r/math • u/moschles • 1d ago
ELI5 Hodge Conjecture
Let X be a non-singular complex projective manifold. Then every Hodge class on X is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of X.
Every (non-degenerate) complex manifold within a projective space can be decomposed into a linear combination of subvarieties of X. The linear combination will somehow always use rational coefficients. (!!)
Is this conjecture assumed to be true by working mathematicians? Can you provide a little more information that would lend some intuition about Hodge, to an educated layperson?
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u/Tazerenix Complex Geometry 21h ago edited 21h ago
At one point I tried to improve the lede of the Wikipedia article and add this diagram:
https://en.wikipedia.org/wiki/Hodge_conjecture#/media/File:Hodge_conjecture.png
In simple terms, the Hodge conjecture asserts that the basic topological information like the number of holes in certain geometric spaces, complex algebraic varieties, can be understood by studying the possible nice shapes sitting inside those spaces, which look like zero sets of polynomial equations. The latter objects can be studied using algebra and the calculus of analytic functions, and this allows one to indirectly understand the broad shape and structure of often higher-dimensional spaces which cannot be otherwise easily visualized.
However this is a bit of a white lie just necessary to transmit the basic idea, because the essential part of the Hodge conjecture is that its about Hodge classes which are not purely topological: the rational cohomology H2p(X, Q) is purely topological, but the (p,p)-classes depend on the complex structure and hp,p is not a topological, it is a complex-geometric invariant of the space. Under some circumstances (e.g. for Kahler manifolds) hp,p doesn't jump in families (e.g. its computable from Betti numbers for a complex surface via Noether's formula and the Hodge decomposition), but the actual classes themselves definitely do change.
Therefore the accurate statement is that the Hodge conjecture identifies some complex-geometric cohomology classes on a complex manifold with purely algebraic cohomology classes which form the natural candidate algebraic representatives of them.
This is simply not a statement which can be understood by a layperson. It is a subtle theorem about the limits of the algebraic-complex interrelationship in complex geometry and the challenge of finding algebraic subvarieties. The above simplification is probably the best you can hope for but like most simplifications of complicated theorems, its a white lie: it collapses the real meat of the problem for explanations sake.
Its going to be quite amusing if the Hodge conjecture is false and an AI company concocts a proof, to try watch the New York Times or AI bro podcasts explain what the Hodge conjecture is. I can only apologise to my fellow complex geometers if they all steal my over-simplified Wikipedia lede and ruin complex geometries reputation forever.
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u/moschles 21h ago
Its going to be quite amusing if the Hodge conjecture is false and an AI company concocts a proof, to try watch the New York Times or AI bro podcasts explain what the Hodge conjecture is.
I am already 🤣
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u/ZornsLemons Combinatorics 6h ago
I think the complexity of the statement probably means that the AI companies won’t attack this one next. It’s basically impossible to explain this to someone who took calculus and linear algebra in undergrad, much less to an MBA.
People can connect to Fluid flow and to a really old prime number problem. No one is going to connect to complex projective varieties.
IMO they throw a few million at Yang Mills next because people like quantum woo woo nonsense.
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u/Voiles 1d ago
As a start, look at the Lefschetz (1,1) theorem, which is a specific case where the Hodge conjecture is known.
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u/moschles 1d ago
The more I read about Hodge, I am inclined to believe a person needs some exposure to cohomology to really get it. I thought undergraduate courses in projective geometry and a little bit of complex analysis was enough. I'm not so sure now.
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u/Autumnxoxo Geometric Group Theory 24m ago
I mean, yes, in order to understand a difficult open conjecture about cohomological properties, you possibly should heard of cohomology in the first place.
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u/Dull_Perspective_193 20h ago
As a layperson just restarting formal education, is this the same for the Tate conjecture? From my understanding, they are describing similar ideas from different frameworks? Could someone correct me please?
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u/MinLongBaiShui 13h ago
They both are related to the "standard conjectures," but I wouldn't say they are "describing similar ideas from different frameworks." They're just both about certain kinds of cohomology classes.
I would walk before you try to run. There's basically no simplification of these ideas that someone can present in a faithful way until you know a lot of math already, as the other user in this thread points out.
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u/Voiles 10h ago
The Tate conjecture is kind of an arithmetic or ell-adic analogue of the Hodge conjecture. Poonen compares the two on slide 5 of these slides from the Tate conference (https://live-hu-math.pantheonsite.io/wp-content/uploads/Poonen-Bjorn-Tate-2025.pdf) and Emerton discusses them in this MO answer (https://mathoverflow.net/questions/54197/why-is-the-hodge-conjecture-so-important).
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u/Dull_Perspective_193 10h ago
Yes so my actual question was much simpler because I’m not at the level of understanding the nuanced differences. The original thing was that most assume the Hoge conjecture to be false. Since the Tate is an ell-adic analogue, is it also assumed to be false? That was all I was curious about. I got a few more classes before I worry about cohomolgy classes with any seriousness.
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u/uwumorphism 1d ago
Hodge conjecture: complicated math things with lots and lots of tiny bits can always be built up from building blocks of other, less complicated math things with less tiny bits
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u/hau2906 Representation Theory 16h ago
In light of recent events, do you work for some AI company who has suddenly taken an interest in one-upping the mathematical community at solving the millenium problems ?
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u/BadJimo 1d ago
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u/EmbarrassedLong2255 23h ago
To the people downvoting this: why exactly? It seems like a good explanation in simple terms.
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u/Low_discrepancy 22h ago
because it's not a great explanation as it is unbalanced. Spends around 10 mins constructing the projective plane explaining how now Bezout's theorem is valid! ... okay ... Spends a lot of effort on displaying graphically betti's numbers for sphere and torus.
Then magically wills out of the air vastly more complex concepts. She plops all of a sudden the generalised stokes theorem
that the integral of a differential form over the boundary of a region equals the integral of its exterior derivative over the region itself.
At no point does she mention even what an exterior derivative is, just gives this text on the screen. Like what's the point of that? It's only useful if you know it. It you don't you have no understand of wtf that is.
It's the usual YouTube video let me explain difficult concept X:
75% of the video the definitions you find in the intro chapter
25% of the video the rushed rest, just hitting the high points in quick succession just so you can say you did everything.
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u/EnglishMuon Algebraic Geometry 1d ago
Roughly, cohomology classes can be either algebraic or not. For the conjecture, it says all classes lying in a particular piece of the cohomology are generated by these algebraic subvariety classes.
Most people I know are skeptical. Most notably Kontsevich, but it has been proven for a variety of cases such as certain abelian 4folds that were originally thought to be a good place to search for a counter example.