All Questions
Tagged with hodge-structures or hodge-structure
59 questions
18
votes
0
answers
511
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 ...
- 4,111
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 ...
- 507
12
votes
0
answers
861
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 ...
- 6,920
8
votes
1
answer
980
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 ...
- 21.8k
8
votes
0
answers
270
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
7
votes
2
answers
667
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 ...
- 44.7k
7
votes
1
answer
458
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
7
votes
1
answer
888
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 ...
- 4,484
7
votes
0
answers
148
views
Is the $\ell$-adic cohomology ring of a cubic threefold a complete invariant?
The only interesting $\ell$-adic cohomology of a smooth cubic threefold $X$ is $H^3(X,\mathbb{Z}_{\ell}(2))$, which is isomorphic as a $\mathrm{Gal}_k$-module to $H^1(JX,\mathbb{Z}_{\ell}(1))^{\vee}$ ...
- 679
6
votes
1
answer
434
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
6
votes
0
answers
196
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).
...
- 507
6
votes
0
answers
264
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
5
votes
1
answer
237
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 ...
- 53
5
votes
1
answer
273
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 ...
- 1,170
5
votes
0
answers
250
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
5
votes
0
answers
185
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
5
votes
0
answers
189
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 ...
- 743
5
votes
0
answers
189
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 ...
- 1,207
4
votes
1
answer
276
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{...
- 5,901
4
votes
1
answer
2k
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 ...
- 21.8k
4
votes
1
answer
608
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 ...
- 1,017
4
votes
0
answers
308
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 \...
- 2,323
4
votes
0
answers
225
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 \...
- 523
4
votes
0
answers
477
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
3
votes
2
answers
396
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
3
votes
1
answer
298
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 ...
- 4,418
3
votes
1
answer
217
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
3
votes
2
answers
275
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 ...
- 1,066
3
votes
1
answer
221
views
When does the rational Hodge structure determine the integral Hodge structure?
Take a smooth complex projective variety $X$, consider $H^k(X,\mathbb Z)$, and take the global period domain as described, for example, in Voisin's Hodge theory book, 10.1.3: it's a subset of a flag ...
- 683
3
votes
1
answer
171
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 ...
- 1,716
3
votes
1
answer
427
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
3
votes
1
answer
557
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 ...
- 1,081
3
votes
1
answer
494
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}
...
- 359
3
votes
1
answer
207
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 ...
- 1,270
3
votes
0
answers
183
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,...
- 498
3
votes
0
answers
116
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.
- 21.8k
3
votes
0
answers
196
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 ...
- 21.8k
3
votes
0
answers
171
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
3
votes
0
answers
981
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^{...
- 349
3
votes
0
answers
141
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 ...
- 385
2
votes
1
answer
225
views
Prefactor $2\pi i$ for Tate-Hodge structure
A rather basic question. What was the original reason to consider the underlying $\mathbb{Z}$-module of the - as canonical object regarded - Tate-Hodge structure $\mathbb{Z}(1)$ to be given as $2 \pi ...
- 6,028
2
votes
1
answer
325
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
294
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}...
- 2,032
2
votes
1
answer
183
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 ...
- 1,716
2
votes
0
answers
170
views
Hodge bundles associated to a family of complex manifolds
I'm reading Voisin's books on Hodge theory. In the first volume she claimed but didn't prove this theorem:
Theorem 10.10 (Voisin) Let $\varphi:\chi\rightarrow B$ be a family of compact complex ...
- 21
2
votes
0
answers
119
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
196
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 ...
- 279
2
votes
0
answers
169
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
2
votes
0
answers
129
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 ...
- 4,164
2
votes
0
answers
222
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 ...
- 2,824