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For any smooth proper rigid space $X$ over a complete algebraically closed extension $K$ of $\mathbb {Q}_p$ we give a geometrisation of the $p$ -adic Simpson correspondence of rank one in terms of analytic moduli spaces: the $p$ -adic character variety is canonically an étale twist of the moduli space of topologically torsion Higgs line bundles over the Hitchin base. This also eliminates the choice of an exponential. The key idea is to relate both sides to moduli spaces of $v$ -line bundles. As an application, we study a major open question in $p$ -adic non-abelian Hodge theory raised by Faltings, namely which Higgs bundles correspond to continuous representations under the $p$ -adic Simpson correspondence. We answer this question in rank one by describing the essential image of the continuous characters $\pi ^{{\mathrm {\acute {e}t}}}_1(X)\to K^\times$ in terms of moduli spaces: for projective $X$ over $K=\mathbb {C}_p$ , it is given by Higgs line bundles with vanishing Chern classes like in complex geometry. However, in general, the correct condition is the strictly stronger assumption that the underlying line bundle is a topologically torsion element in the topological group $\operatorname {Pic}(X)$. [ABSTRACT FROM AUTHOR]

Let $G$ be a split reductive group over the ring of integers in a $p$ -adic field with residue field $\mathbf {F}$. Fix a representation $\overline {\rho }$ of the absolute Galois group of an unramified extension of $\mathbf {Q}_p$ , valued in $G(\mathbf {F})$. We study the crystalline deformation ring for $\overline {\rho }$ with a fixed $p$ -adic Hodge type that satisfies an analog of the Fontaine–Laffaille condition for $G$ -valued representations. In particular, we give a root theoretic condition on the $p$ -adic Hodge type which ensures that the crystalline deformation ring is formally smooth. Our result improves on all known results for classical groups not of type A and provides the first such results for exceptional groups. [ABSTRACT FROM AUTHOR]

We give a new proof of Faltings's $p$ -adic Eichler–Shimura decomposition of the modular curves via Bernstein–Gelfand–Gelfand (BGG) methods and the Hodge–Tate period map. The key property is the relation between the Tate module and the Faltings extension, which was used in the original proof. Then we construct overconvergent Eichler–Shimura maps for the modular curves providing 'the second half' of the overconvergent Eichler–Shimura map of Andreatta, Iovita and Stevens. We use higher Coleman theory on the modular curve developed by Boxer and Pilloni to show that the small-slope part of the Eichler–Shimura maps interpolates the classical $p$ -adic Eichler–Shimura decompositions. Finally, we prove that overconvergent Eichler–Shimura maps are compatible with Poincaré and Serre pairings. [ABSTRACT FROM AUTHOR]

We study horizontal semistable and horizontal de Rham representations of the absolute Galois group of a certain smooth affinoid over a $p$ -adic field. In particular, we prove that a horizontal de Rham representation becomes horizontal semistable after a finite extension of the base field. As an application, we show that every de Rham local system on a smooth rigid analytic variety becomes horizontal semistable étale locally around every classical point. We also discuss potentially crystalline loci of de Rham local systems and cohomologically potentially good reduction loci of smooth proper morphisms. [ABSTRACT FROM AUTHOR]

Let $\mathbb {V}$ be a motivic variation of Hodge structure on a $K$ -variety $S$ , let $\mathcal {H}$ be the associated $K$ -algebraic Hodge bundle, and let $\sigma \in \mathrm {Aut}(\mathbb {C}/K)$ be an automorphism. The absolute Hodge conjecture predicts that given a Hodge vector $v \in \mathcal {H}_{\mathbb {C}, s}$ above $s \in S(\mathbb {C})$ which lies inside $\mathbb {V}_{s}$ , the conjugate vector $v_{\sigma } \in \mathcal {H}_{\mathbb {C}, s_{\sigma }}$ is Hodge and lies inside $\mathbb {V}_{s_{\sigma }}$. We study this problem in the situation where we have an algebraic subvariety $Z \subset S_{\mathbb {C}}$ containing $s$ whose algebraic monodromy group $\textbf {H}_{Z}$ fixes $v$. Using relationships between $\textbf {H}_{Z}$ and $\textbf {H}_{Z_{\sigma }}$ coming from the theories of complex and $\ell$ -adic local systems, we establish a criterion that implies the absolute Hodge conjecture for $v$ subject to a group-theoretic condition on $\textbf {H}_{Z}$. We then use our criterion to establish new cases of the absolute Hodge conjecture. [ABSTRACT FROM AUTHOR]

We construct examples of smooth proper rigid-analytic varieties admitting formal models with projective special fibers and violating Hodge symmetry for cohomology in degrees ${\geq }3$. This answers negatively the question raised by Hansen and Li. [ABSTRACT FROM AUTHOR]

We generalise Simpson's nonabelian Hodge correspondence to the context of projective varieties with Kawamata log terminal (klt) singularities. The proof relies on a descent theorem for numerically flat vector bundles along birational morphisms. In its simplest form, this theorem asserts that given any klt variety X and any resolution of singularities, any vector bundle on the resolution that appears to come from X numerically, does indeed come from X. Furthermore, and of independent interest, a new restriction theorem for semistable Higgs sheaves defined on the smooth locus of a normal, projective variety is established. [ABSTRACT FROM AUTHOR]

We construct the ∧-adic crystalline and Dieudonné analogues of Hida’s ordinary ∧-adic étale cohomology, and employ integral p-adic Hodge theory to prove ∧-adic comparison isomorphisms between these cohomologies and the ∧-adic de Rham cohomology studied in Cais [The geometry of Hida families I: ∧-adic de Rham cohomology, Math. Ann. (2017), doi: 10.1007/s00208-017-1608-1 ] as well as Hida’s $∧-adic étale cohomology. As applications of our work, we provide a ‘cohomological’ construction of the family of (φ,Γ)-modules attached to Hida’s ordinary ∧-adic étale cohomology by Dee [ Φ – Γ modules for families of Galois representations, J. Algebra 235 (2001), 636–664], and we give a new and purely geometric proof of Hida’s finiteness and control theorems. We also prove suitable ∧-adic duality theorems for each of the cohomologies we construct. [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]

A technical ingredient in Faltings’ original approach to $p$-adic comparison theorems involves the construction of $K({\it\pi},1)$-neighborhoods for a smooth scheme $X$ over a mixed characteristic discrete valuation ring with a perfect residue field: every point $x\in X$ has an open neighborhood $U$ whose generic fiber is a $K({\it\pi},1)$ scheme (a notion analogous to having a contractible universal cover). We show how to extend this result to the logarithmically smooth case, which might help to simplify some proofs in $p$-adic Hodge theory. The main ingredient of the proof is a variant of a trick of Nagata used in his proof of the Noether normalization lemma. [ABSTRACT FROM PUBLISHER]

We show that the zero locus of an admissible normal function on a smooth complex algebraic variety is algebraic. In Part II of the paper, which is an appendix, we compute the Tannakian Galois group of the category of one-variable admissible real nilpotent orbits with split limit. We then use the answer to recover an unpublished theorem of Deligne, which characterizes the ${\mathrm{sl} }_{2} $-splitting of a real mixed Hodge structure. [ABSTRACT FROM AUTHOR]

We determine the structure of the Hodge ring, a natural object encoding the Hodge numbers of all compact Kähler manifolds. As a consequence of this structure, there are no unexpected relations among the Hodge numbers, and no essential differences between the Hodge numbers of smooth complex projective varieties and those of arbitrary Kähler manifolds. The consideration of certain natural ideals in the Hodge ring allows us to determine exactly which linear combinations of Hodge numbers are birationally invariant, and which are topological invariants. Combining the Hodge and unitary bordism rings, we are also able to treat linear combinations of Hodge and Chern numbers. In particular, this leads to a complete solution of a classical problem of Hirzebruch’s. [ABSTRACT FROM AUTHOR]

We prove that if W and W′ are non-zero B-pairs whose tensor product is crystalline (or semi-stable or de Rham or Hodge–Tate), then there exists a character μ such that W(μ−1) and W′(μ) are crystalline (or semi-stable or de Rham or Hodge–Tate). We also prove that if W is a B-pair and if F is a Schur functor (for example Sym n or Λn) such that F(W) is crystalline (or semi-stable or de Rham or Hodge–Tate) and if the rank of W is sufficiently large, then there is a character μ such that W(μ−1) is crystalline (or semi-stable or de Rham or Hodge–Tate). In particular, these results apply to p-adic representations. [ABSTRACT FROM AUTHOR]

In [M. Kontsevich and Y. Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson–Thomas invariants, Preprint (2011), arXiv:1006.2706v2[math.AG]], the authors, in particular, associate to each finite quiver Q with a set of vertices I the so-called cohomological Hall algebra ℋ, which is ℤI≥0-graded. Its graded component ℋγ is defined as cohomology of the Artin moduli stack of representations with dimension vector γ. The product comes from natural correspondences which parameterize extensions of representations. In the case of a symmetric quiver, one can refine the grading to ℤI≥0×ℤ, and modify the product by a sign to get a super-commutative algebra (ℋ,⋆) (with parity induced by the ℤ-grading). It is conjectured in [M. Kontsevich and Y. Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson–Thomas invariants, Preprint (2011), arXiv:1006.2706v2[math.AG]] that in this case the algebra (ℋ⊗ℚ,⋆) is free super-commutative generated by a ℤI≥0×ℤ-graded vector space of the form V =Vprim ⊗ℚ[x] , where x is a variable of bidegree (0,2)∈ℤI≥0×ℤ, and all the spaces ⨁ k∈ℤVprimγ,k, γ∈ℤI≥0. are finite-dimensional. In this paper we prove this conjecture (Theorem 1.1). We also prove some explicit bounds on pairs (γ,k) for which Vprimγ,k≠0 (Theorem 1.2). Passing to generating functions, we obtain the positivity result for quantum Donaldson–Thomas invariants, which was used by Mozgovoy to prove Kac’s conjecture for quivers with sufficiently many loops [S. Mozgovoy, Motivic Donaldson–Thomas invariants and Kac conjecture, Preprint (2011), arXiv:1103.2100v2[math.AG]]. Finally, we mention a connection with the paper of Reineke [M. Reineke, Degenerate cohomological Hall algebra and quantized Donaldson–Thomas invariants for m-loop quivers, Preprint (2011), arXiv:1102.3978v1[math.RT]]. [ABSTRACT FROM AUTHOR]

We construct motivic invariants of a subvariety of an algebraic torus from its tropicalization and initial degenerations. More specifically, we introduce an invariant of a compactification of such a variety called the ‘tropical motivic nearby fiber’. This invariant specializes in the schön case to the Hodge–Deligne polynomial of the limit mixed Hodge structure of a corresponding degeneration. We give purely combinatorial expressions for this Hodge–Deligne polynomial in the cases of schön hypersurfaces and matroidal tropical varieties. We also deduce a formula for the Euler characteristic of a general fiber of the degeneration. [ABSTRACT FROM PUBLISHER]

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]

AbstractFor the p-adic Galois representation associated to a Hilbert modular form, Carayol has shown that, under a certain assumption, its restriction to the local Galois group at a finite place not dividing pis compatible with the local Langlands correspondence. Under the same assumption, we show that the same is true for the places dividing p, in the sense of p-adic Hodge theory, as is shown for an elliptic modular form. We also prove that the monodromy-weight conjecture holds for such representations. [ABSTRACT FROM AUTHOR]

AbstractWe use Hodge theoretic methods to study homotopy types of complex projective manifolds with arbitrary fundamental groups. The main tool we use is the schematization functor$X \mapsto (X\otimes \mathbb {C})^{\mathrm {sch}}$, introduced by the third author as a substitute for the rationalization functor in homotopy theory in the case of non-simply connected spaces. Our main result is the construction of a Hodge decompositionon $(X\otimes \mathbb {C})^{\mathrm {sch}}$. This Hodge decomposition is encoded in an action of the discrete group $\mathbb {C}^{\times \delta }$on the object $(X\otimes \mathbb {C})^{\mathrm {sch}}$and is shown to recover the usual Hodge decomposition on cohomology, the Hodge filtration on the pro-algebraic fundamental group, and, in the simply connected case, the Hodge decomposition on the complexified homotopy groups. We show that our Hodge decomposition satisfies a purity property with respect to a weight filtration, generalizing the fact that the higher homotopy groups of a simply connected projective manifold have natural mixed Hodge structures. As applications we construct new examples of homotopy types which are not realizable as complex projective manifolds and we prove a formality theorem for the schematization of a complex projective manifold. [ABSTRACT FROM AUTHOR]

AbstractFor p?3 an odd prime and a nonnegative integer r?p?2, we prove a conjecture of Breuil on lattices in semi-stable representations, that is, the anti-equivalence of categories between the category of strongly divisible lattices of weight rand the category of Galois stable $\mathbb {Z}_p$-lattices in semi-stable p-adic Galois representations with Hodge?Tate weights in {0,?,r}. [ABSTRACT FROM AUTHOR]