The hodge-theory tag has no usage guidance.

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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 $ ...

**2**

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**1**answer

204 views

### mixed Hodge structure on Cohomology with compact support

Let $X$ be a smooth quasi-projective variety of dimension $n$ over the complex numbers (let us assume it is the complement of a normal crossings divisor in a smooth projective variety).
Then how does ...

**3**

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**1**answer

147 views

### on a family of CM Hodge structures

I am having a look at the paper "Hodge structures of type $(n, 0, \ldots, 0, n)$" by Totaro. At the very beginning he says that a Hodge structure of type $(1, 0, 1)$ has always complex multiplication. ...

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957 views

### 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 $Y$?

**2**

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79 views

### How does Cnops' operator terminology correspond to standard terminology?

I have been reading Jan Cnops' book, "An Introduction to Dirac Operators on Manifolds" (Birkhaeuser Boston, 2002) and various more standard texts on both Dirac operators and differential geometry in ...

**8**

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267 views

### 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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173 views

### Hodge structure of abelian surfaces

In my case, I have an abelian surface $A$ of (2,8)-polarization, and I have some finite group (may not be abelian group) $G$ acting on $A$ without fixed point. I want to understand when there is a ...

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**1**answer

173 views

### Relation of Hodge Theorem to Eigenfunction Basis of Laplacian

The classical Hodge theorem I know of relates the de Rham cohomology groups isomorphically to the space of harmonic forms and shows that $Id=\pi+\Delta G$, where $\pi$ is the harmonic projection of ...

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153 views

### on the automorphisms of the transcendental Hodge structure of a K3 surface

Let $S$ be a complex projective K3 surface and consider the sub-Hodge structure
$$
T(S) \subset H^2(S, \mathbb{Q})
$$ consisting of transcendental cycles. Let $\varphi$ be an automorphism of Hodge ...

**2**

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70 views

### CM Hodge structures

There are several definitions of Hodge structures with complex multiplication. One of them is the following:
Definition. A Hodge structure $H$ is CM if it is polarizable and its Mumford-Tate group ...

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432 views

### 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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212 views

### Questions on the Hodge Dual of the Kähler Class

Let $M$ be a compact complex surface, i.e., a complex manifold with complex dimension two. Also, let us denote its Kähler metric by $g$ and its Kähler form by $K$. Let us denote the Kähler class by ...

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45 views

### How to diagonalise the Laplace-de Rham operator on the 5-forms of AdS5 x S5?

I am considering the manifold $M=AdS_5 \times S^5$. I would like to diagonalize the Laplace-de Rham operator $\Delta=(d+\delta)^2$ on the 5-forms of $M$. This yields complicated differential ...

**3**

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**1**answer

136 views

### Hodge numbers of symmetric squares

Let $X$ be a projective variety.
Consider $Sym^2X$, the quotient of $X \times X$ by the involution $(x, x') \mapsto (x', x)$.
What is the relation between the (mixed) Hodge numbers of $Sym^2 X$ ...

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**1**answer

79 views

### Is the Hodge Map Unitary?

Let $(V,< \cdot, \cdot >)$ be an inner product space over a field ${K}$. As usual, we can extend $< \cdot, \cdot >$ to a mapping on the exterior algebra of $V$ using the usual matrix ...

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156 views

### The associated graded of a mixed Hodge module

Unfortunately this question will be a bit vague, since the question revolves around a memory of something I may have heard in a talk (long time ago).
Let $X$ be a smooth complex variety. Let $MHM(X)$ ...

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354 views

### 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 ...

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219 views

### lefschetz hyperplane theorem in positive characteristic

The proof of the Lefschetz Hyperplane theorem over $\mathbb{C}$ is well-known, and relies on Hodge theory in an important way. Does there exist an analogue of this theorem in positive characteristic, ...

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972 views

### 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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565 views

### Does anyone know this seemingly simple result in mixed Hodge theory?

Let $f:X\to Y$ be a proper surjection of complex algebraic varieties. Let $H_i$ denote Borel-Moore homology. Then $$ \mathrm{Gr}^W_{-k} H_k(X) \to \mathrm{Gr}^W_{-k} H_k(Y) $$ is surjective.
...

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**1**answer

147 views

### harmonic forms with respect to different metrics

Given a smooth manifold M with two different metric, we can consider U and V, the spaces of harmonic k-forms on M with respect to the first metric and second metric respectively. The question is ...

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**4**answers

2k views

### 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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224 views

### 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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177 views

### Is the Mumford-Tate group determined by its $\mathbb{Q}$-points?

Let $V$ be a $\mathbb{Q}$-Hodge structure and $MT \subset GL_V$ its Mumford-Tate group. Let $I \subset \mathcal{O}_{GL_V}$ be the ideal of functions vanishing on $MT(\mathbb{Q})$.
Is ...

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**1**answer

478 views

### 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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289 views

### Local System and Gauss-Manin connection

Fix a complex manifold $X$. Then if we have a line bundle $L=\mathcal{O}(D)$ together with Gauss-Manin connection $\nabla: L \rightarrow L \otimes \Omega^{1}_X$, we get the locally constant sheaf $F$ ...

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180 views

### dimension of compact support cohomology

Let $X$ be a smooth complex algebraic variety and let $\overline{X}$ be a compactification by a divisor $D$ with normal crossings. Then there is a non-canonical isomorphism
\begin{equation}
(1) ...

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136 views

### Counter-examples for the quasi-unipotence of monodromy over an annulus?

The question is inspired by Carnahan's comment in the following MO question
What is the intuition for $\mathbb{Q}^{ab}$ having cohomological dimension $1$?
Precisely, Borel proved (not Grothendieck, ...

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141 views

### Polarizations of Hodge structures

Let $V$ be a rational pure Hodge structure of weight $n$ and assume that $V$ is a Hodge sub-structure of the cohomology of some smooth projective complex algebraic variety $X$, that is
$V \subset ...

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943 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 ...

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321 views

### Surjectivity of the Gysin morphism

Let $f:X \to Y$ be a closed immersion between smooth projective complex varieties. Suppose that the codimension of (the image of) $Y$ in $X$ is equal to $r \ge 1$. This induces the Gysin morphism ...

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306 views

### mixed Hodge polynomial

Let $X$ be a smooth projective algebraic variety over a field of characteristic zero. Let $U$ be the complement in $X$ of a simple normal crossings divisor $D$. For each degree $k$, put
$h^{p, ...

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303 views

### Constructing a Sobolev space containing the differential k-forms of a Riemannian manifold

I am currently writing a paper about the Hodge theorem for an algebraic topology course. The specific formulation I am proving can be stated thus. Let $M$ be a compact, orientable n-dimensional ...

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247 views

### Hodge classes and Leray filtration

Let $f :X \to Y$ be a submersion between smooth projective varieties over $\mathbb{C}$ and let $\alpha \in Z^k(X)$ be an algebraic cycle of $X$. Is is true that for all odd numbers $p$ and $q$ such ...

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249 views

### a question on Euler characteristic of normal crossing divisors

Let $X$ be a smooth, projective complex algebraic variety. Let $D$ be a simple normal crossings divisor on $X$, with irreducible components $D_i$, for $i \in I$. For each non-empty subset $J \subset ...

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435 views

### 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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271 views

### $H^1(X,O_X)$, holomorphic $1$-forms, and $b_1(X)/2$ for normal $X$.

Suppose $X$ is a normal projective variety over $\mathbb C$. In the case $X$ is smooth according to Hodge theory $h^1(X,O(X))$ is the dimension of the space of holomorphic $1$-forms on $X$ and this ...

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185 views

### Homogeneous Spaces and Equivariant Hodge Maps

For a homogeneous space $G/H$, endowed with a $H$-equivariant metric $g$, let $\ast$ be the corresponding Hodge star map. It seems that $\ast$ must also be $\ast$-equivariant, but I can't see how one ...

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218 views

### Hodge structures as complex structures

Let $H$ be a finite dimensional $\mathbb{Q}$-vector space. I've heard that to give an effectif Hodge structure of weight 1 on $H$ is the same as to give a complex structure on $H_\mathbb{R}$. Why is ...

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241 views

### 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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500 views

### maxwell's equations and hodge theory

How is Hodge theory of harmonic forms related to maxwell's equations.Atiyah says that Hodge was directly motivated by considerations of maxwell's equations while commenting on donaldson.

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978 views

### what is Deligne's cohomological descent (and what are some examples)

As far as I understand Deligne's far reaching generalisation of Čech cohomology is called cohomological descent and is used to endow any variety with a (mixed) Hodge structure.
Again, AFAIU, the idea ...

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386 views

### 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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250 views

### Determing Hodges Maps by their Essential Algebraic Properties

Let $M$ be a complex manifold of real dimension $2N$, equipped with a Hermitian metric. The corresponding Hodge map $\ast$ has the following properties:
(i) It is a ${\bf C}$-linear map ...

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144 views

### VHS for universal family of false elliptic curves

If $f: X \rightarrow C$ is the universal family of false elliptic curves (i.e. abelian surfaces $A$ such that $End(A) \otimes \mathbb{Q}$ is a totally indefinite quaternion algebra over $\mathbb{Q}$), ...

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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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149 views

### sign in the polarization of Hodge structures

I have been trying to understand signs and conventions in the Hodge theory. Clearly, I am no expert in this area. I apologize if I ask stupid question(s) and make wrong comment(s).
In Deligne's ...

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396 views

### 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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1k 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 ...

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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 $i^* \star \omega = 0$ ...