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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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 ...
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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 ...
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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)....
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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 $Y$ ...
user505967
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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 ...
user508433
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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 ...
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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 ...
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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'...
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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 ...
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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 ...
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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" ...
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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. (...
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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 ...
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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
$$\Omega^{k}(\mathcal{M})\cong\mathcal{H}^{k}(\mathcal{M})\oplus\mathrm{ran}(\...
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Cohomology classes fixed by algebraic automorphism subgroups
Let $X$ be a smooth projective complex variety and $t\in H^{2p}(X,\mathbf{Q})(p)$ a rational cohomology class.
Assume that there exist
$$t_1,\ldots,t_N\in H^{2*}(X,\mathbf{Q})(*)$$
algebraic classes (...
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Period calculation for elliptic curve
In the paper "Hodge cycles on Abelian Varieties" (Proposition 1.5), Deligne proves the following theorem:
Let $X$ be a smooth projective variety over $\overline{\mathbf{Q}}$ of dimension $n$...
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Hodge decomposition for non-elliptic complexes
It is a well-known result that there is a bijective correspondence between harmonic sections and cohomology classes of an elliptic complex in a Riemannian/Hermitian manifold. Now consider Riemannian/...
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Finding a Hodge theoretic condition to measure the rank of isogeny of product abelian surfaces
Let $A$ be an abelian surface over $\mathbb{C}$, then there is a condition on $H^0\left(\Omega^1_A\right)$
to determine if $A$ contains an elliptic curve $E$ as a subvariety. If $A$ were to contain
...
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Rank $2$ motivic local systems on a curve
This question is about the article "Motivic local systems on curves and Maeda's conjecture" by Yeuk Hay Joshua Lam.
In the proof of Theorem 1.1 it is claimed (on lines 4-5 of p. 7) that any ...
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Minimal Betti numbers of simply-connected threefolds with trivial canonical class
By a threefold, I mean a compact complex manifold of dimension three.
For a simply-connected threefold with trivial canonical class, its Betti numbers satisfy:
$$b_2 \ge 0, b_3 \ge 2.$$
I am wondering ...
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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 ...
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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$-...
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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{...
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Hodge theory of the AJS category (proof of Lusztig's conjectures in positive characteristic)
Recently, in this survey paper (https://arxiv.org/abs/1212.0791) Elias-Williamson describe a Hodge theoretic approach to the proof of Kazhdan-Lusztig conjectures; it is essentially equivalent to the ...
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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, ...
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Is there a proof of Hodge theory using condensed mathematics?
As is well known, many results in complex geometry "feel" algebraic (and often have statements which are "completely algebraic") but only have "transcendental" proofs (i....
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Spin 6-fold with signature $\pm 16$
Does there exist a spin (i.e. $\frac{c_{1}}{2} \in H^{2}(M,\mathbb{Z})$) smooth complex projective $6$-fold with signature $\pm 16$?
The motivation is the Rochlin-Ochanine theorem, which says that $16$...
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Example motivating mixed Hodge structures
The suggested intuition behind mixed Hodge structures - developed
in particular to generalize Hodge decomposition of cohomology
groups from complex smooth complete varieties to more general algebraic ...
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Hodge conjecture for generic points
I was reading the following paper: "Beilinson’s Hodge Conjecture For Smooth Varieties". They study the cycle class map $cl_{m,r}: H^{2r-m}_{\mathcal{M}}(U, \mathbb{Q}(r))\rightarrow \text{...
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Beilinson-Hodge conjecture and generation of cohomology ring by $H^1$
Beilinson's version of Hodge conjecture has the following form. For any quasi-projective smooth complex variety $X$ the following map is surjective:
$$H^i_{\mathcal{M}}(X, \mathbb{Q}(j))\rightarrow \...
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The dual of the Lefschetz operator under a perturbation
Let $(X, \omega)$ be a compact Kähler, or more generally, Hermitian manifold. Let $L_{\omega} : \Omega^k(X) \to \Omega^{k+2}(X)$ denote the Lefschetz operator given by $$L_{\omega}(\alpha) : = \omega \...
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What does does the monodromy weight filtration represent?
I'm trying to understand variations of Hodge structure. I understand that this is a very broad field, and that many of the concepts have been extended to algebraic geometry over fields other than $\...
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Modular forms and Petersson inner product via De Rham cohomology, Hodge filtration and cup products
I'm looking for an explanation on how and why you can define modular forms through De Rham cohomology via the Hodge filtration and especially how the Petersson inner product is related to the cup ...
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Algebraic and homological equivalence relations for $0$-cycles
Let $X$ be a connected smooth projective variety. Let $Z_0(X)_{alg}$ be the group of $0$-cycles algebraically equivalent to $0$ and $Z_0(X)_{\hom}$ be the group of $0$-cycles homologically equivalent ...
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Hodge theory beyond Riemannian and Kahler manifolds
Recently I read about graph-theoretic Hodge theory, which has uses in graph theory, topological data analysis, and more generally machine learning. I knew only the basics of Riemannian Hodge theory, ...
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Regularity of Gauss Manin connection
I want to understand the "Regularity of Gauss Manin connection" from the most basic example.
Suppose we have a family of projective manifold $X\rightarrow \mathbb C^*$ with full rank, then ...
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variation of Hodge structure and singularity of period map
Let $X\rightarrow D^*$ be fiberation of projective manifold, here $D^*$ means a punctured disk. Then it induces a variation of hodge structure, i.e a holomorphic vector bundle $H_{\mathbb C}$ over $D^*...
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Any manifold in Fujiki class $\mathcal C$ admits a Kähler deformation?
It is known that small deformations of a manifold $X$ in Fujiki class $\mathcal C$ might no longer be in $\mathcal C$, see This paper for example.
However, the following question is still open:
For ...
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Is the union of Fujiki cones open in $\mathcal H^{1,1}_{\mathbb R}$?
Let $\mathcal X\to B$ be a holomorphic family of compact Kähler manifolds, let $\mathcal K_t$ denote the Kähler cone of the fiber $X_t$, then the union $\cup_{t\in B}\mathcal K_t$ forms an open set in ...
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Interpretation of $\mathcal H^k$
For a holomorphic family $\pi:\mathcal X\to B$ between complex manifolds, the map $\pi$ is a proper holomorphic submersion, $X_t:=\pi^{-1}(t)$, $t\in B$, $X=X_0$, we have the isomorphism $H^k(X,\...
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Nilpotent orbits and mixed Hodge structures
Let $(H_\mathbb{Z}, \{h^{p,q}\}_{p+q=n}, \phi)$ be the datum to define weight $n$ polarized Hodge structures on $H_\mathbb{Z}$, $h^{p,q}$ is the Hodge numbers, $\phi$ is a polarization. Let $D$ be the ...
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Gysin homomorphism of an inclusion to Kähler tubular neighborhood
Let $Z\subset U$ be a Kähler tubular neighborhood of a compact manifold $Z$ of codimension $r$.
Consider de Rham complexes of smooth differential forms $\Lambda^{*,*}(Z),\Lambda^{*,*}(U)$, let $\...
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Simpson's motivicity conjecture
My question is about Simpson's motivicity conjecture, that is the conjecture that for any (cohomogically) rigid irreducible connection $(M,\nabla)$ on a smooth complex scheme $X$ is of geometric ...
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Path spaces vs arc spaces
Let $X$ be a smooth projective variety over $\mathbf{C}$ and denote by $\mathcal{L}_m(X)$ the $m$-th jet space, a smooth $\mathbf{C}$-scheme representing the functor on $\mathbf{C}$-algebras
$$A\...
user178246
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Exterior power of Hodge structures
Let $V$ be a $\mathbb{Q}$-vector space and suppose there is a decomposition of $V_{\mathbb{C}}:=V \otimes_{\mathbb{Q}} \mathbb{C}$ into two $\mathbb{C}$-sub-vector spaces i.e., $V_{\mathbb{C}} \cong V^...
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Reference request: deforming a G-local system to a variation of Hodge structure
Let $X$ be a smooth connected quasiprojective variety over $\mathbb{C}$ and let $G$ be a complex reductive group. Let $$\iota: G\to GL_N$$ be a representation and let $$\rho: \pi_1(X(\mathbb{C}))\to G(...
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The Hodge number $h^{2,0}$ of (finite) quotient variety of a K3 surface
Let $X$ be an (algebraic) K3 surface, then we have $H^{2,0}(X)=\langle \omega_X\rangle$, where $\omega_X$ is the period. Suppose $G=\langle g\rangle$ is a finite group acting on $X$ and $g$ as an ...
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How does intersection form on vanishing cohomology determine hodge type?
In the paper "Complete intersections with middle picard number 1 defined over Q" by Tomohide Terasoma (1985), page 295, line 7 from the bottom, we are in the following situation:
We have a ...
user39380
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Relation between the cohomology group of a curve and the cohomology group of its jacobian
Let $J_C$ be the Jacobian of a smooth projective curve $C$ over $\mathbb{C}$. I would like understand the isomorphism between $H^1(J_C,\mathbb{C})$ and $H^1(C,\mathbb{C})$. I read in a paper that ...
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Reference for the Hodge diamond of the Iwasawa threefold
Let $X = G/\Gamma$ denote the Iwasawa threefold, where
$$G = \left\{\begin{pmatrix} 1 & z_1 & z_3\\ 0 & 1 & z_2\\ 0 & 0 & 1\end{pmatrix} : z_1, z_2, z_3 \in \mathbb{C} \right\},...
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