The hodge-theory tag has no usage guidance.

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### Why is the Hodge Conjecture so important?

The Hodge Conjecture states that every Hodge class of a non singular projective variety over $\mathbf{C}$ is a rational linear combination of cohomology classes of algebraic cycles: Even though I'm ...

**31**

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**1**answer

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### 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 $ ...

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### Proving Hodge decomposition without using the theory of elliptic operators?

In the common Hodge theory books, the authors usually cite other sources for the theory of elliptic operators, like in the Book about Hodge Theory of Claire Voisin, where you find on page 128 the ...

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### Down-to-earth expositions of Hodge theory

What are nice expositions of Hodge theory not using advanced language of algebraic geometry or category theory?
Of course, since I haven't found a (for me) readable introduction, I don't know what I ...

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### Examples of Mixed Hodge Structures

Does anyone know a user-friendly, example-laden introduction to mixed Hodge structures? I get from Wikipedia how to calculate for a punctured and pinched curve ...

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### 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 ...

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**1**answer

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### 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 ...

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2k views

### Comparing cohomology over ${\mathbb C}$ and over ${\mathbb F}_q$

I have the following (probably well-known) question: let $X$ be a regular scheme over
$\mathbb Z$. Let $p$ be a prime and Let us denote the reduction of $X$ mod $p$ by $X_p$.
Let also $X_{\mathbb ...

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### Exercises in Hodge Theory

I was wondering: is there a good place to find exercises in Hodge theory? Mostly computations and proving small (preferably nifty) theorems, is what I have in mind. Something roughly like the ...

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### Bounding failures of the integral Hodge and Tate conjectures

It is well know that the integral versions of the Hodge and Tate conjectures can fail. I once heard an off hand comment however that they should only fail by a "bounded amount". My question is what ...

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**1**answer

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### Semistable filtered vector spaces, a Tannakian category.

Let $k$ be a field (char = 0, perhaps). Let $(V,F)$ be a pair, where $V$ is a finite-dimensional $k$-vector space, and $F$ is a filtration of $V$, indexed by rational numbers, satisfying:
$F^i V ...

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### When does the Torelli Theorem hold?

The Torelli theorem states that the map $\mathcal{M}_g(\mathbb{C})\to \mathcal{A}_g(\mathbb{C})$ taking a curve to its Jacobian is injective. I've seen a couple of proofs, but all seem to rely on the ...

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### Mixed Hodge structure on the rational homotopy type

A mixed Hodge structure (mHs) on a commutative differential graded algebra (cgda) over $\mathbf{Q}$ is a mixed Hodge structure on the underlying vector space such that the product and the differential ...

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888 views

### Why the similarity between Hodge theory for compact Riemannian and complex manifolds?

I'm aware to varying extents of the existence of certain decompositions of the space of $k$-forms on a compact complex or compact Riemannian manifold that split into closed, co-closed, and harmonic ...

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### What exactly does the weight filtration in Hodge theory have to do with the Weil conjectures?

Let $X$ be a variety over $\mathbb{C}$, say separated. According to Deligne's results, there is a "mixed Hodge structure" on the total cohomology $H^\bullet(X(\mathbb{C}), \mathbb{Z})$. One component ...

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### is the Hodge conjecture birationally invariant?

Let $X$ and $Y$ be two birational smooth projective varieties over the complex numbers. Assume $X$ satisfies the Hodge conjecture.
Is it known that the Hodge conjecture holds for $Y$?

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### The prerequisites for Deligne's Théorie de Hodge I, II, III

I am an undergraduate student. I am not sure if it's OK to ask this question here.
I want to learn Hodge theory. But I do not know how to start it, and how much mathematics I should need before I ...

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### Can one find the hodge number by counting points over finite fields?

Given a proper smooth variety $X$ of dimension $n$ over $\mathbb{C}$, assume it has a model over a DVR of mixed characteristic $(0,p)$ with residue field $\mathbb{F}_q$, and assume the closed fiber ...

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### Are there known cases of the Mumford–Tate conjecture that do not use Abelian varieties?

(For a formulation of the Mumford–Tate conjecture, see below.)
The question
As far as I know, all non-trivial known cases of the Mumford–Tate conjecture more or less depend on the Mumford–Tate ...

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### 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 ...

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### Hodge numbers in a family

Let $X \to Y$ be a smooth projective morphism of smooth varieties over $\mathbb{C}$. Then the fibers $X_y, y \in Y$ have locally constant Hodge numbers $H^q(X_y, \Omega^p_{X_y})$. Namely, one can ...

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### When is a homogeneous space a variety?

Let $G$ be a Lie group and let $H$ be a closed subgroup of $G$. Then $G/H$ may not be a group, but it will be a homogeneous space for $G$ with stabilizers conjugate to $H$. Sometimes, this is a ...

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### A_infinity structure on cohomology and the weight filtration

Let $X$ be a complex algebraic variety. The rational cohomology of $X(\mathbb{C})$ carries a canonical filtration called the weight filtration. It also carries a canonical equivalence class of ...

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### Hodge structures on algebraic spaces

Let $X$ be a proper smooth algebraic space over $\mathbb C$ (which amounts, due to Artin, to giving a Moishezon space: a compact complex manifold whose dimension equals the transcendence degree of its ...

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### Mixed Hodge structure on configuration spaces

Let $X$ be a smooth complex projective variety. Let $F(X,n)$ be the configuration space parametrizing $n$ distinct ordered points in $X$. The cohomology groups $H^k(F(X,n),\mathbf Q)$ carry a mixed ...

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### Primitive Cohomology Useful?

In her book, after proving the hodge decomposition, Voisin spends time discussing primitive cohomology $H^r(X, \mathbb{C})_{prim} = \ker L^{n-r+1} \subset H^r(X, \mathbb{C})$ (where $L$ is the ...

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### Witten's proof of Morse inequalities, question on eigenvalues?

See here. I present Theorem 6 and Corollary 7 as follows.
Theorem 6. For large $s$, $\Delta_s$ and $H$ have the same number of eigenvalues in the interval $[0, s^{-2/5}]$.
Corollary 7. $\dim ...

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### Weight filtration and Hodge theory for tropical varieties

Many concepts is algebraic geometry have tropical analogues.
Question: Is there an analogue of the weight filtration or Hodge filtration for tropical varieties?
A tropical curve ends up being ...

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**1**answer

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### Why is the kth cohomology group of the DM-compactification of the moduli space of curves pure of weight k?

I'm trying to understand the paper
Arbarello, Enrico, Cornalba, Maurizio,
Calculating cohomology groups of moduli spaces of curves via algebraic geometry.
Inst. Hautes Études Sci. Publ. Math. No. 88 ...

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### Formal adjoint of the covariant derivative

Let $E \to M$ be a vector bundle over some Riemannian metric $(M, g)$ and endow it with some fibre metric. Assume that covariant derivative $\nabla$ is compatible with the metric.
It is essentially ...

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### Hodge numbers of compactifications

Let $X$ be a smooth complex quasi-projective variety. We can find good compactification: a smooth proper variety $\bar{X}$ such that ${\bar X} \setminus X$ is a divisor with normal crossing. The ...

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### Betti numbers of Proper nonprojective varieties

This is a question about pathologies.
Let $X/\mathbb{C}$ be an irreducible projective variety smooth over $\mathbb{C}$. Then, the singular cohomology groups $H^i(X, \mathbb{C})$ have a hodge ...

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### Is there a yoga of effectivity for motives and their realizations?

Inside the 'usual' categories of motives (Chow, Voevodsky ones) there are the categories of effective motives. Similarly, there are effective Hodge structures; they seem to be closely related with ...

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### Intrinsic definition of the weight filtration

Let $X$ be a smooth quasiprojective complex variety. Then Deligne (Theorie de Hodge II) defined a weight filtration on the Betti cohomology of $X$. The general philosophy is quite simple: express the ...

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### Can I compute the cohomology of the complement of a log canonical divisor as if it were normal crossings?

Let $X$ be a smooth projective variety and $D$ a log-canonical divisor and let $U = X \setminus D$. I have heard the slogan "log-canonical is just as good as normal crossings for Hodge theory". This ...

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### Is there a Hodge isomorphism theorem for part-tangential, part-normal, harmonic differential forms?

Let $M$ be an oriented compact Riemannian $n$-manifold with boundary $\partial M$. A differential $p$-form $\omega$ on $M$ is normal if $i^* \omega = 0$ holds, tangential if $i^* \star \omega = 0$ ...

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### Weight filtration over the integers

This is a follow up question to Weight filtration for smooth analytic manifolds
As mentioned in that question, the integral cohomology of some smooth complex analytic manifolds is equipped with a ...

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### Does there exist a family of curves (or abelian varieties) on the punctured line with specified monodromy on H^1?

Suppose I have a finite set of points in $\mathbb{P}^1$ (over the complex numbers), and suppose that at each point, I am given a [Edit: quasi-unipotent] conjugacy class in $Sp(2g,\mathbb{Z})$ for $g$ ...

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### 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 ...

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### On polarized (pure) Hodge structures

Some simple questions, for which I know no precise reference (and would be deeply grateful for a nice one!):
Is it true that the category of (pure) polarized Hodge structures is abelian semi-simple, ...

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### Duality between singularities and non-compactness in the yoga of weights

According to Deligne's "yoga of weights", the cohomology of an algebraic variety should have a weight filtration. For concreteness we can consider the rational cohomology of complex varieties, with ...

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### Hodge numbers of diffeomorphic complete intersections

Is there an example of two complex projective complete intersections that are diffeomorphic but have different Hodge numbers?
Edit: as written by Daniel Loughran in the comments below, complete ...

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### Cubical cohomology and de Rham cohomology

Qiaochu's question on a discrete analogue of harmonic function theory reminded me of some thoughts I had a long time ago about the relationship between cubical cohomology and de Rham cohomology.
The ...

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### Carleson's Theorem on Manifolds

Let $M$ be an oriented, compact, differentiable manifold with some Riemmanian metric $g$, so that $(M,g)$ has a nice volume form and one can define $L^2(M,g)$ as the completion of $C^\infty(M)$ under ...

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### Hodge structure versus Weight structure

This is a naive question.
One is told that, somehow, Hodge theory for varieties over complex numbers, is an analog of weight theory for varities over finite fields. In weight theory, one considers ...

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### Is there an integral version of Faltings' isomorphism in p-adic Hodge theory between etale and Hodge cohomologies

Let $K$ be a $p$-adic field, that is a complete discrete valuation ring of characteristic $0$ with a perfect residue field $k$ of characteristic $p > 0$ (to simplify one could also take $K$ to be a ...

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### Adjoint of a Connection Using the Hodge Map?

For a Riemannian manifold $(M,g)$ with exterior derivative d, the codifferential d$^\ast$ is defined to be the unique map for which
$$
g(\omega,d\omega') = g(d^* \omega,\omega'), ~~~ \omega,\omega' ...

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### Motives over the complex numbers versus mixed Hodge structures

Let $\mathsf{MM}(\mathbf C)$ be the hypothetical category of mixed motives over the complex numbers, and consider the realization functor $\Phi : \mathsf{MM}( \mathbf C) \to \mathsf{MHS}$ to integral ...

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### Splitting of the weight filtration

All varieties are over $\mathbb{C}$. Notions related to weights etc. refer to mixed Hodge structures (say rational, but I would be grateful if the experts would point out any differences in the real ...

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### Higher degree generalizations of the Hard Lefschetz Theorem

Let $M$ be a $2d$-dimensional manifold. We say that $\omega \in H^2(M)$ has the Hard Lefschetz Property (HLP) if multiplication with $\omega^j$ is an isomorphism $H^{d-j} \to H^{d+j}$. This holds for ...