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This article introduces a new and general construction of discrete Hodge operator in the context of Discrete Exterior Calculus (DEC). This discrete Hodge operator permits to circumvent the well-centeredness limitation on the mesh with the popular diagonal Hodge. It allows a dual mesh based on any interior point, such as the incenter or the barycenter. It opens the way towards mesh-optimized discrete Hodge operators. In the particular case of a well-centered triangulation, it reduces to the diagonal Hodge if the dual mesh is circumcentric. Based on an analytical development, this discrete Hodge does not make use of Whitney forms, and is exact on piecewise constant forms, whichever interior point is chosen for the construction of the dual mesh. Numerical tests oriented to the resolution of fluid mechanics problems and thermal transfer are carried out. Convergence on various types of mesh is investigated. Flat and non-flat domains are considered.

Discrete exterior calculus (DEC) offers a coordinate–free discretization of exterior calculus especially suited for computations on curved spaces. In this work, we present an extended version of DEC on surface meshes formed by general polygons that bypasses the need for combinatorial subdivision and does not involve any dual mesh. At its core, our approach introduces a new polygonal wedge product that is compatible with the discrete exterior derivative in the sense that it satisfies the Leibniz product rule. Based on the discrete wedge product, we then derive a novel primal–to–primal Hodge star operator. Combining these three ‘basic operators’ we then define new discrete versions of the contraction operator and Lie derivative, codifferential and Laplace operator. We discuss the numerical convergence of each one of these proposed operators and compare them to existing DEC methods. Finally, we show simple applications of our operators on Helmholtz–Hodge decomposition, Laplacian surface fairing, and Lie advection of functions and vector fields on meshes formed by general polygons.

We present a few types of integral transforms and integral representations that are very useful for extending to supergeometry many familiar concepts of differential geometry. Among them we discuss the construction of the super Hodge dual, the integral representation of picture changing operators of string theories and the construction of the super-Liouville form of a symplectic supermanifold.

Hodge’s conjecture on algebraic cycles is proved for a smooth projective model X of the fiber product X1 ×CX2 of nonisotrivial one-parameter families of K3 surfaces (possibly with degeneracies) under certain constraints on the ranks of the transcendental cycle lattices of the general geometric fibers Xks and representations of the Hodge groups Hg(Xks).

We study the Hodge structure of elliptic surfaces which are canonically defined from bielliptic curves of genus three. We prove that the period map for the second cohomology has one dimensional fibers, and the period map for the total cohomology is of degree twelve, and moreover, by adding the information of the Hodge structure of the canonical divisor, we prove a generic Torelli theorem for these elliptic surfaces. Finally, we give explicit examples of the pair of non-isomorphic elliptic surfaces which have the same Hodge structure on themselves and the same Hodge structure on their canonical divisors., Comment: 30 pages

Given a variation of Hodge structure over $\mathbb{P}^1$ with Hodge numbers $(1,1,\dots,1)$, we show how to compute the degrees of the Deligne extension of its Hodge bundles, following Eskin-Kontsevich-M\"oller-Zorich, by using the local exponents of the corresponding Picard-Fuchs equation. This allows us to compute the Hodge numbers of Zucker's Hodge structure on the corresponding parabolic cohomology groups. We also apply this to families of elliptic curves, K3 surfaces and Calabi-Yau threefolds.

We hope to understand the Hodge theoretic aspect of mirror symmetry in the framework of the fundamental diagram of log mixed Hodge theory. We give a formulation of mirror conjecture for Calabi–Yau threefolds as the coincidence of log period maps with specified sections under the mirror map. Since a variation of Hodge structure with unipotent monodromies on a product of punctured discs uniquely extends over the puncture to a log Hodge structure, we can work on the boundary point over which the log Riemann–Hilbert correspondence exists, and we can observe clearly in high resolution the behavior of Z-structure over the boundary point (cf. notes in Introduction below). This is an advantage of log Hodge theory.

We show that the Suslin forms are themselves a product of Suslin matrices (over the integers) and that their corresponding linear transformation is the Hodge star operator. The Fossum–Foxby–Iversen theory of acyclic complexes, as developed by Suslin, plays a vital role in this analysis.

Let G = C n ⋉ ϕ C m with a semi-simple action ϕ : C n → G L m ( C ) (not necessarily holomorphic). Suppose that G has a lattice Γ . Then we show that under some conditions on G and Γ , G / Γ admits a Hermitian metric such that the space of harmonic forms satisfies the Hodge symmetry and decomposition. By this result we give many examples of non-Kahler Hermitian solvmanifolds satisfying the Hodge symmetry and decomposition.

We define signed dual volumes at all dimensions for circumcentric dual meshes. We show that for pairwise Delaunay triangulations with mild boundary assumptions these signed dual volumes are positive. This allows the use of such Delaunay meshes for Discrete Exterior Calculus (DEC) because the discrete Hodge star operator can now be correctly defined for such meshes. This operator is crucial for DEC and is a diagonal matrix with the ratio of primal and dual volumes along the diagonal. A correct definition requires that all entries be positive. DEC is a framework for numerically solving differential equations on meshes and for geometry processing tasks and has had considerable impact in computer graphics and scientific computing. Our result allows the use of DEC with a much larger class of meshes than was previously considered possible., Comment: Corrected error in Figure 1 (columns 3 and 4) and Figure 6 and a formula error in Section 2. All mathematical statements (theorems and lemmas) are unchanged. The previous arXiv version v3 (minus the Appendix) appeared in the journal Computer-Aided Design

We review a version of the mirror correspondence for smooth toric varieties with a numerically effective anticanonical bundle. We give a precise description of the so-called B-model, which involves the Gaus-Manin system of a family of Laurent polynomials. We show how to derive from these data a variation of non-commutative Hodge structures and describe general results on period maps and classifying spaces for these generalized Hodge structures. Finally, we explain a version of mirror symmetry as an isomorphism of Frobenius manifolds.

We construct the Hodge dual for supermanifolds by means of the Grassmannian Fourier transform of superforms. In the case of supermanifolds it is known that the superforms are not sufficient to construct a consistent integration theory and that the integral forms are needed. They are distribution-like forms which can be integrated on supermanifolds as a top form can be integrated on a conventional manifold. In our construction of the Hodge dual of superforms they arise naturally. The compatibility between Hodge duality and supersymmetry is exploited and applied to several examples. We define the irreducible representations of supersymmetry in terms of integral and superforms in a new way which can be easily generalised to several models in different dimensions. The construction of supersymmetric actions based on the Hodge duality is presented and new supersymmetric actions with higher derivative terms are found. These terms are required by the invertibility of the Hodge operator., LateX2e, 51 pages. Corrected some further misprints

We describe an equivalence of categories between the category of mixed Hodge structures and a category of equivariant vector bundles on a toric model of the complex projective plane which verify some semistability condition. We then apply this correspondence to define an invariant which generalizes the notion of R-split mixed Hodge structure and give calculations for the first group of cohomology of possibly non smooth or non-complete curves of genus 0 and 1. Finally, we describe some extension groups of mixed Hodge structures in terms of equivariant extensions of coherent sheaves. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

These notes discuss Hodge theory in the algebraic and Kahler context. We introduce the notion of (polarized) Hodge structure on a cohomology algebra and show how to extract from it topological restrictions on compact Kahler manifolds, and stronger topological restrictions on projective complex manifolds. The second part of the notes is devoted to the discussion of the Hodge conjecture, showing in particular that there is no way to extend it to the Kahler context. We will also discuss algebraic de Rham cohomology which is specific to projective complex manifolds and allows to formulate a number of arithmetic questions related to the Hodge conjecture.

We show that closures of families of unitary local systems on quasiprojective varieties for which the dimension of a graded component of Hodge filtration has a constant value can be identified with a finite union of polytopes. We also present a local version of the theorem. This yields the "Hodge decomposition" of the set of unitary local systems with a non-vanishing cohomology extending Hodge decomposition of characteristic varieties of links of plane curves studied by the author earlier. We consider a twisted version of the characteristic varieties generalizing the twisted Alexander polynomials. Several explicit calculations for complements to arrangements are made., Comment: Final version. To appear in Manuscripta Mathematica

Let $X$ be a smooth projective curve of genus $g\geq 2$ over the complex numbers. A holomorphic triple $(E_1,E_2,\phi)$ on $X$ consists of two holomorphic vector bundles $E_{1}$ and $E_{2}$ over $X$ and a holomorphic map $\phi \colon E_{2} \to E_{1}$. There is a concept of stability for triples which depends on a real parameter $\sigma$. In this paper, we determine the Hodge polynomials of the moduli spaces of $\sigma$-stable triples with $\mathrm{rk}(E_1)=\mathrm{rk}(E_2)=2$, using the theory of mixed Hodge structures (in the cases that they are smooth and compact). This gives in particular the Poincar{\'e} polynomials of these moduli spaces. As a byproduct, we also give the Hodge polynomial of the moduli space of even degree rank 2 stable vector bundles., Comment: 28 pages

Discrete exterior calculus (DEC) attempts to mimic basic operations on differential forms in a discrete setting. This paper considers two concrete instances of how the completion of the DEC program is unachievable, and outlines the practical implications for computational electromagnetics. The two problems are the Commutative Cochain Problem (CCP) and the Discrete Star Localization Problem (DSLP). This paper elaborates on how the CCP is related to problems in computational magnetohydrodynamics, and how DSPL is related to material modeling and constructing discrete Hodge star operators whose matrix representations are maximally sparse in some concrete sense. A final section elaborates on the problem of Poincare duality on the level of cochains, and how the metric-free Whitney form finite element discretization of helicity functionals provides a model of how some of these theoretical obstacles can be side-stepped in three dimensions and, more generally, in dimensions.

In this work we introduce an accurate definition of the curl operator on weighted networks that completes the discrete vector calculus developed by the authors. This allows us to define the circulation of a vector field along a curve and to characterize the conservative fields. In addition, we obtain an adequate discrete version of the De Rham cohomology of a compact manifold, giving in particular discrete analogues of the Poincar? and Hodge's decomposition theorems.

In this paper we develop several algebraic structures on the simplicial cochains of a triangulated manifold and prove they converge to their differential-geometric analogues as the triangulation becomes small. The first such result is for a cochain cup product converging to the wedge product on differential forms. Moreover, we show any extension of this product to a C ∞ -algebra also converges to the wedge product of forms. For cochains equipped with an inner product, we define a combinatorial star operator and show that for a certain cochain inner product this operator converges to the smooth Hodge star operator.

Formal (mixed) Hodge structures FHS are introduced in such a way that the Hodge realization of Deligne's 1-motives extends to a realization from Laumon's 1-motives to formal Hodge structures of level 1, providing an equivalence of categories. For the sake of exposition we here confine our study to level 1 mixed Hodge structures. However, it is conceivable and suitable to consider formal mixed Hodge structures with arbitrary Hodge numbers: generalizing our definition herebelow it's not that difficult and we will treat such a matter nextly., 12 pages

We describe our conjecture about the irreducible unitary representations of reductive Lie groups, in the special case of \(\mathrm{SL}(2, \mathbb{R})\).

We give a brief treatment of these topics. The crucial concept here is that of a differential k-form on a manifold. But before discussing that, we have to present a bit of exterior algebra.

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)

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.

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.

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.

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.

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

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

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 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

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.

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

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 ).

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 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

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

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

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.

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.

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.

The endomorphism algebra of a K3 type Hodge structure is a totally real field or a CM field. In this paper we give a low brow introduction to the case of a totally real field. We give existence results for the Hodge structures, for their polarizations and for certain K3 surfaces. We consider the Kuga Satake variety of these Hodge structures and we discuss some examples. Finally we indicate various open problems related to the Hodge conjecture.

A formula which connects the determinant of the Demjanenko matrix with the relative class number of the cyclotomic field is given. A close relation between this matrix and the Hodge group of some Abelian variety is also revealed.

We recall the construction of the Hodge character and we show, us-ing a result due to F. Bittner, that these can be constructed usingclassical pure Hodge theory only, sideskipping Deligne’s constructionof functorial mixed Hodge structures on the cohomology of algberaicvarieties. We then give a proof, based on the weak factorization theo-rem, that the nearby fibre is well-defined in the Grothendieck group ofvarieties. Finally, various formulas for numerical invariants of Hodge-theoretic origin are presented which are associated to one-parameterdegenerations of projective manifolds.Keywords: Hodge structure, Hodge Euler polynomial, Hodge number polynomial,nearby and vanishing cohomologyMSC2000 classification: 14D07, 32G20 1 Introduction The behaviour of the cohomology of a degenerating family of complex pro-jective manifolds has been intensively studied in the nineteen-seventies byClemens, Griffiths, Schmid and others. See [Gr] for a nice overview. Re-cently, the theory of motivic integration, initiated by Kontsevich and de-veloped by Denef and Loeser, has given a new impetus to this topic. Inparticular, in the case of a one-parameter degeneration it has produced anobject ψ

In this article we introduce algorithms which compute iterations of Gauss-Manin connections, Picard-Fuchs equations of Abelian integrals and mixed Hodge structure of affine varieties of dimension n in terms of differential forms. In the case n = 1 such computations have many applications in differential equations and counting their limit cycles. For n > 3, these computations give us an explicit definition of Hodge cycles.