The hodge-theory tag has no wiki summary.

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### Linkage between homotopy equivalence and identification of algorithms

I vaguely recall that someone says there is linkage between homotopy equivalence and identification of algorithms which may be isomorphic or morphism or something like that,the algorithm may be ...

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### Hodge modules and Deligne-Beilinson cohomology of function fields

Let $K$ be a function field over complex numbers i.e. the fraction field of a complex variety. Then one can define the Deligne-Beilinson cohomology and mixed Hodge modules for $K$ as the direct limit ...

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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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### Surjectivity of certain cohomology groups on hypersurfaces of high degree

I had been reading an article by Spencer Bloch. There is a remark in this text which states the surjectivity of a particular map between cohomology groups without explaining further. I had been trying ...

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### splitting of the Hodge filtration

Let $X$ be a smooth projective variety over a subfield $k$ of $\mathbb{C}$. Then one has a short exact sequence
$$
0 \to H^0(X, \Omega^1_X) \to H^1_{dR}(X/k) \to H^1(X, \mathcal{O}_X) \to 0
$$
One ...

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### in which sense is a mixed Hodge structure an extension of pure ones?

I have heard several times that mixed Hodge structures are iterated extensions of pure ones. What does it mean? Here is what I figured out.
A mixed Hodge structure $H$ comes with an increasing ...

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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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### 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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### What implications would a solution of the *Standard Conjectures* have on the *Hodge Conjecture*?

I'm new to the field, so I just would like to know what implications would have a solution of the Standard Conjectures on the Hodge Conjecture. I read somewhere they are related in some way, but I ...

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### extensions of mixed Hodge structures

Let MHS be the category of rational mixed Hodge structures. In particular, it contains extensions of Tate objects $\mathbb{Q}(n)$ for each integer $n$. Here $\mathbb{Q}(n)$ is the only one dimensional ...

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### Tangent bundles of period domains of higher weight Hodge structures

Considering $A_{g}$ the moduli space of principally polarized abelian varieties, there is a variation of Hodge structures $\mathbb{V}=E^{1,0}\oplus E^{0,1}$ of weight $1$ on $A_{g}$. It is well-known ...

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### Hodge isometry sending the Kahler class to its opposite

i would like to ask you a question i can not answer myself, i hope this is not too trivial and i'm not missing something too basic.
Let's suppose we have $X$ and $Y$ Kahler manifolds and ...

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### Question about the “middle” intermediate Jacobian

Suppose $Y$ is a smooth projective variety of dimension $2p-1$ over $\mathbb{C}$. I have a few questions about the $p^{~\text{ th}}$ intermediate Jacobian $J^p(Y)$ of $Y$.
Does it come from (i.e. is ...

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### How to prove this Weitzenbock formula?

In Hutchings and Taubes lecture note on Seiberg-Witten equation HERE, above equation (4.20) the authors claim that there is a version of Weitzenbock formula reads (where $\beta \in \Omega^{0,2}(M, ...

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### Is it possible to compute the mixed Hodge structure of a unramified covering space?

Assume that we know the mixed Hodge structure of a complex manifold $X$. Is it possible to compute the mixed Hodge structure of unramified covering $Y$ of $X$ if we know the deck transformation of the ...

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### Is there a “natural” reformulation of Hodge conjecture in terms of L-functions?

I just glanced at the Wikipedia article about the Hodge conjecture, and a (probably very naive, due to my huge lack of knowledge of the subject) question just came to my mind: can one associate ...

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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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### Rational Hodge Theory

I am currently learning about the formality result for Kaehler manifolds proved in DGMS. Their result uses some basic Hodge theory, and so only gives formality over $\mathbf{R}$. Later, however, ...

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### Local system over $\mathcal A_{g,[n]}$ with unipotent monodromy

Let $\mathcal A_{g,[n]}$ denote the moduli space of principal polarized abelian varieties with level-[n] structure and
$\bar {\mathcal A}_{g,[n]}\supset \mathcal A_{g,[n]}$ a smooth Toroida ...

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### Smooth mixed hodge modules - representations of fundamental group?

I do not know much about mixed Hodge modules. I would like to ask: Let $X$ be a smooth connected algebraic complex variety, with a chosen point. Could one describe smooth mixed Hodge modules on $X$ as ...

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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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### 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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### 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 ...

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### 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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### 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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### 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 ...

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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 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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### 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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### 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 ...

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### 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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### 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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### 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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### 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 ...

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...