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.
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Mumford-Tate groups of abelian surfaces
For elliptic curves, one may easily compute Mumford-Tate groups; there are just two cases:
1) $E$ has no complex multiplication, and the Mumford-Tate group of $E$ is $GL_2$
2) $E$ has complex ...
user94490
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Explicit algebraic cycles
Fix a smooth sextic curve curve $C = \{f_6(x,y,z) = 0\}$ in $\mathbb{P}^2$, and consider the double cover $X_{f_6}$ defined by $z^2 = f_6$ in the appropriate weighted projective space. This is known ...
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Tate twists and cohomology of $\mathbf{P}^1$
I was wondering if anyone could give me some intuition as to why, for a smooth projective variety $X$ over $\mathbf{C}$ of complex dimension $d$, the Tate twist on $H^n(X(\mathbf{C}),\mathbf{Z})$ to ...
user113452
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Full lattice images and Hodge decomposition
Let $X$ be a smooth and proper complex analytic space, or a Kahler complex manifold. Is the image $\Lambda$ of the finitely generated abelian group $H^{2i}(X,\mathbf{Z})$ into $J := H^{2i}(X,\mathbf{C}...
user97068
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Finiteness aspects of Deligne cohomology
Say $X$ is a smooth projective variety over $\mathbf{C}$, and $\mathcal{X} = X^{\rm an}$ its $\mathbf{C}$-analytic space.
For what integers $i,d$ is the Deligne cohomology $H^i_{\mathcal{D}}(\mathcal{...
user97068
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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 $\...
user113453
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Smooth proper fibration of complex projective varieties
Let $X$ be a smooth projective algebraic variety over the complex numbers.
(a) Do there exist:
a smooth proper map $\pi : \mathcal{X}\to S$ of algebraic varieties over the complex numbers, such that ...
user113393
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de Rham closed harmonic form on a Kähler manifold
For a compact Kähler manifold, we say that a form is primitive if it is contaned in the kernel of the dual Lefschetz operator, or the co-Lefschetz operator. For all examples I know, a primitive form $\...
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Non-snc locus relative to a smooth morphism
Let $f\colon X\rightarrow Y$ be a smooth morphism between smooth varieties and $B\subset Y$ a simple normal crossing divisor such that $f^*(B)$ is simple normal crossing as well. Consider a semiample ...
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Important consequences of the Hodge Index Theorem
The Hodge Index Theorem for compact Kaehler manifolds seems to be a big deal in complex geometry. See here for the surface version of the result.
https://en.wikipedia.org/wiki/Hodge_index_theorem
I ...
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Intuition for polarized Hodge structures
A Hodge structure can be defined as a real, algebraic representation of the Deligne torus ${Res}^\mathbb{C}_{\mathbb{R}}\mathbb{G}_m$. Coming from Kahler manifolds the intuition for this is clear. The ...
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Lagrangian up to Hamiltonian in cotangent bundle
I want to understand the folklore conjecture that, in a CY manifold, Lagrangians up to Hamiltonian isotopies are represented by special Lagrangians by examining cotangent bundle and Hodge theory.
...
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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 ...
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Corlette-Simpson correspondence over nodal curves
As we know that over a smooth projective variety $X/\mathbb{C}$, there is a correspondence between semi-simple flat bundles and polystable Higgs bundles with vanishing Chern classes, through ...
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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 ...
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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 ...
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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 ...
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Interesting geometric application of Hitchin Fibration
Let $X$ be a smooth complex projective variety. Let $M_{Higgs}(X, P)$ be the coarse moduli which universally corepresents the functor:
$$M^{\#}_{Higgs}(X, P): Sch/\mathbb{C}\longrightarrow \mathcal{...
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Mixed Hodge modules of product spaces
Let $X$ be an algebraic varietiy (as good as you want, say affine and smooth) and let us denote by $MHM(X)$ the category of mixed Hodge modules as descrived by Saito (see for example this or this).
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Reference request: If the local system extends, then the variation of Hodge structures extends
I'm looking for a precise reference for the following theorem.
Let $C$ be a smooth curve over $\mathbb{C}$ and let $S$ be a finite set of closed points of $C$. Let $\ V$ be a polarized variation ...
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Higgs bundles and Variation of mixed Hodge structures
As we know that: A polystable Higgs bundles $(E, \theta)$ with vanishing Chern classes is fixed by $\mathbb{C}^*$-action iff it comes from variation of Hodge structures.
Q: Is there a conterexample ...
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Second Chern class of Stable Higgs sheaf
Let $X$ be a smooth projective surface, $\mathcal{E}$ a stable torsion free Higgs sheaf of degree 0. Consider the following short exact sequence:
$$0\rightarrow \mathcal{E}\rightarrow \mathcal{E}^{**...
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Analytical point of view of Kawamata's Unipotent reduction condition for Calabi-Yau family
Motivation: Unipotent reduction condition is very important for study of family of algebraic varieties. For example for algebraic fiber space if we have such condition then the direct image of ...
user21574
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Generalised Hodge Conjecture
Further to my question,
A Naive Question on Mixed Motives and Mixed Hodge Structures
that has received very good replies and suggestions, and I really appreciate it. I am going to ask a question on ...
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A Naive Question on Mixed Motives and Mixed Hodge Structures
As a physicist, I have some naive questions about mixed motives and its mixed Hodge structure (MHS) realization. Any references, comments, answers will be appreciated!
The category of mixed motives ...
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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 ...
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Arakelov Motivic Cohomology and Hodge Theory
Lately I have been studying these two papers (first and second) that introduce a new cohomology theory called Arakelov motivic cohomology. While most of the applications presented in the papers are ...
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Variations of Hodge structures of Calabi-Yau threefolds
It is a question about R. Bryant and P. Griffiths's paper Some observations on the Infinitesimal Period Relations for Regular Threefolds with Trivial Canonical Bundle. I am going to use the language ...
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If monodromy representation is unipotent then special fiber is snc?
We know, if $π : X → S$ is a generically smooth family of complex projective
varieties, such that $X_0 := π^{−1}(0)$ is an snc divisor in $X$, then the
monodromy representation is unipotent. Now ...
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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 $\...
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Examples of Maximal degeneration of Deligne on Calabi-Yau degeneration
Pierre Deligne in his celebrated paper entitling "Local behavior of Hodge structures at infinity" introduced Maximal degenerations of Calabi-Yau manifolds.
Let $\pi:X\to \mathbb C^*$ be a family of ...
user21574
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Arbitrarily non-degenerate Hodge to de Rham spectral sequence
It is true that for any $n$ there exists a compact complex manifold which Frolicher spectral sequence does not degenerate at the $n$-th page(https://arxiv.org/pdf/0709.0481.pdf).
Does the analogous ...
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Refinement of Hodge conjecture
This question deals with the classic Hodge conjecture on projective non-singular complex varieties, or in other words, projective Kähler manifolds. In Deligne's writeup for the Clay Foundation he says ...
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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 ...
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MU and the integral Hodge Conjecture
Totaro proved that the cycle class map for an algebraic variety factors through complex cobordism. In other words, we have a factorization
$$CH^i(X) \rightarrow MU^{2i}(X) \rightarrow H^{2i}(X; \...
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Are induced morphisms on cohomology strict with respect to the hodge filtration in the non Kähler case?
For a complex manifold $X$ there is the Hodge filtration on cohomology, induced by the filtration on the complex of holomorphic forms given by:
$$ F^r\Omega_X^p:=\begin{cases}\{0\}\qquad\text{if }r&...
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Reference for Hodge loci on moduli space of principally polarised abelian varieties
Can someone suggest a reference to study Hodge locus, period mappings and period domains on moduli space of principally polarised abelian varieties?
More precisely, consider the moduli space of ...
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Interesting implications on the theory of motives if the Hodge conjecture holds
For example,
Under the Hodge conjecture the Motivic galois group coincides with Mumford-Tate group.
The Hodge conjecture implies the Lefschetz and Kunneth standard conjectures, as well as ...
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Hodge to de Rham spectral sequence for stacks
For some work I'm doing, I need a version of the Hodge to de Rham spectral sequence for stacks. I am not at all an expert on stacks, so please excuse me if I make minor technical mistakes in stating ...
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Hodge standard conjecture for étale cohomology
It is known that Hodge standard conjecture is true for étale cohomology for a field $k$ of characteristic zero. It means that the following pairing
$$
(x,y)\mapsto (-1)^{i}\langle L^{r-2i}(x),y\...
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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.
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Reference - Generalized Hodge conjecture for triangulated motives
GHC for triangulated motives: The Hodge conjecture holds and an object $\rm M \in Dmg$ is effective if and only if its Hodge realization is effective.
I would like to know some references on GHC ...
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Alternative construction of the first Chern class map
Let $X$ be a compact Kähler manifold. Consider the exponential sequence $0 \to \mathbf Z \to \mathscr O_X \to \mathscr O_X^* \to 0$. The boundary map gives a map $H^1(X, \mathscr O_X^*) \to H^2(X, \...
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Hodge decomposition for Bott-Chern cohomology
$\DeclareMathOperator{im}{im}$
I want to prove that Bott-Chern cohomology group $H^{p,q}_{BC}=\frac{\ker\partial\cap \ker\bar{\partial}}{\im\partial\bar{\partial}}$ has finite dimension via Hodge ...
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harmonic differential form integer class
Let $(M,g)$ be a compact Riemannian three-fold such that $H_2(M,\mathbb{Z}) = \mathbb{Z}$ and $S$ any surface representing 1. By Hodge theory there exist a harmonic differential one-form $\eta$ dual ...
user50806
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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 ...
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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^{...
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Families of abelian varieties on the line (or more generally simply connected varieties)
I'm curious whether the following is true:
Question 1: Let $V/\mathbb{C}$ be a smooth connected variety such that $V^\text{an}$ is simply connected. Then, is every abelian scheme $f:\mathscr{A}\to ...
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votes
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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 ...
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Symplectic invariance of Hodge numbers?
Let $(X,\omega)$ be a compact symplectic manifold. If $J$ is a $\omega$-compatible complex structure on $X$ then $(X,\omega,J)$ is a compact Kähler manifold and so has Hodge numbers $h^{p,q}$.
My ...
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