Category Archives: opinions

Why believe the Hodge Conjecture?

Tom Graber recently asked me why people believe the Hodge conjecture, given the sparse evidence for its truth. I didn’t have time then to answer fully (I was giving a talk), but it’s a question that deserves a full answer. So I’ve sketched below what I feel are the reasons for believing the Hodge conjecture.

The Hodge conjecture is perhaps the most famous problem in algebraic geometry. But progress on the Hodge conjecture is slow, and a lot of algebraic geometry goes in different directions from the Hodge conjecture. Why should we believe the Hodge conjecture? How important will it be to solve the problem?

The Hodge conjecture is about the relation between topology and algebraic geometry. The cohomology with complex coefficients of a smooth complex projective variety splits as a direct sum of linear subspaces, the Hodge decomposition H i(X,C) = Σj = 0i H j,i-j(X). The cohomology class of a complex subvariety of codimension p lies in the middle piece H p,p(X) of H 2p(X,C). The Hodge conjecture asserts that any element of H 2p(X,Q) which lies in the middle piece of the Hodge decomposition is the class of an algebraic cycle, meaning a Q‑linear combination of complex subvarieties.

The main evidence for the Hodge conjecture is the Lefschetz (1,1)-theorem, which implies the Hodge conjecture for codimension-1 cycles. Together with the hard Lefschetz theorem, this also implies the Hodge conjecture of cycles of dimension 1. These results are part of algebraic geometers’ good understanding of line bundles and codimension-one subvarieties.

Not much is known about the Hodge conjecture in other cases, starting with 2‑cycles on 4‑folds. For example, it holds for uniruled 4‑folds (Conte-Murre, 1978). That includes 4‑fold hypersurfaces of degree at most 5, but the Hodge conjecture remains unknown for smooth 4‑fold hypersurfaces of degree at least 6. Why should we believe the conjecture?

One reason to believe the Hodge conjecture is that it suggests a close relation between Hodge theory and algebraic cycles, and this hope has led to a long series of discoveries about algebraic cycles. For example, Griffiths used Hodge theory to show that homological and algebraic equivalence for algebraic cycles can be different. (That is, an algebraic cycle with rational coefficients can represent zero in cohomology without being connected to zero through a continuous family of algebraic cycles.) Mumford used Hodge theory to show that the Chow group of zero-cycles modulo rational equivalence can be infinite-dimensional. There are many more discoveries in the same spirit, many of them summarized in Voisin’s book Hodge Theory and Complex Algebraic Geometry.

Another reason for hope about the Hodge conjecture is that it is part of a wide family of conjectures about algebraic cycles. These conjectures add conviction to each other, and some of them have been proved, or checked for satisfying families of examples.

The closest analog is the Tate conjecture, which describes the image of algebraic cycles in etale cohomology for a smooth projective variety over a finitely generated field, as the space of cohomology classes fixed by the Galois group. The Tate conjecture is not known even for codimension-1 cycles. But Tate proved the Tate conjecture for codimension-1 cycles on abelian varieties over finite fields. Faltings proved the Tate conjecture for codimension-1 cycles on abelian varieties over number fields by a deep argument, part of his proof of the Mordell conjecture. An important piece of evidence for the Hodge conjecture is Deligne’s theorem that Hodge cycles on abelian varieties are “absolute Hodge”, meaning that they satisfy the arithmetic properties (Galois invariance) that algebraic cycles would satisfy. This means that the Hodge and Tate conjectures for abelian varieties are closely related.

The Tate conjecture belongs to a broad family of conjectures about algebraic cycles in an arithmetic context. These include the Birch–Swinnerton-Dyer conjecture, on the arithmetic of elliptic curves, and a vast generalization, the Bloch–Kato conjecture on special values of zeta functions. One relation among these conjectures is that the Birch–Swinnerton-Dyer conjecture for elliptic curves over global fields of positive characteristic is equivalent to the Tate conjecture for elliptic surfaces, by Tate. Some of the main advances in number theory over the past 30 years, by Kolyvagin and others, have proved the Birch–Swinnerton-Dyer conjecture for elliptic curves over the rationals of analytic rank at most 1.

The Hodge conjecture belongs to several other families of conjectures. There is Bloch’s conjecture that the Hodge theory of an algebraic surface should determine whether the Chow group of zero cycle is finite-dimensional. There is the Beilinson–Lichtenbaum conjecture, recently proved by Voevodsky and Rost, which asserts that certain motivic cohomology groups with finite coefficients map isomorphically to etale cohomology.

This web of conjectures mutually support each other. Mathematicians continually make progress on one or the other of them. Trying to prove them has led to a vast amount of progress in number theory, algebra, and algebraic geometry. For me, this is the best reason to believe the Hodge conjecture.

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I am not a radical, but I am boycotting Elsevier

I’ve signed the Elsevier boycott declaration at the costofknowledge site inspired by Tim Gowers’ blog post on the many problems with Elsevier’s behavior.

I am not a publishing reform radical. I value the traditional model of subscription journal publishing and have invested time and effort in making peer review work.* I suspect that lots of people in similar positions are among the majority of mathematicians who have not signed the boycott declaration yet.

So, why do I think we should boycott Elsevier?

The mathematical scholarly community operates under a strong social compact — this is one reason so many of us do so much for free. With very rare exceptions, mathematicians hold themselves to a higher standard than the minimal criterion of what they can get away with. So should the publishers we deal with. Scholarly society and university press publishers, for the most part, do. Springer used to but is in transition. Elsevier does not. Elsevier has demonstrated again and again that it will cross the boundaries of acceptable behavior on pricing, on editorial integrity, on legislative lobbying.

None of what Elsevier does is illegal. There is no law against running a journal as Chaos, Solitons and Fractals was run. There is no law against political donations to elected officials who bring forward advantageous legislation. There is, apparently, no law against bundling and ruthless pricing that produces a profit margin in line with what monopolies achieve.

There is, however, social sanction — if we’ll use it. I won’t deal with Elsevier as if it’s part of the mathematical community when it shows little commitment to the standards of behavior that membership implies.

*I was co-editor of Proceedings of the LMS from 2003 to 2008, and am currently a managing editor of Compositio Mathematica.

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Why you should care about positivity

I started this blog about a year ago briefly recommending Rob Lazarsfeld’s book Positivity in Algebraic Geometry, which gives bite-size treatments of many topics hard to find elsewhere.

I’d like to make a stronger case now because it’s an important book. People often give me credit for knowing a lot just because I know what’s in it. It’s rarely on my shelves because it’s almost always in a stack near where I’m working. When I lost my copies in transit between MSRI and Cambridge, I replaced them immediately.*

The title might sound, on the face of it, like something specialized or technical. In fact, positivity is arguably the fundamental difference between algebraic geometry and topology. For example, the intersection multiplicity of two distinct complex curves which meet at a point in a complex algebraic surface S is always positive. As a result, if you know the homology classes of the two curves, then you know the total intersection number N from the cohomology ring of S, and that implies that the “physical” number of intersection points is at most N. This is completely false in topology: you can push around one submanifold to meet another submanifold in as many points as you like. The result is that just knowing the homology class of an algebraic curve controls its geometric properties (it can’t wiggle too much). Much of algebraic geometry builds on this kind of rigidity.

*Not at all painful because Lazarsfeld insisted Springer publish in paperback and keep the price down. Losing Kollár’s Rational Curves on Algebraic Varieties, on the other hand…

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How to like schemes in the right way

Some people like schemes too much. Others don’t like them at all. Here is my view on how to like schemes for the right reasons.

There are several good things about schemes. I usually study varieties over a field, but there are various ways one can encounter more general schemes in the course of a given problem.

One good thing is the way schemes automatically include information about “multiplicities”. For example, given a polynomial f in one variable over a field k, we would like to say that the number of zeros of f is equal to the degree of f. But there are obvious obstacles to that statement. For example, the polynomial x2+1 over the real numbers actually has no roots in R, so we are really talking about roots of f in some algebraic closure of k. But even then, you have to keep track of “multiplicities” for the statement about deg(f) to be true; e.g., the polynomial (x-1)2 has a root at x=1 “with multiplicity 2”.

Both these complications are avoided by the language of schemes. In that language, the zero set of a polynomial f(x) of degree d over a field k is always a 0-dimensional closed subscheme of the affine line A1 over k, and that subscheme always has degree d. This shows two different good aspects of the notion of a scheme. First, a subscheme Y of a variety X determines not only a subset Y(k) of X(k) (where X(k) means the set of solutions in k of the equations defining a variety X), it also determines a subset Y(E) of X(E) for every extension field E of k. (For example, the subscheme Y of the affine line over the real numbers defined by the equation x2+1 = 0 is not the empty scheme, even though Y(R) is the empty set, because we can see that Y(C) is not empty.) Second, a subscheme Y of a variety X contains more subtle “nilpotent” information than just the subsets Y(E) of X(E) for field extensions E of k. For example, the subscheme Y of the affine line over the real numbers defined by the equation (x-1)2=0 is different from the point Z defined by x-1=0, even though Y(E)=Z(E) for all field extensions E of the real numbers. We picture Y as a “fat point”, a point together with an “infinitesimal neighborhood of length 1”.

In this simple situation of polynomials in one variable, one could live without the geometric language of schemes. But for the same phenomena of “multiplicities” and so on in higher dimensions, the language of schemes is unavoidable (and useful). For example, consider the intersection of two smooth curves at a point in a smooth surface. If they intersect transversely, then the intersection (as a scheme) is “reduced”, i.e., it’s just the point. But if the intersection is not transverse, then the intersection is a 0-dimensional subscheme of degree greater than 1, i.e. it’s a “fattened” or “non-reduced” version of the point. (This is relevant even in the familiar situation of varieties over the complex numbers.)

The standard textbook on schemes is Hartshorne’s Algebraic Geometry. Unfortunately the actual definition of schemes is done in a very dry way there. Still, there is a huge amount of illuminating stuff in that book: the informal essay introducing schemes on pp. 55-59 is good, many exercises are interesting, many good examples and so on. By the end of chapter II on schemes, he gets to crucial geometric ideas like divisors (section 6) and differentials (i.e., the tangent bundle, or more precisely its dual). Chapters IV and V on curves and surfaces are appealing and (I think) can be read before going through all the earlier parts of the book. Appendices A, B, C introducing more advanced topics in an informal way are even better: don’t miss these.

There are by now lots of other books introducing schemes, although none has displaced Hartshorne. I have heard good things about Mumford’s Red Book of Varieties and Schemes, but I haven’t read it myself.

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Books for beginning research

I don’t remember exactly when or why I wrote the list below. I suspect it was a couple of years ago for Part III algebraic geometry students.

If you want to be a mathematician, there’s no substitute for knowing some math. You might as well learn it from great writers.

“To learn to write well, one should read Serre, Bott, Milnor,…” (I think I’m quoting Steve Hurder here, but I believe this too.)

The following list includes both short, readable books that everyone should read and longer reference books.

Undergraduate-level books

The standard topics in pure mathematics are: real analysis including Lebesgue integration (I recommend Royden, Real Analysis); complex analysis; topology (I recommend Armstrong, Basic Topology); and algebra including Galois theory. Fourier series are also fundamental; I recommend Dym and McKean, Fourier Series and Integrals, with a variety of applications in a short space.

Part III-level books

Representation theory: Serre, Linear Representations of Finite Groups. Fulton-Harris, Representation Theory (of semisimple Lie algebras, or equivalently of compact Lie groups). By concentrating on examples, Fulton-Harris’s book is wonderfully readable although somewhat long.

Commutative algebra: Atiyah and Macdonald, Introduction to Commutative Algebra. Very clear in a short space.

Number theory: Serre, A Course in Arithmetic. Cassels, Local Fields.

Topology: Bott and Tu’s Differential Forms in Algebraic Topology is a very readable introduction to smooth manifolds and goes far; everyone should read it. Hatcher, Algebraic Topology.

Riemannian geometry: Gallot-Hulin-Lafontaine’s Riemannian Geometry is one of several gentle introductions. Warner’s Foundations of Differentiable Manifolds and Lie Groups is heavier, but is indispensable for giving the only understandable proof of the Hodge theorem for a Riemannian manifold.

Analysis: Royden, Real Analysis. Lieb and Loss, Analysis.

Graduate-level books

Algebraic geometry: Hartshorne, Algebraic Geometry. Griffiths and Harris, Principles of Algebraic Geometry. These are long references, indispensable for the working algebraic geometer (emphasizing algebraic and analytic approaches, respectively). Huybrechts’s Complex Geometry is a good simplification of Griffiths-Harris.

On more specific topics in algebraic geometry, some outstanding books are Mukai, An Introduction to Invariants and Moduli, and Mumford, Abelian Varieties. There are several other great books (both easier and harder) by Mumford. Borel, Linear Algebraic Groups.

Topology: Milnor’s Characteristic Classes and Morse Theory are magnificent books: short, readable, with a tremendous range of applications. Everyone should read them. There are several other great books by Milnor.

McCleary’s A User’s Guide to Spectral Sequences covers a lot of algebraic topology beyond the basics. Thurston, Three-Dimensional Geometry and Topology.

Symplectic geometry: Arnold, Mathematical Methods in Classical Mechanics.

Homological algebra: Brown, Cohomology of Groups, is an excellent book applying topological ideas to algebra. Weibel, An Introduction to Homological Algebra. Benson, Representations and Cohomology (2 vols.) S. MacLane, Categories for the Working Mathematician.

Number theory: Serre, Local Fields, among several other great books. Lang, Algebraic Number Theory. Miyake, Modular Forms. Silverman, The Arithmetic of Elliptic Curves.

Geometric group theory: Serre, Trees. De la Harpe, Topics in Geometric Group Theory, gives quick treatments of a rich variety of topics.

Analysis: Zimmer, Basic Results in Functional Analysis, treats the fundamental topics and applications in a very short space. Krylov, Lectures on Elliptic and Parabolic Equations in H¨older Spaces, is one of the few graduate-level introductions to serious PDE theory. The big reference books are Gilbarg-Trudinger, Elliptic Partial Differential Equations of Second Order, and Evans, Partial Differential Equations.

Dynamical systems: Walters, An Introduction to Ergodic Theory, is a standard short introduction. Hasselblatt and Katok, An Introduction to the Modern Theory of Dynamical Systems, is the standard big reference book.

Collected papers

Everyone interested in algebraic geometry, number theory, and many aspects of topology and group theory should look at Serre’s Oeuvres: Collected Papers. Atiyah’s Collected Papers are fundamental for topology, with links to analysis and differential geometry.

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Lazarsfeld’s good book

Why not think about reading Rob Lazarsfeld’s Positivity in Algebraic Geometry I. Classical Setting: Line Bundles and Linear Series? Two or three people in the past couple of weeks have asked me questions that this book sheds light upon, and I’ve decided that more people need to know about this excellent book.

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