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We solve Prill's problem, originally posed by David Prill in the late 1970s and popularized in Arbarello, Cornalba, Griffiths and Harris's "Geometry of Algebraic Curves." That is, for any curve Y of genus 2, we produce a finite 'etale degree 36 connected cover f: X → Y where, for every point y ∈ Y, the preimage f-1(y) moves in a pencil. [ABSTRACT FROM AUTHOR]

We give a formula comparing the E-series of the moduli stacks of rank 2 degree 0 semistable Higgs bundles in genus g ≥ 2 to intersection E-polynomials of its coarse moduli space. A parallel formula holds in various 2-Calabi–Yau settings, for example for sheaves on K3 surfaces, or preprojective algebras of g -loop quivers. As a consequence we provide evidence for a conjecture of Davison on the BPS cohomology of Higgs bundles, which has implications for non-abelian Hodge theory for stacks. We apply the formula to cohomological χ -independence tests for BPS cohomology of Higgs bundles and K3 surfaces. [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 prove a filtered version of the Homotopy Transfer Theorem which gives an A-infinity algebra structure on any page of the spectral sequence associated to a filtered dg-algebra. We then develop various applications to the study of the geometry and topology of complex manifolds, using the Hodge filtration, as well as to complex algebraic varieties, using mixed Hodge theory. [ABSTRACT FROM AUTHOR]

We describe the E-infinity algebra structure on the complex of singular cochains of a topological space, in the context of sheaf theory. As a first application, for any algebraic variety we define a weight filtration compatible with its E-infinity structure. This naturally extends the theory of mixed Hodge structures in rational homotopy to p-adic homotopy theory. The spectral sequence associated to the weight filtration gives a new family of algebraic invariants of the varieties for any coefficient ring, carrying Steenrod operations. As a second application, we promote Deligne's intersection complex computing intersection cohomology, to a sheaf carrying E-infinity structures. This allows for a natural interpretation of the Steenrod operations defined on the intersection cohomology of any topological pseudomanifold. [ABSTRACT FROM AUTHOR]

This is an elementary introduction to the Hodge Laplacian on a graph, a higher-order generalization of the graph Laplacian. We will discuss basic properties including cohomology and Hodge theory. The main feature of our approach is simplicity, requiring only knowledge of linear algebra and graph theory. We have also isolated the algebra from the topology to show that a large part of cohomology and Hodge theory is nothing more than the linear algebra of matrices satisfying AB = 0. For the remaining topological aspect, we cast our discussion entirely in terms of graphs as opposed to less familiar topological objects like simplicial complexes. [ABSTRACT FROM AUTHOR]

We show that odd order transcendental elements of the Brauer group of a K3 surface can obstruct the Hasse principle. We exhibit a general K3 surface Y of degree 2 over Q together with a 3-torsion Brauer class α that is unramified at all primes except for 3, but ramifies at all 3-adic points of Y. Motivated by Hodge theory, the pair (Y, α) is constructed from a cubic fourfold X of discriminant 18 birational to a fibration into sextic del Pezzo surfaces over the projective plane. Notably, our construction does not rely on the presence of a central simple algebra representative for α. Instead, we prove that a sufficient condition for such a Brauer class to obstruct the Hasse principle is insolubility of the fourfold X (and hence the fibers) over Q3 and local solubility at all other primes. [ABSTRACT FROM AUTHOR]

The homological conjectures, which date back to Peskine, Szpiro and Hochster in the late 60's, make fundamental predictions about syzygies and intersection problems in commutative algebra. They were settled long ago in the presence of a base field and led to tight closure theory, a powerful tool to investigate singularities in characteristic p. Recently, perfectoid techniques coming from p-adic Hodge theory have allowed us to get rid of any base field; this solves the direct summand conjecture and establishes the existence and weak functoriality of big Cohen-Macaulay algebras, which solve in turn the homological conjectures in general. This also opens the way to the study of singularities in mixed characteristic. We sketch a broad outline of this story, taking lastly a glimpse at ongoing work by L. Ma and K. Schwede, which shows how such a study could build a bridge between singularity theory in characteristic p and in characteristic 0. [ABSTRACT FROM AUTHOR]

Let $k=\mathbb{C}$ be the field of complex numbers. In this article we construct Hodge realization functors defined on the triangulated categories of étale motives with rational coefficients. Our construction extends to every smooth quasi-projective $k$-scheme, the construction done by Nori over a field, and relies on the original version of the basic lemma proved by Beĭlinson. As in the case considered by Nori, the realization functor factors through the bounded derived category of a perverse version of the Abelian category of Nori motives. [ABSTRACT FROM AUTHOR]

We study rank 1 flat bundles over solvmanifolds whose cohomologies are nontrivial. By using Hodge theoretical properties for all topologically trivial rank 1 flat bundles, we represent the structure theorem of Kähler solvmanifolds as extensions of Hasegawa's result and Benson–Gordon's result for nilmanifolds. [ABSTRACT FROM AUTHOR]

For a class of co-chain complexes in the category of pre-Hilbert $$A$$ -modules, we prove that their cohomology groups equipped with the canonical quotient topology are pre-Hilbert $$A$$ -modules, and derive the Hodge theory and, in particular, the Hodge decomposition for them. As an application, we show that $$A$$ -elliptic complexes of pseudodifferential operators acting on sections of finitely generated projective $$A$$ -Hilbert bundles over compact manifolds belong to this class if the images of the continuous extensions of their associated Laplace operators are closed. Moreover, we prove that the cohomology groups of these complexes share the structure of the fibers, in the sense that they are also finitely generated projective Hilbert $$A$$ -modules. [ABSTRACT FROM AUTHOR]

For a unital $$C^{*}$$ -algebra $$A$$ , we prove that the cohomology groups of $$A$$ -elliptic complexes of pseudodifferential operators in finitely generated projective $$A$$ -Hilbert bundles over compact manifolds are finitely generated $$A$$ -modules and Banach spaces provided the images of certain extensions of the so-called associated Laplacians are closed. We also prove that under this condition, the cohomology groups are isomorphic to the kernels of the associated Laplacians. This establishes a Hodge theory for these structures. [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]

Abstract: We prove a formula expressing the motivic integral (Loeser and Sebag, 2003) of a K3 surface over with semi-stable reduction in terms of the associated limit mixed Hodge structure. Secondly, for every smooth variety over a complete discrete valuation field we define an analogue of the monodromy pairing, constructed by Grothendieck in the case of abelian varieties, and prove that our monodromy pairing is a birational invariant of the variety. Finally, we propose a conjectural formula for the motivic integral of maximally degenerate K3 surfaces over an arbitrary complete discrete valuation field and prove this conjecture for Kummer K3 surfaces. [Copyright &y& Elsevier]

Abstract: We generalize the theorem of E. Cattani, P. Deligne, and A. Kaplan to admissible variations of mixed Hodge structure. [Copyright &y& Elsevier]

We introduce in this paper a hypercohomology version of the resonance varieties and obtain some relations to the characteristic varieties of rank one local systems on a smooth quasi-projective complex variety M. A logarithmic resonance variety is also considered and, as an application, we determine the first characteristic variety of the configuration space of n distinct labeled points on an elliptic curve. Finally, for a logarithmic 1-form α on M we investigate the relation between the resonance degree of α and the codimension of the zero set of α on a good compactification of M. This question was inspired by the recent work by Cohen, Denham, Falk and Varchenko. [ABSTRACT FROM AUTHOR]

We show that an algebra over a cyclic operad supplied with an additional linear algebra datum called Hodge decomposition admits a minimal model whose structure maps are given in terms of summation over trees. This minimal model is unique up to homotopy. [ABSTRACT FROM AUTHOR]

Abstract: This short Note proves a generalization of the Hirzebruch Riemann–Roch theorem equivalent to the Cardy condition. This is done using an earlier result that explicitly describes what the Mukai pairing on Hochschild homology descends to in Hodge cohomology via the Hochschild–Kostant–Rosenberg map twisted by the root Todd genus. To cite this article: A.C. Ramadoss, C. R. Acad. Sci. Paris, Ser. I 347 (2009). [Copyright &y& Elsevier]

We demonstrate that the four-dimensional (4D) ((3+1)-dimensional) free Abelian 2-form gauge theory presents a tractable field-theoretical model for the Hodge theory where the well-defined symmetry transformations correspond to the de Rham cohomological operators of differential geometry. The conserved charges, corresponding to the above continuous symmetry transformations, obey an algebra that is reminiscent of the algebra obeyed by the cohomological operators. The discrete symmetry transformation of the theory represents the realization of the Hodge duality operation that exists in the relationship between the exterior and co-exterior derivatives of differential geometry. Thus, we provide the realizations of all the mathematical quantities, associated with the de Rham cohomological operators, in the language of the symmetries of the present 4D free Abelian 2-form gauge theory. [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]

Let V c pS be a reduced and irreducible threefold of degree s, complete intersection on a smooth hypersurface of degree t, with s > t² - t. In this paper, we prove that if the singular locus of V consists of δ < 3s/8t ordinary double points, then any projective surface contained in V is a complete intersection on V. In particular, V is Q-factorial. [ABSTRACT FROM AUTHOR]

We exhibit a direct correspondence between the potential defining the H1,1 small quantum module structure on the cohomology of a Calabi-Yau manifold and the asymptotic data of the A-model variation of Hodge structure. This is done in the abstract context of polarized variations of Hodge structure and Frobenius modules. [ABSTRACT FROM AUTHOR]

This paper gives a necessary criterion in terms of Hodge theory for representability by quaternion algebras of certain 2-torsion classes in the unramified Brauer group of a complex function field. This criterion is used to give examples of threefolds with unramified Brauer group elements which are the classes of biquaternion division algebras. [ABSTRACT FROM PUBLISHER]

In substitution for Dirac monopoles with string (and for topological monopoles), ‘‘monopoles without string’’ have recently been introduced on the basis of a generalized potential, the sum of a vector A, and a pseudovector γ5 B potential. By making recourse to the Clifford bundle C(τM,g) [(TxM,g)=R1,3; C(TxM,g)=R1,3], which just allows adding together for each x∈M tensors of different ranks, in a previous paper a Lagrangian and Hamiltonian formalism was constructed for interacting monopoles and charges that can be regarded as satisfactory from various points of view. In the present article, after having ‘‘completed’’ the formalism, a purely geometrical interpretation of it is put forth within the Kähler–Clifford bundle K(τ*M,g) of differential forms, essential ingredients being a generalized curvature and the Hodge decomposition theorem. Thus the way is paved for the extension of our ‘‘monopoles without string’’ to non-Abelian gauge groups. The analogy with supersymmetric theories is apparent. [ABSTRACT FROM AUTHOR]