Hodge-de Rham Research Articles

Spectral Torsion

We introduce a trilinear functional of differential one-forms for a finitely summable regular spectral triple with a noncommutative residue. We demonstrate that for a canonical spectral triple over a closed spin manifold it recovers the torsion of the linear connection. We examine several spectral triples, including Hodge-de Rham, Einstein-Yang-Mills, almost-commutative two-sheeted space, conformally rescaled noncommutative tori, and quantum SU(2) group, showing that the third one has a nonvanishing torsion if nontrivially coupled.

A note on Hodge–Tate spectral sequences

AbstractWe prove that the Hodge–Tate spectral sequence of a proper smooth rigid analytic variety can be reconstructed from its infinitesimal $\mathbb{B}_{\text{dR}}^+$ -cohomology through the Bialynicki–Birula map. We also give a new proof of the torsion-freeness of the infinitesimal $\mathbb{B}_{\text{dR}}^+$ -cohomology independent of Conrad–Gabber spreading theorem, and a conceptual explanation that the degeneration of Hodge–Tate spectral sequences is equivalent to that of Hodge–de Rham spectral sequences.

Persistent de Rham-Hodge Laplacians in Eulerian representation for manifold topological learning

<p>Recently, topological data analysis has become a trending topic in data science and engineering. However, the key technique of topological data analysis, i.e., persistent homology, is defined on point cloud data, which does not work directly for data on manifolds. Although earlier evolutionary de Rham-Hodge theory deals with data on manifolds, it is inconvenient for machine learning applications because of the numerical inconsistency caused by remeshing the involving manifolds in the Lagrangian representation. In this work, we introduced persistent de Rham-Hodge Laplacian, or persistent Hodge Laplacian (PHL), as an abbreviation for manifold topological learning. Our PHLs were constructed in the Eulerian representation via structure-persevering Cartesian grids, avoiding the numerical inconsistency over the multi-scale manifolds. To facilitate the manifold topological learning, we proposed a persistent Hodge Laplacian learning algorithm for data on manifolds or volumetric data. As a proof-of-principle application of the proposed manifold topological learning model, we considered the prediction of protein-ligand binding affinities with two benchmark datasets. Our numerical experiments highlighted the power and promise of the proposed method.</p>

On almost η-Ricci-Bourguignon solitons

We investigate a Riemannian manifold with almost η-Ricci-Bourguignon soliton structure. We use the Hodge-de Rham decomposition theorem to make a link with the associated vector field of an almost η-Ricci-Bourguignon soliton. Moreover, we show that a nontrivial, compact almost η-Ricci-Bourguignon soliton of constant scalar curvature is isometric to the Euclidean sphere. Using some results obtaining from almost η-Ricci Bourguignon soliton, we give the integral formulas for compact orientable almost η-Ricci-Bourguignon soliton.

On the de Rham homology of affine varieties in characteristic 0

Let K be a field of characteristic 0, let S be a complete local ring with coefficient field K, let K[[x1,…,xn]] be the ring of formal power series in variables x1,…,xn with coefficients from K, let K[[x1,…,xn]]→S be a K-algebra surjection and let E••,• be the associated Hodge-de Rham spectral sequence for the computation of the de Rham homology of S. Nicholas Switala [12] proved that this spectral sequence is independent of the surjection beginning with the E2 page, and the groups E2p,q are all finite-dimensional over K.In this paper we extend this result to affine varieties. Namely, let Y be an affine variety over K, let X be a non-singular affine variety over K, let Y⊂X be an embedding over K and let E••,• be the associated Hodge-de Rham spectral sequence for the computation of the de Rham homology of Y. Then this spectral sequence is independent of the embedding beginning with the E2 page, and the groups E2p,q are all finite-dimensional over K.

Hodge-de Rham Laplacian and geometric criteria for gravitational waves

The curvature tensor \(\hat{R}\) of a manifold is called harmonic, if it obeys the condition \(\Delta^{\text{(HR)}}\hat{R}=0\), where \(\Delta^{\text{(HR)}}=DD^{\ast} +
 D^{\ast}D\) is the Hodge–deRham Laplacian. It is proved that all solutions of the Einstein equations in vacuum, as well as all solutions of the Einstein–Cartan theory in vacuum have a harmonic curvature. The statement that only solutions of Einstein’s equations of type \(N\) (describing gravitational radiation) are harmonic is refuted.

Some refinements of the Deligne–Illusie theorem

We extend the results of Deligne and Illusie on liftings modulo $p^2$ and decompositions of the de Rham complex in several ways. We show that for a smooth scheme $X$ over a perfect field $k$ of characteristic $p>0$, the truncations of the de Rham complex in $\max(p-1, 2)$ consecutive degrees can be reconstructed as objects of the derived category in terms of its truncation in degrees at most one (or, equivalently, in terms the obstruction class to lifting modulo $p^2$). Consequently, these truncations are decomposable if $X$ admits a lifting to $W_2(k)$, in which case the first nonzero differential in the conjugate spectral sequence appears no earlier than on page $\max(p,3)$ (these corollaries have been recently strengthened by Drinfeld, Bhatt-Lurie, and Li-Mondal). Without assuming the existence of a lifting, we describe the gerbes of splittings of two-term truncations and the differentials on the second page of the conjugate spectral sequence, answering a question of Katz. The main technical result used in the case $p>2$ belongs purely to homological algebra. It concerns certain commutative differential graded algebras whose cohomology algebra is the exterior algebra, dubbed by us "abstract Koszul complexes", of which the de Rham complex in characteristic $p$ is an example. In the appendix, we use the aforementioned stronger decomposition result to prove that Kodaira-Akizuki-Nakano vanishing and Hodge-de Rham degeneration both hold for $F$-split $(p+1)$-folds.

M-quasi Einstein manifolds with subharmonic potential

The main objective of this paper is to investigate the m-quasi Einstein manifold when the potential function becomes subharmonic. In this article, it is proved that an m-quasi Einstein manifold satisfying some integral conditions with vanishing Ricci curvature along the direction of potential vector field has constant scalar curvature and hence the manifold turns out to be an Einstein manifold. It is also shown that in an m-quasi Einstein manifold the potential function agrees with Hodge-de Rham potential up to a constant. Finally, it is proved that if a complete non-compact and non-expanding m-quasi Einstein manifold has bounded scalar curvature and the potential vector field has global finite norm, then the scalar curvature vanishes.

Hodge theory for elliptic curves and the Hopf element ν$\nu$

AbstractWe show that the vector bundle on the moduli stack of elliptic curves associated to the 2‐cell complex is isomorphic to the de Rham cohomology sheaf of the universal elliptic curve . We use this to calculate the homotopy groups of the ‐quotient of by , called the spectrum of ‘topological quasimodular forms’, by relating its Adams–Novikov spectral sequence to the cohomology of the moduli stack of cubic curves with a chosen splitting of the Hodge–de Rham filtration.

Hodge–de Rham numbers of almost complex 4-manifolds

We introduce and study Hodge–de Rham numbers for compact almost complex 4-manifolds, generalizing the Hodge numbers of a complex surface. The main properties of these numbers in the case of complex surfaces are extended to this more general setting, and it is shown that all Hodge–de Rham numbers for compact almost complex 4-manifolds are determined by the topology, except for one (the irregularity). Finally, these numbers are shown to prohibit the existence of complex structures on certain manifolds, without reference to the classification of surfaces.

Isometries on almost Ricci–Yamabe solitons

The purpose of the present paper is to examine the isometries of almost Ricci–Yamabe solitons. Firstly, the conditions under which a compact gradient almost Ricci–Yamabe soliton is isometric to Euclidean sphere S^n(r) are obtained. Moreover, we have shown that the potential f of a compact gradient almost Ricci–Yamabe soliton agrees with the Hodge–de Rham potential h. Next, we studied complete gradient almost Ricci–Yamabe soliton with alpha ne 0 and non-trivial conformal vector field with non-negative scalar curvature and proved that it is either isometric to Euclidean space E^n or Euclidean sphere S^n. Also, solenoidal and torse-forming vector fields are considered. Lastly, some non-trivial examples are constructed to verify the obtained results.

Integral $p$-adic Hodge filtrations in low dimension and ramification

Given an integral p -adic variety, we observe that if the integral Hodge–de Rham spectral sequence behaves nicely, then the special fiber knows the Hodge numbers of the generic fiber. Applying recent advancements of integral p -adic Hodge theory, we show that such a nice behavior is guaranteed if the p -adic variety can be lifted to an analogue of second Witt vectors and satisfies some bound on dimension and ramification index. This is a (ramified) mixed characteristic analogue of results due to Deligne–Illusie, Fontaine–Messing, and Kato. Lastly, we discuss an example illustrating the necessity of the aforementioned lifting condition, which is of independent interest.

Equivariant splitting of the Hodge\u2013de Rham exact sequence

Let X be an algebraic curve with a faithful action of a finite group G over a field k. We show that if the Hodge–de Rham short exact sequence of X splits G-equivariantly then the action of G on X is weakly ramified. In particular, this generalizes the result of Köck and Tait for hyperelliptic curves. We discuss also converse statements and tie this problem to lifting coverings of curves to the ring of Witt vectors of length 2.

Degenerations of log Hodge de Rham spectral sequences, log Kodaira vanishing theorem in characteristic $$p>0$$ and log weak Lefschetz conjecture for log crystalline cohomologies

We prove that the log Hodge de Rham spectral sequences of certain proper log smooth schemes of Cartier type in characteristic $$p>0$$ degenerate at $$E_1$$ . We also prove that the log Kodaira vanishings for them hold when they are projective. We formulate the log weak Lefschetz conjecture for log crystalline cohomologies and prove that it is true in certain cases.

Some characterizations of gradient Yamabe solitons

In this article we have proved that a gradient Yamabe soliton satisfying some additional conditions must be of constant scalar curvature. Later, we have showed that in a gradient expanding Yamabe soliton with non-negative scalar curvature if the potential function satisfies an integral condition then it is isometric to the Euclidean space Rn, and for steady case with the same condition the potential function becomes harmonic. Also we have proved that, in a compact gradient Yamabe soliton, the potential function agrees with the Hodge-de Rham potential up to a constant.

Sasakian 3-metric as a generalized Ricci-Yamabe soliton

In the present paper, we first investigate a Sasakian 3-metric as a quasi-Yamabe gradient soliton. In the sequel, extending the notions of quasi-Yamabe soliton and Ricci-Yamabe soliton, the notion of generalized Ricci-Yamabe soliton is introduced. It is shown that if (g, V, λ, α, β, γ) is a generalized gradient Ricci-Yamabe soliton on a complete Sasakian 3-manifold M with potential function f , then M is compact Einstein and locally isometric to a unit sphere. Moreover, the potential vector field V is an infinitesimal contact transformation and pointwise collinear with the characteristic vector field ξ. Further, if h is the Hodge-de Rham potential for V, then, upto a constant, f = h.

Littlewood–Paley–Stein Functions for Hodge-de Rham and Schrödinger Operators

We study the Littlewood–Paley–Stein functions associated with Hodge-de Rham and Schrodinger operators on Riemannian manifolds. Under conditions on the Ricci curvature, we prove their boundedness on $$L^p$$ for p in some interval $$(p_1,2]$$ and make a link to the Riesz Transform. An important fact is that we do not make assumptions of doubling measure or estimates on the heat kernel in this case. For $$p > 2$$ , we give a criterion to obtain the boundedness of the vertical Littlewood–Paley–Stein function associated with Schrodinger operators on $$L^p$$ .

Smoothing toroidal crossing spaces

Abstract We prove the existence of a smoothing for a toroidal crossing space under mild assumptions. By linking log structures with infinitesimal deformations, the result receives a very compact form for normal crossing spaces. The main approach is to study log structures that are incoherent on a subspace of codimension 2 and prove a Hodge–de Rham degeneration theorem for such log spaces that also settles a conjecture by Danilov. We show that the homotopy equivalence between Maurer–Cartan solutions and deformations combined with Batalin–Vilkovisky theory can be used to obtain smoothings. The construction of new Calabi–Yau and Fano manifolds as well as Frobenius manifold structures on moduli spaces provides potential applications.

Remarks on automorphism and cohomology of finite cyclic coverings of projective spaces

For a smooth finite cyclic covering over a projective space of dimension greater than one, we show that the group of automorphisms acts faithfully on the cohomology except for a few cases. In characteristic zero, we study the equivariant deformation theory and automorphism groups for complex cyclic coverings. The proof uses the decomposition of the sheaf of differential forms due to Esnault and Viehweg. In positive characteristics, a lifting criterion of automorphisms reduces the faithfulness problem to characteristic zero. To apply this criterion, we prove the degeneration of the Hodge-de Rham spectral sequences for a family of smooth finite cyclic coverings, and the infinitesimal Torelli theorem for finite cyclic coverings defined over an arbitrary field.

Effective actions for dual massive (super) p-forms

In d dimensions, the model for a massless p-form in curved space is known to be a reducible gauge theory for p > 1, and therefore its covariant quantisation cannot be carried out using the standard Faddeev-Popov scheme. However, adding a mass term and also introducing a Stueckelberg reformulation of the resulting p-form model, one ends up with an irreducible gauge theory which can be quantised à la Faddeev and Popov. We derive a compact expression for the massive p-form effective action, {Gamma}_p^{(m)} , in terms of the functional determinants of Hodge-de Rham operators. We then show that the effective actions {Gamma}_p^{(m)} and {Gamma}_{d-p-1}^{(m)} differ by a topological invariant. This is a generalisation of the known result in the massless case that the effective actions Γp and Γd−p−2 coincide modulo a topological term. Finally, our analysis is extended to the case of massive super p-forms coupled to background mathcal{N} = 1 supergravity in four dimensions. Specifically, we study the quantum dynamics of the following massive super p-forms: (i) vector multiplet; (ii) tensor multiplet; and (iii) three-form multiplet. It is demonstrated that the effective actions of the massive vector and tensor multiplets coincide. The effective action of the massive three-form is shown to be a sum of those corresponding to two massive scalar multiplets, modulo a topological term.