# 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.

466
questions

6
votes

0
answers

197
views

### Road map and references for combinatorial hodge theory

I'm a PhD student. I'm familiar with graduate level algebraic geometry and toric varieties.
I wanted to know a road map for getting into combinatorial hodge theory and other prerequisites that I'll ...

2
votes

0
answers

56
views

### Relative Dolbeault cohomology using currents

I need to compute the cohomology groups of some relative holomorphic $i$-forms $H^\bullet(X, \Omega^i_{X/Y})$ for a fibration of complex manifolds $X\to Y$, using a kind of distributional de Rham ...

6
votes

1
answer

328
views

### Different Hodge numbers arising from different holomorphic structures?

Does anyone have an example or know any references for a complex manifold $M$ with two different holomorphic structures that give rise to different Hodge numbers?

1
vote

0
answers

130
views

### How does the Torelli theorem behave with respect to cyclic covering?

Let $Y\xrightarrow{2:1}\mathbb{P}^3$ be the double cover, branched over a quartic K3 surface $S$, known as quartic double solid. Assume $S$ is generic, we know that there is a Torelli theorem for $Y$ ...

1
vote

0
answers

165
views

### How to "eliminate" the log pole of a logarithmic $(p,q)$-form？

Let $X$ be a compact complex manifold, and $D=\sum_{i=1}^{r} D_i$ be a simple normal crossing divisor on $X$. Let $\alpha$ be a logarithmic $(p,q)$-form, namely, on an open subset $U$, we can write
$$\...

2
votes

0
answers

134
views

### Hodge numbers of a complement

Let $Y\subset X$ be an analytic subvariety of codimension $d$ of a smooth compact complex variety $X$. Denote $U = X\setminus Y$. The relative cohomology exact sequence implies that
$$
H^i(X) \to H^i(...

3
votes

1
answer

195
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 ...

2
votes

1
answer

209
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
votes

1
answer

773
views

### Understanding the Hodge filtration

Let $X$ be a smooth quasiprojective scheme defined over $\mathbb{C}$, and let $\Omega^{\bullet}_X$ denote its cotangent complex, explicitly, we have:
$\Omega^{\bullet}_X:=\mathcal{O}_X\longrightarrow \...

0
votes

0
answers

82
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 ...

3
votes

2
answers

519
views

### How does complex conjugation act on the Hodge filtration?

Let $X$ be a $\mathbb{R}$-defined smooth proper scheme, and let $H^i_{\text{dR}}(X)$, denote its algebraic de Rham cohomology. The Hodge filtration gives an $\mathbb{R}$-defined pure Hodge structure ...

2
votes

0
answers

87
views

### Action of monodromy on the $p$-adic period domain in Lawrence-Venkatesh

In here, I asked various questions related to Lawrence and Venkatesh's work on the Mordell-Weil conjecture, which failed to receive any answers.
This is my attempt to try and focus the question.
In ...

3
votes

0
answers

111
views

### What is the correct definition of intermediate Jacobian for this singular threefold?

I am considering blow up of $\mathcal{C}\subset(\mathbb{P}^1)^3$, $X=\operatorname{Bl}_{\mathcal{C}}(\mathbb{P}^1)^3$, where $\mathcal{C}$ is a curve given by $$\{s^2u=0\}\subset\mathbb{P}^1_{s:t}\...

2
votes

0
answers

166
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 ...

2
votes

1
answer

174
views

### Blowup formula for a morphism

Let $f: X\to S$ be a smooth projective morphism between smooth schemes over $\mathbb C$, $i: Z \to X$ a closed subscheme of codimension $c$, also smooth over $S$, and let $g: Y\to S$ be the blowup ...

2
votes

0
answers

221
views

### What are the Hodge and log Hodge groups of $M_{g,n}$?

I would like to know, ideally with a reference, what the Hodge and log Hodge numbers of the moduli space of stable curves $\bar M_{g, n}$ are. At the very least I'd like to know the genus zero case $g ...

3
votes

1
answer

247
views

### Original proof of Lefschetz's theorem on $(1,1)$ classes

Is there a "modern" account of Lefschetz proof of his theorem about $(1,1)$ classes for projective surfaces ?
I believe that would be very interesting to understand the original arguments ...

1
vote

0
answers

52
views

### Using connection form for unknown frame field

I have a way to calculate the connection 1-form $\alpha$ associated to a compact simply connected parallelizable Riemannian surface $(M,g)$ (so, $M$ is topologically a disk) and a special orthonormal ...

2
votes

0
answers

126
views

### Hodge coniveaux of Calabi-Yau manifolds

Let $X$ be a strict compact Calabi-Yau manifold of dimension $n$. By this, I mean that $X$ is a simply connected projective manifold whose holomorphic forms are generated by a nowhere zero top degree ...

2
votes

0
answers

83
views

### Is the Leray projection continuous with respect to the Frechet topology of smooth periodic vector fields in $3$ dimensions?

Let $\mathbb{T}^3:=(\mathbb{R}/\mathbb{Z})^3$ be the $3$-torus and $C^\infty(\mathbb{T}^3,\mathbb{R}^3)$ be the Frechet space of smooth periodic vector fields on $\mathbb{T}^3$.
By Helmholtz ...

3
votes

1
answer

401
views

### On Simpson's motivicity conjecture

Simpson's motivicity conjecture says that for any rigid, flat irreducible connection $(V,\nabla)$ on a smooth complex variety $M$, there exists a proper smooth morphism $f:X \to M$ s.t. $(V,\nabla)$ ...

2
votes

0
answers

42
views

### Torelli theorem for veronese double cone(reference needed)

Let $Y$ be a smooth Veronese double cone, which is a smooth del Pezzo threefold of degree one, which can be regarded as a weighted hypersurface of degree 6 in $\mathbb{P}(1,1,1,2,3)$. I was wondering ...

4
votes

0
answers

109
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 ...

6
votes

0
answers

152
views

### Fourier transform and Hodge-$*$ operator

Suppose I have a full-rank lattice $\Lambda\subset\mathbf{C}$. Then the classical Poisson summation formula says
$$\sum_{\lambda\in\Lambda}f(\lambda)=\sum_{\lambda\in\Lambda'}\widehat{f}(\lambda)$$
...

11
votes

1
answer

1k
views

### Hodge conjecture as the equality of arithmetic and algebraic weights of motivic L-functions

Recently I became aware of the following statement given on page 13 of this paper. First, let us recall the following definitions:
Definition 4.1. Suppose $L(s)$ is an analytic $L$-function with ...

3
votes

0
answers

192
views

### Hodge symmetry without $\mathbb{C}$ [duplicate]

If $k$ is a field of characteristic zero and $X$ is a smooth irreducible projective variety over $k$, then $X$ satisfy Hodge symmetry, meaning that
$$\dim H^p(X, \Omega_{X/k}^q) = \dim H^q(X, \Omega_{...

1
vote

0
answers

144
views

### Intermediate Jacobian for small resolution of a singular Fano threefold?

I am mainly interested in the nodal Gushel-Mukai threefold. Let $X$ be a Gushel-Mukai threefold with one node, then by page 21 of the paper https://arxiv.org/pdf/1004.4724.pdf there is a short exact ...

4
votes

0
answers

173
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 ...

4
votes

1
answer

234
views

### Eigenforms of the Laplacian on Lie groups

I am a bit rusty in my differential geometry and I would like to confirm that my reasoning below holds, and I have some related questions (and all references to related concepts are of interest to me)....

1
vote

1
answer

315
views

### Cohomology of singular curves

Suppose $X$ is a singular quasi-projective curve over the complex numbers, and $X'$ is a good nonsingular compactification of a resolution of singularities $Y\to X$. Let $D$ be the complement of $Y$ ...

2
votes

1
answer

295
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 ...

12
votes

2
answers

1k
views

### Does Poincaré duality preserve algebraic cycles?

Let $X$ be a smooth, projective (complex) variety of dimension $n$ and $Z \subset X$ be a subvariety of codimension $k$ (if necessary assume $Z$ is non-singular). We know the cohomology class $[Z]$ of ...

1
vote

0
answers

105
views

### Confusion about notations in limit mixed Hodge structure

I am reading the paper Monodromy at infinity and Fourier transform by Claude Sabbah and got some confusions about notations. (note first that I am not specialized in mixed Hodge theory but, I am ...

43
votes

2
answers

6k
views

### Clausen's modified Hodge Conjecture

In a recent talk at the University of Geneve, Dustin Clausen presented a "modified Hodge Conjecture". I found the abstract intriguing but couldn't find videos or notes available online.
If I'...

2
votes

1
answer

265
views

### Triangulated structure on complexes of mixed Hodge structures

I'm trying to read parts of the Peters Tata Lectures on "Motivic Aspects of Mixed Hodge structures"
One aspect I don't really understand is the construction of the ''mixed cone'' for ...

5
votes

1
answer

285
views

### Formula in non-Abelian Hodge theory - Hodge-Riemann bilinear relations

I am currently reading about the non-Abelian Hodge correspondence. Let $(X,\omega)$ be a compact Kahler manifold. Given a Higgs bundle $(E, D_0)$ on $X$, we want to construct the corresponding flat ...

3
votes

1
answer

289
views

### Given a smooth hyperplane section Y of a variety X there exists a Lefschetz pencil of hyperplane sections of X containing Y

Let $X$ be a variety contained in $\mathbb{P}^N$ and let $Y$ be a smooth hyperplane section of $X$. I have read in page 54 of Voisin's book "Hodge theory and complex algebraic geometry II" ...

1
vote

0
answers

99
views

### Hodge-Helmholtz decomposition for 1-form of strategic game

This question is motivated by the attempt of decomposing (in a direct sum sense) a strategic game into components in the spirit of Hodge decomposition.
Preamble
Combinatorial setting
Candogan et al. (...

4
votes

0
answers

165
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 ...

4
votes

0
answers

320
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}(\...

1
vote

1
answer

139
views

### Cohomology classes fixed by algebraic automorphism subgroups

Let $X$ be a smooth projective complex variety and $t\in H^{2p}(X,\mathbf{Q})(p)$ a rational cohomology class.
Assume that there exist
$$t_1,\ldots,t_N\in H^{2*}(X,\mathbf{Q})(*)$$
algebraic classes (...

1
vote

0
answers

199
views

### Period calculation for elliptic curve

In the paper "Hodge cycles on Abelian Varieties" (Proposition 1.5), Deligne proves the following theorem:
Let $X$ be a smooth projective variety over $\overline{\mathbf{Q}}$ of dimension $n$...

2
votes

1
answer

217
views

### Hodge decomposition for non-elliptic complexes

It is a well-known result that there is a bijective correspondence between harmonic sections and cohomology classes of an elliptic complex in a Riemannian/Hermitian manifold. Now consider Riemannian/...

2
votes

0
answers

119
views

### Finding a Hodge theoretic condition to measure the rank of isogeny of product abelian surfaces

Let $A$ be an abelian surface over $\mathbb{C}$, then there is a condition on $H^0\left(\Omega^1_A\right)$
to determine if $A$ contains an elliptic curve $E$ as a subvariety. If $A$ were to contain
...

6
votes

0
answers

246
views

### Rank $2$ motivic local systems on a curve

This question is about the article "Motivic local systems on curves and Maeda's conjecture" by Yeuk Hay Joshua Lam.
In the proof of Theorem 1.1 it is claimed (on lines 4-5 of p. 7) that any ...

2
votes

0
answers

145
views

### Minimal Betti numbers of simply-connected threefolds with trivial canonical class

By a threefold, I mean a compact complex manifold of dimension three.
For a simply-connected threefold with trivial canonical class, its Betti numbers satisfy:
$$b_2 \ge 0, b_3 \ge 2.$$
I am wondering ...

5
votes

0
answers

480
views

### Can Hodge symmetry and invariance of Hodge numbers in smooth families be proven purely algebraically?

Let $k$ be an algebraically closed field of characteristic 0.
I am wondering if there are proofs of the following facts that do not use the analytic topology over $\mathbb{C}$:
Let $X$ be a smooth ...

1
vote

1
answer

113
views

### Intermediate Jacobian under group action

Let $X$ be a smooth Fano threefold with a finite group $G$ action. Assume that the orbit space $X/G$ is smooth. Is it true that $J(X/G)\cong J(X)^G$ As an abelian variety? Here, $J(X)^G$ is the $G$-...

2
votes

0
answers

88
views

### Hard Lefschetz for perverse sheaves on Kähler manifolds

Let $(X,\omega)$ be a compact Kähler manifold, $k\ge0$, $P\in Perv(X)$ be a semisimple object, then do we have the hard Lefschetz isomorphism between perverse cohomology sheaves $\omega^k:{}^p\mathcal{...

6
votes

1
answer

169
views

### Properties of non-integer powers of the Hodge Laplacian

Consider a complete smooth Riemannian manifold $(M,g)$.
I think that it is not difficult to prove that the $k$ Hodge Laplacian is essentially selfadjoint in the relevant $L^2$ space of $k$ forms, ...