Math Student Struggling with Griffiths Harris Algebraic Geometry

[Kalidor]

Hi everyone I'm a math student trying to go through Griffiths Harris Principles of algebraic geometry. I'm especially interested in complex tori and K3 surfaces but I confess I'm having a hard time. Can anyone suggest a reference (online notes or books, preferably a single book) that covers the prerequisites?

 

[lavinia]

Hi everyone I'm a math student trying to go through Griffiths Harris Principles of algebraic geometry. I'm especially interested in complex tori and K3 surfaces but I confess I'm having a hard time. Can anyone suggest a reference (online notes or books, preferably a single book) that covers the prerequisites?

what do you already know?

 

[Kalidor]

On the calculus side most of what is needed up to complex analysis (but just in 1 variable) and just the very first facts about real manifolds. Some algebraic geometry, say the content of hartshorne from the beginning up to proper schemes in the 2nd chapter.

 

[lavinia]

On the calculus side most of what is needed up to complex analysis (but just in 1 variable) and just the very first facts about real manifolds. Some algebraic geometry, say the content of hartshorne from the beginning up to proper schemes in the 2nd chapter.

This is not my area but I sat in on an undergraduate course for a semester and the book was quite simple and well written. I will try to find out the name of it for you..

To help I also got the book Algebraic curves and Riemann Surfaces by Rick Miranda. i found it extremely well written and though quite long, it goes from the basic to the sublime. There is a bunch of stuff in there about complex tori.

Another book Algebraic Curves and One dimensional fields by two russian sounding names - Bogomolov and Petrov - seems very good.

These books are for curves only.

I have a book on elliptic curves that is super - one of the best teaching books I have ever found - the authors make you figure everything out for yourself and just guide you along. they also have a historical outlook which is interesting by itself. It is Elliptic Curves by Mckean and Moll

 

[Kalidor]

Thanks a lot lavinia

 

[mathwonk]

what are you having a hard time with?

 

[Kalidor]

what are you having a hard time with?

Well, at first I tried to read directly the chapter about complex tori and abelian varieties, but I got brainraped by terms such as "Kahler metric" , "hodge form" and "Kodaira embedding theorem" within just a few lines.
So I went back to chapter 0 and specifically where the notion of complex manifold is introduced but I felt that what was written there was a brief summary and that I honestly couldn't be expected to get familiar with all those definitions (of tangent space, of de Rham cohomology etc) explained in just a few pages. Anyway those old Harvard folks managed to make me feel very dumb so here I came.

 

[Landau]

Perhaps you should just study manifolds first? That seems like the most common order, and you will become familiar with tangent space, de Rham cohomology, etc.

 

[Kalidor]

Perhaps you should just study manifolds first? That seems like the most common order, and you will become familiar with tangent space, de Rham cohomology, etc.

Yeah I thought so too, anyway I started browsing amazon today looking for a book on the subject.

 

[mathwonk]

have you tried:

https://www.amazon.com/s/ref=nb_sb_...s=swinnerton+dyer&x=16&y=19&tag=pfamazon01-20

and maybe the tata lectures on theta by mumford, volume I?

or lectures on riemann surfaces by robert gunning, or

algebraic curves and riemann surfaces by rick miranda,

or roy smith's article in lectures on riemann surfaces by cornalba, gomez- mont and verjovsky.

or griffiths' china lectures on algebraic curves.

 

[lavinia]

Yeah I thought so too, anyway I started browsing amazon today looking for a book on the subject.

the books I suggested develop the idea on manifolds from the point of view of complex analysis. they are very good.

A book on basic geometry and topology, Lecture Notes on Elementary Topology and Geometry, by Singer and Thorpe 2 well known mathematicians covers basic topology, the fundamental geoup and covering spaces, manifolds and calculus on manifolds, De Rham's theorem, and differential geometry of surfaces from the modern view point. It is an undergraduate text, it is well written, and its approach is modern.

 

[Kalidor]

the books I suggested develop the idea on manifolds from the point of view of complex analysis. they are very good.

A book on basic geometry and topology, Lecture Notes on Elementary Topology and Geometry, by Singer and Thorpe 2 well known mathematicians covers basic topology, the fundamental geoup and covering spaces, manifolds and calculus on manifolds, De Rham's theorem, and differential geometry of surfaces from the modern view point. It is an undergraduate text, it is well written, and its approach is modern.

And it's available for 10$, wow!

 

[Kalidor]

have you tried:

https://www.amazon.com/s/ref=nb_sb_...s=swinnerton+dyer&x=16&y=19&tag=pfamazon01-20

and maybe the tata lectures on theta by mumford, volume I?

or lectures on riemann surfaces by robert gunning, or

algebraic curves and riemann surfaces by rick miranda,

or roy smith's article in lectures on riemann surfaces by cornalba, gomez- mont and verjovsky.

or griffiths' china lectures on algebraic curves.

Luckily I have found miranda somewhere in my room (probably a leftover from a course on plane algebraic curves) so I'll try and take a look at that.

 

[mathwonk]

beauville's book on complex algebraic surfaces is very well written and has a nice chapter on complex tori.