64
votes
Accepted
Clausen's modified Hodge Conjecture
It's a bit of a long story, but I can at least give the idea. Let $X$ be a smooth projective variety over $\mathbb{C}$. The Hodge conjecture says that for all $p\geq 0$, the cycle class map
$$Ch^p(X)...
27
votes
What is prismatic cohomology?
I just take a quick opportunity to share what a prism is, and why it is called like that (as I learned from Lars Hesselholt). All the theory is developed relatively to a fixed prime $p \in \mathbb{N}$....
22
votes
Hodge theory (after Deligne)
A brief answer.
First of all, the results are miraculous. Deligne's Hodge II and Hodge III give just a few example applications of the kind of results you can prove using mixed Hodge theory; these ...
20
votes
Accepted
Mixed Hodge structure on sheaf cohomology of a variation of Hodge structures
The answer to your question is yes (provided the VHS is polarized). This is due to Morihiko Saito. It is implicitly contained his two long papers on (mixed) Hodge modules, and there is an explicit ...
17
votes
Mixed Hodge structure on the rational homotopy type
I think these questions are very satisfactorially answered in a paper of joana cirici on arxiv 2013.
this paper was published in 2015
dennis sullivan
16
votes
Accepted
Tate twists and cohomology of $\mathbf{P}^1$
The Tate twist is what we need to express Poincaré duality without making any choice. Such a choice appears in the choice of an orientation of the affine line minus the origin, and have shadows in the ...
15
votes
Diffeomorphic Kähler manifolds with different Hodge numbers
A few years after this post appeared, the question of oriented diffeomorphism invariance of Hodge and Chern numbers of smooth, projective varieties was settled completely by Kotschick and Schreieder ...
Community wiki
15
votes
Accepted
Hodge to de Rham spectral sequence for stacks
Short answer: yes.
As I recall, Teleman constructs such a spectral sequence for fairly general stacks. You can look at his paper The quantization conjecture revisited Annals 2000. But this may fall ...
15
votes
Hodge theory (after Deligne)
The laudatio for the Wolf prize explains it like this:
Central to modern algebraic geometry is the theory of moduli, i.e.,
variation of algebraic or analytic structure. This theory was
...
14
votes
Accepted
Hodge standard conjecture for étale cohomology
I'm very interested myself on a better answer to this question, but let me point out the obvious: the main problem is that there is no Hodge theory on positive characteristic.
The proof in ...
14
votes
Accepted
How to think about infinite generatedness of motivic cohomology
While waiting for someone more competent than me to answer, let me turn the question right back to you. Why should motivic cohomology be finitely generated?
The answer is, of course, that there's no ...
14
votes
Accepted
Relation between the cohomology group of a curve and the cohomology group of its jacobian
$\def\Alb{\text{Alb}}\def\Pic{\text{Pic}}\def\CC{\mathbb{C}}\def\ZZ{\mathbb{Z}}\def\RR{\mathbb{R}}\def\cO{\mathcal{O}}$There are two abelian varieties associated to a smooth projective connected $n$-...
13
votes
Accepted
Motivic vs Deligne cohomology
The existence of a cycle class map from motivic cohomology is a general fact to every cohomology satisfying certain axioms. For example, to every mixed Weil theory in the terminology of Cisinski-...
13
votes
Accepted
What is Kontsevich's Hodge theory of path integrals?
The most usual Hodge theory is related to integrals, periods of algebraic differential forms on algebraic varieties, whereas the physics path integral usually involves integration of an exponential ...
13
votes
Accepted
Is $h^{0,k}$ a topological invariant?
The answer is no.
There are counterexamples already for surfaces, due to X. Gang and F. Campana (unpublished). The link to Campana's article is here, and the relevant result is
Proposition 0.1 ...
12
votes
Accepted
What makes a Kähler manifold projective?
If I understand the question correctly, I think that the answer is given by the main result in
S. Ji: Currents, metrics and Moishezon manifolds, Pac. J. Math. 158, No. 2, 335-351 (1993). ZBL0785.32011....
11
votes
Accepted
How to think about $\mathbf{Z}(n)_{\mathcal{M}}$
[All cohomology will be reduced cohomology for ease of notation].
There is no analog for classical homotopy theory. This is related to the fact that the Picard group of the category of spectra is $\...
11
votes
Accepted
Simpson's motivicity conjecture
At the time Simpson formulated his conjecture, he had proved that rigid local systems correspond to rational (in a suitable sense) variations of Hodge structure. And, as you point out, we now know ...
11
votes
Accepted
Does Poincaré duality preserve algebraic cycles?
A positive answer to your question* would imply one of Grothendieck's standard conjectures (conjecture A) for complex varieties. See his note: Standard conjectures on algebraic cycles. It's safe to ...
10
votes
Accepted
Interesting geometric application of Hitchin Fibration
You may know this already, but I'm not sure if you'll get many other answers. A nice result of Brunebarbe, Klingler and Totaro [Symmetric differentials and the fundamental group, Duke 2013] is that if ...
10
votes
What is prismatic cohomology?
The preprint has been released to the general public now:
Bhargav Bhatt and Peter Scholze, Prisms and Prismatic Cohomology, arXiv:1905.08229.
10
votes
Hodge structure not coming from the cohomology of a manifold
The existence of non-motivic (not coming from algebraic geometry) polarized Hodge structures is currently non-constructive. For period domains which are not Hermitian symmetric, motivic Hodge ...
10
votes
Accepted
Log-concavity of matroids: characterization of equality?
I think the following shows it's never possible for there to be equality.
Indeed, Ardila-Denham-Huh https://arxiv.org/abs/2004.13116 recently showed for any matroid $M$ that $T_M(x,0)$ has log-concave ...
9
votes
Equivalent descriptions of Hodge conjecture?
[...] "Hodge conjecture can be reformulated by saying that the Hodge realization of the algebraically defined $\mathbb{Q}$-vector space of codimension p algebraic cycles modulo numerical (or ...
Community wiki
9
votes
Accepted
Variations of Hodge structures over the line
See Theorem 11 on page 191 of these notes. A special case is as follows.
Theorem of the fixed part. Let $S$ be a smooth quasiprojective variety, and $V$ a variation of $\mathbb{Q}$-Hodge structures ...
9
votes
Important consequences of the Hodge Index Theorem
Let $S$ be a smooth projective surface and $H$ a $\mathbb{Q}$-divisor with $H^2>0$ (for instance, an ample divisor).
If $D$ is another $\mathbb{Q}$-divisor such that $HD=0$, then $D^2 \leq 0$ and ...
9
votes
Accepted
Reference for flatness in complex-analytic geometry
A reference for the fact that flat maps are open in the complex-analytic category is Theorem 2.12, p. 180 in
C. Bănică, O. Stănăşilă: Algebraic methods in the global theory of complex spaces.
See ...
9
votes
Hodge decomposition and degeneration of the spectral sequence
I am far from being an expert in this area, but I will try to present my understanding of this subject.
First of all, this is true that Hodge decomposition holds for smooth proper varieties over $\...
9
votes
When do flat holomorphic connections exist?
The comment of HYL should be an answer. Since the OP has asked for explicit counterexamples, I will give an example that 1) does not imply 2):
Consider a compact Riemann surface $\Sigma$ of genus $g\...
8
votes
Hodge map and the Cohomology Ring of a Riemannian Manifold
As has been observed in the comments, it suffices to construct a closed differential form $\omega$ for which $\star\omega$ is not closed. Here is an explicit example.
Let $M$ be the circle $S^1$ for ...
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