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29
votes
1answer
791 views

What is the meaning of $(h^{11},h^{21})\to (h^{11}-240,h^{21}+240)$ in Calabi-Yau threefolds?

By browsing through the Hodge data of known Calabi-Yau threefolds, I stumbled upon an observation that frequently enough a pair of Hodge numbers $(h^{11},h^{21})$ comes together with the pair $ ...
18
votes
4answers
3k views

Why do people think that abelian varieties are the hardest case for the Hodge conjecture?

Today, I heard that people think that if you can prove the Hodge conjecture for abelian varieties, then it should be true in general. Apparently this case is important enough (and hard enough) that ...
19
votes
1answer
1k views

Diffeomorphic Kähler manifolds with different Hodge numbers

This question made me wonder about the following: Are there orientedly diffeomorphic Kähler manifolds with different Hodge numbers? It seems that this would require that those manifolds are not ...
8
votes
2answers
841 views

Torsion in cohomology of smooth manifolds

I've been interested in the possible (singular) cohomology groups of complex projective algebraic varieties, and there are lots of theorems that give various restrictions on these (Hodge ...
10
votes
2answers
675 views

Hodge decomposition in Minkowski space

This question is motivated by the physical description of magnetic monopoles. I will give the motivation, but you can also jump to the last section. Let us recall Maxwell’s equations: Given a ...
7
votes
0answers
347 views

Fontaine-Mazur for Hodge Structures

Is there a conjecture, or known result, describing which integral Hodge structures are composition factors in the Hodge structure on the cohomology groups of smooth proper algebraic varieties over ...
6
votes
1answer
541 views

Mumford-Tate group and Galois representations

Could someone please point me towards a proof of why the image of a Galois representation on the Tate-module of an abelian variety is limited by its Mumford-Tate group?
3
votes
1answer
324 views

Is it written anywhere that open subvarieties of affine spaces have “completely impure” cohomology?

Consider complex affine space $\mathbb{C}^n$ and let $U$ be a Zariski open subset of $\mathbb{C}^n$. By a celebrated result of Deligne, the cohomology $H^i(U)$ has a canonical Hodge structure. In ...