Has there been any recent progress in this area? What are the obstacles?
Progress on the Hodge conjecture (as of 2023)
The Hodge conjecture per se, not so much. Maybe ten years ago people were working on various special cases for products of surfaces. But there's certainly progress on Hodge theory more broadly, in the sense of proving that certain a priori analytic objects in Hodge theory are in fact algebraic. For example, Bakker-Brunebarbe-Tsimerman recently proved a conjecture of Griffiths that the image of the period map is quasi-projective:
https://arxiv.org/abs/1811.12230
Concretely what this means is that if you have a family of smooth projective varieties, then you can look at the periods/Hodge structures on the cohomology on the members of the family as defining a subset of the "period domain" of Hodge structures of that type. While whole period domain is something analytic (it's the upper half plane in the simplest classical case of families of elliptic curves), we now know that in fact that the subset of "periods" for a fixed family of smooth projective varieties has the natural structure of quasi-projective variety, that is to say, can be described by finitely many polynomial equations.
By the way, the proof uses the same circle of ideas as used in the proofs of Andre-Oort and Manin-Mumford, that is to say o-minimal geometry.
ChatGPT, how can I prove the Hodge conjecture?
The Hodge conjecture is a famous open problem in algebraic geometry, which asserts that for a projective algebraic variety over the complex numbers, the Hodge numbers determine the rational cohomology ring. While there has been significant progress towards the conjecture in recent years, it remains an unsolved problem.
Right...
The Hodge conjecture per se, not so much. Maybe ten years ago people were working on various special cases for products of surfaces. But there's certainly progress on Hodge theory more broadly, in the sense of proving that certain a priori analytic objects in Hodge theory are in fact algebraic. For example, Bakker-Brunebarbe-Tsimerman recently proved a conjecture of Griffiths that the image of the period map is quasi-projective:
https://arxiv.org/abs/1811.12230
Concretely what this means is that if you have a family of smooth projective varieties, then you can look at the periods/Hodge structures on the cohomology on the members of the family as defining a subset of the "period domain" of Hodge structures of that type. While whole period domain is something analytic (it's the upper half plane in the simplest classical case of families of elliptic curves), we now know that in fact that the subset of "periods" for a fixed family of smooth projective varieties has the natural structure of quasi-projective variety, that is to say, can be described by finitely many polynomial equations.
By the way, the proof uses the same circle of ideas as used in the proofs of Andre-Oort and Manin-Mumford, that is to say o-minimal geometry.
This direction is unrelated to the Hodge conjecture, so why mention it?
The Hodge conjecture per se, not so much. Maybe ten years ago people were working on various special cases for products of surfaces. But there's certainly progress on Hodge theory more broadly, in the sense of proving that certain a priori analytic objects in Hodge theory are in fact algebraic. For example, Bakker-Brunebarbe-Tsimerman recently proved a conjecture of Griffiths that the image of the period map is quasi-projective:
https://arxiv.org/abs/1811.12230
Concretely what this means is that if you have a family of smooth projective varieties, then you can look at the periods/Hodge structures on the cohomology on the members of the family as defining a subset of the "period domain" of Hodge structures of that type. While whole period domain is something analytic (it's the upper half plane in the simplest classical case of families of elliptic curves), we now know that in fact that the subset of "periods" for a fixed family of smooth projective varieties has the natural structure of quasi-projective variety, that is to say, can be described by finitely many polynomial equations.
By the way, the proof uses the same circle of ideas as used in the proofs of Andre-Oort and Manin-Mumford, that is to say o-minimal geometry.
This direction is unrelated to the Hodge conjecture, so why mention it?
Just to annoy lemons like you.
The Hodge conjecture per se is one of many statements, conjectural or proved, that are similar in spirit: certain a priori topological or analytic objects associated to algebraic varieties satisfy strong constraints that should force them to be algebraic.
[...]
This direction is unrelated to the Hodge conjecture, so why mention it?
Just to annoy lemons like you.
The Hodge conjecture per se is one of many statements, conjectural or proved, that are similar in spirit: certain a priori topological or analytic objects associated to algebraic varieties satisfy strong constraints that should force them to be algebraic.
I disagree with you that the Hodge conjecture and the Griffiths conjecture can be grouped together. The Hodge conjecture is about topological information encoding the data of cycles in the algebraic context, and the reason why it should hold is quite naive. The Griffiths conjecture on the other hand happens in the analytic context, so it is more like a GAGA type statement. The whole setup of the Griffiths conjecture is holomorphic, not merely topological. Also the moral reason why it should hold is very clear: you want to use Chow's theorem, and you expect it to be applicable because 1. you have some positivity by positivity and horizontality of Griffiths bundle and 2. you think it can be compactified because you have a control of singularities of the period map.
[...]
Just to annoy lemons like you.
The Hodge conjecture per se is one of many statements, conjectural or proved, that are similar in spirit: certain a priori topological or analytic objects associated to algebraic varieties satisfy strong constraints that should force them to be algebraic.
I disagree with you that the Hodge conjecture and the Griffiths conjecture can be grouped together. The Hodge conjecture is about topological information encoding the data of cycles in the algebraic context, and the reason why it should hold is quite naive. The Griffiths conjecture on the other hand happens in the analytic context, so it is more like a GAGA type statement. The whole setup of the Griffiths conjecture is holomorphic, not merely topological. Also the moral reason why it should hold is very clear: you want to use Chow's theorem, and you expect it to be applicable because 1. you have some positivity by positivity and horizontality of Griffiths bundle and 2. you think it can be compactified because you have a control of singularities of the period map.
How is the Hodge conjecture about topological information? Rather, it is about the intersection of topological and holomorphic information implying algebraicity.
However, I agree that the Griffiths conjecture is more of a GAGA statement, and indeed that's how Bakker-Brunebarbe-Tsimerman frame it.
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