All Questions
Tagged with hodge-structures or hodge-structure
32 questions with no upvoted or accepted answers
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
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
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
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
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
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
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
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
195
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
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
1
vote
0
answers
88
views
Unique polarization on a very general curve with Mumford-Tate
I try to understand why a very general curve (smooth, projective over $\mathbb{C}$) has an unique polarization up to scalar on the $H^1(X,\mathbb{Q})$.
I was advised to look at the maximality of the ...
1
vote
0
answers
163
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 \...
- 5,901
1
vote
0
answers
116
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
1
vote
0
answers
213
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^...
- 1,593
1
vote
0
answers
96
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
1
vote
0
answers
440
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 ...
- 507
1
vote
0
answers
125
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 ...
- 4,484
0
votes
0
answers
99
views
Hodge filtration vs Hodge structure on algebraic de Rham cohomology
I have a basic question on the relation between the definitions of the Hodge structure on the algebraic de Rham of a smooth proper scheme defined over a subfield of $\mathbb{C}$ and the Hodge ...
- 2,907