# Tagged Questions

The study of harmonic differential forms on complex projective varieties, their invariantly defined filtrations, their integrals over topological cycles, especially over subvarieties, the deformations of these integrals and filtrations in families, and a multitude of generalizations.

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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 don'...
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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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### 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 $k$...
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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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### 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 (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, ...
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 H^...