Search

The method of exact and spectral sequences is applied to compute the Hodge numbers of all known complete intersection Calabi–Yau manifolds. When plotted on a two-dimensional chart, the distribution of their values reveals an interesting periodic pattern. Next, we introduce a seven-dimensional crystal-like structure which seems to be relevant to study the systematics of the Hodge number distribution.

We show that for a general fibre X s of the Hessian family X of elliptic curves the mixed Hodge structure on the cohomology group H 2 ( X,X s ) is a non-splitting extension of ℤ(−2) 4 by H 1 ( X s ).

Each finite dimensional irreducible rational representation V of the symplectic group Sp2g(Q) determines a generically defined local system V over the moduli space Mg of genus g smooth projective curves. We study H2 (Mg; V) and the mixed Hodge structure on it. Specifically, we prove that if g ≥ 6, then the natural map IH2(M~g; V) → H2(Mg; V) is an isomorphism where M~_g is tfhe Satake compactification of Mg. Using the work of Saito we conclude that the mixed Hodge structure on H2(Mg; V) is pure of weight 2+r if V underlies a variation of Hodge structure of weight r. We also obtain estimates on the weight of the mixed Hodge structure on H2(Mg; V) for 3 ≤ g < 6. Results of this article can be applied in the study of relations in the Torelli group Tg.

Using Hodge theory and L2-cohomology we study the singularities and topology of configuration and moduli spaces of polygonal linkages in the 2-sphere. As a consequence we describe the local deformation space of a folded paper cone in R3.

The dyadic Green function for an anisotropic, inhomogenous medium is reformulated as a double differential form by embedding material properties in the Hodge star operator. This formalism simplifies the manipulation of differential operators in the anisotropic case. An integral equation is obtained which relates the Green form for the electric field to a generalization of the usual free space scalar Green function.

The space D of Hodge structures on a fixed polarized lattice is known as Griffiths period domain, and its quotient by the isometry group of the lattice is the moduli of polarized Hodge structures of a fixed type. When D is a Hermitian symmetric domain, then we have automorphic forms on D, which according to Baily–Borel theorem, they give an algebraic structure to the mentioned moduli space. In this article we slightly modify this picture by considering the space U of polarized lattices in a fixed complex vector space with a fixed Hodge filtration and polarization. It turns out that the isometry group of the filtration and polarization, which is an algebraic group, acts on U and the quotient is again the moduli of polarized Hodge structures. This formulation leads us to a notion of quasi-automorphic forms which generalizes quasi-modular forms attached to elliptic curves.

In this paper we introduce the q-deformed Hodge star operator, q-deformed inner product, and q-deformed adjoint of the q-deformed exterior derivative and investigate their properties. Using this mathematical background, we construct the q-deformed self-dual Yang-Mills theory.

For smooth projective varietiesX over ℂ, the Hodge Conjecture states that every rational Cohomology class of type (p, p) comes from an algebraic cycle. In this paper, we prove the Hodge conjecture for some moduli spaces of vector bundles on compact Riemann surfaces of genus 2 and 3.

For teleparallel gravity in four dimensions, Lucas and Pereira have shown that its action can be constructed via a generalized Hodge dual for torsion tensor. In this paper, we demonstrate that a direct generalization of this approach to other dimensions fails due to the fact that no generalized Hodge dual operator could be given in general dimensions. Furthermore, if one enforces the definition of a generalized Hodge dual to be consistent with the action of teleparallel gravity in general dimensions, the basic identity for any sensible Hodge dual would require an ad hoc definition for the second Hodge dual operation which is totally unexpected. Therefore, we conclude that at least for the torsion tensor, the observation of Lucas and Pereira only applies to four dimensions.

We compute the behaviour of Hodge data by tensor product with a unitary rank-one local system and middle convolution by a Kummer unitary rank-one local system for an irreducible variation of polarized complex Hodge structure on a punctured complex affine line. We give applications of these formulas to local systems with G_2-monodromy., Comment: 34 pages. Changes in v2 only consist of appropriate references to previous results of Green-Griffiths-Kerr and to a preprint of Kerr-Pearlstein. Revised version v3, 36 pages. A restricted version of v3, without considerations of irregular singularities, published in Publ. RIMS, Kyoto Univ. V4: correction post publication in the statement of Theorem 3.2.3

We present a noncommutative (NC) version of the action for vielbein gravity coupled to gauge fields. Noncommutativity is encoded in a twisted star product between forms, with a set of commuting background vector fields defining the (abelian) twist. A first order action for the gauge fields avoids the use of the Hodge dual. The NC action is invariant under diffeomorphisms and twisted gauge transformations. The Seiberg-Witten map, adapted to our geometric setting and generalized for an arbitrary abelian twist, allows to re-express the NC action in terms of classical fields: the result is a deformed action, invariant under diffeomorphisms and usual gauge transformations. This deformed action is a particular higher derivative extension of the Einstein-Hilbert action coupled to Yang-Mills fields, and to the background vector fields defining the twist. Here noncommutativity of the original NC action dictates the precise form of this extension. We explicitly compute the first order correction in the NC parameter of the deformed action, and find that it is proportional to cubic products of the gauge field strength and to the symmetric anomaly tensor D_{IJK}., 18 pages, LaTeX

We study the generalized Hodge conjecture for certain sub-Hodge structure defined as the kernel of the cup product map with a big cohomology class, which is of Hodge coniveau at least 1. As predicted by the generalized Hodge conjecture, we prove that the kernel is supported on a divisor, assuming the Lefschetz standard conjecture., 23 pages. V2: Typos corrected. Comments still welcome. To appear in Math.Res.Lett

This chapter is devoted to \((0,r)-\) skew-symmetric tensors. After determining their strict components, it is proved that they form a vector space \(\varLambda _r(E_n)\). Then, the dimension of this space is determined together with the bases of this space and the transformation properties of strict components. Finally, the exterior algebra is formulated and oriented vector spaces are introduced.

A mixed Hodge structure in a vector space is two filtrations of the space, satisfying the axioms indicated below. In the space of cohomologies, vanishing at the critical point of the holomorphic function, there is a natural mixed Hodge structure. The role of the above-mentioned filtrations is played by the weight and Hodge filtrations, introduced in Chapter 13. The weight filtration is constructed from the Jordan structure of the monodromy operator and reflects the behaviour of integrals over vanishing cycles under analytic continuation of the integrals round critical values of the parameter.

Accurate simulation of atmospheric flow in weather and climate prediction models requires the discretization of the governing equations to have a number of desirable properties. Although these properties can be achieved relatively straightforwardly on a latitude-longitude grid, they are much more challenging on the various quasi-uniform spherical grids that are now under consideration. A recently developed scheme—called TRiSK—has these desirable properties on grids that have an orthogonal dual. The present work extends the TRiSK scheme into a more general framework suitable for grids that have a nonorthogonal dual, such as the equiangular cubed sphere. We also show that this framework fits within the wider framework of mimetic discretizations and discrete exterior calculus. One key ingredient is the definition of certain mapping operators that are discrete analogues of the Hodge star operator, enabling the definition of a compatible inner product. Discrete Coriolis terms are also included within the mimet...

Using a Mathematica code, we present a straightforward numerical analysis of the 384-dimensional solution space of signed permutation 4x4 matrices, which in sets of four provide representations of the GR(4,4) algebra, closely related to the N=1 (simple) supersymmetry algebra in 4-dimensional spacetime. Following after ideas discussed in previous papers about automorphisms and classification of adinkras and corresponding supermultiplets, we make a new and alternative proposal to use equivalence classes of the (unsigned) permutation group S4 to define distinct representations of higher dimensional spin bundles within the context of adinkras. For this purpose, the definition of a dual operator akin to the well-known Hodge star is found to partition the space of these GR(4,4) representations into three suggestive classes., Comment: 23 pages, 6 figures (v.2: numerical data corrections in appendix C)

On the exterior algebra over the quantum SU(2) coming from the four dimensional bicovariant calculus \`a la Woronowicz we introduce, using sesquilinear contraction maps, a class of metrics and Hodge duality operators, and compare this formulation with the previous ones presented in the literature., Comment: v2, 2 references added. 18 pages

Give the smooth subset of a normal singular complex projective surface the metric induced from the ambient projective space. The L 2 {L^2} cohomology of this incomplete manifold is isomorphic to the surface’s intersection cohomology, which has a natural Hodge decomposition. This paper identifies Dolbeault complexes whose ∂ ¯ \bar \partial -closed and ∂ ¯ \bar \partial -coclosed forms represent the classes of pure type in the corresponding Hodge decomposition of L 2 {L^2} cohomology.

Let $a>b>0$ and $f$ be a conformal map from $B_a\setminus B_b\subseteq R^2$ into $\R^n$, with $|\nabla f|^2=2e^{2u}$. Then $(e_1, e_2)$ with $e_1=e^{-u}\frac{\partial f}{\partial r},$ and $e_2=r^{-1}e^{-u}\frac{\partial f}{\partial\theta}$ is a moving frame on $f(B_a\setminus B_b)$. It satisfies the following equation $$d\star=0,$$ where $\star$ is the Hodge star operator on $R^2$ with respect to the standard metric. We will study the Dirichret energy of this frame and give some applications., Comment: Some typos have been polished and I find that the best constant for L^infinity norm in Wente inequality has been established by Topping in a paper dated to 1997

In this paper, we prove a general theorem concerning the analyticity of the closure of a subspace defined by a family of variations of mixed Hodge structures, which includes the analyticity of the zero loci of degenerating normal functions. For the proof, we use a moduli of the valuative version of log mixed Hodge structures.

This article gives a survey of recent results on a generalization of the notion of a Hodge structure. The main example is related to the Fourier-Laplace transform of a variation of polarizable Hodge structure on the punctured affine line, like the Gauss-Manin systems of a proper or tame algebraic function on a smooth quasi-projective variety. Variations of non-commutative Hodge structures often occur on the tangent bundle of Frobenius manifolds, giving rise to a tt* geometry., 32 pages

We study Maxwell's equations on a 4-manifold where the electromagnetic medium is described by an antisymmetric $2\choose 2$-tensor $\kappa$. In this setting, the Tamm-Rubilar tensor density determines a polynomial surface of fourth order in each cotangent space. This surface is called the Fresnel surface and acts as a generalisation of the light-cone determined by a Lorentz metric; the Fresnel surface parameterises electromagnetic wave-speed as a function of direction. Favaro and Bergamin have recently proven that if $\kappa$ has only a principal part and if the Fresnel surface of $\kappa$ coincides with the light cone for a Lorentz metric $g$, then $\kappa$ is proportional to the Hodge star operator of $g$. That is, under additional assumptions, the Fresnel surface of $\kappa$ determines the conformal class of $\kappa$. The purpose of this paper is twofold. First, we provide a new proof of this result using Gr\"obner bases. Second, we describe a number of cases where the Fresnel surface does not determine the conformal class of the original $2\choose 2$-tensor $\kappa$. For example, if $\kappa$ is invertible we show that $\kappa$ and $\kappa^{-1}$ have the same Fresnel surfaces., Comment: 23 pages, 1 figure

Mixed finite element methods solve a PDE involving two or more variables. In typical problems from electromagnetics and electrodiffusion, the degrees of freedom associated to the different variables are stored on both primal and dual domain meshes and a discrete Hodge star is used to transfer information between the meshes. We show through analysis and examples that the choice of discrete Hodge star is essential to the model and numerical stability of a finite element method. We also show how to define interpolation functions and discrete Hodge stars on dual meshes which can be used to create previously unconsidered mixed methods.

We study the mixed Hodge theoretic aspects of the B-model side of local mirror symmetry. Our main objectives are to define an analogue of the Yukawa coupling in terms of the variations of the mixed Hodge structures and to study its properties. We also describe a local version of Bershadsky-Cecotti-Ooguri-Vafa's holomorphic anomaly equation., Comment: 39 pages, 2 figures; v2: typos corrected; v3: added references

If f is a smooth function on a Hodge manifold, we construct a canonical sequence of real algebraic functions that converge to f in the smooth topology. The definition of of the approximants is inspired by Berezin-Toeplitz quantization. The proof follows quickly from known results of Fine, Liu and Ma., Comment: Small corrections. Some remarks added

Differential forms and exterior calculus are important theories in mathematics. Exterior calculus have found wide applications in fields such as general relativity, theory of electromagnetic fields, thermodynamics, theory of elasticity, differential geometry, topology and nonlinear differential equations. Differential forms are the most natural language for expressing electromagnetic and gauge fields mathematically. This language is independent of coordinates. Exterior calculus of differential forms give an alternative to vector calculus, which is ultimately simpler and more nature.

In this chapter, we introduce the concepts of multiforms, extensors, the canonical and metric Clifford algebras. Given a real vector space V with dim V = n, multiforms are elements of a space denoted ∧V, each multiform being a sum of non homogeneous k-forms (0 ≤ k ≤ n). We define the exterior product of those objects. Next we introduce a Euclidean product (called canonical product) in V and extend it to the space ∧V, defining also the right and left contracted products of multiforms. Those concepts permit us to introduce the canonical Clifford product in ∧V. We, thus, get a very powerful algebraic structure, the canonical Clifford algebra of ∧V, denoted \(\mathcal{C}\ell(V,\cdot )\) that is utilized as a basic calculational tool in the rest of the text. Next, we introduce the concept of extensors, linear mappings from Cartesian products \(\wedge V \times....\times \wedge V \rightarrow \wedge V\) and study with details the properties of so-called (1,1)-extensors and some important extensors related to it (in particular its extensions), which will appear in different occasions in the following chapters. In particular, they will play a crucial role in our formulation of the differential geometry of manifolds ( Chapter 4) and our theory of the gravitational field ( Chapter 5). Equipped with the concept of extensor, we introduce in ∧V a metrical extensor g of arbitrary signature (an object directly associated to a metric tensor g of the same signature) and extend it to ∧V. Given a g, we can define a corresponding metric Clifford algebra. We introduce a pseudo-Euclidean metric extensor ηin V and find a (1,1)-extensor h which relates η to g (the deformation of η by h). Moreover, we derive an important formula which permits obtaining h (modulus a Lorentz transformation) in terms of a given g. Several remarkable results are then presented, such as the golden rule relating the scalar, contracted and Clifford products associated to different metric extensors. With the golden rule, it becomes possible also to relate easily thorough a nice formula, the Hodge star operators associated with two different metric extensors. The chapter ends with a section presenting a set of useful identities that are used several times in the remainder of the book.

We construct toroidal partial compactifications of the moduli spaces of mixed Hodge structures with polarized graded quotients. They are moduli spaces of log mixed Hodge structures with polarized graded quotients. We construct them as the spaces of nilpotent orbits.

We show that the limit, by rescaling, of the 'new supersymmetric index' attached to the Fourier–Laplace transform of a polarized variation of Hodge structure on a punctured affine line is equal to the spectral polynomial attached to the same object. We also extend the definition by Deligne of a Hodge filtration on the de Rham cohomology of a exponentially twisted polarized variation of complex Hodge structure and prove a $E_1$-degeneration property for it.

This article describes features of a natural representation of the Clifford algebra of the space of bivectors of a four-dimensional vector space, the representation space being the vector space plus its dual. Vectors and covectors are then pure spinors. The natural map taking a pure spinor to a totally null 3-bivector is shown to be intertwined with the operation of lowering or raising by a metric with the action of the Hodge dual operator. New formulae exhibiting a metric in terms of its dial operator are presented, as are a few applications.

International audience; We define a new theory of discrete Riemann surfaces and present its basic results. The key idea is to consider not only a cellular decomposition of a surface, but the union with its dual. Discrete holomorphy is defined by a straightforward discretisation of the Cauchy-Riemann equation. A lot of classical results in Riemann theory have a discrete counterpart, Hodge star, harmonicity, Hodge theorem, Weyl's lemma, Cauchy integral formula, existence of holomorphic forms with prescribed holonomies. Giving a geometrical meaning to the construction on a Riemann surface, we define a notion of criticality on which we prove a continuous limit theorem. We investigate its connection with criticality in the Ising model. We set up a Dirac equation on a discrete universal spin structure and we prove that the existence of a Dirac spinor is equivalent to criticality.

Author(s): Mulase, Motohico; Zhang, Naizhen | Abstract: We establish a polynomial recursion formula for linear Hodge integrals. It is obtained as the Laplace transform of the cut-and-join equation for the simple Hurwitz numbers. We show that the recursion recovers the Witten-Kontsevich theorem when restricted to the top degree terms, and also the combinatorial factor of the lambda_g formula as the lowest degree terms.

We use a result of van Geemen to determine the endomorphism algebra of the Kuga--Satake variety of a K3 surface with real multiplication. This is applied to prove the Hodge conjecture for self-products of double covers of $\PP^2$ which are ramified along six lines., 24 pages

Let $X$ be a smooth projective curve of genus $g\geq 2$ over the complex numbers. Fix $n\geq 2$, and an integer $d$. A pair $(E,\phi)$ over $X$ consists of an algebraic vector bundle $E$ of rank $n$ and degree $d$ over $X$ and a section $\phi$. There is a concept of stability for pairs which depends on a real parameter $\tau$. Let $M_\tau(n,d)$ be the moduli space of $\tau$-semistable pairs of rank $n$ and degree $d$ over $X$. We prove that the cohomology groups of $M_\tau(n,d)$ are Hodge structures isomorphic to direct summands of tensor products of the Hodge structure $H^1(X)$. This implies a similar result for the moduli spaces of stable vector bundles over $X$., Comment: 23 pages

In 1999 M. Eastwood has used the general construction known as the Bernstein-Gelfand-Gelfand (BGG) resolution to prove, at least in smooth situation, the equivalence of the linear elasticity complex and of the de Rham complex in $\mathbf{R}^{3}$. The main objective of this paper is to study the linear elasticity complex for general Lipschitz domains in $\mathbf{R}^{3}$ and deduce a complete Hodge orthogonal decomposition for symmetric matrix fields in $L^{2}$, counterpart of the Hodge decomposition for vector fields. As a byproduct one obtains that the finite dimensional terms of this Hodge decomposition can be interpreted in homological terms as the corresponding terms for the de Rham complex if one takes the homology with value in $RIG\cong \mathbf{R}^{6}$ as in the (BGG) resolution.

Given a special Kahler manifold M , we give a new, direct proof of the relationship between the quaternionic structure on T ∗ M and the variation of Hodge structures on T C M .

This paper is a survey of the subject of variations of Hodge structure (VHS) considered as exterior differential systems (EDS). We review developments over the last twenty-six years, with an emphasis on some key examples. In the penultimate section we present some new results on the characteristic cohomology of a homogeneous Pfaffian system. In the last section we discuss how the integrability conditions of an EDS affect the expected dimension of an integral submanifold. The paper ends with some speculation on EDS and Hodge conjecture for Calabi-Yau manifolds.

×Twhich we call the preferred Goldman-Millson isomorphism (definition 2.11). Themain theorem of [GolMil88] asserts that an abstract isomorphism between these twoformal germs exists. The original construction is not quite canonical, and we havetried to make the choices made there as explicit as possible. This is the rationalefor the length of section 2. It turns out that there is no good reason to give any kindof privilege to this preferred isomorphism. Two other very nice Goldman-Millson

Let \(\mathcal {X}\) be a proper scheme over the field F of functions meromorphic in an open neighborhood of zero in the complex plane. The scheme \(\mathcal {X}\) gives rise to a proper morphism of complex analytic spaces \({\mathcal X}^h \to D^\ast\) and, if the radius of the open disc D is sufficiently small, the cohomology groups of the fibers \({\mathcal X}^h_t\) at points \(t\in D^\ast\) form a variation of mixed Hodge structures on \(D^\ast\), which admits a limit mixed Hodge structure. The purpose of the paper is to construct a canonical isomorphism between the weight zero subspace of this limit mixed Hodge structure and the rational cohomology group of the non-Archimedean analytic space \({\mathcal X}^{\rm an}\) associated to the scheme \(\mathcal {X}\) over the completion of the field F.

A Hodge decomposition theorem is proved for the leafwise Laplacian of a Riemannian foliation on a closed Riemannian manifold. © 1991.

Wu-Yang ambiguities are related to torsion free driebeins. A general solution of the non-Abelian Gauss law is written down and non-Abelian analogs of Hodge decomposition are worked out. These results are used to carry out a duality transformation for 3+1 dimensional Yang-Mills theory.