Hodge-de Rham Research Articles

Equivariant Hodge theory and noncommutative geometry

We develop a version of Hodge theory for a large class of smooth formally proper quotient stacks $X/G$ analogous to Hodge theory for smooth projective schemes. We show that the noncommutative Hodge-de Rham sequence for the category of equivariant coherent sheaves degenerates. This spectral sequence converges to the periodic cyclic homology, which we canonically identify with the topological equivariant K-theory of $X$ with respect to a maximal compact subgroup of $G$, equipping the latter with a canonical pure Hodge structure. We also establish Hodge-de Rham degeneration for categories of matrix factorizations for a large class of equivariant Landau-Ginzburg models.

Nash Embedding, Shape Operator and Navier-Stokes Equation on a Riemannian Manifold

What is the suitable Laplace operator on vector fields for the Navier-Stokes equation on a Riemannian manifold? In this note, by considering Nash embedding, we will try to elucidate different aspects of different Laplace operators such as de Rham-Hodge Laplacian as well as Ebin-Marsden's Laplacian. A probabilistic representation formula for Navier-Stokes equations on a general compact Riemannian manifold is obtained when de Rham-Hodge Laplacian is involved. MSC 2010: 35Q30, 58J65

Results Related to the Chern–Yamabe Flow

Let $$(X,\omega _0)$$ be a compact complex manifold of complex dimension n endowed with a Hermitian metric $$\omega _0$$ . The Chern–Yamabe problem is to find a conformal metric of $$\omega _0$$ such that its Chern scalar curvature is constant. As a generalization of the Chern–Yamabe problem, we study the problem of prescribing Chern scalar curvature. We then estimate the first nonzero eigenvalue of Hodge–de Rham Laplacian of $$(X,\omega _0)$$ . On the other hand, we prove a version of conformal Schwarz lemma on $$(X,\omega _0)$$ . All these are achieved by using geometric flows related to the Chern–Yamabe flow. Finally, we prove the backwards uniqueness of the Chern–Yamabe flow.

On the blow-up formula of twisted de Rham cohomology

We derive a blow-up formula for the de Rham cohomology of a local system of complex vector spaces on a compact complex manifold. As an application, we obtain the blow-up invariance of $$E_{1}$$ -degeneracy of the Hodge–de Rham spectral sequence associated with a local system of complex vector spaces.

$$L^p$$-Estimates for the Heat Semigroup on Differential Forms, and Related Problems

We consider a complete non-compact Riemannian manifold satisfying the volume doubling property and a Gaussian upper bound for its heat kernel (on functions). Let $$\overrightarrow{\Delta }_k $$ be the Hodge–de Rham Laplacian on differential k-forms with $$k \ge 1$$. By the Bochner decomposition formula, $$\overrightarrow{\Delta }_k = \nabla ^* \nabla + R_k$$, where $$\nabla $$ denotes the Levi-Civita connection and $$R_k$$ is a symmetric section of $$\mathrm{End}(\Lambda ^kT^*M)$$. Under the assumption that the negative part $$R_k^-$$ is in an enlarged Kato class, we prove that for all $$p \in [1, \infty ]$$, $$\Vert e^{-t\overrightarrow{\Delta }_k}\Vert _{p-p} \le C ( t \log t)^{\frac{D}{4}(1- \frac{2}{p})}$$ (for large t), where D is a homogeneous “dimension” appearing in the volume doubling property. This estimate can be improved if $$R_k^-$$ is strongly sub-critical. In general, $$(e^{-t\overrightarrow{\Delta }_k})_{t>0}$$ is not uniformly bounded on $$L^p$$ for any $$p \not = 2$$. We also prove the gradient estimate $$\Vert \nabla e^{-t\Delta }\Vert _{p-p} \le C t^{-\frac{1}{p}}$$, where $$\Delta $$ is the Laplace–Beltrami operator (acting on functions). Finally we discuss heat kernel bounds on forms and the Riesz transform on $$L^p$$ for $$p > 2$$.

Spectral analysis of Morse-Smale gradient flows

On a smooth, compact and oriented manifold without boundary, we give a complete description of the correlation function of a Morse-Smale gradient flow satisfying a certain nonresonance assumption. This is done by analyzing precisely the spectrum of the generator of such a flow acting on certain anisotropic spaces of currents. In particular, we prove that this dynamical spectrum is given by linear combinations with integer coefficients of the Lyapunov exponents at the critical points of the Morse function. Via this spectral analysis and in analogy with Hodge-de Rham theory, we give an interpretation of the Morse complex as the image of the de Rham complex under the spectral projector on the kernel of the generator of the flow. This allows us to recover classical results from differential topology such as the Morse inequalities and Poincare duality.

Some results on [formula omitted]-almost Ricci solitons

The purpose of this paper is to investigate some equations of structure for h-almost Ricci soliton which are a natural generalization for almost Ricci solitons. As a result we obtain that a compact nontrivial h-almost Ricci soliton is isometric to a Euclidean sphere with some conditions using the Hodge–de Rham decomposition theorem. Moreover, we acquire an integral formula for the compact h-almost Ricci solitons.

Asymptotics for the wave equation on differential forms on Kerr–de Sitter space

We study asymptotics for solutions of Maxwell's equations, in fact of the Hodge-de Rham equation $(d+\delta)u=0$ without restriction on the form degree, on a geometric class of stationary spacetimes with a warped product type structure (without any symmetry assumptions), which in particular include Schwarzschild-de Sitter spaces of all spacetime dimensions $n\geq 4$. We prove that solutions decay exponentially to $0$ or to stationary states in every form degree, and give an interpretation of the stationary states in terms of cohomological information of the spacetime. We also study the wave equation on differential forms and in particular prove analogous results on Schwarzschild-de Sitter spacetimes. We demonstrate the stability of our analysis and deduce asymptotics and decay for solutions of Maxwell's equations, the Hodge-de Rham equation and the wave equation on differential forms on Kerr-de Sitter spacetimes with small angular momentum.

The $$S^1$$ S 1 -Equivariant Signature for Semi-free Actions as an Index Formula

Lott (Math Ann 316(4):617–657, 2000) defined an integer-valued signature $$\sigma _{S^1}(M)$$ for the orbit space of a compact orientable manifold with a semi-free $$S^1$$ -action but he did not construct a Dirac-type operator which has this signature as its index. We construct such operator on the orbit space and we show that it is essentially unique and that its index coincides with Lott’s signature, at least when the stratified space satisfies the so-called Witt condition. For the non-Witt case, this operator remains essentially self-adjoint (in contrast to the Hodge-de Rham operator) and it has a well-defined index which we conjecture will also compute $$\sigma _{S^1}(M)$$ .

Induced Dirac-Schrödinger operators on $S^1$-semi-free quotients

John Lott has computed an integer-valued signature for the orbit space of a compact orientable $(4k+1)$ manifold with a semi-free $S^1$-action, which is a homotopy invariant of that space, but he did not construct a Dirac type operator which has this signature as its index. In this Thesis, we construct such operator on the orbit space, a Thom-Mather stratified space with one singular stratum of positive dimension, and we show that it is essentially unique and that its index coincides with Lott's signature, at least when the stratified space satisfies the so called Witt condition. We call this operator the induced Dirac-Schrodinger operator. The strategy of the construction is to push down an appropriate $S^1$-invariant first order transversally elliptic operator to the quotient space. The Witt condition, a topological condition which in this case depends on the codimension of the fixed point set, has various analytic consequences. In particular, when not satisfied, the Hodge-de Rham operator on the quotient space does not need to be essentially self-adjoint and therefore a choice of boundary conditions is required. This choice freedom is not natural in view of the fact that Lott's signature is well defined independently of the Witt condition. The Dirac-Schrodinger operator constructed in this Thesis differs from the Hodge-de Rham operator by a zero order term which ensures it to be essentially self-adjoint. Moreover, this zero order term anti-commutes with the chirality involution allowing the whole operator to split so that the index can be computed even if the Witt condition is not satisfied.

A Hodge–de Rham Dirac operator on the quantum SU(2)

We describe how it is possible to define a Hodge–de Rham Dirac operator associated to a suitable Cartan-Killing metric form upon the exterior algebra over the quantum spheres [Formula: see text] equipped with a three-dimensional left covariant calculus.

Lagrangian theory of structure formation in relativistic cosmology. IV. Lagrangian approach to gravitational waves

The relativistic generalization of the Newtonian Lagrangian perturbation theory is investigated. In previous works, the perturbation and solution schemes that are generated by the spatially projected gravitoelectric part of the Weyl tensor were given to any order of the perturbations, together with extensions and applications for accessing the nonperturbative regime. We here discuss more in detail the general first-order scheme within the Cartan formalism including and concentrating on the gravitational wave propagation in matter. We provide master equations for all parts of Lagrangian-linearized perturbations propagating in the perturbed spacetime, and we outline the solution procedure that allows one to find general solutions. Particular emphasis is given to global properties of the Lagrangian perturbation fields by employing results of Hodge-de Rham theory. We here discuss how the Hodge decomposition relates to the standard scalar-vector-tensor decomposition. Finally, we demonstrate that we obtain the known linear perturbation solutions of the standard relativistic perturbation scheme by performing two steps: first, by restricting our solutions to perturbations that propagate on a flat unperturbed background spacetime and, second, by transforming to Eulerian background coordinates with truncation of nonlinear terms.

On the de Rham homology and cohomology of a complete local ring in equicharacteristic zero

Let $A$ be a complete local ring with a coefficient field $k$ of characteristic zero, and let $Y$ be its spectrum. The de Rham homology and cohomology of $Y$ have been defined by R. Hartshorne using a choice of surjection $R\rightarrow A$ where $R$ is a complete regular local $k$-algebra: the resulting objects are independent of the chosen surjection. We prove that the Hodge–de Rham spectral sequences abutting to the de Rham homology and cohomology of $Y$, beginning with their $E_{2}$-terms, are independent of the chosen surjection (up to a degree shift in the homology case) and consist of finite-dimensional $k$-spaces. These $E_{2}$-terms therefore provide invariants of $A$ analogous to the Lyubeznik numbers. As part of our proofs we develop a theory of Matlis duality in relation to ${\mathcal{D}}$-modules that is of independent interest. Some of the highlights of this theory are that if $R$ is a complete regular local ring containing $k$ and ${\mathcal{D}}={\mathcal{D}}(R,k)$ is the ring of $k$-linear differential operators on $R$, then the Matlis dual $D(M)$ of any left ${\mathcal{D}}$-module $M$ can again be given a structure of left ${\mathcal{D}}$-module, and if $M$ is a holonomic ${\mathcal{D}}$-module, then the de Rham cohomology spaces of $D(M)$ are $k$-dual to those of $M$.

Dirac operators on the S3 and S2 spheres

We describe both the Hodge–de Rham and the spin manifold Dirac operator on the spheres [Formula: see text] and [Formula: see text], following the formalism introduced by Kähler, and exhibit a complete spectral resolution for them in terms of suitably globally defined eigenspinors.

$E_1$-degeneration and $d'd''$-lemma

For a double complex $(A, d', d'')$, we show that if it satisfies the $d'd''$-lemma and the spectral sequence $\{E^{p, q}_r\}$ induced by $A$ does not degenerate at $E_0$, then it degenerates at $E_1$. We apply this result to prove the degeneration at $E_1$ of a Hodge-de Rham spectral sequence on compact bi-generalized Hermitian manifolds that satisfy a version of $d'd''$-lemma.

On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for C<sup>*</sup>-Dynamical Systems

The analog of the Chern-Gauss-Bonnet theorem is studied for a $C^*$-dynamical system consisting of a $C^*$-algebra $A$ equipped with an ergodic action of a compact Lie group $G$. The structure of the Lie algebra $\mathfrak{g}$ of $G$ is used to interpret the Chevalley-Eilenberg complex with coefficients in the smooth subalgebra $\mathcal{A} \subset A$ as noncommutative differential forms on the dynamical system. We conformally perturb the standard metric, which is associated with the unique $G$-invariant state on $A$, by means of a Weyl conformal factor given by a positive invertible element of the algebra, and consider the Hermitian structure that it induces on the complex. A Hodge decomposition theorem is proved, which allows us to relate the Euler characteristic of the complex to the index properties of a Hodge-de Rham operator for the perturbed metric. This operator, which is shown to be selfadjoint, is a key ingredient in our construction of a spectral triple on $\mathcal{A}$ and a twisted spectral triple on its opposite algebra. The conformal invariance of the Euler characteristic is interpreted as an indication of the Chern-Gauss-Bonnet theorem in this setting. The spectral triples encoding the conformally perturbed metrics are shown to enjoy the same spectral summability properties as the unperturbed case.

Riesz transforms of the Hodge-de Rham Laplacian on Riemannian manifolds

Let $M$ be a complete non-compact Riemannian manifold satisfying the doubling volume property. Let $\overrightarrow{\Delta}$ be the Hodge-de Rham Laplacian acting on $1$-differential forms. According to the Bochner formula, $\overrightarrow{\Delta}=\nabla^*\nabla+R_+-R_-$ where $R_+$ and $R_-$ are respectively the positive and negative part of the Ricci curvature and $\nabla$ is the Levi-Civita connection. We study the boundedness of the Riesz transform $d^*(\overrightarrow{\Delta})^{-\frac{1}{2}}$ from $L^p(\Lambda^1T^*M)$ to $L^p(M)$ and of the Riesz transform $d(\overrightarrow{\Delta})^{-\frac{1}{2}}$ from $L^p(\Lambda^1T^*M)$ to $L^p(\Lambda^2T^*M)$. We prove that, if the heat kernel on functions $p_t(x,y)$ satisfies a Gaussian upper bound and if the negative part $R_-$ of the Ricci curvature is $\epsilon$-sub-critical for some $\epsilon\in[0,1)$, then $d^*(\overrightarrow{\Delta})^{-\frac{1}{2}}$ is bounded from $L^p(\Lambda^1T^*M)$ to $L^p(M)$ and $d(\overrightarrow{\Delta})^{-\frac{1}{2}}$ is bounded from $L^p(\Lambda^1T^*M)$ to $L^p(\Lambda^2T^* M)$ for $p\in(p_0',2]$ where $p_0>2$ depends on $\epsilon$ and on a constant appearing in the doubling volume property. A duality argument gives the boundedness of the Riesz transform $d(\Delta)^{-\frac{1}{2}}$ from $L^p(M)$ to $L^p(\Lambda^1T^*M)$ for $p\in [2,p_0)$ where $\Delta$ is the non-negative Laplace-Beltrami operator. We also give a condition on $R_-$ to be $\epsilon$-sub-critical under both analytic and geometric assumptions.

Partial collapsing and the spectrum of the Hodge–de Rham operator

The goal of the present paper is to calculate the limit spectrum of the Hodge-de Rham operator under the perturbation of collapsing one part of a manifold obtained by gluing together two manifolds with the same boundary. It appears to take place in the general problem of blowing up conical singularities as introduced in \cite{Maz} and \cite{Row1,Row2}.

The Hodge–de Rham Laplacian and [formula omitted]-boundedness of Riesz transforms on non-compact manifolds

Let M be a complete non-compact Riemannian manifold satisfying the volume doubling property as well as a Gaussian upper bound for the corresponding heat kernel. We study the boundedness of the Riesz transform dΔ−12 on both Hardy spaces Hp and Lebesgue spaces Lp under two different conditions on the negative part R− of the Ricci curvature. First we prove that if R− is α-subcritical for some α∈[0,1), then the Riesz transform d∗Δ⃗−12 on differential 1-forms is bounded from the associated Hardy space HΔ⃗p(Λ1T∗M) to Lp(M) for all p∈[1,2]. As a consequence, dΔ−12 is bounded on Lp for all p∈(1,p0) where p0>2 depends on α and the constant appearing in the doubling property. Second, we prove that if∫01‖|R−|12v(⋅,t)1p1‖p1dtt+∫1∞‖|R−|12v(⋅,t)1p2‖p2dtt<∞, for some p1>2 and p2>3, then the Riesz transform dΔ−12 is bounded on Lp for all 1<p<p2. Furthermore, we study the boundedness of the Riesz transform of Schrödinger operators A=Δ+V on Lp for p>2 under conditions on R− and the potential V. We prove both positive and negative results on the boundedness of dA−12 on Lp.

Comparative Analysis of Spectral Properties of the Hodge–De Rham and Tachibana Operators

In this paper, we perform a comparative analysis of spectral properties of the Hodge-de Rham and Tachibana operators on compact Riemannian manifolds with sign-definite bounded cur- vature operator. We also find bounds of spectra of these operators and obtain estimates of their multiplicities.