimed at differential geometers? (Note: this is a pretty serious book, so will take some time. But then algebraic geometry is a very rich subject, so it's not so easy to learn it without a substantial commitment.) – R. van Dobben de Bruyn Sep 1 '20 at 4:28
• Not being a geometer of any kind, I found my brief attempts to use G+H as a gateway rather like running into a brick wall. (A more sober and informed take on the book can be found here projecteuclid.org/euclid.bams/1183545213 ) – Yemon Choi Sep 1 '20 at 4:36
• Given the background you list, you might find it more interesting/accessible to start with complex algebraic curves a.k.a. Riemann surfaces, and then one text which seems quite readable is Fulton's book, a PDF version of which can be found on his website at math.lsa.umich.edu/~wfulton/CurveBook.pdf Alternatively, maybe Miranda's book Algebraic Curves and Riemann Surfaces but I don't know too much about the content of that one. – Yemon Choi Sep 1 '20 at 4:45
• @R.vanDobbendeBruyn Thank you for your recommendation! I am working on Kähler manifolds, so GH's book may actually be a good reference for me. – yll_4302 Sep 1 '20 at 4:47
• Griffiths and Harris is a classic, but I kind of agree with Yemon Choi, that it is a difficult read. For basic facts about Kähler manifolds and Hodge theory, I would recommend Wells, Differential Analysis on Complex Manifolds as an alternative. – Donu Arapura Sep 1 '20 at 6:02

I started some years ago from this lecture note: Notes Algebraic Geometry 2 by by Karen Smith which is about Sheaf theory and Schemes

You may need to read her first lecture note: Notes Algebraic Geometry 1 which is nessesary for her second lecture note. With these two lecture notes you spend much less Time to cover all the nessesary concepts instead to read a Book.

The Book of J. P. Demailly : Analytic methods in algebraic Geometry is also useful

If you are interested in Complex Geometry and Kähler Geometry , I read this Book when I was a master student ; Lectures on Kähler Manifolds - Werner Ballmann And also Canonical Metrics in Kähler Geometry , Gang Tian

Also this Book is useful Complex Geometry, Daniel Huybrechts

The Book of Hodge Theory and Complex Algebraic Geometry I and II of Claire Voisin also very useful. I read them some years ago.

• Thank you very much for your reply, all the notes you provided are extremely useful for me, even for what I am lacking right now as a student working on the analytic aspect of Kähler geometry. – yll_4302 Sep 1 '20 at 7:22
• There is also the other book by J.-P. Demailly, where he intruduces tools like sheaves, locally ringed spaces, cohomology, and Chern classes, from a more analytic perspective. There is a section on algebraic varieties, defined via the maximal spectrum of a ring, meaning that the notion of generic point is not treated. Overall, it might be a good bridge towards algebraic geometry. – Aurelio Sep 1 '20 at 9:09
• @Aurelio I actually read Demailly's book before when I started Kähler geometry, back then I skipped most of the algebraic parts as I was told that I should focus on the analytic aspect. Perhaps it is time for me to pick the book up again. Thanks! – yll_4302 Sep 1 '20 at 10:36
• When I was initially trying to learn complex geometry, my advisor had me read Complex Analytic and Differential Geometry. Unfortunately, I don't remember much of it, but it was a good way to start learning the material. Also, he had written a book on complex geometry himself, so it was pretty high praise for him to recommend Demailly. – Gabe K Sep 1 '20 at 17:35