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Given two smooth projective varieties X and Y over a field, we say that X motivates Y if the (suitably defined) motive of Y is contained in the category generated from X by taking sums, summands and products. This notion has appeared implicitly in many places, but it seems useful to make it explicit. Some techniques are given for checking this condition, but in a nutshell it involves building a correspondence which "dominates" Y. Among the more interesting examples dealt with are moduli spaces. We show that in number of cases moduli spaces of sheaves over curves or surfaces are motivated by underly curve or surface. This allows us to check (or recheck) the (generalized) Hodge and Lefschetz standard conjectures for some of these examples. The dominating correspondence in these examples is built from the universal sheaf, and can, in some instances, be realized as a kind of Fourier-Mukai transform., Comment: 18 pages, latex; Final revision (to appear in Adv. in Math)

In recent publications, we have defined complexes of differential forms on analytic spaces which are resolutions of the constant sheaf. These complexes were used to prove the existence of a mixed Hodge structure on the cohomology of analytic spaces which possess kahlerian hypercoverings, in particular, projective algebraic varieties. We define an exterior product on these forms, which induces the cup product on the cohomology of analytic spaces. The main difficulty is to prove that this exterior product is functorial with respect to morphisms of analytic spaces. This exterior product can be used to prove that the cup product is compatible with the mixed Hodge structure on the cohomology.

This paper is intended to investigate Grassmann and Clifford algebras over Peano spaces, introducing their respective associated extended algebras, and to explore these concepts also from the counterspace viewpoint. The exterior (regressive) algebra is shown to share the exterior (progressive) algebra in the direct sum of chiral and achiral subspaces. The duality between scalars and volume elements, respectively under the progressive and the regressive products is shown to have chirality, in the case when the dimension n of the Peano space is even. In other words, the counterspace volume element is shown to be a scalar or a pseudoscalar, depending on the dimension of the vector space to be respectively odd or even. The de Rham cochain associated with the differential operator is constituted by a sequence of exterior algebra homogeneous subspaces subsequently chiral and achiral. Thus we prove that the exterior algebra over the space and the exterior algebra constructed on the counterspace are only pseudoduals each other, when we introduce chirality. The extended Clifford algebra is introduced in the light of the periodicity theorem of Clifford algebras context, wherein the Clifford and extended Clifford algebras Cl(p,q) can be embedded in Cl(p+1,q+1), which is shown to be exactly the extended Clifford algebra. Clifford algebras are constructed over the counterspace, and the duality between progressive and regressive products is presented using the dual Hodge star operator. The differential and codifferential operators are also defined for the extended exterior algebras from the regressive product viewpoint, and it is shown they uniquely tumble right out progressive and regressive exterior products of 1-forms., Comment: 17 pages, to appear in Adv. Appl. Clifford Algebras 16 (3) (2006)

We offer an intrinsic theoretical framework to reveal the inner relationships among three theories for Euler characteristic number, including Gauss–Bonnet–Chern theorem, Hopf–Poincare theorem and Morse theory. Moreover, we consider the Gauss–Bonnet–Chern (GBC) form imbedded in arbitrary higher-dimensional manifold, which suggests a Hodge dual tensor current. We show the brane structure inherent in the GBC tensor current and obtain the generalized Nambu action for the multi branes with quantized topological charge.

Hodge integrals over moduli spaces of curves appear naturally during the localization procedure in computation of Gromov-Witten invariants. A remarkable formula of Marino-Vafa expresses a generation function of Hodge integrals via some combinatorial and algebraic data seemingly unrelated to these apriori algebraic geometric objects. We prove in this paper by directly expanding the formula and estimating the involved terms carefully that except a specific type all the other Hodge integrals involving up to three Hodge classes can be calculated from this formula. This implies that amazingly rich information about moduli spaces and Gromov-Witten invariants is encoded in this complicated formula. We also give some low genus examples which agree with the previous results in literature. Proofs and calculations are elementary as long as one accepts Mumford relations on the reductions of products of Hodge classes.

We discuss a discrete analogue of the Dirac-K\"{a}hler equation in which chiral properties of the continual counterpart are captured. We pay special attention to a discrete Hodge star operator. To build one a combinatorial construction of double complex is used. We describe discrete exterior calculus operations on a double comlex and obtain the discrete Dirac-K\"{a}hler equation using these tools. Self-dual and anti-self-dual discrete inhomogeneous forms are presented. The chiral invariance of the massless discrete Dirac-K\"{a}hler equation is shown. Moreover, in the massive case we prove that a discrete Dirac-K\"{a}hler operator flips the chirality., Comment: 18 pages. arXiv admin note: substantial text overlap with arXiv:1307.1220

The Hodge star mean curvature flow on a 3-dimension Riemannian or pseudo-Riemannian manifold, the geometric Airy flow on a Riemannian manifold, the Schrodingier flow on Hermitian manifolds, and the shape operator curve flow on submanifolds are natural non-linear dispersive curve flows in geometric analysis. A curve flow is integrable if the evolution equation of the local differential invariants of a solution of the curve flow is a soliton equation. For example, the Hodge star mean curvature flow on $R^3$ and on $R^{2,1}$, the geometric Airy flow on $R^n$, the Schrodingier flow on compact Hermitian symmetric spaces, and the shape operator curve flow on an Adjoint orbit of a compact Lie group are integrable. In this paper, we give a survey of these results, describe a systematic method to construct integrable curve flows from Lax pairs of soliton equations, and discuss the Hamiltonian aspect and the Cauchy problem of these curve flows., 51 pages

If M is a smooth compact oriented Riemannian manifold of dimension n=4k+2, with or without boundary, and F is a vector bundle on M with an inner product and a flat connection, we construct a modification of the Hodge star operator on the parabolic cohomology H^{2k+1}_{par}(M;F). The operator gives a canonical complex structure on H^{2k+1}_{par}(M;F) compatible with the symplectic form \omega given by the wedge product of forms in the middle dimension. In case when k=0 that gives a canonical almost complex structure on the non-singular part of the moduli space of flat connections on a Riemann surface with or without boundary and monodromies along boundary components belonging to fixed conjugacy classes. The almost complex structure is compatible with the standard symplectic form \omega on the moduli space., Comment: 9 pages

The central aim of this paper is the study of the spectrum of the Hodge Laplacian on differential forms of any order k in L p . The underlying space is a C ∞ -smooth open manifold M N with Ricci Curvature bounded below and uniformly subexponential volume growth. It will be demonstrated that on such manifolds the L p spectrum of the Hodge Laplacian on differential k -forms is independent of p for 1 ⩽ p ⩽ ∞ , whenever the Weitzenbock Tensor on k -forms is also bounded below. It follows as a corollary that the isolated eigenvalues of finite multiplicity are L p independent. The proof relies on the existence of a Gaussian upper bound for the Heat kernel of the Hodge Laplacian. By considering the L p spectra on the Hyperbolic space H N + 1 we conclude that the subexponential volume growth condition is necessary in the case of one-forms. As an application, we will show that the spectrum of the Laplacian on one-forms has no gaps on certain manifolds with a pole or that are in a warped product form. This will be done under less strict curvature restrictions than what has been known so far and it was achieved by finding the L 1 spectrum of the Laplacian.

The Kuga-Satake construction associates to a K3 type polarized weight 2 Hodge structure H an abelian variety A such that H is a quotient Hodge structure of H^2(A). The first step is to consider the Clifford algebra of H. It turns out that it is endowed with a weight 2 Hodge structure compatible with the algebra structure. We show more generally that a weight 2 polarized Hodge structure which carries a compatible (unitary, associative) algebra structure is a quotient of the H^2 of an abelian variety.

The Hodge operator ("star" operator) plays an important role in the theory of differential forms, where it serves as a tool for the switching between the exterior derivative and co-derivative. In the theory of many-electron systems involving a finite-dimensional fermionic Fock space, one can define the Hodge operator as a unique (i.e., invariant with respect to linear transformations of the spin-orbital basis set) antilinear operator. The similarity transformation based on the Hodge operator results in the switching between the fermion creation and annihilation operators. The present paper gives a self-contained account on the algebraic structures which are necessary for the construction of the Hodge operator: the fermionic Fock space, the corresponding Grassmann algebra, and the generalized creation and annihilation operators. The Hodge operator is then defined, and its properties are reviewed. It is shown how the notion of the Hodge operator can be employed in a construction of the electronic time-reversal operator.

Lin and Sjamaar have used symplectic Hodge theory to obtain canonical equivariant extensions for Hamiltonian actions on closed symplectic manifolds that have the strong Lefschetz property. Here we obtain canonical equivariant extensions much more generally by means of classical Hodge theory., 10 pages

The Hodge conjecture is shown to be equivalent to a question about the homology of very ample divisors with ordinary double point singularities. The infinitesimal version of the result is also discussed.

Let Δ(Bn) be the order complex of the Boolean algebra and let B(n, k) be the part of Δ(Bn) where all chains have a gap at most k between each set. We give an action of the symmetric group Sl on the l-chains that gives B(n, k) a Hodge structure and decomposes the homology under the action of the Eulerian idempontents. The Sn action on the chains induces an action on the Hodge pieces and we derive a generating function for the cycle indicator of the Hodge pieces. The Euler characteristic is given as a corollary. We then exploit the connection between chains and tabloids to give various special cases of the homology. Also an upper bound is obtained using spectral sequence methods. Finally we present some data on the homology of B(n, k).

In this paper, we define and study the Weil–Petersson geometry. Under the framework of the Weil–Petersson geometry, we study the Weil–Petersson metric and the Hodge metric. Among the other results, we represent the Hodge metric in terms of the Weil–Petersson metric and the Ricci curvature of the Weil–Petersson metric for Calabi–Yau fourfold moduli. We also prove that the Hodge volume of the moduli space is finite. Finally, we proved that the curvature of the Hodge metric is bounded if the Hodge metric is complete and the dimension of the moduli space is 1.

For a smooth complex projective variety X defined over a number field, we have filtrations on the Chow groups depending on the choice of realizations. If the realization consists of mixed Hodge structure without any additional structure, we can show that the obtained filtration coincides with the filtration of Green and Griffiths, assuming the Hodge conjecture. In the case the realizations contain Hodge structure and etale cohomology, we prove that if the second graded piece of the filtration does not vanish, it contains a nonzero element which is represented by a cycle defined over a field of transcendence degree one. This may be viewed as a refinement of results of Nori, Schoen, and Green-Griffiths-Paranjape. For higher graded pieces we have a similar assertion assuming a conjecture of Beilinson and Grothendieck’s generalized Hodge conjecture.

Let A be a general member of a PEL-family of abelian varieties with endomorphisms by an imaginary quadratic number field k, and let E be an elliptic curve with complex multiplications by k. We show that the usual Hodge conjecture for products of A with powers of E implies the general Hodge conjecture for all powers of A. We deduce the general Hodge conjecture for all powers of certain 5-dimensional abelian varieties.

Using a functional inequality, the essential spectrum and eigenvalues are estimated for Laplace-type operators on Riemannian vector bundles. Consequently, explicit upper bounds are obtained for the dimension of the correspondingL 2-harmonic sections. In particular, some known results concerning Gromov's theorem and theL 2-Hodge decomposition are considerably improved.

We review the notion of the deformation of the exterior wedge product. This allows us to construct the deformation of the algebra of exterior forms over a vector space and also over an arbitrary manifold. We relate this approach to the generalized statistics. We study quons, as a particular case of these generalized statistics. We also give their statistical properties. A large part of the work is devoted to the problem of constructing coherent states for the deformed oscillators. We give a review of all the approaches existing in the literature concerning this point and enforce it with many examples.

We give a new proof of a result due to Y. Shalom: if the fundamental group of a compact real hyperbolic manifold of dim n is a free product of its subgroups A and B over the amalgamated subgroup C , then the critical exponent of C is not smaller than n −2. The proof, which is geometric, allows one to treat the equality case and an extension to variable curvature. To cite this article: G. Besson et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Abelian varieties of dimension 2n on which a definite quaternion algebra acts are parametrized by symmetrical domains of dimension n(n-1)/2. Such abelian varieties have primitive Hodge classes in the middle dimensional cohomology group. In general, it is not clear that these are cycle classes. In this paper we show that a particular 6-dimensional family of such 8-folds are Prym varieties and we use the method of C. Schoen to show that all Hodge classes on the general abelian variety in this family are algebraic. We also consider Hodge classes on certain 5-dimensional subfamilies and relate these to the Hodge conjecture for abelian 4-folds., Comment: 18 pages, LaTeX

The spectral problem associated with the linearization about solitary waves of the generalized fifth-order KdV equation is formulated in terms of the Evans function, a complex analytic function whose zeros correspond to eigenvalues. A numerical framework, based on a fast robust shooting algorithm on exterior algebra spaces is introduced. The complete algorithm has several new features, including a rigorous numerical algorithm for choosing starting values, a new method for numerical analytic continuation of starting vectors, the role of the Grassmannian G 2 ( C 5 ) in choosing the numerical integrator, and the role of the Hodge star operator for relating ⋀ 2 ( C 5 ) and ⋀ 3 ( C 5 ) and deducing a range of numerically computable forms for the Evans function. The algorithm is illustrated by computing the stability and instability of solitary waves of the fifth-order KdV equation with polynomial nonlinearity.

Let X=G/P be a homogeneous space of a complex semisimple Lie group G equipped with a hermitian metric. We study the action of the Hodge star operator on the space of harmonic differential forms on X. We obtain explicit combinatorial formulas for this action when X is an irreducible hermitian symmetric space of compact type., Comment: 15 pages, LaTeX, 4 figures

The compound matrix method, which was first proposed for numerically integrating systems of differential equations in hydrodynamic stability on k=2,3 dimensional subspaces of \({\mathbb C}^n\), by using compound matrices as coordinates, is reformulated in a coordinate-free way using exterior algebra spaces, \(\bigwedge^{k}({\mathbb C}^n)\).

The calculus of generalized p-forms, −1 ≤ p ≤ n, is developed further. On an n-dimensional manifold with metric, M, a (Hodge) star operator, inner product, co-differential and Laplacian operators are introduced. The inner product and Lie derivative with respect to vector fields on M are defined. Applications are made to Hamiltonian mechanics, field theories and Einstein's vacuum equations.

We consider basic equations for a deformed medium including a defect field on the basis of differential forms. To make our analysis, we extend three basic equations: (I) an incompatibility equation; (II) the Peach-Kohler equation; (III) the Navier equation based on the Hodge duality of the deformed medium. By combining two exterior differential operators, we derive (I) an incompatibility equation that extends the compatibility equation to include a defect field. The Hodge dual of the incompatibility equation becomes a generalized stress function, which includes previously derived stress functions such as Beltrami's, Morera's, Maxwell's and Airy's stress functions. By applying homotopy operators, we extend (II) the Peach-Kohler equation to include disclinations. In this case, we can define the basic quantities of stress space by analogy with the monopole theory. By combining exterior differential operators and star operators, we extend (III) the Navier equation to include a defect field. In this analysis, we define a Navier operator that is related to the Laplace operator through Hodge duality. We consider gauge conditions for a defect field based on the differential geometry of a deformed medium. This suggests a duality between yielding and fatigue fractures. The gauge condition in strain space-time is interpreted as basic relations in polycrystalline plastic deformation.

We consider two basic potential theoretic problems in Riemannian manifolds: Hodge decompositions and Maxwell's equations. Here we are concerned with smoothness and integrability assumptions. In the context of L p forms in Lipschitz domains, we show that both are well posed provided that 2− e p e , for some e >0, depending on the domain. Our approach is constructive (in the sense that we produce integral representation formulas for the solutions) and emphasizes the intimate connections between the two problems at hand. Applications to other related PDEs, such as boundary problems for the Hodge Dirac operator, are also presented.

The complement of an arrangement of hyperplanes is a good example of a mixed Hodge structure on the fundamental group of an algebraic variety. We compute its isomorphism class using iterated integrals and then show that the cross ratio of a arrangement is a combinatorial and projective invariant. So cross ratio equivalent arrangements are isomorphic. Moreover, an isomorphism of polarized mixed Hodge structures on the fundamental group induces cross ratio equivalent arrangements.

We identify the optimal range of coefficients s, p for which differential forms with coefficients in the Sobolev space L p s (Ω) admit natural Hodge decompositions in arbitrary two and three dimensional Lipschitz domains Ω . To cite this article: D. Mitrea, M. Mitrea, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 109–112

For an arbitrary semisimple Frobenius manifold we construct Hodge integrable hierarchy of Hamiltonian partial differential equations. In the particular case of quantum cohomology the tau-function of a solution to the hierarchy generates the intersection numbers of the Gromov--Witten classes and their descendents along with the characteristic classes of Hodge bundles on the moduli spaces of stable maps. For the one-dimensional Frobenius manifold the Hodge hierarchy is a deformation of the Korteweg--de Vries hierarchy depending on an infinite number of parameters. Conjecturally this hierarchy is a universal object in the class of scalar Hamiltonian integrable hierarchies possessing tau-functions., 49 pages

We describe a charge-conserving scatter–gather algorithm for particle-in-cell simulations on unstructured grids. Charge conservation is obtained from first principles, i.e., without the need for any post-processing or correction steps. This algorithm recovers, at a fundamental level, the scatter–gather algorithms presented recently by Campos-Pinto et al. (2014) (to first-order) and by Squire et al. (2012), but it is derived here in a streamlined fashion from a geometric viewpoint. Some ingredients reflecting this viewpoint are (1) the use of (discrete) differential forms of various degrees to represent fields, currents, and charged particles and provide localization rules for the degrees of freedom thereof on the various grid elements (nodes, edges, facets), (2) use of Whitney forms as basic interpolants from discrete differential forms to continuum space, and (3) use of a Galerkin formula for the discrete Hodge star operators (i.e., “mass matrices” incorporating the metric datum of the grid) applicable to generally irregular, unstructured grids. The expressions obtained for the scatter charges and scatter currents are very concise and do not involve numerical quadrature rules. Appropriate fractional areas within each grid element are identified that represent scatter charges and scatter currents within the element, and a simple geometric representation for the (exact) charge conservation mechanism is obtained by such identification. The field update is based on the coupled first-order Maxwell’s curl equations to avoid spurious modes with secular growth (otherwise present in formulations that discretize the second-order wave equation). Examples are provided to verify preservation of discrete Gauss’ law for all times.

We review the application of twist deformation formalism and the construction of noncommutative gauge theory on $\kappa$-Minkowski space-time. We compare two different types of twists: the Abelian and the Jordanian one. In each case we provide the twisted differential calculus and consider ${U}(1)$ gauge theory. Different methods of obtaining agauge invariant action and related problems are thoroughly discussed.

We construct a hermitian metric on the classifying spaces of graded-polarized mixed Hodge structures and prove analogs of the strong distance estimate between an admissible period map and the approximating nilpotent orbit. We also consider the asymptotic behavior of the biextension metric, the norm estimates and the asymptotics of the reduced limit Hodge filtration., Comment: 31 pages

By the results of G. Forni and of R. Trevi\~no, the Lyapunov spectrum of the Hodge bundle over the Teichm\"uller geodesic flow on the strata of Abelian and of quadratic differentials does not contain zeroes even though for certain invariant submanifolds zero exponents are present in the Lyapunov spectrum. In all previously known examples, the zero exponents correspond to those PSL(2,R)-invariant subbundles of the real Hodge bundle for which the monodromy of the Gauss-Manin connection acts by isometries of the Hodge metric. We present an example of an arithmetic Teichm\"uller curve, for which the real Hodge bundle does not contain any PSL(2,R)-invariant, continuous subbundles, and nevertheless its spectrum of Lyapunov exponents contains zeroes. We describe the mechanism of this phenomenon; it covers the previously known situation as a particular case. Conjecturally, this is the only way zero exponents can appear in the Lyapunov spectrum of the Hodge bundle for any PSL(2,R)-invariant probability measure., Comment: 47 pages, 10 figures. Final version (based on the referee's report). A slightly shorter version of this article will appear in Commentarii Mathematici Helvetici. A pdf file containing a copy of the Mathematica routine "FMZ3-Zariski-numerics_det1.nb" is available at this link here: http://w3.impa.br/~cmateus/files/FMZ3-Zariski-numerics_det1.pdf

Many linear boundary value problems arising in computational physics can be formulated in the calculus of differential forms. Discrete differential forms provide a natural and canonical approach to their discretization. However, much freedom remains concerning the choice of discrete Hodge operators, that is, discrete analogues of constitutive laws. A generic discrete Hodge operator is introduced and it turns out that most finite element and finite volume schemes emerge as its specializations. We reap the possibility of a unified convergence analysis in the framework of discrete exterior calculus.

In this paper, we give an expression and some estimates of the curvature tensor of the Hodge metric over the moduli space of a polarized Calabi-Yau threefold. The symmetricity of the Yukawa coupling is also studied. In the last section of this paper, an extra restriction of the limiting Hodge structure for the degeneration of Calabi-Yau threefolds is given., We corrected a typo error in the title and corrected the Journal-ref

We show the existence of a co(dual)-BRST symmetry for the usual BRST invariant Lagrangian density of an Abelian gauge theory in two dimensions of space-time where a U(1) gauge field is coupled to the Noether conserved current (constructed by the Dirac fields). Under this new symmetry, it is the gauge-fixing term that remains invariant and the symmetry transformations on the Dirac fields are analogous to the chiral transformations. This interacting theory is shown to provide a tractable field theoretical model for the Hodge theory. The Hodge dual operation is shown to correspond to a discrete symmetry in the theory and the extended BRST algebra for the generators of the underlying symmetries turns out to be reminiscent of the algebra obeyed by the de Rham cohomology operators of differential geometry., 12 pages, LaTeX, no figures, journal ref. given

We first prove local versions of the Poincare inequality for solutions to the A -harmonic equation. Then, as applications of the local results, we obtain the global versions of the Poincare inequality for solutions to the A -harmonic equation in L s (μ, 0)-averaging domains and L s -averaging domains, respectively. These results can be considered as generalizations of the classical Poincare theorem.

As generalizations of the original Volkov-Akulov action in four-dimensions, actions are found for all space-time dimensions D invariant under N non-linear realized global supersymmetries. We also give other such actions invariant under the global non-linear supersymmetry. As an interesting consequence, we find a non-linear supersymmetric Born-Infeld action for a non-Abelian gauge group for arbitrary D and N, which coincides with the linearly supersymmetric Born-Infeld action in D=10 at the lowest order. For the gauge group U({\cal N}) for M(atrix)-theory, this model has {\cal N}^2-extended non-linear supersymmetries, so that its large {\cal N} limit corresponds to the infinitely many (\aleph_0) supersymmetries. We also perform a duality transformation from F_{\mu\nu} into its Hodge dual N_{\mu_1...\mu_{D-2}}. We next point out that any Chern-Simons action for any (super)groups has the non-linear supersymmetry as a hidden symmetry. Subsequently, we present a superspace formulation for the component results. We further find that as long as superspace supergravity is consistent, this generalized Volkov-Akulov action can further accommodate such curved superspace backgrounds with local supersymmetry, as a super p-brane action with fermionic kappa-symmetry. We further elaborate these results to what we call `simplified' (Supersymmetry)^2-models, with both linear and non-linear representations of supersymmetries in superspace at the same time. Our result gives a proof that there is no restriction on D or N for global non-linear supersymmetry. We also see that the non-linear realization of supersymmetry in `curved' space-time can be interpreted as `non-perturbative' effect starting with the `flat' space-time.

We introduce the notion of mixed Hodge complex on an algebraic variety, improving Du Bois' filtered complex, and relate Deligne's theory of mixed Hodge structure with the theory of mixed Hodge module. This was supposed to be true, but is quite nontrivial, because the theory of mixed Hodge module does not work on a simplicial scheme. Some application to the Du Bois singularity is given., AMSTeX v2.1, 47 pages

The intent of this paper is first to provide a comprehensive and unifying development of Sobolev spaces of differential forms on Riemannian manifolds with boundary. Second, is the study of a particular class of nonlinear, first order, elliptic PDEs, called Hodge systems. The Hodge systems are far reaching extensions of the Cauchy-Riemann system and solutions are referred to as Hodge conjugate fields. We formulate and solve the Dirichlet and Neumann boundary value problems for the Hodge systems and establish the ℒp for such solutions. Among the many desirable properties of Hodge conjugate fields, we prove, in analogy with the case of holomorphic functions on the plane, the compactness principle and a strong theorem on the removability of singularities. Finally, some relevant examples and applications are indicated.

In this article we prove that locally Griffiths' horizontal distribution on the period domain is given by a generalized version of the familiar contact differential system. As a consequence of this description we obtain strong local rigidity properties of maximal dimensional variations of Hodge structure. For example, we prove that if the weight is odd then there is a unique germ of maximal dimensional variation of Hodge stucture though every point of the period domain. Similar results hold if the weight is even with the exception of one case., Comment: 24 pages, latex2e uses amsart.cls

A fermionic analogue of the Hodge star operation is shown to have an explicit operator representation in models with fermions, in spacetimes of any dimension. This operator realizes a conjugation (pairing) not used explicitly in field-theory, and induces a metric in the space of wave-function(al)s just as in exterior calculus. If made real (Hermitian), this induced metric turns out to be identical to the standard one constructed using Hermitian conjugation; the utility of the induced complex bilinear form remains unclear., 11 pages, plain TeX + harvmac

In this paper, some structures which underlie the numerical treatment of second-order boundary value problems are studied using magnetostatics as an example. The authors show that the construction of a discrete Hodge is a central problem. In this light, they interpret finite element techniques as a realization of the discrete Hodge operator in the Whitney complex. This enables one to view the Galerkin method as a way to set up circuit equations, the metric of space being encoded in the values of branch impedances.

Two generalisations of the Evans function, for the analysis of the linearisation about solitary waves, are shown to be equivalent. The generalisation introduced by Alexander, Gardner and Jones [J. reine Angew. Math. 410 (1990) 167] is based on exterior algebra and the generalisation introduced by Swinton [Phys. Lett. A 163 (1992) 57] is based on a matrix formulation and adjoint systems. In regions of the complex plane where both formulations are defined, the equivalence is geometric: we show that the formulations are dual and the duality can be made explicit using Hodge duality and the Hodge star operator. Swinton's formulation excludes potential branch points at which the Alexander, Gardner and Jones formulation is well-defined. Therefore we consider the implications of equivalence on the analytic continuation of the two formulations.

We define the notion of a morphism of generalized semi-stable type, which is a generalization of the notion of a semistable degeneration over a curve. We partially generalize Steenbrink's results on the limit of Hodge structures to the case of such a morphism. As an application we prove the E1-degeneration of the relative Hodge–De Rham spectral sequence for this case.

We study the Hodge decomposition of L 1-(and measure-) differential forms over a compact manifold without boundary, giving positive results and counterexamples. The theory is then applied to the relaxation and minimization, in cohomology classes, of convex functionals with linear growth. This corresponds to a non-linear version of the Hodge theory, in the spirit of L. M. Sibner and R. J. Sibner [SS].