Questions tagged [hodge-theory]

For question about Hodge theory, which is a method for studying the cohomology groups of a smooth manifold using partial differential equations.

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Toroidal/polodial decomposition of a solenoid vector field in cartesian coordinates

I am trying to understand some aspects of the toroidal/polodial decomposition of a solenoid vector field in cartesian coordinates. We have a divergence-free ('solenoid') vector field $u$ on a compact ...
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Characterization of harmonic $(1,1)$-forms

Let $(X,\Omega)$ be a compact Kähler manifold. Then there is the "usual" definition of the vector space $\mathcal H^{p,q}_{\bar\partial}$ which is the space of $\bar\partial$-harmonic $(p,q)$...
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Hodge * Operation on a non-coordinate basis

I was reading Nakahara's Geometry, Topology and Physics, where I saw a claim that for a non-coordinate basis $\{\theta^\alpha\}=\{e^{\alpha}_{\ \ \mu}dx^\mu\}$, the Hodge * operation becomes $$*(\...
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Connection between Abel Jacobi map and Period domains of curves.

Let $M_g$ be the space of all smooth curves of genus $g$. Then there is a period map $M_g\to D$ where $D$ is the “period domain”. As I understand it, in this case $D$ should be a subspace of the ...
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Voisin 3.1.1 Hermitian Geometry

I am a bit confused about what appears to be basic linear algebra in Voisin's book. I suspect I am misunderstanding maybe some of the notations. My first confusion is from this passage. $W_{\mathbb{C}...
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Almost closed forms

Suppose we have a closed (compact, without boundary) manifold $M$. Let's assume that it is orientable, although it might play no role in the question. Now, De Rham cohomology measures how far a closed ...
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Combinatorial Laplacian for homology with $Z_2$ coefficients

Consider I have boundary operators $\partial_1$, $\partial_2$: $\partial_1 \circ \partial_2 = 0$. Then if interested in $\text{ker}\,\partial_1 / \text{im}\,\partial_2$ one can study $\text{ker}\,(\...
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Why is the image of $H_1(X,\mathbb{Z})\rightarrow (\Omega^1)^*$ a lattice in $(\Omega^1)^*$? (for Albanese varieties)

Albanese varieties are described here, and provide motivation for this question, although are not central to it. Let $X$ be an algebraic variety and $\Omega^1$ the space of everywhere regular ...
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Definition of a period using cohomology.

In algebraic geometry, a period is an integral of an algebraic function over an algebraic domain (for instance, $2i\pi=\int_{\partial D(0,1)}\frac{dz}{z}$). It is said in my notes that such a period ...
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Interpretation of the Higher Order Laplace-Hodge Operator

As an operator on functions, one intuitive way to think about the Laplacian is as an operator that returns the average difference between a function's value at a point and the values of its ...
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Query about Laplace Eqution and Harmonic functions

In the text below, I am unable to understand how Eq 2.5 represents the Laplace Equation because I don't see any partial derivatives etc. Does Eq 2.5 somehow reduce to $\nabla^2 u=0$? The snippet is ...
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Why consider $dx/x$ on a complex curve?

In a paper I'm reading, the author considers a compact Riemann surface -- or smooth algebraic curve, you pick -- $X$ given by the equation $y^d=x^n-1$ for some natural numbers $n,d$. My understanding ...
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Is monodromy group perfect?

Given a smooth proper morphism $f:X\to S$ between (integral) smooth algebraic varieties over $\mathbb{C}$ and we consider the associated monodromy representation $$\rho:\pi_1(S^{an},s_0)\to \mathrm{GL}...
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Equivalent condition for 3-form being primitive in 6 dimensions

Let $(V, \omega)$ be a symplectic vector space of dimension 6, with a compatible metric $g$. Let $\varphi$ be a 3-form on $V$. Then $\phi$ is primitive, meaning $\star \varphi \wedge \omega = 0$, if ...
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Characterizing strict morphisms in the category of bifiltered vector spaces

Let $k$ be a field, and let $C$ be the category whose objects are finite dimensional $k$-vector spaces endowed with two finite filtrations $W$ and $F$, the former being ascending and the latter ...
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What does the $0$ in "Hodge number is the dimension of $H^{0}(V,\Omega^{n})$" mean please?

Here is the page I was reading.. I tried reading more into what Hodge numbers are and it was too complex for the short amount of time I had but I figured the $0$ probably indicated something like a &...
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A coordinate-free criterion for ellipticity of a linear differential operator

In Chapter 6 of Warner's Foundations of Differentiable Manifolds and Lie Groups, he develops a self-contained theory of local elliptic operators to establish the Hodge theorem. I got a bit stuck on a ...
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If $M$ is a compact Riemann surface, then $H^1(M,\mathbb R)\cong H^1(M,\mathcal O)$

I am wondering why we have the isomorphism stated in the title. Concretely, we have the following exact sequences of sheaves: $$0\rightarrow\mathbb R\rightarrow\mathcal O\rightarrow\mathcal O/\mathbb ...
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Why are 1,0 differential forms $\mathbb C$-linear?

I am struggling to understand a passage from Claire Voisin’s book on Hodge Theory. In page 53, at the beginning of the section 2.3.1, there is the following assertion: “…the bundle $\Omega_X^{1,0}$ ...
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Why is the Hodge Decomposition profound?

The Hodge Decomposition is the following result: Let $X$ be a smooth projective variety over $\mathbf{C}$. For every integer $k \geq 0$, we have a direct sum decomposition $$H^k(X, \mathbf{C}) \cong \...
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The definition of special cubic fourfolds

According to the basic paper "Special Cubic Fourfolds" (https://www.math.brown.edu/bhassett/papers/cubics/cubiclong.pdf, [BH98]) by Brendan Hasset, a special cubic fourfold is defined as ...
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Sign in definition of polarization of Hodge structure

Let $H_{\mathbb{Z}}$ be an integral Hodge structure of weight $n$ with Hodge decomposition $H_{\mathbb{C}} = \sum_{p+q=n}H^{p, q}$. In the definition of a polarized Hodge structure, I have come across ...
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Proof of discrete Hodge decomposition

In this survey by Lubotzky, he has the following: Proposition 2.1 (Hodge decomposition): The following are true: $C^i=B^i\oplus\mathcal{H}^i\oplus\mathcal{B}_i$, $\mathcal{H}^i\cong H^i(X;\mathbb{R})...
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Hodge decomposition for complex manifold

Let $X$ be a compact oriented Riemannian manifold. By Hodge decomposition, we can decompose $$\Omega^k(X)=\mathrm{im}(d)\oplus\mathrm{im}(d^*)\oplus\ker(\Delta).$$ Now, if further $X$ has a complex ...
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A real vector space with an almost complex structure vs its complexification

Suppose $(V, J)$ is a real vector space of dimension $2n$ equipped with an almost complex structure $J$ such that $J^2 = -Id$. $V$ can be realized as a complex vector space by setting $iv = J(v)$ for ...
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Bound for ratio of multivariable polynomials

Let $\xi = (\xi_1, \ldots ,\xi_n)$ is multi-index $(\xi_i \in \mathbb{Z})$. If we have multi-indexes $\xi, \alpha$ then $\xi^{\alpha} = \xi_1^{\alpha_1} \xi_2^{\alpha_2}\cdots \xi_n^{\alpha_n}$ ($0^0$ ...
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Connection between polarization of K3 surface, as a morphism of Hodge structures, and polarization of K3 surface as an ample line bundle

I'm currently reading through Daniel Huybrecht's Lectures on K3 Surfaces and have come across (what seems at first) two different interpretations of a polarization. At the end of chapter 2 (§2.4 ...
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Hodge star operator on complex manifold

The Hodge star is defined as \begin{equation} \alpha\wedge\overline{\star\beta} = \langle\alpha,\beta\rangle(\star1),\quad \alpha,\beta\in\Omega^{p,q}(M). \end{equation} Here the inner product $\...
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Hodge star operator commutes with the harmonic projection

the basic setup is a dimension 4 compact oriented Riemannian manifold, consider we have a closed 2-form $a$ on it with the cohomology class $[a]$. Suppose now we have a restriction on the self-dual ...
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Nowhere vanishing harmonic 1-forms on 3-manifolds

Consider $(S^1 \times \Sigma^2, g)$, where $g$ is any Riemannian metric on the compact and closed $3$-manifold $S^1 \times \Sigma^2$. Question: Does there always exist a nowhere vanishing harmonic $1$...
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Analytification of a smooth projective variety is a compact Kähler manifold.

I am reading “Fourier-Mukai transforms in algebraic geometry” by Daniel Huybrecht. On page 130 it is written that by Hodge theory there is a natural direct sum decomposition $$H^n(X,\mathbb{C})=\...
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fundamental elliptic complex in Hodge theory

I was trying to understand the following fundamental elliptic complex in Hodge theory (Here $M$ is a compact oriented four dimensional Riemannian manifold): $$0\rightarrow \Omega^0(M) \stackrel{d}{\...
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Signature is independent of the Riemannian metric chosen

The signature of a 4-manifold is defined to be the dimension of harmonic self-dual two forms minus the dimension of harmonic anti-self-dual two forms, where the self or anti-self-dual is defined in ...
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estimation use Hodge theory

I'm confused by how we get the following estimation from Hodge theory It seems to me that it is using the fundamental inequality of elliptic operators, but I could not see what this operator is.
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Why Hodge decomposition reflect the analytic structure

In the book Principles of Algebraic Geometry by Griffiths and Harris, the authors state that the Hodge decomposition reflects the analytic structure but that the Lefschetz decomposition is essentially ...
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Calabi Yau threefolds

Let be $X$ a Calabi-Yau manifold (i.e. Kahler, compact with trivial canonical bundle) with $\dim(X) = 3$. Let $\phi : A \mapsto B$ a proper holomoprhic submersion such that $X_{t_{0}} := \phi^{-1}(t_{...
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Pullback of $S^n$ volume form

I am working on problems from Nakahara's book Geometry, Topology and Physics and I am struggling with exercise 9.2 Let $\Omega_n$ be the volume form of $S^n$ normalized to 1, $f:S^{2n-1}\to S^n$ be a ...
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Hodge star without a metric

The standard definition of the Hodge star is as follows: the Hodge dual of a differential $p$-form defined on an $n$ dimensional manifold $M$, $\alpha \in \Omega^p(M)$, is the unique form $\star \...
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Hodge star duality and the metric

Let $X$ be a smooth compact Riemannian manifold of even dimension $2n$. Using the Hodge star $*: \Omega^r(X) \to \Omega^{2n-r}(X)$ one can define self-dual and anti-self-dual $n$-forms on $X$, $$ \...
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Hodge decomposition.

Let $ X $ be a smooth complex projective algebraic variety of complex dimension $ n $. Then, there exists the following Hodge decomposition: $$ H^{2k} (X, \mathbb{C}) = \displaystyle \bigoplus_{p + q =...
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Counter-example for complex version of Hodge conjecture.

Hodge conjecture claims that every rational Hodge class is in the $\mathbb{Q}$-span of the image of the algebraic cycles in the cohomology. I believe the complexifed version of the conjecture should ...
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Showing $\Delta(\Omega^p M)=d(\Omega^{p-1}M)\oplus \delta(\Omega^{p+1}M)$ [duplicate]

Let $M$ be a compact Riemannian manifold, $*$ be the Hodge star operation, $d$ the exterior derivative, $\delta$ the codifferential $(\delta\omega=(-1)^{n(k+1)+1} *d*\omega$ where $n=\dim M$ and $k=\...
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$\partial \bar \partial$-lemma for $d$-exact differential form.

This is the $\partial \bar \partial$-lemma as stated in [1]: Proposition 6.17 Let $X$ be a Kähler manifold, and let $\omega$ be a form which is both $\partial$ and $\bar \partial$-closed. Then if $\...
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1 vote
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Determinant and trace of Hodge Star operators

The Hodge star operator in differential geometry is a map from $*: \Omega^{k}(\mathcal{M}) \to \Omega^{m-k}(\mathcal{M})$ where $\Omega^{l}(\mathcal{M})$ is the space of $l$-forms on $m$-dimensional ...
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Computing Hodge Laplacian

I have a confusion regarding computation of the Hodge Laplacian on an $m$-dimensional manifold. Conisder $$\Delta f = d^{\dagger}d f = -*d*(\partial_{\mu}f ~dx^{\mu})\\=-*d\left(\frac{\sqrt{g}}{(m-1)!}...
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Deformations of complex manifolds: Why is $T_t^{0,1} = \{v - \alpha_t(v) \,|\, v \in T^{0,1}\}$?

Let $X$ be a compact complex manifold, and let $\phi: \mathcal X \to B$ be a deformation of $X$, i.e. $\phi$ is a proper submersive morphism of complex manifolds, with central fiber $X_0 = X$. By the ...
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Question on specific properties of inner product on complex manifolds

I am having trouble understanding part of a proof within Principles of Algebraic Geometry; GRIFFITHS / HARRIS. The proof I am struggling with is located at page 112 under the subitem The Hodge ...
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Bloch's construction of the de Rham class of subvarieties

Let $X$ be a smooth projective variety (over $\mathbb C$ to fix ideas) and $Z\subset X$ a locally complete intersection of codimension $k$. We fix the notation $U:=X-Z$ for the complement and $j: U\to ...
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Equivalent definitions of a Hodge structure

Generally, a Hodge structure of weight $k$ on a finitely generated abelian group $H$ is defined as a decomposition of the complexification: $$ H\otimes \mathbb C = \bigoplus_{p+q=k} H^{p,q}, $$ where ...
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1 vote
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Condition $\overline{V^{p,q}}= V^{q,p}$ in the definition of a Hodge structure

Suppose that $V$ is a finitely generated $\mathbb Z$-module. A Hodge structure of wight $k$ on $V$ is a decomposition of the complexification of $V$ into complex vector spaces $V^{p,q}$ such that $\...
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