Search

We extend the Ax–Schanuel theorem recently proven for Shimura varieties by Mok–Pila–Tsimerman to all varieties supporting a pure polarizable integral variation of Hodge structures. In fact, Hodge theory provides a number of conceptual simplifications to the argument. The essential new ingredient is a volume bound for Griffiths transverse subvarieties of period domains. [ABSTRACT FROM AUTHOR]

In this article, we consider L 2 harmonic forms on a complete non-compact Riemannian manifold X with a nonzero parallel form ω . The main result is that if (X , ω) is a complete G 2 - (or Spin (7) -) manifold with a d(linear) G 2 - (or Spin (7) -) structure form ω , then the L 2 harmonic 2-forms on X vanish. As an application, we prove that the instanton equation with square-integrable curvature on (X , ω) only has trivial solution. We would also consider the Hodge theory on the principal G-bundle E over (X , ω) . [ABSTRACT FROM AUTHOR]

Given a complex variety X , a linear algebraic group G and a representation ρ of the fundamental group π 1 (X , x) into G , we develop a framework for constructing a functorial mixed Hodge structure on the formal local ring of the representation variety of π 1 (X , x) into G at ρ using mixed Hodge diagrams and methods of L ∞ algebras. We apply it in two geometric situations: either when X is compact Kähler and ρ is the monodromy of a variation of Hodge structure, or when X is smooth quasi-projective and ρ has finite image. [ABSTRACT FROM AUTHOR]

The notion of Eliashberg–Gromov convex ends provides a natural restricted setting for the study of analogs of Moser's symplectic stability result in the noncompact case, and this has been significantly developed in work of Cieliebak–Eliashberg. Retaining the end structure on the underlying smooth manifold, but dropping the convexity and completeness assumptions on the symplectic forms at infinity, we show that symplectic stability holds under a natural growth condition on the path of symplectic forms. The result can be straightforwardly applied as we show through explicit examples. [ABSTRACT FROM AUTHOR]

The signature of closed oriented manifolds is well-known to be multiplicative under finite covers. This fails for Poincaré complexes as examples of C. T. C. Wall show. We establish the multiplicativity of the signature, and more generally, the topological L-class, for closed oriented stratified pseudomanifolds that can be equipped with a middle-perverse Verdier self-dual complex of sheaves, determined by Lagrangian sheaves along strata of odd codimension. This class of spaces, called L-pseudomanifolds, contains all Witt spaces and thus all pure-dimensional complex algebraic varieties. We apply this result in proving the Brasselet–Schürmann–Yokura conjecture for normal complex projective 3-folds with at most canonical singularities, trivial canonical class and positive irregularity. The conjecture asserts the equality of topological and Hodge L-class for compact complex algebraic rational homology manifolds. [ABSTRACT FROM AUTHOR]

Hodge correlators are complex numbers given by certain integrals assigned to a smooth complex curve. We show that they are correlators of a Feynman integral, and describe the real mixed Hodge structure on the pronilpotent completion of the fundamental group of the curve. We introduce motivic correlators, which are elements of the motivic Lie algebra and whose periods are the Hodge correlators. They describe the motivic fundamental group of the curve. We describe variations of real mixed Hodge structures on a variety by certain connections on the product of the variety by twistor plane. We call them twistor connections. In particular, we define the canonical period map on variations of real mixed Hodge structures. We show that the obtained period functions satisfy a simple Maurer–Cartan type non-linear differential equation. Generalizing this, we suggest a DG-enhancement of the subcategory of Saito's Hodge complexes with smooth cohomology. We show that when the curve varies, the Hodge correlators are the coefficients of the twistor connection describing the corresponding variation of real MHS. Examples of the Hodge correlators include classical and elliptic polylogarithms, and their generalizations. The simplest Hodge correlators on the modular curves are the Rankin–Selberg integrals. Examples of the motivic correlators include Beilinson's elements in the motivic cohomology, e.g. the ones delivering the Beilinson–Kato Euler system on modular curves. [ABSTRACT FROM AUTHOR]

Mixed boundary conditions are introduced to finite element exterior calculus. We construct smoothed projections from Sobolev de Rham complexes onto finite element de Rham complexes which commute with the exterior derivative, preserve homogeneous boundary conditions along a fixed boundary part, and satisfy uniform bounds for shape-regular families of triangulations and bounded polynomial degree. The existence of such projections implies stability and quasi-optimal convergence of mixed finite element methods for the Hodge Laplace equation with mixed boundary conditions. In addition, we prove the density of smooth differential forms in Sobolev spaces of differential forms over weakly Lipschitz domains with partial boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, we study the symplectic cohomologies and symplectic harmonic forms which introduced by Tseng and Yau. Based on this, we get if (M 2 n, ω) is a closed symplectic parabolic manifold which satisfies the hard Lefschetz property, then its Euler number satisfies the inequality (− 1) n χ (M 2 n) ≥ 0 ⁠. [ABSTRACT FROM AUTHOR]

Let X be a complete intersection inside a variety M with finite-dimensional motive and for which the Lefschetz-type conjecture B (M) holds. We show how conditions on the niveau filtration on the homology of X influence directly the niveau on the level of Chow groups. This leads to a generalization of Voisin's result. The latter states that if M has trivial Chow groups and if X has non-trivial variable cohomology parametrized by c-dimensional algebraic cycles, then the cycle class maps A k (X) → H 2 k (X) are injective for k < c ⁠. We give variants involving group actions, which lead to several new examples with finite-dimensional Chow motives. [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]

Abstract Any irreducible Dynkin diagram Δ is obtained from an irreducible Dynkin diagram Δ h of type ADE by folding via graph automorphisms. For any simple complex Lie group G with Dynkin diagram Δ and compact Riemann surface Σ , we give a Lie-theoretic construction of families of quasi-projective Calabi–Yau threefolds together with an action of graph automorphisms over the Hitchin base associated to the pair (Σ , G). They induce families of global quotient stacks with trivial canonical class, referred to as Calabi–Yau orbifolds, over the same base. Their intermediate Jacobian fibration, constructed in terms of equivariant cohomology, is isomorphic to the Hitchin system of the same type away from singular fibers. [ABSTRACT FROM AUTHOR]

Abstract The cyclotomic p -adic multi-zeta values are the p -adic periods of π 1 u n i (G m ∖ μ M , ⋅) , the unipotent fundamental group of the multiplicative group minus the M -th roots of unity. In this paper, we compute the cyclotomic p -adic multi-zeta values at all depths. This paper generalizes the results in [9] and [10]. Since the main result gives quite explicit formulas we expect it to be useful in proving non-vanishing and transcendence results for these p -adic periods and also, through the use of p -adic Hodge theory, in proving non-triviality results for the corresponding p -adic Galois representations. [ABSTRACT FROM AUTHOR]

Abstract Let M be a 2 n -dimensional smooth manifold with a symplectic pair which is a pair of closed 2-forms of constant ranks with complementary kernel foliations. Similar to Moser's stability theorem for symplectic forms, one desires to establish a stability theorem for symplectic pairs. Some sufficient and necessary conditions are obtained by Bande, Ghiggini and Kotschick. In this article, we consider a technical problem relating to the stability theorem. To complete the proof of the stability theorem for symplectic pairs, we verify the smoothness of the isotopy which is ignored in the literature. The Hodge theory for Riemannian foliation is crucial to our discussion. [ABSTRACT FROM AUTHOR]

Abstract We study the Milnor monodromies of non-isolated hypersurface singularities and show that the reduced cohomology groups of the Milnor fibers are concentrated in the middle degree for some eigenvalues of the monodromies. As an application of this result, we give an explicit formula for some parts of their Jordan normal forms. [ABSTRACT FROM AUTHOR]

Abstract In [13] , initially motivated by questions about the Hodge line bundle of a Hodge-type Shimura variety, we singled out a generalization of the notion of minuscule character which we termed quasi-constant. Here we prove that the character of the Hodge line bundle is always quasi-constant. Furthermore, we classify the quasi-constant characters of an arbitrary connected, reductive group over an arbitrary field. As an application, we observe that, if μ is a quasi-constant cocharacter of an F p -group G , then our construction of group-theoretical Hasse invariants in loc. cit. applies to the stack G - Zip μ , without any restrictions on p , even if the pair (G , μ) is not of Hodge type and even if μ is not minuscule. We conclude with a more speculative discussion of some further motivation for considering quasi-constant cocharacters in the setting of our program outlined in loc. cit. [ABSTRACT FROM AUTHOR]

Abstract We transform the positive-genus real Gromov–Witten invariants of many real-orientable symplectic threefolds into signed counts of curves. These integer invariants provide lower bounds for counts of real curves of a given genus that pass through conjugate pairs of constraints. We conclude with some implications and related conjectures for Hodge integrals. [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]

In the 80's H. Masur and W. Veech defined two numerical invariants of strata of abelian differentials: the volume and the Siegel-Veech constant. Based on numerical experiments, A. Eskin and A. Zorich proposed a series of conjectures for the large genus asymptotics of these invariants. By a careful analysis of the asymptotic behavior of quasi-modular forms, D. Chen, M. Moeller, and D. Zagier proved that this conjecture holds for strata of differentials with simple zeros. Here, with a mild assumption of existence of a good metric, we show that the conjecture holds for the other extreme case, i.e. for strata of differentials with a unique zero. Our main ingredient is the expression of the numerical invariants of these strata in terms of Hodge integrals on moduli spaces of curves. [ABSTRACT FROM AUTHOR]

We extend the study of the de Rham operator with ideal boundary conditions from the case of isolated conic singularities, as analyzed by Cheeger, to the case of arbitrary stratified pseudomanifolds. We introduce a class of ideal boundary conditions and the notion of mezzoperversity, which intermediates between the standard lower and upper middle perversities in intersection theory, as interpreted in this de Rham setting, and show that the de Rham operator with these boundary conditions is Fredholm and has compact resolvent. We also prove an isomorphism between the resulting Hodge and L 2 L^{2} de Rham cohomology groups, and that these are independent of the choice of iterated edge metric. On spaces which admit ideal boundary conditions of this type which are also self-dual, which we call 'Cheeger spaces', we show that these Hodge/de Rham cohomology groups satisfy Poincaré duality. [ABSTRACT FROM AUTHOR]

The standard mixed finite element approximations of Hodge Laplace problems associated with the de Rham complex are based on proper discrete subcomplexes. As a consequence, the exterior derivatives, which are local operators, are computed exactly. However, the approximations of the associated coderivatives are nonlocal. In fact, this nonlocal property is an inherent consequence of the mixed formulation of these methods, and can be argued to be an undesired effect of these schemes. As a consequence, it has been argued, at least in special settings, that more local methods may have improved properties. In the present paper, we construct such methods by relying on a careful balance between the choice of finite element spaces, degrees of freedom, and numerical integration rules. Furthermore, we establish key convergence estimates based on a standard approach of variational crimes. [ABSTRACT FROM AUTHOR]

We introduce two Hopf algebroids associated to a proper and holomorphic Lie group action on a complex manifold. We prove that the cyclic cohomology of each Hopf algebroid is equal to the Dolbeault cohomology of invariant differential forms. When the action is cocompact, we develop a generalized complex Hodge theory for the Dolbeault cohomology of invariant differential forms. We prove that every cyclic cohomology class of these two Hopf algebroids can be represented by a generalized harmonic form. This implies that the space of cyclic cohomology of each Hopf algebroid is finite dimensional. As an application of the techniques developed in this paper, we generalize the Serre duality and prove a Kodaira type vanishing theorem. [ABSTRACT FROM AUTHOR]

Let D be a simple normal crossing divisor in a smooth complex projective variety X. We show that the existence on X - D X-D of a non-trivial polarized complex variation of Hodge structures with integral monodromy implies that the pair (X , D) (X,D) has a non-zero logarithmic symmetric differential (a section of a symmetric power of the logarithmic cotangent bundle). When the corresponding period map is generically immersive, we show more precisely that the logarithmic cotangent bundle is big. [ABSTRACT FROM AUTHOR]

We study how the geometry of a projective variety X is reflected in the positivity properties of the diagonal Δ X considered as a cycle on X × X . We analyze when the diagonal is big, when it is nef, and when it is rigid. In each case, we give several implications for the geometric properties of X . For example, when the cohomology class of Δ X is big, we prove that the Hodge groups H k , 0 ( X ) vanish for k > 0 . We also classify varieties of low dimension where the diagonal is nef and big. [ABSTRACT FROM AUTHOR]

Let K be a finite extension of Q p . Let A , B be abelian varieties over K with good reduction. For any integer m ≥ 1 , we consider the Galois symbol K ( K ; A , B ) / m → H 2 ( K , A [ m ] ⊗ B [ m ] ) , where K ( K ; A , B ) is the Somekawa K -group attached to A , B . This map is a generalization of the Galois symbol K 2 M ( K ) / m → H 2 ( K , μ m ⊗ 2 ) of the Bloch–Kato conjecture, where K 2 M ( K ) is the Milnor K -group of K . In this paper we give a geometric description of the image of this generalized Galois symbol by looking at the Tate duality pairing H 2 ( K , A [ m ] ⊗ B [ m ] ) × Hom G K ( A [ m ] , B ⋆ [ m ] ) → Z / m , where B ⋆ is the dual abelian variety of B . Under this perfect pairing we compute the exact annihilator of the image of the Galois symbol in terms of an object of integral p -adic Hodge theory. In this way we generalize a result of Tate for H 1 . Moreover, our result has applications to zero cycles on abelian varieties. [ABSTRACT FROM AUTHOR]

Stankova and West showed that for any non-negative integer s and any permutation γ of { 4 , 5 , … , s + 3 } there are as many permutations that avoid 231 γ as there are that avoid 312 γ . We extend this result to the setting of words. [ABSTRACT FROM AUTHOR]

We introduce the notion of effective resistance for a simplicial network ( X , R ) where X is a simplicial complex and R is a set of resistances for the top simplices, and prove two formulas generalizing previous results concerning effective resistance for resistor networks. Our approach, based on combinatorial Hodge theory, is to assign a unique harmonic class to a current generator σ , an extra top-dimensional simplex to be attached to X . We will show that the harmonic class gives rise to the current I σ and the voltage V σ for X ∪ σ , satisfying Thomson's energy-minimizing principle and Ohm's law for simplicial networks. The effective resistance R σ of a current generator σ shall be defined as a ratio of the σ -components of V σ and I σ . By introducing potential for voltage vectors, we present a formula for R σ via the inverse of the weighted combinatorial Laplacian of X in codimension one. We also derive a formula for R σ via weighted high-dimensional tree-numbers for X , providing a combinatorial interpretation for R σ . As an application, we generalize Foster's Theorem, and discuss various high-dimensional examples. [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]

We prove that the Beilinson regulator, which is a map from K -theory to absolute Hodge cohomology of a smooth variety, admits a refinement to a map of E ∞ -ring spectra in the sense of algebraic topology. To this end we exhibit absolute Hodge cohomology as the cohomology of a commutative differential graded algebra over R . The associated spectrum to this CDGA is the target of the refinement of the regulator and the usual K -theory spectrum is the source. To prove this result we compute the space of maps from the motivic K -theory spectrum to the motivic spectrum that represents absolute Hodge cohomology using the motivic Snaith theorem. We identify those maps which admit an E ∞ -refinement and prove a uniqueness result for these refinements. [ABSTRACT FROM AUTHOR]

We derive the continuous nilpotent symmetries of the four ( 3 + 1 ) -dimensional (4 D ) model of the Hodge theory (i.e. 4 D Abelian 2-form gauge theory) by exploiting the beauty and strength of the symmetry invariant restrictions on the (anti-)chiral superfields. The above off-shell nilpotent symmetries are the Becchi–Rouet–Stora–Tyutin (BRST), anti-BRST and (anti-)co-BRST transformations which turn up beautifully due to the (anti-)BRST and (anti-)co-BRST invariant restrictions on the (anti-)chiral superfields that are defined on the (4, 1)-dimensional (anti-)chiral super-submanifolds of the general (4, 2)-dimensional supermanifold on which our ordinary 4 D theory is generalized. The latter supermanifold is characterized by the superspace coordinates Z M = ( x μ , θ , θ ̄ ) where x μ ( μ = 0 , 1 , 2 , 3 ) are the bosonic coordinates and a pair of Grassmannian variables θ and θ ̄ are fermionic in nature as they obey the standard relationships: θ 2 = θ ̄ 2 = 0 , θ θ ̄ + θ ̄ θ = 0 ). The derivation of the proper (anti-)co-BRST symmetries and proof of the absolute anticommutativity property of the conserved (anti-)BRST and (anti-) co-BRST charges are novel results of our present investigation (where only the (anti-)chiral superfields and their super-expansions have been taken into account). [ABSTRACT FROM AUTHOR]

We give a sufficient condition for a class of tame compactified Landau-Ginzburg models in the sense of Katzarkov-Kontsevich-Pantev to satisfy some versions of their conjectures. We also give examples that satisfy the condition. The relations to the quantum D-modules of Fano manifolds and the original conjectures are explained in the appendices. [ABSTRACT FROM AUTHOR]

We compute the Alexander polynomial of a nonreduced nonirreducible complex projective plane curve with mutually coprime orders of vanishing along its irreducible components in terms of certain multiplier ideals. [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 prove results concerning the behavior of Hodge ideals under restriction to hypersurfaces or fibers of morphisms, and addition. The main tool is the description of restriction functors for mixed Hodge modules by means of the V-filtration. [ABSTRACT FROM AUTHOR]

In this paper we study the existence of ghost classes in the cohomology of the Shimura variety associated to the group of symplectic similitudes GSp4. The existence of ghost classes for the trivial coefficient system is known. We show that ghost classes only exist for the trivial coefficient system and they lie in the cohomology group in degree 2. Moreover we prove that the weight of the mixed Hodge structure associated to the space of ghost classes is the middle weight. [ABSTRACT FROM AUTHOR]

We complete the construction of the fundamental diagram of various partial compactifications of themoduli spaces of mixedHodge structures with polarized graded quotients. The diagram includes the space of nilpotent orbits, the space of SL(2)-orbits, and the space of Borel-Serre orbits.We give amplifications of this fundamental diagram and amplify the relations of these spaces.We describe how this work is useful in understanding asymptotic behaviors of Beilinson regulators and of local height pairings in degeneration.We discuss mild degenerations in which regulators converge. [ABSTRACT FROM AUTHOR]

We determine the image of the (strongly) parabolic Hitchin map for all parabolics in classical groups and G2. Surprisingly, we find that the image is isomorphic to an affine space in all cases, except for certain 'bad parabolics' in type D, where the image can be singular. [ABSTRACT FROM AUTHOR]

In this paper, we develop symplectic Hodge theory on transversely symplectic foliations. In particular, we establish the symplectic dδ-lemma for any such foliations with the (transverse) s-Lefschetz property. As transversely symplectic foliations include many geometric structures, such as contact manifolds, co-symplectic manifolds, symplectic orbifolds and symplectic quasifolds as special examples, our work provides a unifying treatment of symplectic Hodge theory in these geometries. As an application, we show that on compact K-contact manifolds, the s-Lefschetz property implies a general result on the vanishing of cup products, and that the cup length of a 2n + 1 dimensional compact K-contact manifold with the (transverse) s-Lefschetz property is at most 2n - s. For any even integer s≥2, we also apply our main result to produce examples of K-contact manifolds that are s-Lefschetz, but not (s + 1)-Lefschetz. [ABSTRACT FROM AUTHOR]

In this paper we study the existence of ghost classes in the cohomology of the Shimura variety associated to the group of symplectic similitudes GSp4. The existence of ghost classes for the trivial coefficient system is known. We show that ghost classes only exist for the trivial coefficient system and they lie in the cohomology group in degree 2. Moreover we prove that the weight of the mixed Hodge structure associated to the space of ghost classes is the middle weight. [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]

Lagrangians of the field-theory model of a scalar field are considered as 4-forms on a Riemannian manifold. The model is constructed on the basis of the Hodge inner product, this latter being an analog of the scalar product of two functions. Including the basis fields in the action of the terms with tetrads makes it possible to reproduce the Klein-Gordon equation and the Maxwell equations, and also the Einstein-Hilbert action. We conjecture that the principle of construction of the Lagrangians as 4-forms can give a criterion restricting possible forms of the field-theory models. [ABSTRACT FROM AUTHOR]

In this paper, we establish some formulas on the Mordell-Weil rank and the Hodge number h1'1 for a fibration. [ABSTRACT FROM AUTHOR]

We give a way of constructing real variations of mixed Hodge structures over compact Kähler manifolds by using mixed Hodge structures on Sullivan’s 1-minimal models of certain differential graded algebras associated with real variations of Hodge structures. [ABSTRACT FROM AUTHOR]

In this article, we give a complex-geometric proof of the Alexandrov–Fenchel inequality without using toric compactifications. The idea is to use the Legendre transform and develop the Brascamp–Lieb proof of the Prékopa theorem. New ingredients in our proof include an integration of Timorin's mixed Hodge–Riemann bilinear relation and a mixed norm version of Hörmander's L 2 -estimate, which also implies a non-compact version of the Khovanskiĭ–Teissier inequality. [ABSTRACT FROM AUTHOR]