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.
390 questions
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Understanding modular curves as "moduli of Hodge structures"
It is well-known that modular curves parametrize elliptic curves with level structures. For the purpose of this question, I will work complex-analytically and describe analytically the moduli space $$\...
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Composing Hodge star and contraction (interior product)
I came across the following construct
$$\star i_X \star i_X \alpha$$
where $\alpha$ is a $p$ form and $X$ a vector field. The first thing i noted is that the result will again be a $p$ form. Up to ...
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Symmetry-invariant (local) Hodge decomposition
Consider a differential 2-form field $F$ on a 4d oriented smooth spacetime manifold $M$ endowed with a Lorentzian metric $g$. We additionally have an infinitesimal group action $\Gamma$ on $M$ of a ...
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Exercise 2.1 of Birkenhake: Complex Abelian Varieties
I am working through Birkenhake and got stuck on an exercise:
My attempt was to do this with a diagram chase: a 2-cocycle $F\in H²(\Lambda, \mathbb{C})$ is just a map $F:\Lambda\times \Lambda \to \...
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(Co)Homology Groups of Finite Quotient Complex Manifolds & Compatibility with Hodge Theory
Let $X$ be a (complex) compact manifold and $G$ a finite group acting (holomorphically) freely on it such that one can canonically endow the quotient $Y:=X/G$ with (complex) manifold structure.
(Here ...
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Hodge star of a $(0,q)$-form wedged with the symplectic form on a symplectic manifold with a compatible almost complex structure
Let $(X,\omega)$ be a symplectic manifold of dimension $2n$ and $J$ be an almost complex structure compatible with $\omega$. In particular, $g(v,w):=\omega(v,Jw)$ defines a Riemannian metric, and ...
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How to show that Laplacian operator commute with Hodge star operator
I am trying to show that $\star\Delta=\Delta\star$. As we know that, $\Delta=\delta d+d\delta$ where $\delta$ is the co-differential operator. Then,
\begin{align}
\star\Delta&=\star(\delta d+d\...
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Why does the Green's operator of a self-adjoint elliptic differential operator extend to a bounded operator on all Sobolev spaces?
In the book "Differential analysis on complex manifolds" Well's asserts in Theorem 4.12 that for any self-adjoint elliptic differential operator $L:\Gamma(E)\to \Gamma(E)$ of degree $m$, ...
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Hodge structure on intersection cohomology
Im trying to understand the hodge structure on the intersection cohomology. In Saito's article "Mixed Hodge Modules" from 1990 he says:
"we
get a natural Hodge structure on $IH^*(X, L)$ ...
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Polarization on $H^1$ of abelian variety
Let $X$ be an $n$-dimensional abelian variety over the complex numbers. Choose a polarization for $X$, and let $\omega \in H^2(X, \mathbb{Z})$ be the corresponding class.
There seem to be two ways to ...
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When is the Hodge group of a simple abelian variety over $\mathbb C$ semisimple?
The Hodge group (or Special Mumford-Tate group) for a rational Hodge structure $(V,h)$, $h:\text{Res}_{\mathbb C/\mathbb R}\mathbb G_{m}\to GL(V)$ is defined as the minimal $\mathbb Q$ algebraic ...
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Question on definition of Hodge Star operator for Complex manifolds
I understand that for an Riemanian manifold (M, g), given an orthonormal basis $\{v_1, ... v_n\}$ of $T_p M$ we can get an metric on $T_p^*M$ by requiring that the dual basis $\{v^1, ..., v^n \}$ is ...
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Mixed Hodge numbers of hypersurfaces
Let $X\subseteq \mathbb{P}_{\mathbb{C}}^3$ be a projective hypersurface of degree $d$, i.e. the zero locus of some homogeneous polynomial of degree $d$. When $X$ is non-singular, the Hodge numbers of ...
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Non-abelian Hodge theory generalises the abelian one?
Question: Does non-abelian Hodge theory really generalise the usual Hodge theory?
More detail: (I'm learning these, so some details below might be slightly wrong) A version of non-abelian Hodge theory ...
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Griffiths and Harris Regularity Lemma II
I'm trying to make my way through Griffiths and Harris's proof of the Hodge Theorem in Principles of Algebraic Geometry. I've gotten through most of it just fine, but I've been stuck on the estimates ...
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