Among the so-called Millenium Prize Problems – seven notoriously difficult mathematical problems, each open for decades, and each now carrying, courtesy of the Clay Mathematics Institute, a million-dollar prize – Kieth J. Devlin places the Hodge Conjecture last. “[A]n author should delay as long as possible introducing anything is likely to make his reader give up in despair,” Devlin writes in his book, The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of our Time. “There is no… path even to the problem’s front door.”
The Hodge Conjecture fascinated me even before I began studying mathematics. Its sheer inscrutability surely played a role. I sensed, behind the incomprehensible words and symbols of its Wikipedia page, a bafflingly deep, and coherent, world. I had to understand.
The problem’s formidability was only partly to blame. This world – which I envisioned – was not just expansive, but beautiful. I perceived, there, something like Dante’s “music of the heavenly spheres”.
My graduate study has taken me into a field of math relatively close to that which the Hodge Conjecture occupies. As I’ve explored the rich foothills of this towering mountain, its peak has become even more stunning and mysterious.
I will try to explain the conjecture to this blog’s lay readers.
Smooth complex projective algebraic varieties
We will introduce this mathematical concept through a few successive steps.
Let’s imagine a few things in our standard 3-dimensional space:
- A point, or two or three.
- A curve: start with a straight line, and then bend it or curve it.
- A surface: start with a flat plane, and then bend it in a smooth way, letting it wave and curve.
Actually, the 3-dimensional ambient space is not important here. Let’s take an n-dimensional ambient space instead (where n is, say, bigger than 3), and some number k smaller than n, and imagine:
- A smooth, curved “k-dimensional object”: take a k-dimensional space, and then bend it.
Pattern? Each object’s dimension is indicated by the bullet point next to it.
It’s easy enough to see what it would mean to curve a point (nothing), a straight line, or a flat plane. Yet even three-dimensional space can become curved in mathematics – though we couldn’t see it – as can four-dimensional space (General Relativity posits that this is exactly what gravity consists in). This works for space of any dimension k.
I often hear protestations at this point: “When you start getting into higher dimensions, I get off.” I don’t know what these things look like any more than you do! I’m just willing to accept that they exist and can be studied mathematically. You can be too.
Thus, we have a:
Smooth, curved, k-dimensional “object” in n-dimensional space.
We’ll call this a smooth algebraic variety. The story is not over yet.
Real numbers suck. The equation doesn’t even have any solutions! Among the many (apparently) pointless things you learned in high-school math, perhaps the deepest was that by passing to a new, larger number system – the complex numbers, in which has a square root, namely – we endow equations like the above with solutions. (What are they?)
We now work in an n-dimensional space in which each point is described by n complex coordinates, instead of the standard n real (Cartesian) coordinates. This changes the “material” of the space we work in. Just as the “missing solutions” to the equation (they’re and ) appear once we pass to the complex numbers, the “missing parts” of an algebraic variety of any dimension now arise with our passage to n-dimensional complex space.
Any complex number requires two (real) dimensions to describe (recall that each complex number lives in the complex plane, which consists of both a real and an imaginary axis). Precisely because a single complex number requires two dimensions to describe, the (real) dimension of our entire space will thus also double now, as will that of our points, curves, surfaces, and k-dimensional varieties. This makes everything harder to visualize. But it also makes them more complete.
“Yet there have been and still are geometricians and philosophers, and even some of the most distinguished,” argues Ivan, of Dostoevsky’s 1880 The Brothers Karamazov, “who… dare to dream that two parallel lines, which according to Euclid can never meet on earth, may meet somewhere in infinity. I have come to the conclusion that, since I can’t understand even that, I can’t expect to understand about God.”
Projective space is a non-Euclidean geometry in which, among other things, two parallel lines meet at infinity. Though its exact mathematical construction is challenging and perhaps uninstructive, the mathematical universe of projective space extends, and in some sense completes, our standard Euclidean space. The passage to projective space, indeed, offers another sense in which the naïve Euclidean picture represents only an artificial part of the whole.
This becomes evident, for one, in the characteristic eventual meeting of two parallel lines. In Euclidean space, two distinct lines usually intersect in a point, except in the special or “degenerate” case in which the lines become parallel and they don’t intersect at all. This pathology is actually an artificial artifact of the Euclidean setting. These lines, in some sense, actually always meet in exactly one point – provided, that is, that we work with the “whole” projective space. Parallel itself actually ceases to have meaning, here; whether two lines are in fact parallel depends only on the (arbitrary) way in which we ultimately truncate them using a restricted Euclidean lens.
Many situations, in fact, feature apparently disparate phenomena which unite upon passage to projective space. For example, a so-called conic curve in projective space reveals the disparate conic sections that we know only after it is artificially truncated using various Euclidean perspectives.
Complex projective space furnishes the most natural setting in which to study algebraic varieties.
It’s become part of folk-physics lore that if you head off into space in some fixed direction, you’ll eventually land back where you started. The universe, in other words, is without boundary, in the sense that you’ll never come up, as you go along, against some edge.
We want our algebraic varieties to be the same way. What does this look like?
- A point should… Any point is automatically closed.
- A curve shouldn’t just stop at a point; it should loop back upon itself.
- A surface should never cut off at an edge; it should wrap back upon itself, like (the exterior of) a blob in space.
- You get the idea.
Varities thus represent k-dimensional smooth closed objects living inside an n-dimensional complex projective space.
Cohomology and Algebraic Cycles
Smooth, complex, projective algebraic varieties are elegant objects. The Hodge Conjecture asserts something deep about them.
Cohomology and Hodge classes
Cohomology is a mathematical tool that extracts information about the shape of an algebraic variety and assembles it into a package, called a cohomology group, of entities called cohomology classes. Each cohomology group provides structural information about the variety.
Each dimension of any given variety, in fact, gets its own cohomology group, so that, for a fixed variety X of complex dimension k (and so real dimension 2k), we have cohomology groups which record the behavior of X in each of the separate dimensions i = 0, 1, … , 2k.
W. V. D. Hodge, in the 30s, discovered – among the many fantastic phenomena which arise when we work with the complex numbers – a new one, now termed the Hodge decomposition, whereby each cohomology group of a complex projective variety decomposes into a combination of smaller groups which reflect, in even more detail, the structure of that algebraic variety. Each of the even-dimensional cohomology groups i = 0, 2, … , 2k, in particular, contains a singularly special sub-group whose constituent cohomology classes are known as Hodge classes.
Any algebraic variety admits, meanwhile, so-called subvarieties, which are exactly what they sound like: algebraic varieties of lower dimension embedded within the original variety. (Imagine, for example, a curve inscribed on/in a surface.)
There’s a natural process which associates, to any i-dimensional subvariety of a k-dimensional variety, a cohomology class of dimension 2k – 2i. This cohomology class, called an algebraic cycle, in some sense represents the contribution of that subvariety to the variety’s cohomology. More is true: This algebraic cycle is necessarily a Hodge class.
The Hodge conjecture is a question about the extent to which the cohomology of an algebraic variety is informed by its own subvarieties. Every subvariety determines a Hodge class. The conjecture asks the converse. Roughly:
Is every Hodge class the cohomology class of a subvariety?
The Hodge Conjecture, if true, would permit one to deduce deep structural information about an algebraic variety merely by understanding its cohomology groups, which are often easier to compute in practice.
Beyond the Hodge Conjecture
The Hodge Conjecture (see [1, Conjecture 2.25] for a scholarly reference) is actually just the (admittedly redoubtable) gateway to an entire field of math, which could acceptably be termed Hodge Theory. This field is rich with marvelous results and outstanding conjectures. It’s also one of the hardest extant fields of math.
The standard Hodge Conjecture, for one, appears in a much deeper form in the mathematical titan Grothendieck’s legendary Generalized Hodge Conjecture (see [1, Conjecture 2.40]). The so-called GHC is so deep, in fact, that within it the original Hodge Conjecture, itself no easy assertion to grasp, becomes again buried in profound generality (this evokes also Grothendieck’s generalization of the Riemann-Roch theorem). This deeeper form is even more aesthetically fascinating.
Also worth mentioning is the GHC’s great cousin, the Generalized Bloch Conjecture (see [1, Conjecture 1.9]). The Generalized Bloch Conjecture’s assertion about smooth projective varieties is just as inspiring, if just as difficult to state, as is that of the Generalized Hodge Conjecture.
Beyond these two, one could point to a litany of fantastic papers by Claire Voisin, Hodge Theory’s greatest living servant   . Indeed, Hodge Theory has no greater force for progress, precision, and beauty than Claire Voisin. This informal article is dedicated to her.
- Voisin: Chow Rings, Decomposition of the Diagonal, and the Topology of Families
- Voisin: The Generalized Hodge and Bloch Conjectures are Equivalent for General Complete Intersections
- Voisin: The Generalized Hodge and Bloch Conjectures are Equivalent for General Complete Intersections, II
- Voisin: Hodge Loci and Absolute Hodge Classes