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In this paper, we propose a weak version of quotient for the algebraic action of a group on a variety, which we shall call a pseudo-quotient. They arise when we focus on the purely topological properties of good Geometric Invariant Theory (GIT) quotients regardless of their algebraic properties. The flexibility granted by their topological nature enables an easier identification in geometric constructions than classical GIT quotients. We obtain several results about the interplay between pseudo-quotients and good quotients. Additionally, we show that in characteristic zero pseudo-quotients are unique up to virtual class in the Grothendieck ring of algebraic varieties. As an application, we compute the virtual class of S L 2 (k) -character varieties for free groups and surface groups as well as their parabolic counterparts with punctures of Jordan type. [ABSTRACT FROM AUTHOR]

We extend the analogy between the extended Robba rings of p-adic Hodge theory and the one-dimensional affinoid algebras of rigid analytic geometry, proving some fundamental properties that are well known in the latter case. In particular, we show that these rings are regular and excellent. The extended Robba rings are of interest as they are used to build the Fargues–Fontaine curve. [ABSTRACT FROM AUTHOR]

We show that if V and V ′ are two p -adic representations of Gal ( Q ¯ p / Q p) whose tensor product is trianguline, then V and V ′ are both potentially trianguline. [ABSTRACT FROM AUTHOR]

Differential geometry, especially the use of curvature, plays a central role in modern Hodge theory. The vector bundles that occur in the theory (Hodge bundles) have metrics given by the polarizations of the Hodge structures, and the sign and singularity properties of the resulting curvatures have far reaching implications in the geometry of families of algebraic varieties. A special property of the curvatures is that they are 1 st order invariants expressed in terms of the norms of algebro-geometric bundle mappings. This partly expository paper will explain some of the positivity and singularity properties of the curvature invariants that arise in the Hodge theory with special emphasis on the norm property. [ABSTRACT FROM AUTHOR]

In this paper, we extend the Hodge theory to complex Lie algebroids, introduce the Laplacian operators and decompose the cohomological groups with respect to these operators, and generalize Kahler and Nakano identities to Kähler Lie algebroids as well. Our main purpose is to extend Kodaira vanishing theorem to Kahler Lie algebroids. [ABSTRACT FROM AUTHOR]

We derive the nilpotent (anti-)BRST and (anti-)co-BRST symmetry transformations for the system of a toy model of Hodge theory (i.e. a rigid rotor) by exploiting the (anti-)BRST and (anti-)co-BRST invariant restrictions on the (anti-)chiral supervariables that are defined on the appropriately chosen (1 , 1) -dimensional super-submanifolds of the general (1 , 2) -dimensional supermanifold on which our system of a one (0 + 1) -dimensional (1D) toy model of Hodge theory is considered within the framework of the augmented version of the (anti-)chiral supervariable approach (ACSA) to Becchi–Rouet–Stora–Tyutin (BRST) formalism. The general (1 , 2) -dimensional supermanifold is parametrized by the superspace coordinates (t , 𝜃 , 𝜃 ̄) , where t is the bosonic evolution parameter and (𝜃 , 𝜃 ̄) are the Grassmannian variables which obey the standard fermionic relationships: 𝜃 2 = 𝜃 ̄ 2 = 0 , 𝜃 𝜃 ̄ + 𝜃 ̄ 𝜃 = 0. We provide the geometrical interpretations for the symmetry invariance and nilpotency property. Furthermore, in our present endeavor, we establish the property of absolute anticommutativity of the conserved fermionic charges which is a completely novel and surprising observation in our present endeavor where we have considered only the (anti-)chiral supervariables. To corroborate the novelty of the above observation, we apply this ACSA to an 𝒩 = 2 SUSY quantum mechanical (QM) system of a free particle and show that the 𝒩 = 2 SUSY conserved and nilpotent charges do not absolutely anticommute. [ABSTRACT FROM AUTHOR]

We discuss the Hard Lefschetz Condition (HLC) on various cohomology groups and verify them for the Nakamura manifold of completely solvable type and the Kodaira-Thurston manifold. A general Demailly-Griffiths-Kähler identity is also given. [ABSTRACT FROM AUTHOR]

For a particular class of pseudo manifolds, we show that the intersection cohomology groups for any perversity may be naturally represented by extended weighted L 2 harmonic forms for a complete metric on the regular stratum with respect to some weight determined by the perversity. Extended weighted L 2 harmonic forms are harmonic forms that are almost in the given weighted L 2 space for the metric in question, but not quite. This result is akin to the representation of absolute and relative cohomology groups for a manifold with boundary by extended harmonic forms on the associated manifold with cylindrical ends. In analogy with that setting, in the unweighted L 2 case, the boundary values of the extended harmonic forms define a Lagrangian splitting of the boundary space in the long exact sequence relating upper and lower middle perversity intersection cohomology groups. [ABSTRACT FROM AUTHOR]

We describe an algorithm to compute the reduction modulo p of a crystalline Galois representation of dimension 2 of Gal ( ℚ ¯ p / ℚ p ) with distinct Hodge–Tate weights via the semi-simple modulo p Langlands correspondence. We give some examples computed with an implementation of this algorithm in SAGE. [ABSTRACT FROM AUTHOR]

A recent theorem of [E. Gasparim, L. Grama and L. A. B. San Martin, Lefschetz fibrations on adjoint orbits, Forum Math. 28(5) (2016) 967-980.] showed that adjoint orbits of semisimple Lie algebras have the structure of symplectic Lefschetz fibrations. We investigate the behavior of their fiberwise compactifications. Expressing adjoint orbits and fibers as affine varieties in their Lie algebra, we compactify them to projective varieties via homogenization of the defining ideals. We find that their Hodge diamonds vary wildly according to the choice of homogenization, and that extensions of the potential to the compactification must acquire degenerate singularities. [ABSTRACT FROM AUTHOR]

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

We consider the Dirichlet to Neumann operator for Abelian Yang-Mills boundary conditions. The aim is constructing a complex structure for the symplectic space of boundary conditions of Euler-Lagrange solutions modulo gauge for space-time manifolds with smooth boundary. Thus we prepare a suitable scenario for geometric quantization within the reduced symplectic space of boundary conditions of Abelian gauge fields. [ABSTRACT FROM AUTHOR]

We study variations of Hodge structures over a Picard modular surface, and compute the weights and types of their degenerations through the cusps of the Baily-Borel compactification. These computations are one of the key inputs which allow Wildeshaus [On the interior motive of certain Shimura varieties: the case of Picard surfaces, Manuscripta Math. 148(3) (2015) 351-377] to construct motives associated with Picard modular forms. [ABSTRACT FROM AUTHOR]

We discuss a set of novel discrete symmetry transformations of the supersymmetric quantum mechanical model of a charged particle moving on a sphere in the background of Dirac magnetic monopole. The usual five continuous symmetries (and their conserved Noether charges) and two discrete symmetries together provide the physical realizations of the de Rham cohomological operators of differential geometry. We have also exploited the supervariable approach to derive the nilpotent SUSY transformations and provided the geometrical interpretation in the language of translational generators along the Grassmannian directions and onto -dimensional supermanifold. [ABSTRACT FROM AUTHOR]

We propose a Hodge theory for the spaces featuring at the second step either in the Frölicher spectral sequence of an arbitrary compact complex manifold or in the spectral sequence associated with a pair of complementary regular holomorphic foliations on such a manifold. The main idea is to introduce a Laplace-type operator associated with a given Hermitian metric on whose kernel in every bidegree is isomorphic to in either of the two situations discussed. The surprising aspect is that this operator is not a differential operator since it involves a harmonic projection, although it depends on certain differential operators. We then use this Hodge isomorphism for to give sufficient conditions for the degeneration at of the spectral sequence considered in each of the two cases in terms of the existence of certain metrics on . For example, in the Frölicher case, we prove degeneration at if there exists an SKT metric (i.e. such that ) whose torsion is small compared to the spectral gap of the elliptic operator defined by . In the foliated case, we obtain degeneration at under a hypothesis involving the Laplacians and associated with the splitting induced by the foliated structure. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we demonstrate the existence of a set of novel discrete symmetry transformations in the case of an interacting supersymmetric quantum mechanical model of a system of an electron moving on a sphere in the background of a magnetic monopole and establish its interpretation in the language of differential geometry. These discrete symmetries are, over and above, the usual three continuous symmetries of the theory which together provide the physical realizations of the de Rham cohomological operators of differential geometry. We derive the nilpotent SUSY transformations by exploiting our idea of supervariable approach and provide geometrical meaning to these transformations in the language of Grassmannian translational generators on a -dimensional supermanifold on which our SUSY quantum mechanical model is generalized. We express the conserved supercharges and the invariance of the Lagrangian in terms of the supervariables (obtained after the imposition of the SUSY invariant restrictions) and provide the geometrical meaning to (i) the nilpotency property of the supercharges, and (ii) the SUSY invariance of the Lagrangian of our SUSY theory. [ABSTRACT FROM AUTHOR]

We consider the toy model of a rigid rotor as an example of the Hodge theory within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism and show that the internal symmetries of this theory lead to the derivation of canonical brackets amongst the creation and annihilation operators of the dynamical variables where the definition of the canonical conjugate momenta is not required. We invoke only the spin-statistics theorem, normal ordering and basic concepts of continuous symmetries (and their generators) to derive the canonical brackets for the model of a one -dimensional (1D) rigid rotor without using the definition of the canonical conjugate momenta anywhere. Our present method of derivation of the basic brackets is conjectured to be true for a class of theories that provide a set of tractable physical examples for the Hodge theory. [ABSTRACT FROM AUTHOR]

It is shown that the Hodge theory for metric spaces based on the Alexander Spanier coboundary operator, in the presence of a measure previously developed in [4], holds for the class of compact Alexandrov spaces with curvature bounded from above. In particular, the real cohomology of the space is isomorphic to the corresponding space of harmonic co-chains. [ABSTRACT FROM AUTHOR]

We provide a construction of associating a de Rham subbundle to a Higgs subbundle in characteristic p in the geometric case. As applications, we obtain a Higgs semistability result and a W2-unliftable result. [ABSTRACT FROM AUTHOR]

Taking the simple examples of an Abelian 1-form gauge theory in two (1+1)-dimensions, a 2-form gauge theory in four (3+1)-dimensions and a 3-form gauge theory in six (5+1)-dimensions of space-time, we establish that such gauge theories respect, in addition to the gauge symmetry transformations that are generated by the first-class constraints of the theory, additional continuous symmetry transformations. We christen the latter symmetry transformations as the dual-gauge transformations. We generalize the above gauge and dual-gauge transformations to obtain the proper (anti-)BRST and (anti-)dual-BRST transformations for the Abelian 3-form gauge theory within the framework of BRST formalism. We concisely mention such symmetries for the 2D free Abelian 1-form and 4D free Abelian 2-form gauge theories and briefly discuss their topological aspects in our present endeavor. We conjecture that any arbitrary Abelian p-form gauge theory would respect the above cited additional symmetry in D = 2p(p = 1, 2, 3, ...) dimensions of space-time. By exploiting the above inputs, we establish that the Abelian 3-form gauge theory, in six (5+1)-dimensions of space-time, is a perfect model for the Hodge theory whose discrete and continuous symmetry transformations provide the physical realizations of all aspects of the de Rham cohomological operators of differential geometry. As far as the physical utility of the above nilpotent symmetries is concerned, we demonstrate that the 2D Abelian 1-form gauge theory is a perfect model of a new class of topological theory and 4D Abelian 2-form as well as 6D Abelian 3-form gauge theories are the field theoretic models for the quasi-topological field theory. [ABSTRACT FROM AUTHOR]

We establish a relation between the Hodge structure of the double cover of a nonsingular cubic surface branched along its Hessian and the Hodge structure of the triple cover of P3 branched along the cubic surface. Then we introduce a method to study the infinitesimal variations of Hodge structure of the double cover of the cubic surface. Using these results, we compute the Néron-Severi lattices for the double cover of a generic cubic surface and the Fermat cubic surface. [ABSTRACT FROM AUTHOR]

We classify the subgroups of the automorphism group of the product of four projective lines admitting an invariant anticanonical smooth divisor on which the action is free. As a first application, we describe new examples of Calabi-Yau 3-folds with small Hodge numbers. In particular, the Picard number is 1 and the number of moduli is 5. Furthermore, the fundamental group is nontrivial. We also construct a new family of minimal surfaces of general type with geometric genus zero, K2 = 3 and fundamental group of order 16. We show that this family dominates an irreducible component of dimension 4 of the moduli space of the surfaces of general type. [ABSTRACT FROM AUTHOR]

The nilpotent Becchi-Rouet-Stora-Tyutin (BRST), anti-BRST, dual-BRST and anti-dual-BRST symmetry transformations are constructed in the context of noncommutative (NC) 1-form as well as 2-form gauge theories. The corresponding Noether's charges for these symmetries on the Moyal plane are shown to satisfy the same algebra, as by the de Rham cohomological operators of differential geometry. The Hodge decomposition theorem on compact manifold is also studied. We show that noncommutative gauge theories are field theoretic models for Hodge theory. [ABSTRACT FROM AUTHOR]

Let G be a reductive affine algebraic group and let X be an affine algebraic G-variety. We establish a (poly)stability criterion for points x ∈ X in terms of intrinsically defined closed subgroups Hx of G and relate it with the numerical criterion of Mumford and with Richardson and Bate-Martin-Röhrle criteria, in the case X = GN. Our criterion builds on a close analogue of a theorem of Mundet and Schmitt on polystability and allows the generalization to the algebraic group setting of results of Johnson-Millson and Sikora about complex representation varieties of finitely presented groups. By well established results, it also provides a restatement of the non-abelian Hodge theorem in terms of stability notions. [ABSTRACT FROM AUTHOR]

In previous work, with Bartholdi and Schick [1], the authors developed a Hodge-de Rham theory for compact metric spaces, which defined a cohomology of the space at a scale α. Here, in the case of Riemannian manifolds at a small scale, we construct explicit chain maps between the de Rham complex of differential forms and the L2 complex at scale α, which induce isomorphisms on cohomology. We also give estimates that show that on smooth functions, the Laplacian of [1], when appropriately scaled, is a good approximation of the classical Laplacian. [ABSTRACT FROM AUTHOR]

In view of a raging controversy on the topic of dual-Becchi-Rouet-Stora-Tyutin (dual-BRST/co-BRST) and anti-co-BRST symmetry transformations in the context of four (3+1)-dimensional (4D) Abelian two-form and 2D (non-)Abelian one-form gauge theories, we attempt, in our present short note, to settle the dust by taking the help of mathematics of differential geometry, connected with the Hodge theory, which was the original motivation for the nomenclature of "dual-BRST symmetry" in our earlier set of works. It has been claimed, in a recent set of papers, that the co-BRST symmetries are not independent of the BRST symmetries. We show that the BRST and co-BRST symmetries are independent symmetries in the same fashion as the exterior and co-exterior derivatives are independent entities belonging to the set of de Rham cohomological operators of differential geometry. [ABSTRACT FROM AUTHOR]

Let X be a smooth projective curve of genus g ≥ 2 over ℂ. Fix n ≥ 2, d ∈ ℤ. A pair (E, ϕ) over X consists of an algebraic vector bundle E of rank n and degree d over X and a section ϕ ∈ H0(E). There is a concept of stability for pairs which depends on a real parameter τ. Let ${\mathfrak M}_\tau(n, d)$ be the moduli space of τ-semistable pairs of rank n and degree d over X. Here we prove that the cohomology groups of ${\mathfrak M}_\tau(n, d)$ are Hodge structures isomorphic to direct summands of tensor products of the Hodge structure H1(X). This implies a similar result for the moduli spaces of stable vector bundles over X. [ABSTRACT FROM AUTHOR]

We investigate a method of construction of Calabi–Yau manifolds, that is, by smoothing normal crossing varieties. We develop theories for calculating the Picard groups of the Calabi–Yau manifolds obtained in this method. As an application, we show that many examples of Kawamata and Namikawa are different Calabi–Yau three-folds although they have same Hodge numbers. [ABSTRACT FROM AUTHOR]

We investigate Calabi–Yau three folds which are small resolutions of fiber products of elliptic surfaces with section admitting reduced fibers. We start by the classification of all fibers that can appear on such varieties. Then, we find formulas to compute the Hodge numbers of obtained three folds in terms of the types of singular fibers of the elliptic surfaces. Next, we study Kummer fibrations associated to these fiber products. [ABSTRACT FROM AUTHOR]

In this paper, we continue the investigation of coherent systems of type (n, d, k) on the projective line which are stable with respect to some value of a parameter α. We consider the case k = 1 and study the variation of the moduli spaces with α. We determine inductively the first and last moduli spaces and the flip loci, and give an explicit description for ranks 2 and 3. We also determine the Hodge polynomials explicitly for ranks 2 and 3 and in certain cases for arbitrary rank. [ABSTRACT FROM AUTHOR]

This is the first paper in a series (of four) designed to show how to use geometric algebras of multivectors and extensors to a novel presentation of some topics of differential geometry which are important for a deeper understanding of geometrical theories of the gravitational field. In this first paper we introduce the key algebraic tools for the development of our program, namely the euclidean geometrical algebra of multivectors $\mathcal{C}\ell(V,G_{E})$ and the theory of its deformations leading to metric geometric algebras $\mathcal{C}\ell(V,G)$ and some special types of extensors. Those tools permit obtaining, the remarkable golden formula relating calculations in $\mathcal{C}\ell(V,G)$ with easier ones in $\mathcal{C}\ell(V,G_{E})$ (e.g. a noticeable relation between the Hodge star operators associated to G and GE). Several useful examples are worked in details for the purpose of transmitting the "tricks of the trade". [ABSTRACT FROM AUTHOR]

We demonstrate that the two (1+1)-dimensional (2D) free 1-form Abelian gauge theory provides an interesting field theoretical model for the Hodge theory. The physical symmetries of the theory correspond to all the basic mathematical ingredients that are required in the definition of the de Rham cohomological operators of differential geometry. The conserved charges, corresponding to the above continuous symmetry transformations, constitute an algebra that is reminiscent of the algebra obeyed by the de Rham cohomological operators. The topological features of the above theory are discussed in terms of the BRST and co-BRST operators. The super-de Rham cohomological operators are exploited in the derivation of the nilpotent (anti-)BRST, (anti-)co-BRST symmetry transformations and the equations of motion for all the fields of the theory, within the framework of the superfield formulation. The derivation of the equations of motion, by exploiting the super-Laplacian operator, is a completely new result in the framework of the superfield approach to BRST formalism. In an Appendix, the interacting 2D Abelian gauge theory (where there is a coupling between the U(1) gauge field and the Dirac fields) is also shown to provide a tractable field theoretical model for the Hodge theory. [ABSTRACT FROM AUTHOR]

Let X be a smooth projective curve of genus g ≥ 2 over the complex numbers. A holomorphic pair on X is a couple (E, ϕ), where E is a holomorphic bundle over X of rank n and degree d, and ϕ ∈ H0(E) is a holomorphic section. In this paper, we determine the Hodge polynomials of the moduli spaces of rank 2 pairs, using the theory of mixed Hodge structures. We also deal with the case in which E has fixed determinant. [ABSTRACT FROM AUTHOR]

In this article we find connections between the values of Gauss hypergeometric functions and the dimension of the vector space of Hodge cycles of four-dimensional cubic hypersurfaces. Since the Hodge conjecture is well-known for those varieties we calculate values of hypergeometric series on certain CM points. Our methods are based on the calculation of the Picard–Fuchs equations in higher dimensions, reducing them to the Gauss equation and then applying the Abelian Subvariety Theorem to the corresponding hypergeometric abelian varieties. [ABSTRACT FROM AUTHOR]

An appropriate definition of the Hodge duality ⋆ operation on any arbitrary dimensional supermanifold has been a long-standing problem. We define a working rule for the Hodge duality ⋆ operation on the (2+2)-dimensional supermanifold parametrized by a couple of even (bosonic) space–time variables xμ(μ = 0, 1) and a couple of odd (fermionic) variables θ and $\bar\theta$ of the Grassmann algebra. The Minkowski space–time manifold, hidden in the supermanifold and parametrized by xμ(μ = 0, 1), is chosen to be a flat manifold on which a two (1+1)-dimensional (2D) free Abelian gauge theory, taken as a prototype field theoretical model, is defined. We demonstrate the applications of the above definition (and its further generalization) for the discussion of the (anti-)co-BRST symmetries that exist for the field theoretical models of 2D and 4D free Abelian gauge theories considered on the four (2+2)- and six (4+2)-dimensional supermanifolds, respectively. [ABSTRACT FROM AUTHOR]

On promoting the type IIA side of the $\mathcal{N}=1$ heterotic/type IIA dual pairs of Ref. 1 to M-theory on a "barely G2 manifold" of Ref. 2, by spectrum-matching we show a possible triality between heterotic, M and F theories. The manifolds that figure in this triality are a self-mirror Calabi–Yau, a barely G2 manifold and an elliptically fibered Calabi–Yau four-fold with trivially rationally ruled CP1× Enriques surface base, respectively. We raise an apparent puzzle on the F-theory side, namely, the Hodge data of the 4-fold derived cannot be obtained by a naive freely acting orbifold of CY3(3,243)×T2 as one might have guessed on the basis of arguments related to dualities involving string, M and (definition of) F theories. There are some interesting properties of the antiholomorphic involution used in Ref. 1 for constructing the type IIA orientifold and by us in constructing the "barely G2 manifold," that we also study. [ABSTRACT FROM AUTHOR]

We define the notion of mirror of a Calabi-Yau manifold with a stable bundle in the context of type II strings in terms of supersymmetric cycles on the mirror. This allows us to relate the variation of Hodge structure for cohomologies arising from the bundle to the counting of holomorphic maps of Riemann surfaces with boundary on the mirror side. Moreover it opens up the possibility of studying bundles on Calabi-Yau manifolds in terms of supersymmetric cycles on the mirror. [ABSTRACT FROM AUTHOR]