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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5
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299 views

Families of abelian varieties on the line (or more generally simply connected varieties)

I'm curious whether the following is true: Question 1: Let $V/\mathbb{C}$ be a smooth connected variety such that $V^\text{an}$ is simply connected. Then, is every abelian scheme $f:\mathscr{A}\to ...
1
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1answer
164 views

Monodromy theorem of degeneration of smooth projective varieties to non-reduced central fiber

Given a family $\pi: \mathcal{X}\rightarrow\Delta$ smooth away from $0\in\Delta$, Where $\mathcal{X}$ is a smooth complex manifold, $\Delta$ is a small disk, the general fiber of $\pi$ is smooth ...
5
votes
0answers
128 views

Symplectic invariance of Hodge numbers?

Let $(X,\omega)$ be a compact symplectic manifold. If $J$ is a $\omega$-compatible complex structure on $X$ then $(X,\omega,J)$ is a compact Kähler manifold and so has Hodge numbers $h^{p,q}$. My ...
8
votes
1answer
213 views

Gauss-Manin connection via crystals

Let $f \colon X \to S$ be a proper smooth morphism between smooth varieties over $\mathbb{C}$. Then each relative de Rham cohomology group $R^if_*(\Omega_{X/S}^\bullet)$ is a vector bundle equipped ...
9
votes
2answers
270 views

Lower central series quotients in terms of (co)homology

Let $G$ be a group. It is well-known that $H_1(G,\mathbb{Z})=G/[G,G]$. Also (at least up to torsion) $[G,G]/[G,[G,G]]=\Lambda^2H^1(G,\mathbb{Z})/H_2(G,\mathbb{Z})$ as explained, for example, in this ...
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0answers
97 views

Monodromy of Geometric Variation of Hodge Structures over punctured disc

We know that the monodromy action $T$ is a morphism of limiting MHS on cohomology of nearby fiber $H^n(X_{\infty})$ derived from the Geometric variation of hodge structure $\pi: \mathcal{X}\rightarrow ...
6
votes
1answer
209 views

Regularity of Hodge Laplacian on bounded domains

I need a reference for the $W^{s,p}$ regularity of the Hodge boundary value problem on bounded domains. I need estimates $\lVert \omega \rVert_{W^{s+2,p}} \leq c \lVert f \rVert _{W^{s,p}}$, for ...
1
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0answers
117 views

Is the category of mixed Hodge modules bi-filtered?

Let $X$ be a smooth complex algebraic variety and let $MHM(X)$ be the category of mixed Hodge modules on $X$, as defined in (Saito, "Mixed Hodge Modules", 1990), (Peters-Steenbrink, "Mixed Hodge ...
2
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0answers
131 views

Question regarding a lemma in Principles of Algebraic Geometry

My question is regarding the lemma on page 81 of the book by Griffiths and Harris. The lemma says the following: A $\bar{\partial}$-closed form $\psi\in Z^{p,q}_{\bar{\partial}}(M)$ is of minimal ...
3
votes
0answers
87 views

Symplectic Hodge Maps and Mirror Symmetry

The notion of Hodge theory for symplectic manifolds seems to be getting more and more attentions in these days. See the series of papers by Yau: http://arxiv.org/abs/1011.1250 ...
11
votes
1answer
302 views

Mixed Hodge structure on sheaf cohomology of a variation of Hodge structures

I'm new here. I hope to do it right! I am interested in studying mixed Hodge structures over complex algebraic surfaces and their generalizations. Let us take a smooth complex variety $X$ and a ...
3
votes
1answer
254 views

Hypersurfaces without variable cohomology

Let $X$ be a smooth projective variety over $\mathbb C$ of dimension $n+1$. If $Y$ is a smooth very ample hypersurface, we know that except $H^n(X;\mathbb Q)\rightarrow H^n(Y;\mathbb Q)$, the ...
4
votes
0answers
61 views

The metric gives the optimal element in a class

In geometry there is plenty of examples in which the following happens: Some elements are considered equivalent, in some topological or algebraic sense We take the quotient The metric is usually not ...
6
votes
2answers
251 views

Hodge map and the Cohomology Ring of a Riemannian Manifold

For a compact Riemannian manifold $M$, we know that the Hodge map $\ast$ and Laplacian $\Delta$ commute. From Hodge decomposition and its implied isomorphism between harmonic forms and cohomology ...
3
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0answers
175 views

some terminologies on limiting mixed hodge structures or rather Derived categories

$f: X\rightarrow S$ is proper surjective homomorphism map from connected complex manifold to unite disk. $Y=f^{-1}(0)$ is algebraic and normal crossing in X, f is smooth away from 0, $X^*=X\setminus ...
4
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0answers
100 views

Deligne-Hitchin twistor space

Let $M$ be the moduli space of stable holomorphic Higgs bundles on a Riemann surface $\Sigma$. By Hitchin s work it is equipped with a hyper-kaehler structure, and one can define its twistor space, a ...
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1answer
124 views

Looking for a good exposition - Rees Construction

Can someone point me in the direction of a good exposition of the Rees Construction in Hodge Theory? Thanks!
11
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1answer
405 views

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 ...
3
votes
1answer
255 views

Is there a description of variation of (mixed) Hodge Structures in terms of a Deligne operator?

A complex Mixed Hodge Structure is given by a complex vector space $V$ together with a descending filtration $W$ and two ascending filtrations $F,\bar{F}$ that satisfy the condition \begin{equation} ...
5
votes
2answers
405 views

A systematic canonical construction of the Hodge star operator

I'm struggling to make sense of the Hodge star as a global canonical object. Here are my struggles so far and some questions: Let $M$ be a finitely generated projective $R$-module (hence locally free ...
5
votes
1answer
210 views

Birational Invariants

Let $X$ be a smooth rational variety of dimension $n$. We have $\dim H^0(X,\Omega_X^p) = \dim H^0(\mathbb{P}^n,\Omega_{\mathbb{P}^n}^p)$ for any $p$. These are Hodge numbers. I know that we can not ...
5
votes
1answer
691 views

Serre's Analogue of the Weil Conjectures for Non-Compact Kahler Manifolds

The classical Riemann Hypothesis concerns the locations of zeroes of the Riemann zeta-function, or more generally the Dedekind zeta-functions of number fields. Its analogue for varieties defined over ...
6
votes
1answer
747 views

Hodge Standard Conjecture in Positive Characteristic

In the Wikipedia article on the Hodge Standard Conjecture it is written (note: it has since been fixed): In characteristic zero the Hodge standard conjecture holds, being a consequence of Hodge ...
4
votes
1answer
201 views

How do non-trivial global differentials give non-trivial cohomology classes in positive characteristic

Let $k$ be an algebraically closed field and let $X$ be an $n$-dimensional smooth projective variety over $k$. If $k= \mathbb C$, there is a natural injective morphism of vector spaces ...
2
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1answer
138 views

Connection between the p and q Laplacians

I'm just looking for some quick and dirty intuition(and/or reading material) about the following: I read that Hodge duality provides a way to interchange the p-Laplacian $ \Delta_p = \nabla\cdot( ...
9
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0answers
319 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 ...
2
votes
0answers
113 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 ...
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votes
0answers
48 views

What is an induced Abel Jacobi map?

Let $X$ be a compact Kahler threefold, $C\subset X$ be a smooth curve, $\tau\colon\tilde{X}\to X$ be the blow up of $X$ along $C$, $j\colon E\to \tilde{X}$ be rthe exceptional divisor, $\tau_E$ be the ...
10
votes
2answers
589 views

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 ...
10
votes
2answers
1k 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 ...
0
votes
0answers
102 views

higher direct images of relative canonical sheaf plus a fractional divisor

For a map $f: Y \rightarrow X$ branched over simple normal crossing divisor $B=\sum_iB_i$, do we know of similar local freeness property for higher direct image of relative canonical sheaf plus a ...
6
votes
1answer
158 views

How small can the Mumford-Tate group of hypersurface be?

Is there some way of giving a lower bound on the dimension of the Mumford-Tate group of a hypersurface? Let's say it's of general type, say, of degree $10$ inside $ \mathbb{P}^3$. (Edited from here ...
4
votes
1answer
175 views

Does this extension of Hodge structures split over $\mathbb{Q}$?

Let $X$ be a smooth projective curve of genus $\geq 1$ over $\mathbb{C}$, $H^\cdot=H^\cdot(X)$, and $K$ be the kernel of cup product $\cup: H^1\otimes H^1\rightarrow H^2$. Consider the extension of ...
4
votes
1answer
349 views

Tate twist and comparison between Betti and de Rham cohomology

In Deligne's paper "Hodge cycles on abelian varieties" (see page 11 of http://jmilne.org/math/Documents/Deligne82.pdf) he says that the following diagram fails to commute by a factor of $(2 \pi i)^m$, ...
2
votes
0answers
77 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 ...
5
votes
0answers
247 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 ...
3
votes
2answers
241 views

What are the easiest examples of irreducible, but not big, monodromy representations

Let $\rho: \pi_1(S,s_0) \to GL(V)$ be the monodromy representation associated to a local system of $\mathbb Q$-modules $\mathbb V$ with $\mathbb V_{s_0} = V$. Let $H$ be the Zariski closure of the ...
4
votes
0answers
128 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 ...
4
votes
1answer
293 views

Mixed Hodge structure and cup product

I'm looking for a reference for the answer to the following questions. Let $X$ be an algebraic variety over C. When is the cup product a morphism of Mixed Hodge structures? Does $X$ have to be ...
3
votes
2answers
372 views

Why can we not always take a Kähler class to be in rational cohomology?

Given a Kähler manifold $(X,\omega)$ we know that its Kähler class lies in an open cone of $H^{1,1}(X) \cap H^2 (X,\mathbb{R})$. Since $\mathbb{Q}$ is dense in $\mathbb{R}$ we should be able to find a ...
0
votes
1answer
170 views

Hodge structure of relative cohomology groups

I need a hint or a good reference for definition of mixed Hodge structure on the relative cohomology groups ($\mathrm{H}^*(X,Y)$, $Y\subset X$ a closed subvariety of a comolex quasiprojective variety ...
3
votes
1answer
682 views

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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vote
0answers
114 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 ...
7
votes
1answer
396 views

periods of Mixed Hodge Structures

Two Questions: First. As I know the notion of periods comes when one has two vector spaces over a subfield $k$ of $\mathbb{C}$ (usually given by two cohomology theories) and an isomorphism between ...
6
votes
0answers
154 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 ...
12
votes
2answers
633 views

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 ...
4
votes
0answers
118 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 ...
9
votes
0answers
198 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 $i^* \star \omega = 0$ ...
13
votes
2answers
1k 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
votes
0answers
115 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 $S=\overline{C}-C$. ...