The hodge-theory tag has no usage guidance.

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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 ...

**14**

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1k views

### 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 ...

**11**

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290 views

### 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 ...

**9**

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294 views

### 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 ...

**9**

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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 ...

**9**

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191 views

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

**9**

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827 views

### 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 ...

**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 ...

**7**

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305 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, ...

**7**

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390 views

### Fontaine-Mazur for Hodge Structures

Is there a conjecture, or known result, describing which integral Hodge structures are composition factors in the Hodge structure on the cohomology groups of smooth proper algebraic varieties over ...

**7**

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519 views

### How does complex conjugation act on the Hodge decomposition?

Let $A$ be a principally polarized
abelian variety over $\mathbf{Q}$. Let $G$
be the Mumford--Tate group of $A$.
The action of complex
conjugation on $A(\mathbf{C})$ induces an involution on
the de ...

**6**

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**0**answers

142 views

### Logarithmic view on the classical Eichler–Shimura isomorphism

[Apologies ahead of time that this question is (1) quite heavy in laying out notations, and (2) of the can-you-spot-my-error variety.]
I am trying to recast the Eichler–Shimura isomorphism as a ...

**6**

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**0**answers

173 views

### 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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270 views

### Cohomology of Zariski neighborhoods

Do there exist smooth compact (=complete) connected complex algebraic varieties $X\subset Y$ and a Zariski neighborhood $U$ of $X$ in $Y$ such that the image of $H^{\ast}(U,\mathbf{Z})$ in ...

**6**

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227 views

### Positivity properties of virtual Hodge numbers of Calabi-Yaus

Let $X$ be a normal, projective complex variety with an anticanonical divisor $D$. Do the virtual Hodge numbers of the noncompact Calabi-Yau variety $X$ \ $D$ enjoy some sort of positivity property?
...

**5**

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226 views

### de Rham cohomology group and Dolbeault cohomology group on compact complex analytic spaces

My question is:
On a compact complex analytic space,since Hodge Theorem becomes invalid,is it true that the de Rham cohomology group $H^p_{DR}(M)$ and the Dolbeault cohomology group ...

**5**

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152 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 ...

**5**

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**0**answers

198 views

### Which pure categories related with Weil cohomology theories are semi-simple?

As far as I understand, the category of pure polarizable Hodge modules is semi-simple, whereas the cohomology of the corresponding schemes is graded polarizable. Is it true that one doesn't have any ...

**5**

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**0**answers

310 views

### Generalized Hodge conjecture for cohomology of smooth non-proper varieties?

Is there such a statement known i.e. does there exist a conjectural description of the coniveau filtration for singular cohomology of a smooth non-proper variety over the field of complex numbers (in ...

**4**

votes

**0**answers

114 views

### Real structure in the mixed Hodge structure associated to an isolated singularity

We know that a mixed Hodge structure on a complex vector space $H$ with an integral lattice $H_{\mathbb Z}$ consists of the weight filtration and the Hodge filtration. For an isolated hypersurface ...

**4**

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68 views

### Archimedean $\varepsilon$-factors

Let $K$ be either $\bf R$ or $\bf C$. Let $p$ and $q$ be integers with $p \leq -1$, $q \geq 0$, and $p+q=-1$. Consider the Hodge structure $M = M(p,q)$ over $K$ with coefficients in $\bf R$, defined ...

**4**

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205 views

### Mixed structures on Hom spaces induced by mixed sheaves

Let $D^b_m(X)$ (resp $D^b(X)$) denote the derived category of mixed Hodge modules (resp. constructible sheaves) on a complex variety $X$. Let
$rat\colon D^b_m(X)\to D^b(X)$
be the `forgetful' ...

**4**

votes

**0**answers

200 views

### Leray degeneration for smooth projective morphisms and formality of families of compact Kähler manifolds

Let $\pi \colon X \to S$ be a smooth projective morphism of algebraic varieties, say over $\mathbf C$. By Deligne's argument ("Théorème de Lefschetz...", 1968), there is for each $i$ an injection
$$ ...

**4**

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327 views

### Which properties of p-adic representations can be recovered from (\varphi,\Gamma)-modules?

In Berger's "An introduction to the Theory of $p$-adic Representations", he mentions that due the the equivalence of categories between etale $(\varphi,\Gamma)$-modules and $p$-adic representations, ...

**4**

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478 views

### Torelli theorem for cubic hypersurfaces

I'm reading the paper of Claire Voisin on the Torelli theorem for cubic hypersurfaces of $\mathbb{P}^5$ (in french : http://www.springerlink.com/content/j8675gn214l17523/). Are there lecture notes, or ...

**4**

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257 views

### Rozansky-Witten class associated to the Theta Graph

Suppose I have a holomorphic symplectic manifold, and a smooth $(1,0)$ connection on the tangent bundle which is compatible with both the complex and the symplectic structures. Say that the ...

**3**

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61 views

### 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 ...

**3**

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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 ...

**3**

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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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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 ...

**3**

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257 views

### Hilbert function of a Hilbert scheme

Given a Hilbert scheme $H$ of curves in $\mathbb{P}^3$ satisfying certain Hilbert polynomial, is there any way of understanding the degree or arithmetic genus of an irreducible component of the ...

**3**

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257 views

### Does Artin's vanishing hold for '$E_2$-weight pieces' for (torsion) cohomology of affine varieties?

Let $X$ be an affine variety of dimension $n$ (say, over complex numbers). Then Artin's vanishing theorem yields: both singular and etale cohomology $H^i(X):=H^i(X,\mathbb{Z}/l\mathbb{Z})=0$ for ...

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314 views

### genus one Gromov-Witten invariants of Calabi-Yau 3-folds

In
http://arxiv.org/PS_cache/hep-th/pdf/9302/9302103v1.pdf
physicists calculate (predict) genus one GW invariants of quintic (Table 1) and some other cases (Table 2).
Can any body explain to me ...

**2**

votes

**0**answers

107 views

### What sorts of weights for perverse sheaves were or can be computed?

I am studying certain weights for (triangulated categories of relative) motives. Those are interesting; yet one can hardly say that they are very much explicit or effectively computable. So, I would ...

**2**

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121 views

### Hodge numbers of non-commutative varieties

Let $(X, \mathcal{A})$ be a non-commutative variety, by this I mean $X$ is a (smooth) algebraic variety and $\mathcal{A}$ is a sheaf of algebra on $X$. One such example in my mind is when $X$ admits ...

**2**

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111 views

### Hodge numbers of l-adic sheaves?

Assume first that $C$ is a curve, say over $\mathbb{Q}$ and $(E, \nabla)$ is a vector bundle with a flat connection. Assume further that $(E, \nabla)$ has regular singularities at $S=\overline{C}-C$. ...

**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 ...

**2**

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**0**answers

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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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 ...

**2**

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**0**answers

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, ...

**2**

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**0**answers

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 ...

**2**

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162 views

### Basis for hodge decomposition of Elliptic K3 Surfaces

We know the hodge numbers of K3 Surfaces. To work out some ideas, I'd like to know an explicit basis for the hodge decomposition $H^{p,q}$ of a smooth Elliptic K3 Surface over $\mathbb{C}$ (for all ...

**2**

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**0**answers

171 views

### A Hodge substructure with ''nice' weight factors that does not correspond to a mixed submotif?

Suppose that we have a mixed motif $M$ with only two (subsequent) weights, and have a subobject for each weight factor. How we can control whether there exists a submotif of $M$ with these two ...

**2**

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**0**answers

132 views

### On 'graded polarizable' triangulated categories; are there any mixed Galois module analogues known? Also on mixed realizations

There is a construction by Beilinson (in section 3 of "Notes on absolute Hodge cohomology") of the derived category of graded polarizable mixed Hodge complexes; he also proved that this category is ...

**1**

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**0**answers

68 views

### Does the monodromy of such VHS have to be trivial

Consider a variation of polarized Hodge structure on a punctured disk. Suppose that connection preserves Hodge filtration (which is much stronger, than Griffiths transversality). Moreover assume that ...

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112 views

### Hodge structures generated by cohomology groups of varities with dimension less than $n$

Let $X$ be a smooth projective variety over $\mathbb{C}$ with dimension $n$. Is it true that for every $i<n$, the Hodge structure on $\mathrm{H}^i(X,\mathbb{Q})$ is generated by Hodge structures of ...

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185 views

### 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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73 views

### 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 ...

**1**

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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 ...

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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 $MHM(X)$ ...