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.

Filter by
Sorted by
Tagged with
3 votes
0 answers
37 views

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, ...
22 votes
0 answers
566 views

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....
  • 1,379
11 votes
0 answers
198 views

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 $16$...
  • 6,508
2 votes
1 answer
336 views

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 ...
4 votes
1 answer
211 views

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{...
  • 5,333
1 vote
0 answers
129 views

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 \...
  • 5,333
2 votes
0 answers
84 views

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 \...
  • 137
2 votes
1 answer
179 views

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 $\...
2 votes
0 answers
108 views

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 ...
2 votes
1 answer
183 views

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 ...
  • 439
1 vote
1 answer
162 views

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, ...
  • 571
2 votes
1 answer
226 views

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 ...
  • 603
3 votes
0 answers
94 views

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^*...
  • 603
2 votes
0 answers
136 views

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 ...
  • 147
1 vote
0 answers
34 views

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 ...
  • 147
1 vote
0 answers
89 views

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,\...
  • 147
2 votes
0 answers
86 views

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 ...
2 votes
0 answers
63 views

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 $\...
12 votes
2 answers
793 views

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 ...
2 votes
0 answers
148 views

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\...
's user avatar
1 vote
0 answers
177 views

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^...
  • 1,573
18 votes
0 answers
1k views

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(...
1 vote
2 answers
254 views

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 ...
  • 199
3 votes
0 answers
164 views

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 ...
  • 5,643
4 votes
1 answer
356 views

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 ...
  • 439
1 vote
1 answer
160 views

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\},...
  • 883
2 votes
0 answers
134 views

How to calculate Gauss Manin connection?

If $f:X\rightarrow B$ is a holomorphic family of compact complex manifold. Fix a $k$, then all the $H^k(X_t,\mathbb{C})$ is the same with respect to $t$. Say take a $d$-closed form $\alpha\in H^k(X_t,...
3 votes
0 answers
153 views

The Lefschetz hyperplane theorem and when there is an even number of vanishing cycles

Let's suppose we have a smooth, complex algebraic surface $S \subset \mathbb{P}^N$ where $N$ is some large positive integer. Then, a generic Lefschetz pencil is a family of hyperplanes $H_t$ and we ...
  • 478
6 votes
0 answers
278 views

Is there a relation on Hodge numbers, weaker than $h^{2,0}=0$, that implies a compact Kähler manifold is projective?

The Kodaira embedding theorem yields as a corollary that a compact Kähler manifold $X$ with $h^{2,0} =0$ is projective. Is there a weaker relation on Hodge numbers that implies that a compact Kähler ...
  • 883
0 votes
0 answers
134 views

Chow countability argument

I would like to know what the "Chow countability argument or HIlbert schemes countability argument" says in order to finish an exercise. Any reference will also be very useful :)!
  • 439
2 votes
0 answers
141 views

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. ...
  • 1,685
1 vote
0 answers
50 views

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 ...
2 votes
1 answer
321 views

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 ...
  • 147
2 votes
0 answers
190 views

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 ...
  • 1,558
0 votes
0 answers
161 views

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 ...
  • 439
9 votes
0 answers
256 views

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 ...
  • 41.7k
0 votes
0 answers
88 views

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 ...
  • 439
8 votes
1 answer
248 views

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 ...
3 votes
0 answers
302 views

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^...
  • 5,333
2 votes
1 answer
298 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....
  • 443
3 votes
1 answer
261 views

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,...
2 votes
0 answers
153 views

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 ...
  • 21
14 votes
3 answers
580 views

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}(...
  • 19.6k
3 votes
1 answer
280 views

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 ...
  • 1,959
3 votes
2 answers
286 views

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 $...
  • 1,909
8 votes
0 answers
132 views

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 ...
6 votes
1 answer
127 views

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 ...
10 votes
2 answers
1k views

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, $A(\mathcal{E}) = 0$....
  • 1,842
4 votes
0 answers
174 views

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 \...
  • 1,959
7 votes
0 answers
149 views

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 ...
  • 2,631

1
2 3 4 5
9