When I first tried to read Griffiths and Harrris’s Principles of Algebraic Geometry, I was baffled by formulas like . The absolute value function wasn’t analytic, so its derivative with respect to wasn’t defined. And what were all these ‘s I was seeing? What were they, and why didn’t they seem to be equal to ?
Maybe I’m the only person who was confused by this. But, if this stuff bothers you too, then this post is for you.
In algebraic geometry, the most important functions are the analytic functions. (In this post, “analytic” means “complex analytic”.) Indeed, much of the progress in algebraic geometry in the last fifty years has been learning how to study the geometry of algebraic varieties using only the algebraic, and hence analytic, functions on those varieties. This is especially necessary to those who want to prove results over fields other than .
Before learning these ideas, though, one should probably learn how to study smooth functions on complex varieties. In particular, the deRham theory is much nicer if we allow all smooth functions, rather than restricting to just analytic ones. (To get a few hints of why, remember that a bounded analytic functions is constant, and nonzero analytic functions never have compact support.)
So, algebraic geometers have developed a notation which allows them to work with smooth functions that are not analytic. At the same time, analytic functions do play a special role in the theory, so the notation is particularly adapted to work well with analytic functions. This can be confusing to the beginner (it was for me!) because it is easy to memorize results which hold only in the analytic case and try to apply them in the smooth case.
In the rest of this post, I will explain this notation. I will assume you are familiar with differential forms; if you are not, I recommend Terry Tao’s PCM article.
To start out with, suppose that we have a smooth function from to . For example: . Then we can take its differential and get a differential form . When we evaluate on a tangent vector , and at a point , we get a measure of how the function changes between and , for real . For example, with as above, we have . Of course, is a complex valued one form, because is a complex valued function, but we can still think of as measuring change along perfectly ordinary tangent vectors.
We could write as . However, some experience shows that it is better to express one forms in terms of and . What do these symbols mean? Well, and are complex valued functions on , so their differentials are one forms. One can check that their differentials are everywhere linearly independent, so every one form can be written uniquely as a linear combination of and . For example, the above function is just , so . Supposing that I had consisidered . Then .
This illustrates the general principle: If is an analytic function, then , where is the derivative you learned in your first complex analysis course. The function will also be analytic. On the other hand, if is a smooth, but not analytic function, then will be of the form . Neither nor will necessarily be analytic.
In general, when you are working with analytic functions, all the rules you learned in single variable calculus work: the sum rule, the product rule, the chain rule and so forth. On the other hand, when you are working with smooth but nonanalytic functions, everything works the way you learned in multivariable calculus. In particular, this explains my confusion above about why isn’t ; it’s the same reason that, writing and for the coordinates on , the one-form isn’t .
One-forms of the form are called forms, while one-forms of the form are called forms. More generally, if we are working with functions of complex variables, we will have -forms, for . In coordinates, a -form is a form that can be written as a sum of terms of the form a smooth function times
More conceptually, a -form is a -form such that
for any vectors , , …, .
This seems like a good point to distinguish two concepts which confused me when I was learning this material. A -form is a sum of terms of the form a smooth function times . A holomorphic -form is a sum of terms of the form an analytic function times . Both of them can intuitively be thought of as “a form which is purely holomorphic”, but they make this concept rigorous in different ways.
Finally, what is ? By definition,
Notice that this equation makes sense: , and are all one forms, whose meaning we know. The expressions and denote complex-valued functions of , which are determined by the above equation. When is analytic, . But, when is smooth, you pretty much have to fall back on the definition.
If you are still confused by all this notation, I recommend trying to read a book which uses a lot of it, thinking back frequently to the definitions to make sure everything makes sense. Pretty soon, everything will seem obvious and second nature. At that point, you’ll be ready to confuse everyone else!