Questions tagged [hodge-theory]

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.

Filter by
Sorted by
Tagged with
4 votes
1 answer
427 views

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 ...
Sasha's user avatar
  • 5,492
4 votes
2 answers
415 views

Is there a $k$-structure for Hodge modules over a $k$-variety?

I am trying to understand (Saito's?) category of mixed Hodge modules as a category (i.e. I am not interested in its construction, just in properties of objects and morphisms). I would be grateful for ...
Mikhail Bondarko's user avatar
4 votes
1 answer
269 views

Hodge conjecture for generic points

I was reading the following paper: "Beilinson’s Hodge Conjecture For Smooth Varieties". They study the cycle class map $cl_{m,r}: H^{2r-m}_{\mathcal{M}}(U, \mathbb{Q}(r))\rightarrow \text{...
user127776's user avatar
  • 5,831
4 votes
1 answer
239 views

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 ...
Kali's user avatar
  • 503
4 votes
1 answer
368 views

Hodge classes and Leray filtration

Let $f :X \to Y$ be a submersion between smooth projective varieties over $\mathbb{C}$ and let $\alpha \in Z^k(X)$ be an algebraic cycle of $X$. Is is true that for all odd numbers $p$ and $q$ such ...
JacobI's user avatar
  • 233
4 votes
1 answer
575 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 ...
Priyavrat Deshpande's user avatar
4 votes
1 answer
229 views

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. ...
hdg90's user avatar
  • 41
4 votes
1 answer
478 views

Special fiber of the Neron Model of an Abelian scheme in terms of Limit Hodge Structure

Let $\mathcal{A}$ be an Abelian scheme over a smooth curve $S^*\subset S$ and let $\mathcal{A_S}$ be the Neron model of $\mathcal{A}$ over $S$. Is it possible to describe the special fiber of the ...
AJ Stewart's user avatar
4 votes
1 answer
624 views

An integral of the Hodge-Neumann Laplacian on a Riemannian manifold

Background Let $M$ be a compact, oriented Riemannian manifold with boundary, with $\text{vol}$ the Riemannian volume form. Let $\nu$ be the outwards unit normal vector field on $\partial M$ and $\nu^\...
Iza_lazet's user avatar
  • 179
4 votes
1 answer
752 views

A contradiction caused by the Kähler identity and the formal adjoint relation

I found a contradiction in the Principle of Algebraic Geometry by G&H, section 1.2. I have post this on MSE but it didn't get enough attention. I couldn't sleep or eat or do anything else due to ...
XT Chen's user avatar
  • 1,064
4 votes
0 answers
92 views

Simpson correspondence for perverse sheaves

Let $X$ be a projective complex manifold. Then Simpson's correspondence from nonabelian Hodge theory shows that the category of semisimple local systems on $X$ is equivalent to the category of ...
Doug Liu's user avatar
  • 433
4 votes
0 answers
141 views

Is $H^*($vanishing cycles$)$ computed by the twisted de Rham complex?

In notes by Sabbah (Theorem 3), it is stated that the cohomology $$\text{H}^*(X,\varphi_f)$$ of the vanishing cycle sheaf of a function $f:X\to \mathbf{A}^1$ for certain $X$ is expected to be the same ...
Pulcinella's user avatar
  • 5,506
4 votes
0 answers
143 views

Balanced manifolds and the $dd^c$-lemma

Let $X$ be a compact complex manifold. A Hermitian metric $\omega$ is balanced if $d\omega^{n-1}=0$, where $n=\dim_{\mathbf{C}} X$. By a theorem of Alessandrini-Basanelli, this class of Hermitian ...
AmorFati's user avatar
  • 1,349
4 votes
0 answers
293 views

Hodge decomposition on non-compact manifolds

Let $(\mathcal{M},g)$ be a compact Riemannian manifold without boundary. Then we have the well-known Hodge decomposition $$\Omega^{k}(\mathcal{M})\cong\mathcal{H}^{k}(\mathcal{M})\oplus\mathrm{ran}(\...
B.Hueber's user avatar
  • 987
4 votes
0 answers
265 views

Criterion for triviality of monodromy in smooth families

Let $\pi: X \to \Delta^*$ be a smooth, projective morphism. We know that for each $k$, there is a natural local system $L:=R^k \pi_*\mathbb{C}$. The associated vector bundle $\mathcal{L}:=L \otimes \...
user45397's user avatar
  • 2,195
4 votes
0 answers
138 views

Computing mixed hodge structure using different extension of the constant sheaf

Let $U$ be a non-compact smooth algebraic variety, and let $U\hookrightarrow X$ be a smooth compactification such that $D=X\setminus U$ is SNC. The connection $(\mathcal{O}_X,d)$ is the canonical ...
user2520938's user avatar
  • 2,768
4 votes
0 answers
222 views

Quasi-unipotent monodromy for variation of Landau-Ginzburg cohomology

A pair, $(X,f)$, consisting of a smooth variety and a global function $f:X\rightarrow\mathbb{A}^{1}$ is called a Landau-Ginzburg model, LG-model for short. The LG-cohomology of the pair, dentoed $H(X,...
user avatar
4 votes
0 answers
227 views

Do Poincaré residue and integrable log connection commute?

Here are some basic notations and definitions: (ignore this part if familiar) 1.Let $(X,\omega)$ be a compact Kähler manifold of dimension $n$, and let $D=\sum_{i=1}^r D_i$ be a simple normal crossing ...
Invariance's user avatar
4 votes
0 answers
319 views

Understanding the $\text{SL}_{2}$-orbit theorem

I am trying to understand a meaning of Schmid's $\text{SL}_{2}$-orbit theorem. The nilpotent orbit theorem now seems quite clear to me (both its proof and why one would want to consider such theorem), ...
GTA's user avatar
  • 1,014
4 votes
0 answers
172 views

Elliptic boundary value problem for vector valued forms

Let $U \subset R^n$ be a regular bounded domain having the topology of a ball. Then, the boundary value problem for $\omega\in \Omega^2(U)$, $$ d\omega = 0 \qquad \delta\omega = \sigma \qquad \...
Raz Kupferman's user avatar
4 votes
0 answers
333 views

Regular functions vs holomorphic functions

Let $X$ be an affine smooth variety over the complex numbers, $X^{an}$ its associated smooth complex analytic space, and $\mathcal{O}$, resp. $\mathcal{O}^{an}$ the respective structure sheaves. Is ...
Ari's user avatar
  • 181
4 votes
0 answers
213 views

Principal bundle analogue for Hodge bundle

Let $X$ be a connected smooth complex projective variety. A holomorphic Higgs bundle is a pair $(E, \theta)$ consists of a holomorphic vector bundle $E$ on $X$ together with a Higgs field $\theta \...
user124771's user avatar
4 votes
0 answers
265 views

Does local system cohomology come equiped with a mixed hodge structure?

Let $X$ be a quasi-projective variety over $\mathbb{C}$, and let $\mathcal{L}$ be a rank one $\mathbb{C}$ local system on $X$. Does $H^*(X,\mathcal{L})$ come with some mixed hodge structure in general?...
user2520938's user avatar
  • 2,768
4 votes
0 answers
127 views

Topological cycles with Lagrangian support

For a compact Kähler manifold of dimension $2n$, is there a classification of the homological $n$-cycles which are supported in a compact Lagrangian submanifold? The main example for this question ...
D. L Garcia's user avatar
4 votes
0 answers
256 views

Motives up to homological equivalence

Let $X$ be a smooth projective variety over a field $k$ finitely generated over its prime field, and $M_{hom}(X)$ the category of motives modulo $\ell$-adic homological equivalence. (1) Is $M_{hom}(...
user avatar
4 votes
0 answers
239 views

Hard Lefschetz for cycles

Let $X$ be a smooth projective variety over a field $k$. It is known by work of Deligne, that the Lefschetz operator: $$ L^k:H^{2n-2k}\left(X_{\overline{k}},\mathbf{Q}_{\ell}\right)\to H^{2n+2k}\left(...
user avatar
4 votes
0 answers
88 views

Locus of Hodge classes

Let $\pi: X\to S$ be a proper smooth morphism of complex analytic spaces, with connected smooth $X$ and $S$ over $\mathbf{C}$, projective fibers, and $$\mathscr{H}_{X/S}^p := R^p\pi_*\Omega^{\bullet}_{...
user avatar
4 votes
0 answers
277 views

Hodge cycles defined over algebraic extensions of $\mathbf{Q}$

Is it true that the Hodge conjecture for all smooth projective varieties over the complex numbers, follows from the Hodge conjecture for smooth projective varieties defined over $\overline{\mathbf{Q}}$...
user avatar
4 votes
0 answers
299 views

About the exponential sequence

For a complex analytic space $X$, we have the exponential sequence $$0\to\mathbf{Z}(1)_X\to\mathcal{O}_X\to\mathcal{O}_X^{\times}\to 1$$ the last map being the exponential $\text{exp}$. For $d>0$ ...
user avatar
4 votes
0 answers
348 views

Homotopical enhancements of cycle class maps

Fix a smooth projective variety $X$ over the complex numbers. We write $H^n(X,\mathbf{Z}(d)) = \text{CH}^d(X, 2d-n)$ for Bloch's higher Chow groups. Notation For a field $k$, recall $\Delta^n_{k} :=...
user avatar
4 votes
0 answers
1k views

Variational Hodge Conjecture vs Hodge Conjecture

Motivation. Let us state the following version of Grothendieck's variational Hodge Conjecture: Conjecture (VHC). Let $\mathcal{X}\to S$ be a proper smooth map of smooth algebraic varieties over $\...
user avatar
4 votes
0 answers
234 views

On the class of Shimura data of Hodge type that cover a given Shimura datum of abelian type

A Shimura datum $(G,X)$ is of Hodge type if there exists an injective morphism of Shimura data $(G,X) \hookrightarrow (\mathrm{GSp}_{2g}, \mathfrak{H}^{\pm})$, for some integer $g$. Let $(G',X')$ be ...
user105976's user avatar
4 votes
0 answers
413 views

Complex manifolds with the same cohomology

Is there an example of two non-homeomorphic projective smooth complex varieties $X$ and $Y$ such that there exists an isomorphism $H^{\ast}(X,\mathbb{C})\rightarrow H^{\ast}(Y,\mathbb{C})$ of ...
Nguyen lan Lee's user avatar
4 votes
0 answers
136 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/...
Jaak van der Smut's user avatar
4 votes
0 answers
253 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, $X^*=X\setminus Y$...
Feng Hao's user avatar
  • 1,071
4 votes
0 answers
164 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 ...
Li Yutong's user avatar
  • 3,362
4 votes
0 answers
92 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 ...
Jay's user avatar
  • 725
4 votes
0 answers
351 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 ...
Mikhail Bondarko's user avatar
4 votes
0 answers
99 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 ...
Darius Math's user avatar
  • 2,181
4 votes
0 answers
269 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 $MHM(X)$ ...
Reladenine Vakalwe's user avatar
4 votes
0 answers
275 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 ...
Liang Xiao's user avatar
4 votes
0 answers
234 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 $$ \...
Dan Petersen's user avatar
  • 39.2k
4 votes
0 answers
379 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, ...
James's user avatar
  • 41
4 votes
0 answers
285 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 ...
Oren Ben-Bassat's user avatar
3 votes
2 answers
1k 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 \ker (\...
user avatar
3 votes
3 answers
403 views

When can Hodge filtrations (decompositions?) be described explicitly in terms of periods?

It seems that there is no chance to explain the Hodge theory (to students) in an hour or so. Yet do there exist any cases when the Hodge filtration (or the Hodge decomposition) of the cohomology of a ...
Mikhail Bondarko's user avatar
3 votes
1 answer
416 views

Is it written anywhere that open subvarieties of affine spaces have "completely impure" cohomology?

Consider complex affine space $\mathbb{C}^n$ and let $U$ be a Zariski open subset of $\mathbb{C}^n$. By a celebrated result of Deligne, the cohomology $H^i(U)$ has a canonical Hodge structure. In ...
Ben Webster's user avatar
  • 43.9k
3 votes
2 answers
365 views

Generalization of the Leray-Hirsch theorem

We know the classical Leray-Hirsch theorem for fibrations. My question is, whether a similar statement also holds for flat, proper morphism? In particular, consider a faithfully flat, proper morphism $...
user43198's user avatar
  • 1,949
3 votes
1 answer
403 views

octic K3s inside cubic 4-folds

From the Thesis of B.Hassett I seem to understand that a smooth cubic 4-fold $X$ containing a $\mathbb{P}^2$ should contain also a octic K3, but I cannot see a natural way by which this K3 octic could ...
IMeasy's user avatar
  • 3,717
3 votes
2 answers
917 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.
DLIN's user avatar
  • 1,905

1
3 4
5
6 7
10