Questions tagged [hodge-structure]
The hodge-structure tag has no usage guidance.
53
questions
2
votes
1
answer
220
views
Geometric Interpretation of absolute Hodge cohomology
$\quad$Let $\mathcal{Sch}/\mathbb C$ denote the category of schemes over $\mathbb C$. For an arbitrary $X\in\mathcal{Ob}(\mathcal{Sch}/\mathbb C)$, Deligne in his Article defined a polarizable Hodge ...
user508433
2
votes
1
answer
186
views
Why do we ask Shimura datum to have Hodge weight $(-1,1),(0,0),(1,-1)$?
Why do we ask Shimura datum to have Hodge weight $(-1,1),(0,0),(1,-1)$?
I know it's related to the decomposition of a complex Lie algebra $\frak{g}_{\mathbb{C}}=\frak{t}\oplus\frak{p}^{+} \oplus \frak ...
- 1,025
4
votes
1
answer
261
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{...
- 5,781
1
vote
0
answers
144
views
Beilinson-Hodge conjecture and generation of cohomology ring by $H^1$
Beilinson's version of Hodge conjecture has the following form. For any quasi-projective smooth complex variety $X$ the following map is surjective:
$$H^i_{\mathcal{M}}(X, \mathbb{Q}(j))\rightarrow \...
- 5,781
1
vote
0
answers
110
views
Abelian subvarieties corresponding to vector subspaces
Let $S$ be a connected smooth projective surface.
Let $C$ a smooth curve on $S$
In page 9 of the paper "https://arxiv.org/abs/1704.04187v1" a read the following:
Let
\begin{equation*}
r: ...
- 519
2
votes
0
answers
106
views
Nilpotent orbits and mixed Hodge structures
Let $(H_\mathbb{Z}, \{h^{p,q}\}_{p+q=n}, \phi)$ be the datum to define weight $n$ polarized Hodge structures on $H_\mathbb{Z}$, $h^{p,q}$ is the Hodge numbers, $\phi$ is a polarization. Let $D$ be the ...
- 21
2
votes
0
answers
175
views
Cohomology of maps between Hilbert schemes
Let $S$ be a smooth complex projective surface. We consider the following two types of Hilbert schemes of $S$.
The Hilbert scheme of an ample curve $D$. Suppose that $D$ is sufficiently ample, then ...
- 259
1
vote
0
answers
188
views
Exterior power of Hodge structures
Let $V$ be a $\mathbb{Q}$-vector space and suppose there is a decomposition of $V_{\mathbb{C}}:=V \otimes_{\mathbb{Q}} \mathbb{C}$ into two $\mathbb{C}$-sub-vector spaces i.e., $V_{\mathbb{C}} \cong V^...
- 1,583
7
votes
2
answers
609
views
Super mixed Hodge structures?
It's common in subjects that have some version of the "yoga of weights" that you have a functor called "Tate twist" and that the most natural version of it seems like it should be ...
- 43.3k
3
votes
2
answers
316
views
Abelian varieties corresponding to Hodge substructures
In an exercise of Voisin book, says:
Let $j:C\rightarrow S$ the inclusion of a smooth curve on a smooth connected projective surface. Set
$H=ker(j_*:H^1(C,\mathbb{Z})\rightarrow H^3(S,\mathbb{Z}))$.
...
- 519
5
votes
0
answers
249
views
Existence of an affine variety with homotopy type of suspension of another affine variety
Let $X$ be an affine variety. My question is does there exist another affine variety with the homotopy type of the suspension of $X$?
- 69
3
votes
0
answers
166
views
Is there a compact Kähler non-projective manifold with polarizable Hodge structures?
Let $V$ be a rational Hodge structure of degree $k$. Precisely, $V$ is a finite dimensional $\mathbb{Q}$-vector space whose complexification admits a decomposition $V_\mathbb{C} = \oplus_{p+q=k} V^{p,...
- 478
2
votes
0
answers
164
views
Is the dimension of the pieces of a mixed Hodge structure constant under smooth deformations?
In the case of a family of compact complex manifolds we have the following:
Theorem. Let $f:X→B$ be a family of complex manifolds and assume that $X_0$ is Kähler for some $0\in B$. Then for $b$ in a ...
- 21
4
votes
0
answers
241
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 \...
- 2,185
4
votes
1
answer
212
views
$\mathbb{Q}$-Zariski Closure not equal to smallest Q-subgroup
I wonder if there is a simple instance of the following phenomena : an abstract subgroup S $\subset GL_n(\mathbb{C})$ whose $\mathbb{Q}$-Zariski closure isn't a group ?
Is there some criteria to ...
- 43
3
votes
0
answers
111
views
Is the category of pure Hodge structures abelian semi-simple? [duplicate]
Apologies if the question in the title is too elementary.
A reference would be helpful.
- 20.5k
3
votes
0
answers
165
views
Hodge structure on intersection cohomology of toric varieties
Given a convex polytope with integer vertices, one can construct a complex projective variety $X$ called toric variety. In general $X$ is not smooth. As I have heard, by the work of M. Saito, the ...
- 20.5k
3
votes
1
answer
250
views
Automorphism of integral Hodge structures
Let $(V,V^{p,q},Q)$ be a polarized integral Hodge strucutre of weight $n$. I would like to understand the automorphism of this datum better. Specifically, I'm wondering if there are good conditions ...
- 4,136
1
vote
0
answers
85
views
Polarization of Prym varieites
I'm trying to understand polarization and rational Hodge structure of spectral curves and Prym varieties.
Excuse me that this is similar to my previous question.
I want to prove the following,
Let $X$...
- 297
3
votes
1
answer
1k
views
Applications of Hodge-Riemann bilinear relations [closed]
I am wondering if the Hodge-Riemann bilinear relations have any further applications/ developments in Kahler or algebraic geometry.
Let me briefly remind the statement.
Given a compact Kahler ...
- 20.5k
2
votes
1
answer
260
views
Middle cohomology of very general hyperplane sections
Let $X$ be a smooth, projective variety over $\mathbb{C}$ of dimension $n$ satisfying the property that for every $i \ge 0$, $H^{i,i}(X,\mathbb{C}) \cap H^{2i}(X,\mathbb{Q})=\mathbb{Q}c_1(\mathcal{O}...
- 2,002
1
vote
1
answer
221
views
Hodge variation
I am reading Milne's online book of Shimura Varieties https://www.jmilne.org/math/xnotes/svi.pdf, I confused by a Definition of Hodge variation. On page 29, it was said something is called Hodge ...
- 397
3
votes
1
answer
212
views
How to cook up an Artin motive from a positive-dimensional variety
I am trying to make sense of the paper "Eigenvalues of Frobenius and Hodge Numbers" (Kisin--Lehrer). I have not succeeded after some hours of intent staring at the screen.
In the proof of Corollary ...
- 31
2
votes
0
answers
122
views
Is there a Hodge structure on $\text{Hom}(V,W)$?
Let $V, W$ be real (pure) Hodge structures of weight $m, n$. Is there a natural Hodge structure on $\text{Hom}(V,W)$?
As I understand, there is one in the case $V = W$, although the definition I ...
- 3,994
5
votes
1
answer
237
views
Failure of Borel-Schmid quasi-unipotency theorem in non-algebraic case
Let $\mathcal{X} \to \Delta^{\times}$ be a smooth family of compact Kähler manifolds over a punctured disc.
When this family is algebraic, the celebrated quasi-unipotency theorem (I think, due to ...
- 1,115
4
votes
0
answers
206
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 \...
- 513
3
votes
0
answers
155
views
Duality of Mixed Hodge Structures without compactness
Let $X$ be a smooth separated algebraic variety over $\mathbb{C}$ and $Z \subset X$ a subvariety of codimension $p$. There are no compactness assumptions. I am looking for an isomorphism of mixed ...
- 195
8
votes
1
answer
792
views
Variations of Hodge structures over the line
Let $f\colon X\to \mathbb{A}^1$ be a smooth projective morphism of complex algebraic manifolds, where the target $\mathbb{A}^1$ is the affine line. Are there any restrictions on the Hodge structures ...
- 20.5k
2
votes
0
answers
203
views
For a mixed hodge structure, what is the exact condition on the graded pieces?
A mixed ($\mathbb{Q}$)-hodge structure is defined to be a vector space $V/\mathbb{Q}$ with an increasing "weight" filtration of $\mathbb{Q}$- vector spaces $0\subset W_0\subset \dots$ and a ...
- 2,756
3
votes
1
answer
162
views
How can I determine the monodromy of this variation of mixed hodge structures?
Consider the variation of mixed hodge structures which generates at the origin:
$$
f:X = \text{Proj}\left( \frac{\mathbb{C}[t][x,y,z]}{(xy(x + y + tz))} \right) \to \mathbb{A}^1_t
$$
How can I compute ...
- 1,696
2
votes
1
answer
171
views
How can I compute the mixed hodge structure for three copies of $\mathbb{P}^1$ intersecting at one point?
I know there is a spectral sequence for a variety with normal crossing singularities $X$ which gives a tool for making the computation of the mixed hodge structure computable. How can I compute the ...
- 1,696
6
votes
0
answers
180
views
Mixed Hodge modules of product spaces
Let $X$ be an algebraic varietiy (as good as you want, say affine and smooth) and let us denote by $MHM(X)$ the category of mixed Hodge modules as descrived by Saito (see for example this or this).
...
- 445
5
votes
0
answers
173
views
Reference request: If the local system extends, then the variation of Hodge structures extends
I'm looking for a precise reference for the following theorem.
Let $C$ be a smooth curve over $\mathbb{C}$ and let $S$ be a finite set of closed points of $C$. Let $\ V$ be a polarized variation ...
- 51
7
votes
1
answer
428
views
Two mixed Hodge structures on equivariant cohomology for actions by finite groups
The answer to the following question might be obvious but I haven’t found a full proof yet (neither by myself nor in the literature). So my apologies if it is trivial.
Let $X$ be a (for simplicity ...
- 143
4
votes
0
answers
459
views
Why is the Hodge conjecture equivalent to the assertion that $ \mathcal{R}_{ \mathrm{Hodge} } $ is fully faithfull?
On pages 17 and 18 of the following document: https://www.math.tifr.res.in/~sujatha/ihes.pdf, we find the following paragraph:
Let $ \mathbb{Q} \mathrm{HS}$ be the category of pure Hodge structures ...
- 325
6
votes
0
answers
253
views
Cohomology theories from Saito's mixed Hodge complexes
The definition of mixed Hodge complexes by Saito is a very interesting one, since it's more a cohomology theoretic than geometric generalization of Hodge structures. Since Saito's motivation for mixed ...
- 341
3
votes
1
answer
374
views
what is the definition of Hodge structure of geometric origin
Let $H$ be a mixed Hodge structure or, more generally, a mixed Hodge structure over a subfield $k$ of $\mathbb{C}$, by which I mean a $k$-vector space with two filtrations (Hodge and weight), a $\...
- 31
8
votes
0
answers
239
views
Mumford-Tate group of the Fermat curve
Let $C$ be the Fermat curve of degree $d$, defined by the equation $x^d+y^d=z^d$ in $\mathbb{P}^2$. The first cohomology group $H^1(C, \mathbb{Q})$ carries a pure Hodge structure, so it has an ...
- 81
4
votes
1
answer
547
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 ...
3
votes
0
answers
841
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^{...
- 351
3
votes
1
answer
489
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 ...
- 1,071
5
votes
0
answers
177
views
Nonabelian Hodge structure for noncompact curves and Hodge structure on the fundamental group
Nonabelian Hodge theory, introduced by C. Simpson and others, may be interpreted as a description of the (real) Hodge structure on the fundamental group (say, of a compact curve) in terms of some ...
- 733
1
vote
0
answers
430
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 ...
- 445
18
votes
0
answers
505
views
Does the "holomorphic spheres-to-continuous spheres" forgetful function respect the mixed Hodge structures on homotopy groups?
For each smooth, projective, complex variety $X$ that is simply connected, John Morgan constructed a natural mixed Hodge structure on the homotopy group $\pi_k(X,x)\otimes \mathbb{Q}$. This was ...
- 4,081
15
votes
1
answer
1k
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 ...
- 445
3
votes
1
answer
472
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}
...
- 349
12
votes
0
answers
794
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 ...
- 6,800
3
votes
0
answers
140
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 ...
- 375
3
votes
2
answers
256
views
Variation of Hodge structures associated to a hermitian symmetric domain
Let $D$ be an irreducible hermitian symmetric domain. Then there exists a variation of Hodge structures $(h_s)_{s\in D}$ on a vector space $V$ satisfying specific conditions which depend on $D$ such ...
- 974
6
votes
1
answer
411
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?...
- 61