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)...
Dustin Clausen's user avatar
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 $p \in \mathbb{N}$....
Riccardo Pengo's user avatar
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 ...
Dan Petersen's user avatar
  • 39.2k
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 ...
Donu Arapura's user avatar
  • 34.2k
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
Dennis Sullivan's user avatar
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 ...
D.-C. Cisinski's user avatar
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 ...
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 ...
Donu Arapura's user avatar
  • 34.2k
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 ...
Carlo Beenakker's user avatar
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 ...
Myshkin's user avatar
  • 17.4k
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 ...
Denis Nardin's user avatar
  • 16.2k
14 votes
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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 $n$-...
David E Speyer's user avatar
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-...
Tintin's user avatar
  • 2,741
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 ...
user25309's user avatar
  • 6,810
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 ...
Francesco Polizzi's user avatar
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....
Francesco Polizzi's user avatar
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 $\...
Denis Nardin's user avatar
  • 16.2k
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 ...
Donu Arapura's user avatar
  • 34.2k
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 ...
Donu Arapura's user avatar
  • 34.2k
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 ...
Donu Arapura's user avatar
  • 34.2k
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.
Squid with Black Bean Sauce's user avatar
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 ...
Ryan Keast's user avatar
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 ...
Hunter Spink's user avatar
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 ...
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 ...
Daniel Litt's user avatar
  • 22.2k
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 ...
Francesco Polizzi's user avatar
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 ...
Francesco Polizzi's user avatar
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 $\...
gdb's user avatar
  • 2,851
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\...
Sebastian's user avatar
  • 6,715
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 ...
Johannes Huisman's user avatar

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