Two Miracles of Algebraic Geometry
Posted by John Baez
In real analysis you get just what you pay for. If you want a function to be seven times differentiable you have to say so, and there’s no reason to think it’ll be eight times differentiable.
But in complex analysis, a function that’s differentiable is infinitely differentiable, and its Taylor series converges, at least locally. Often this lets you extrapolate the value of a function at some faraway location from its value in a tiny region! For example, if you know its value on some circle, you can figure out its value inside. It’s like a fantasy world.
Algebraic geometry has similar miraculous properties. I recently learned about two.
Suppose if I told you:
- Every group is abelian.
- Every function between groups that preserves the identity is a homomorphism.
You’d rightly say I’m nuts. But all this is happening in the category of sets. Suppose we go to the category of connected projective algebraic varieties. Then a miracle occurs, and the analogous facts are true:
- Every connected projective algebraic group is abelian. These are called abelian varieties.
- If $A$ and $B$ are abelian varieties and $f : A \to B$ is a map of varieties with $f(1) = 1$, then $f$ is a homomorphism.
The connectedness is crucial here. So, as Qiaochu Yuan pointed out in our discussion of these issues on MathOverflow, the magic is not all about algebraic geometry: you can see signs of it in topology. As a topological group, an abelian variety is just a torus. Every continuous basepoint-preserving map between tori is homotopic to a homomorphism. But the rigidity of algebraic geometry takes us further, letting us replace ‘homotopic’ by ‘equal’.
This gives some interesting things. From now on, when I say ‘variety’ I’ll mean ‘connected projective complex algebraic variety’. Let $Var_*$ be the category of varieties equipped with a basepoint, and basepoint-preserving maps. Let $AbVar$ be the category of abelian varieties, and maps that preserve the group operation. There’s a forgetful functor
$U: AbVar \to Var_*$
sending any abelian variety to its underlying pointed variety. $U$ is obviously faithful, but Miracle 2 says that it’s is a full functor.
Taken together, these mean that $U$ is only forgetting a property, not a structure. So, shockingly, being abelian is a mere property of a variety.
Less miraculously, the functor $U$ has a left adjoint! I’ll call this
$Alb: Var_* \to AbVar$
because it sends any variety $X$ with basepoint to something called its Albanese variety.
In case you don’t thrill to adjoint functors, let me say what this mean in ‘plain English’ — or at least what some mathematicians might consider plain English.
Given any variety $X$ with a chosen basepoint, there’s an abelian variety $Alb(X)$ that deserves to be called the ‘free abelian variety on $X$’. Why? Because it has the following universal property: there’s a basepoint-preserving map called the Albanese map
$i_X \colon X \to Alb(X)$
such that any basepoint-preserving map $f: X \to A$ where $A$ happens to be abelian factors uniquely as $i_X$ followed by a map
$\overline{f} \colon Alb(X) \to A$
that is also a group homomorphism. That is:
$f = \overline{f} \circ i_X$
Okay, enough ‘plain English’. Back to category theory.
As usual, the adjoint functors
$U: AbVar \to Var_* , \qquad Alb: Var_* \to AbVar$
define a monad
$T = U \circ Alb : Var_* \to Var_*$
The unit of this monad is the Albanese map. Moreover $U$ is monadic, meaning that abelian varieties are just algebras of the monad $T$.
All this is very nice, because it means the category theorist in me now understands the point of Albanese varieties. At a formal level, the Albanese variety of a pointed variety is a lot like the free abelian group on a pointed set!
But then comes a fact connected to Miracle 2: a way in which the Albanese variety is not like the free abelian group! $T$ is an idempotent monad:
$T^2 \cong T$
Since the right adjoint $U$ is only forgetting a property, the left adjoint $Alb$ is only ‘forcing that property to hold’, and forcing it to hold again doesn’t do anything more for you!
In other words: the Albanese variety of the Albanese variety is just the Albanese variety.
(I am leaving some forgetful functors unspoken in this snappy statement: I really mean “the underlying pointed variety of the Albanese variety of the underlying pointed variety of $X$ is isomorphic to the Albanese variety of $X$”. But forgetful functors often go unspoken in ordinary mathematical English: they’re not just forgetful, they’re forgotten.)
Four puzzles:
Puzzle 1. Where does Miracle 1 fit into this story?
Puzzle 2. Where does the Picard variety fit into this story? (There’s a kind of duality for abelian varieties, whose categorical significance I haven’t figured out, and the dual of the Albanese variety of $X$ is called the Picard variety of $X$.)
Puzzle 3. Back to complex analysis. Suppose that instead of working with connected projective algebraic varieties we used connected compact complex manifolds. Would we still get a version of Miracles 1 and 2?
Puzzle 4. How should we pronounce ‘Albanese’?
( I don’t think it rhymes with ‘Viennese’. I believe Giacomo Albanese was one of those ‘Italian algebraic geometers’ who always get scolded for their lack of rigor. If he’d just said it was a bloody monad…)
Re: Two Miracles of Algebraic Geometry
Puzzle 4 is easy to knock off: it is pronounced “alba-nay-say”.
For Puzzle 3: yes, both miracles carry over to connected compact complex manifolds. There are proofs in the first few pages of the book “Complex Abelian Varieties” by Birkenhake–Lange.
The other two seem to require some actual thought.