Tensor structure on perverse Nori motives

Recently, Bondarko constructed a DG category of motives, whose homotopy category is equivalent to Voevodsky's category of effective geometric motives, assuming resolution of singularities. Soon after, Levine extended this idea to construct a DG category whose homotopy category is equivalent to the full subcategory of motives over a base-scheme S generated by the motives of smooth projective S-schemes, assuming that S is itself smooth over a perfect field. In both constructions, the tensor structure requires ℚ-coefficients. In this article, it is shown how to provide a tensor structure on the homotopy category mentioned above, when S is semi-local and essentially smooth over a field of characteristic zero. This is done by defining a pseudo-tensor structure on the DG category of motives constructed by Levine, which induces a tensor structure on its homotopy category.

In this article we further the study of noncommutative numerical motives, initiated in [30, 31]. By exploring the change-of-coefficients mechanism, we start by improving some of the main results of [30]. Then, making use of the notion of Schur-finiteness, we prove that the category NNum (k)_F of noncommutative numerical motives is (neutral) super-Tannakian. As in the commutative world, NNum (k)_F is not Tannakian. In order to solve this problem we promote periodic cyclic homology to a well-defined symmetric monoidal functor \overline{HP_\ast} on the category of noncommutative Chow motives. This allows us to introduce the correct noncommutative analogues C_{NC} and D_{NC} of Grothendieck's standard conjectures C and D . Assuming C_{NC} , we prove that NNum (k)_F can be made into a Tannakian category NNum ^\dagger(k)_F by modifying its symmetry isomorphism constraints. By further assuming D_{NC} , we neutralize the Tannakian category Num ^\dagger(k)_F using \overline{HP_\ast} . Via the (super-)Tannakian formalism, we then obtain well-defined noncommutative motivic Galois (super-)groups . Finally, making use of Deligne-Milne's theory of Tate triples, we construct explicit morphisms relating these noncommutative motivic Galois (super-)groups with the classical ones as suggested by Kontsevich.

We investigate the nature of divergences in quantum field theory, showing that they are organized in the structure of a certain “motivic Galois group” U*, which is uniquely determined and universal with respect to the set of physical theories. The renormalization group can be identified canonically with a one-parameter subgroup of U*. The group U* arises through a Riemann-Hilbert correspondence. Its representations classify equisingular flat vector bundles, where the equisingularity condition is a geometric formulation of the fact that in quantum field theory the counterterms are independent of the choice of a unit of mass. As an algebraic group scheme, U* is a semidirect product by the multiplicative group G_m of a prounipotent group scheme whose Lie algebra is freely generated by one generator in each positive integer degree. There is a universal singular frame in which all divergences disappear. When computed as iterated integrals, its coefficients are certain rational numbers that appear in the local index formula of Connes-Moscovici. When working with formal Laurent series over ℚ, the data of equisingular flat vector bundles define a Tannakian category whose properties are reminiscent of a category of mixed Tate motives.

A new design of space deployable reflector is presented. In particular, we consider closed-chain system (with central network), which as a result of transformation reaches the conical shape. In conformity with the technical specifications, individual parts of the system perform the simultaneous motion in the radial and axial directions. The main motion of the system produced by geometric constraints is studied, i.e., we consider the degree of structural motion. Parametric degrees of freedom caused by technological errors, modes of motion, types of load or deployment velocity are not taken into consideration at this stage. A peculiar feature of the deployable structure presented in the paper is that, as compared with analogous structures, for connecting the sections with one another there is no need of using synchronization devices in both—upper and lower kinematic chains simultaneously. This structural mechanism is a differential lever mechanism, the driving elements of which enable us to obtain the desired law of motion of every characteristic link. The kinematic model represents the whole system. Therefore, we can construct the function of position of the lever mechanism and also the kinematic functions of transmission. For the preliminary investigation of the structure and making possible changes in it, two mathematical models have been constructed by means of the ANSYS software using the Ansys Parametric Design Language. The degrees of freedom of the hinges are simulated in local coordinate systems and are as much as possible approximated to the real model. Calculations are performed for various kinds of loads and appropriate results are obtained.

On neutral Tannakian subcategories of loop quiver representations

This chapter summarizes the classical results of Hodge theory concerning algebraic maps. Hodge theory gives nontrivial restrictions on the topology of a nonsingular projective variety, or, more generally, of a compact Kähler manifold: the odd Betti numbers are even, the hard Lefschetz theorem, the formality theorem, stating that the real homotopy type of such a variety is, if simply connected, determined by the cohomology ring. Similarly, Hodge theory gives nontrivial topological constraints on algebraic maps. This chapter focuses on the latter, as it considers how the existence of an algebraic map f : X → Y of complex algebraic varieties is reflected in the topological invariants of X.

The Hard Lefschetz theorem is known to hold for the intersection cohomology of the toric variety associated to a rational convex polytope. One can construct the intersection cohomology combinatorially from the polytope, hence it is well defined even for nonrational polytopes when there is no variety associated to it. We prove the Hard Lefschetz theorem for the intersection cohomology of a general polytope.

Finite matroids are combinatorial structures that express the concept of linear independence. In 1964, G.-C. Rota conjectured that the coefficients of the "characteristic polynomial" of a matroid $M$, polynomial whose coefficients enumerate its subsets of given rank, form a log-concave sequence. K. Adiprasito, J. Huh et E. Katz have proved this conjecture using methods which, although entirely combinatorial, are inspired by algebraic geometry. From the Bergman fan of the matroid $M$, they define a graded "Chow ring" $A(M)$ for which they prove analogs of the Poincar\'e duality, the Hard Lefschetz theorem, and the Hodge--Riemann relations. The sought for log-concavity inequalities are then analogous to the Khovanskii--Teissier inequalities.

This study adopts Hodge theory as a rigorous mathematical framework to construct a quantitative mapping system between the high-dimensional topological invariants of CalabiYau (CY) manifolds and nuclear physics parameters, thereby establishing a strict logical closure from algebraic geometry to low-energy nuclear physics. First, based on the Hodge decomposition theorem of complex manifolds and the geometric properties of compact Kähler manifolds, the study develops a topological characterization system for 3-dimensional complex CY manifolds, clarifying the algebraic structures of Hodge numbers, Chern classes, and Hodge classes, as well as their corresponding physical interpretations. Second, the proton/neutron distributions of nuclides are abstracted as algebraic cycles in Hodge theory (protons correspond to 1-codimensional cycles 𝒵1𝒵2cl: 𝒵𝑝 → ℋ𝑝ℙ3 , and neutrons correspond to 2-codimensional cycles). Using the surjectivity of the cohomology mapping (Lefschetz (1,1) theorem and its high-dimensional generalization), the quantitative relationship between “Hodge numbers and nucleon numbers” is derived.

We obtain new general results on the structure of the space of translation invariant continuous valuations on convex sets (a version of the hard Lefschetz theorem). Using these and our previous results we obtain explicit characterization of unitarily invariant translation invariant continuous valuations. It implies new integral geometric formulas for real submanifolds in Hermitian spaces generalizing the classical kinematic formulas in Euclidean spaces due to Poincaré, Chern, Santaló, and others.

The paper surveys the mirror symmetry conjectures of Hausel-Thaddeus and Hausel-Rodriguez-Villegas concerning the equality of certain Hodge numbers of SL(n, ℂ) vs. PGL(n, ℂ) flat connections and character varieties for curves, respectively. Several new results and conjectures and their relations to works of Hitchin, Gothen, Garsia-Haiman and Earl-Kirwan are explained. These use the representation theory of finite groups of Lie-type via the arithmetic of character varieties and lead to an unexpected conjecture for a Hard Lefschetz theorem for their cohomology.

We define the quantum exterior product ∧ h \wedge _h and quantum exterior differential d h d_h on Poisson manifolds. The quantum de Rham cohomology, which is a deformation quantization of the de Rham cohomology, is defined as the cohomology of d h d_h . We also define the quantum Dolbeault cohomology. A version of quantum integral on symplectic manifolds is considered and the corresponding quantum Stokes theorem is stated. We also derive the quantum hard Lefschetz theorem. By replacing d d by d h d_h and ∧ \wedge by ∧ h \wedge _h in the usual definitions, we define many quantum analogues of important objects in differential geometry, e.g. quantum curvature. The quantum characteristic classes are then studied along the lines of the classical Chern-Weil theory. The quantum equivariant de Rham cohomology is defined in the similar fashion.

We study the anisotropy theorem for Stanley-Reisner rings of simplicial homology spheres in characteristic 2 by Papadakis and Petrotou. This theorem implies the Hard Lefschetz theorem as well as McMullen’s g-conjecture for such spheres. Our first result is an explicit description of the quadratic form. We use this description to prove a conjecture stated by Papadakis and Petrotou. All anisotropy theorems for homology spheres and pseudo-manifolds in characteristic 2 follow from this conjecture. Using a specialization argument, we prove anisotropy for certain homology spheres over the field ℚ. These results provide another self-contained proof of the g-conjecture for homology spheres in characteristic 2.

Smooth valuations on manifolds are studied by establishing a link with the Rumin-de Rham complex of the co-sphere bundle. Several operations on differential forms induce operations on smooth valuations: signature operator, Rumin-Laplace operator, Euler-Verdier involution and derivation operator. As an application, Alesker’s Hard Lefschetz Theorem for even translation invariant valuations on a finite-dimensional Euclidean space is generalized to all translation invariant valuations. The proof uses Kahler identities, the Rumin-de Rham complex and spectral geometry.

McMullen’s proof of the Hard Lefschetz Theorem for simple polytopes is studied, and a new proof of this theorem that uses conewise polynomial functions on a simplicial fan is provided.