Crash course on algebraic geometry with view to applications in physics

Question

Could you please recommend any good texts on algebraic geometry (just over the complex numbers rather than arbitrary fields) and on complex geometry including Kahler manifolds that could serve as an informal introduction to the subject for a theoretical physicist (having in mind the applications in physics, e.g. in the string theory)?

What I want for a moment is to get some informal picture of the subject rather than being dug up into the gory details of the proofs and lost in higher and higher layers of abstraction of commutative algebra and category theory. The texts I have found so far are all rather dry and almost completely lack this informal streak, and all of them are geared towards pure mathematicians, so if there exists something like "Algebraic geometry for physicists" and "Kahler manifolds for physicists" (of course, they would probably have different titles :)), I would greatly appreciate the relevant references.

Answer 1

Griffiths and Harris' "Principles of Algebraic Geometry" (Wiley) is the best for your purposes (read only the parts on Kahler geometry). The sections on algebraic geometry in "Mirror Symmetry" (Clay/AMS) are essentially a Crib Notes version of that paper and some of the classic CY and special geometry papers referred to above.

What you should keep in mind going in is the following:

Kahler manifolds are complex manifolds with a hermitian inner product on tangent vectors which have a metric that is determined (locally) by a single function. It is the geometry in which the metric and the complex structure "get along very nicely." This simplifies lots of calculations and adds new symmetries. That's why we know so much about them.

Answer 2

I am not sure about generic applications to physics but when I hear about algebraic geometry I immediately think of string theory. And when I hear about Kähler manifolds it's pretty hard not to think of Calabi-Yaus :-)

So if you don't mind this kind of applications, you could find useful a paper by Brian Greene: String Theory on Calabi-Yau Manifolds. It contains also general talk about differential geometry and string theory. But most importantly, you'll find there the basics of Kähler and Calabi-Yau manifolds as well as lots of applications like mirror symmetry and investigation of moduli spaces.

Answer 3

One text that immediately comes to mind is "lectures on complex geometry" by Philip Candelas, an accessible introduction covering the very basics. I think it is hidden in some Trieste proceedings or some such.

Answer 4

Though not particularly aimed at physicist, you could take a look at An Invitation to Algebraic Geometry by Smith, Kahanpää, Kekäläinen and Traves. It's very short and avoids many technicalities and proof. Instead it give a bird's-eye view and succeeds well at transmitting some of the basics.

Answer 5

A new and more popular exposition directly from fields medalist Shing-Tung Yau: The shape of inner space
Find a review here: http://plus.maths.org/content/node/5389
Find a very gentle and short intro here: http://plus.maths.org/content/node/5388
Find a preview here: http://books.google.de/books?id=M40Ytp8Os_gC&lpg=PP1&ots=3dHRt8v3KI&dq=The%20shape%20of%20inner%20space&hl=en&pg=PP1#v=onepage&q&f=false
Find the webpage here: http://www.shapeofinnerspace.com/

Answer 6

One good source my undergraduate adviser recommended to me are the lecture notes of Candelas on Complex Geometry. They are written with string theory in mind and cover a lot of basic ground. I am not sure, if they are available online. Griffiths and Harris is very good, but probably not suitable as your only source for self-study. Just to get an idea what ideas were needed in string theory 25 years ago a look at chapter 12,14,15,16 in the second volume of Green, Schwarz, Witten might be helpful. Especially 14 and 15 should be interesting to you, even if you did not take a course in string theory yet.

By now there are of course a lot of other applications of ideas from algebraic geometry to the study of string theory beyond those ordinarily found in textbooks. For example model building in $F$-theory requires among other things to the study of singularities of elliptic fibrations and the approximate dynamics of certain branes is determined by variations of hodge structure. To actually find interesting examples knowledge of toric varieties is helpful. Most of those topics are actually not discussed in introductory texts.

Answer 7

Mirror Symmetry, especially the first two chapters give a brief introduction to algebric geometry. URL:http://www.claymath.org/library/monographs/cmim01.pdf

Answer 8

Some algebraic geometry with the main purpose of understanding D-branes in the context of mirror symmetry is reviewed in Paul Aspinwall's 'D-branes on Calabi-Yau manifolds' (http://arxiv.org/abs/hep-th/0403166).

I have not begun reading it thoroughly myself, but it seems accessible to physicists, at least those with the basic math background for string theory.

I believe category theory is necessary in understanding algebraic geometry, and a very basic introduction to category theory with the objective of understanding how topological quantum field theories (TQFTs), their observables and their boundary conditions form categories can be found in Ketan Vyas' thesis 'Topics in topological and holomorphic quantum field theory' (http://thesis.library.caltech.edu/5894/). It should be easily readable by those familiar with TQFTs.