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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Integral lattice in noncommutative Hodge theory

Associated to a $DG_{\mathbb{C}}$-category, $\mathcal{C}$, we have some Hodge theoretic data - $HH_{*}(\mathcal{C})$ plays the role of Hodge cohomology and $HP$ plays the role of de Rham cohomology. ...
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Optimality condition of the harmonic form representatives of a homology class

In "Hodge theory on metric spaces, Smale et al." the $d$-th harmonic forms of the Hodge Laplacian $\Delta_d=\delta^* \delta+\delta \delta^*$ satisfying $\Delta_d(f)=0$ are claimed to be ...
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Period map for $\partial\bar\partial$-manifolds

When we talk about the theory of variation of Hodge structures, we always assume that the central fiber is a Kähler manifold $X$, then consider a family of deformations $\pi:\mathcal X\to B$ and the ...
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Fibers of period map

Consider the period map for some smooth varieties, $p:\mathcal{X}\rightarrow\mathcal{A}:X\mapsto J(X)$, where $\mathcal{X}$ is moduli space of certain varieties and $\mathcal{A}$ is moduli space of ...
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Definition of rational equivalence

In the book "C. Voisin. Hodge Theory and Complex Algebraic Geometry. Volume II. Cambridge studies in advanced mathematics 76 (2002)" page 247, definition 9.4, says: Definition 9.4. The ...
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Understanding the mixed Hodge structure on D-modules on $\mathbb{C}$

Consider the category of regular D-modules on $\mathbb{C}$ (let's say as a complex manifold). It's a well-known fact that if you consider the subcategory of these where the singular support is the ...
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Closed algebraic subset dominating a curve

In the book "C. Voisin. Hodge Theory and Complex Algebraic Geometry. Volume II. Cambridge studies in advanced mathematics 76 (2002)" page 228 says: Let $X$ be a smooth projective variety. If ...
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reference to a theorem about a product of harmonic and parallel forms

Let $\alpha$ be an exterior product of a harmonic and a parallel form on a Riemannian manifold. Then $\alpha$ is known to be harmonic. I have heard that this is an old result due to R. Bott, but I ...
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Relations between the morphic cohomology and Hodge theory

The main question can be summarized in the following form: For a smooth projective complex variety $X$, is the cohomology $H^{2p}(X, \tau^{\leq p}\Omega_{alg}^{\bullet})$ supposed to surject onto $(H^...
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215 views

Do we have Hodge symmetry for char $p$?

Let $X$ be a smooth projective variety over a field $k$. Let $h^{p,q}=dim_k H^q(X,\Omega_{X/k}^p)$ be the Hodge numbers. If $k$ is of char $0$, by Lefschetz principle, we always have Hodge symmetry, i....
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Non Abelian Hodge theory: underlying structure holomorphic vector bundles

Let $X$ be a compact Riemann surface. We fix a complex vector bundle $E$ of rank $n$ and degree $d$ (unique up to diffeomorphism). From results coming originally (I think at least) by Simpson,...
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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 ...
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Log-concavity of matroids: characterization of equality?

Let $M$ be a (loopless) matroid of rank $r$. The characteristic polynomial $\chi_M(x)$ is defined by $\chi_M(x)=\sum_{F \in \mathcal{L}(M)}\mu(\hat{0},F) \cdot x^{\mathrm{rk}(F)}$, where $ \mathcal{L}(...
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Cycle class map for singular varieties

I am reading the cycle class map for singular projective varieties as mentioned by Laterveer in this article (see Definition $1$). The article does not define the map but refers to an article of ...
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Generalization of the Leray-Hirsch theorem

We know the classical Leray-Hirsch theorem for fibrations. My question is, whether a similar statement also holds for flat, proper morphism? In particular, consider a faithfully flat, proper morphism $...
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Is there a classification of non-simple Jacobians?

An abelian variety in the interior of the Torelli locus is non-decomposable, but it could possibly be non-simple (i.e. isogenous to a product of abelian varieties with lower dimension). For certain ...
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Tangential harmonic $1$-forms are pullbacks of harmonic functions

This question has also been posted on MSE, but maybe here is the right place to obtain an answer. Let $(M^3,g)$ be a compact connected oriented Riemannian $3$-manifold with nonempty boundary. The ...
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When do flat holomorphic connections exist?

Let $X$ be a smooth projective variety over $\mathbb{C}$. I know that a vector bundle $\mathcal{E}$ on $X$ admits a holomorphic/algebraic connection iff its Atiyah class vanishes, $A(\mathcal{E}) = 0$....
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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 \...
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Does the Hodge decomposition hold for equivariant differential forms?

Let $M$ be a Riemannian manifold. The Hodge decomposition tells that $$ \Omega^*(M) = \mathrm{im} \ d \oplus \mathrm{im} \ d^* \oplus \mathscr H^*(M) $$ where $d^*$ is the adjoint operator of the ...
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Is the abelian category of pure Hodge modules semi-simple?

I am a beginner in the subject, and at the moment I am trying to understand basic properties of the main objects of the M. Saito's theory of the mixed Hodge modules in general.The question in the ...
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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.
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Is the Hodge bundle a holomorphic vector bundle?

I have just started reading through the paper of Cattani--Kaplan--Schmid -- Degeneration of Hodge structures (Annals of Mathematics, 123 (1986), 457--535). For the purposes here, take $f : X \to S$ to ...
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Hodge theoretic properties of intersection cohomology

Let $X$ be a complex projective irreducible reduced variety. It is well known that the intersection cohomology of $X$ satisfies versions of Poincare duality and hard Lefschetz theorem. Does it admit a ...
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What makes a Kähler manifold projective?

Let $X$ be a compact Kähler manifold, I know there are (at least?) 2 ways to make $X$ a projective manifold. (integral condition) If the Kähler class $[\omega]$ is integral, i.e., $[\omega]\in H^2(X,\...
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A contradiction caused by the Kähler identity and the formal adjoint relation

I found a contradiction in the Principle of Algebraic Geometry by G&H, section 1.2. I have post this on MSE but it didn't get enough attention. I couldn't sleep or eat or do anything else due to ...
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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 ...
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Is every $\omega\in H^0(U,\Omega_U^n)$ representable in $H^n(U,\mathbb{C})$ by an element from $H^0(X,\Omega_X^n(\log D))$?

Let $U$ be a smooth variety, and $U\hookrightarrow X$ an smooth compactification with snc boundary $D=X\setminus U$. Suppose that $\omega\in H^0(U,\Omega^n_U)$ is global algebraic $n$-form on $U$. It ...
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Computing mixed hodge structure using different extension of the constant sheaf

Let $U$ be a non-compact smooth algebraic variety, and let $U\hookrightarrow X$ be a smooth compactification such that $D=X\setminus U$ is SNC. The connection $(\mathcal{O}_X,d)$ is the canonical ...
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How does the MHS on $H_Y^i(X)$ behave with respect to the Thom isomorphism?

Let $X$ be a smooth variety and let $Y\subset X$ be a closed smooth subvariety. We have the local cohomology $H^i_Y(X)$, which becomes a MHS via the mixed cone construction, as explained in Section 5....
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Degeneration twisted Hodge to de Rham spectral sequence

Let $X$ be a proper and smooth scheme over $\mathbf{C}$ and let $\mathbb{L}$ be a local system of finite dimensional $\mathbf{C}$-vector spaces. By the Riemann Hilbert correspondence, to $\mathbb{L}$ ...
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Is there a way to describe the image of the $n$-fold residue map from $H^0(Y,\Omega_Y^n(\log E))$?

Let $X$ be an $n$-dimensional smooth algebraic variety, and let $Y$ be a compactification with $E=Y\setminus X$ simple normal crossings. There is the natural quotient map $$\Omega_Y^n(\log E)\to \...
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Does the Jacobian ring of a weighted projective hypersurface determine it up to isomorphism?

Let $V = H^0(\mathbb{P}^{n+1}, \mathcal{O}(1))$. Then the Mather-Yau Theorem states (Proposition 1.1 in Generic Torelli for projective hypersurfaces, Donagi) Theorem. If $f,g \in S^dV$ have the same ...
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Is there a proper smooth variety in characteristic $p$ whose Hodge-to-de Rham spectral sequence does not degenerate at $E_1$?

By Deligne-Illusie, such a variety has no lifting to $W_2(k)$. In their paper they state that they do not know if such an example exists. Has this question been answered since then?
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Quasi-unipotent monodromy for variation of Landau-Ginzburg cohomology

A pair, $(X,f)$, consisting of a smooth variety and a global function $f:X\rightarrow\mathbb{A}^{1}$ is called a Landau-Ginzburg model, LG-model for short. The LG-cohomology of the pair, dentoed $H(X,...
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Using principal polarisation to "cancel" Jacobian summands in isomorphism

I'm working through the sketch proof of irrationality of cubic threefolds in Huybrechts' The geometry of cubic hypersurfaces. Let $J(X)$ denote the intermediate Jacobian of a cubic threefold $X \...
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cycle class in the cohomology of the nearby fibre

Let $f: X \to \Delta \subset \mathbb{C}$ be a projective morphism of a complex manifold to a small disc, smooth away from 0, and such that $Y=f^{-1}(0)=\sum E_i$ is a strictly normal crossing divisor, ...
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rational Hodge structure of spectral curve and Prym variety

I have a problem about rational Hodge structure of spectral curves and Prym varieties. I want to prove the following, Let $X$ be smooth projective curve over $\mathbb{C}$, $\mathscr{M}$ be moduli ...
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Is there a direct way to show Fano surface of lines and conics on the pairs of Fano threefolds isomorphic?

I am considering the following setting: Let $(Y_d, X_{4d+2})$ be the pair of degree $d$ and index 2 Fano threefold $Y_d$ and degree $4d+2$ index 1 Fano threefold and both of them are Picard number 1. ...
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A Green's function for the Laplacian on k-forms

Let $X$ be a compact, oriented, Riemannian $n$-fold. Then we have a Laplacian operator $\Delta = d d^{\ast} + d^{\ast} d$ from $\Omega^k(X)$ to itself. We have the Hodge decomposition $\Omega^k(X) = \...
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Hodge structure and rational coefficients

Suppose $X$ is a complex projective variety with a model $X_\mathbb{Q}$ defined over the rational numbers. Then there is a rational de Rham lattice $H^k_{dR}(X_\mathbb{Q}, \mathbb{Q})\subset H^k(X, \...
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Can every Hodge structure be polarized?

I suspect this is very elementary, but it is not stated anywhere. A Hodge structure of weight $k$ consists of a finite rank lattice $H_{\mathbb{Z}}$ together with a decomposition of its ...
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multiplicative structure on the monodromy weight filtration spectral sequence

Let $f: X \to \Delta \subset \mathbb{C}$ be a projective morphism of a complex manifold to a small disc, smooth away from 0, and such that $Y=f^{-1}(0)$ is a strictly normal crossing divisor, and let $...
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What is the meaning of the monodromy theorem in Hodge theory?

Let $f : X^m \to Y^n$ be an algebraic fiber space (between projective manifolds) whose discriminant locus is denoted by $E$. Let $U$ be a polydisk in $\mathbb{C}^n$ (with coordinates $(y_1, ..., y_n)$)...
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Period map on non-Kähler manifold

Is there a theory of period map on non-Kähler manifolds that has Hodge decomposition? Any reference is helpful. Thank you.
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Do Poincaré residue and integrable log connection commute?

Here are some basic notations and definitions: (ignore this part if familiar) 1.Let $(X,\omega)$ be a compact Kähler manifold of dimension $n$, and let $D=\sum_{i=1}^r D_i$ be a simple normal crossing ...
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Period domain closure and mixed Hodge structures

The moduli space of Hodge structures is the period domain $$D\ = \ \coprod_{V,\psi} \text{Hom}_{\mathbf{R}\text{ alg.gp.}}(\mathbf{C}^*,G_\mathbf{R})/G_\mathbf{Z}$$ where $G\subseteq \text{GL}(V)$ are ...
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Can logarithmic connection operate on currents?

Let $(X,\omega)$ be a compact Kähler manifold of dimension $n$, and let $D=\sum_{i=1}^r D_i$ be a simple normal crossing divisor on it, i.e., a divisor with smooth components $D_i$ intersecting each ...
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Can logarithmic connection on holomorphic vector bundle induce logarithmic connection on dual bundle?

Let $(X,\omega)$ be a compact K"ahler manifold of dimension $n$, and let $D=\sum_{i=1}^r D_i$ be a simple normal crossing divisor on it, i.e., a divisor with smooth components $D_i$ intersecting ...
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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 ...

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