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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1answer
113 views

Hodge decomposition on open manifold

For the open manifold like $X\times \mathbb R$ or $X\times \mathbb R^+$, where $X$ is a closed manifold. Is there any decomposition like (Hodge Decomposition) of the Differential forms on it.
5
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2answers
232 views

Alternative construction of the first Chern class map

Let $X$ be a compact Kähler manifold. Consider the exponential sequence $0 \to \mathbf Z \to \mathscr O_X \to \mathscr O_X^* \to 0$. The boundary map gives a map $H^1(X, \mathscr O_X^*) \to H^2(X, \...
4
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1answer
218 views

Hodge decomposition for Bott-Chern cohomology

$\DeclareMathOperator{im}{im}$ I want to prove that Bott-Chern cohomology group $H^{p,q}_{BC}=\frac{\ker\partial\cap \ker\bar{\partial}}{\im\partial\bar{\partial}}$ has finite dimension via Hodge ...
2
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1answer
77 views

harmonic differential form integer class

Let $(M,g)$ be a compact Riemannian three-fold such that $H_2(M,\mathbb{Z}) = \mathbb{Z}$ and $S$ any surface representing 1. By Hodge theory there exist a harmonic differential one-form $\eta$ dual ...
2
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1answer
175 views

The compatibility of the Gysin sequence with mixed Hodge structures

Let $X$ be a compact complex $n$-manifold and $D$ be a smooth comdimension $1$ submanifold. Also let $U:= X\setminus D$ and $j$ be the inclusion of $U$ in $X$. Then it is well known that the ...
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0answers
128 views

Reference for the Hodge polynomial or the Hodge Characteristic

What is the first work that studies, refers to, or mentions the Hodge characteristic? The Hodge polynomial is the unique ring homomorphism $$ P_{hdg}:K_0(\mathbf{Var}/\mathbb{C)}\to \mathbb{Z}[u,v,u^{...
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3answers
379 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
184 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
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0answers
138 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 ...
10
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2answers
311 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
101 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 ...
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0answers
129 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 ...
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0answers
133 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 ...
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0answers
90 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 http://arxiv.org/abs/...
11
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1answer
340 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 ...
4
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0answers
62 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 ...
3
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1answer
257 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 ...
6
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2answers
263 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
185 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, $X^*=X\setminus Y$...
4
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0answers
109 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
133 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!
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1answer
270 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
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2answers
422 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
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1answer
215 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 ...
8
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1answer
292 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 $...
4
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1answer
203 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 $$H^0(X,\...
2
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1answer
140 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
322 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
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0answers
118 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 ...
0
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0answers
51 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
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2answers
613 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 \...
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0answers
149 views

higher direct images of relative canonical sheaf plus a fractional divisor

For a map $f: Y \rightarrow X$ smooth outside a 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
160 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
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1answer
177 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
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1answer
373 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
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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 ...
6
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0answers
259 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 $H^{p,q}_{\bar{\...
3
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2answers
250 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
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0answers
130 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 ...
5
votes
1answer
300 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 smooth?...
3
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2answers
380 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
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1answer
176 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 $...
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0answers
116 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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1answer
411 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 ...
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0answers
157 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
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2answers
665 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
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0answers
122 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 ...
2
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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 $S=\overline{C}-C$. ...
12
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0answers
355 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 ...
2
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1answer
291 views

When is the Hodge diamond concentrated in $H^{n,n}$'s?

Let $X$ be a smooth projective complex algebraic variety. The Hodge decomposition tells us that $H^n(X, \mathbf C) = \oplus H^{p,q}$. Here is my question: For what kind of $X$ is $H^{2n}(X) = H^{...