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 an afine line. We call them twistor connections. Generalising 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.
We construct a quasi-categorically enhanced Grothendieck six-functor formalism on schemes of finite type over the complex numbers. In addition to satisfying many of the same properties as M. Saito's derived categories of mixed Hodge modules, this new six-functor formalism receives canonical motivic realization functors compatible with Grothendieck's six functors on constructible objects.
This is meant to be a survey article for the Cubo Journal. We discuss the existence and number of rational points over a finite field, the Hodge type over the complex numbers, and the motivic conjectures which are controlling those invariants. We present a geometric version of it.
Following the work of Gangl, Goncharov and Levin in [GGL], we will give a combinatorial framework for motivic study of iterated integrals on the affine line. We will show that under a certain genericity condition these combinatorial objects yield to elements in the motivic Hopf algebra constructed in Bloch-Kriz [BK]. It will be shown that the Hodge realization of these elements coincides with the Hodge structure induced from the fundamental torsor of path of punctured affine line.
Our goal is to prove that the Leray spectral sequence associated to a map of algebraic varieties is motivic in the following sense: If the singular cohomology groups of the category of quasiprojective varieties defined over a subfield of C can be canonically endowed with some additional structure, such as its mixed Hodge structure or its motive (in Nori's sense or as a product of realizations). Then the Leray spectral sequence for a projective map in this category is automatically compatible with this structure. As byproduct, we get a relatively cheap construction of a mixed Hodge structure on…
In this paper we give a proof of the fact, that the motivic Hodge-Chern class transformation MHC_y and Hirzebruch class transformation MHT_y* for mixed Hodge modules and strictly specializable filtered D-modules commute with specialization in the algebraic and in a suitable complex analytic context. Here specialization in the Hodge- and D-module context means the corresponding nearby cycles defined in terms of the $V$-filtration of Malgrange-Kashiwara. This generalizes a corresponding specialization result of Verdier about MacPherson's Chern class transformation c_*.
We introduce the notion of dR-absolutely special subvarieties in motivic variations of Hodge structure as special subvarieties cut out by (de Rham-)absolute Hodge cycles and conjecture that all special subvarieties are dR-absolutely special. This is implied by Deligne's conjecture that all Hodge cycles are absolute Hodge cycles, but is a much weaker conjecture. We prove our conjecture for subvarieties satisfying a simple monodromy condition introduced by Klingler-Otwinowska-Urbanik. We study applications to typical respectively atypical intersections and $\bar{\mathbb{Q}}$-bialgebraic subvarie…
The Hodge correlators ${\rm Cor}_{\mathcal H}(z_0,z_1,\dots,z_n)$ are functions of several complex variables, defined by Goncharov (arXiv:0803.0297) by an explicit integral formula. They satisfy some linear relations: dihedral symmetry relations, distribution relations, and shuffle relations. We found new second shuffle relations. When $z_i\in0\cupμ_N$, where $μ_N$ are the $N$-th roots of unity, they are expected to give almost all relations. When $z_i$ run through a finite subset $S$ of $\mathbb C$, the Hodge correlators describe the real mixed Hodge-Tate structure on the pronilpotent complet…
We define a new type of Hall algebras associated e.g. with quivers with polynomial potentials. The main difference with the conventional definition is that we use cohomology of the stack of representations instead of constructible sheaves or functions. In order to take into account the potential we introduce a generalization of theory of mixed Hodge structures, related to exponential integrals. Generating series of our Cohomological Hall algebra is a generalization of the motivic Donaldson-Thomas invariants introduced in arXiv:0811.2435. Also we prove a new integrality property of motivic Dona…
We make explicit Serre's generalization of the Sato-Tate conjecture for motives, by expressing the construction in terms of fiber functors from the motivic category of absolute Hodge cycles into a suitable category of Hodge structures of odd weight. This extends the case of abelian varietes, which we treated in a previous paper. That description was used by Fite--Kedlaya--Rotger--Sutherland to classify Sato-Tate groups of abelian surfaces; the present description is used by Fite--Kedlaya--Sutherland to make a similar classification for certain motives of weight 3. We also give conditions under…
In this paper we describe a connection between realizations of the action of the motivic Galois group on the motivic fundamental groups of Gaussian and Eisenstein elliptic curves punctured at the $\mathfrak{p}$-torsion points, $π_1^{\rm Mot}(E-E[\mathfrak{p}],v_0)$, and the geometry of the Bianchi hyperbolic threefolds $Γ_1(\mathfrak{p})\setminus\mathbb{H}^3$, where $Γ_1(\mathfrak{p})$ is a congruence subgroup of ${\rm GL}_2({\rm End}(E))$. The first instance of such a connection was found by A.Goncharov (arXiv:math/0510310). In particular, we study the Hodge realization of the image of the ab…
We define Hodge correlators for a compact Kahler manifold X. They are complex numbers which can be obtained by perturbative series expansion of a certain Feynman integral which we assign to X. We show that they define a functorial real mixed Hodge structure on the rational homotopy type of X. The Hodge correlators provide a canonical linear map from the cyclic homomogy of the cohomology algebra of X to the complex numbers. If X is a regular projective algebraic variety over a field k, we define, assuming the motivic formalism, motivic correlators of X. Given an embedding of k into complex numb…
Let (G,X) be a Shimura datum and K a neat open compact subgroup of $G(\mathbb{A}_f)$. Under mild hypothesis on (G,X), the canonical construction associates a variation of Hodge structure on $\textrm{Sh}_K(G,X)(\mathbb{C})$ to a representation of G. It is conjectured that this should be of motivic origin. Specifically, there should be a lift of the canonical construction which takes values in relative Chow motives over $\textrm{Sh}_K(G,X)$ and is functorial in (G,X). Using the formalism of mixed Shimura varieties, we show that such a motivic lift exists on the full subcategory of representation…
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 schon 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 schon hypersurfaces and smooth tropical varieties. We also deduce a formula for the Euler characteristic of a gen…
We prove a conjecture of Milne pertaining to the existence of integral canonical models of Shimura varieties of abelian type in arbitrary unramified mixed characteristic $(0,p)$. As an application we prove for $p=2$ a motivic conjecture of Milne pertaining to integral canonical models of Shimura varieties of Hodge type.
Given a smooth projective variety V of dimension n, one may say that V has motivic dimension less than d+1 if the cohomology of V comes from varieties of dimensions less than d+1 in some geometric way. In this paper, we show that a smooth complete interesection of k quadrics has a motivic dimension less than k.
We define motivic analogues of Igusa's local zeta functions. These functions take their values in a Grothendieck group of Chow motives. They specialize to p-adic Igusa local zeta functions and to the topological zeta functions we introduced several years ago. We study their basic properties, such as functional equations, and their relation with motivic nearby cycles. In particular the Hodge spectrum of a singular point of a function may be recovered from the Hodge realization of these zeta functions.
The Hodge Galois group is the Tannakian Galois group of the category of real mixed Hodge structures. It has a subgroup, called the twistor Galois group, which is the Galois group of the category of mixed twistor structures, defined by C. Simpson. It is isomorphic to a semidirect product of C* and the unipotent radical of the Hodge Galois group. We define a natural action of the twistor Galois group by A-infinity autoequivalences of the derived category of complexes of sheaves with smooth cohomology on a compact smooth Kahler manifold X. The action of C* is provided by Simpson's action on irred…
The motivic nearby fiber is an invariant obtained from degenerating a complex variety over a disc. It specializes to the Euler characteristic of the original variety but also contains information on the variation of Hodge structure associated to the degeneration which is encoded as a limit mixed Hodge structure. However, this invariant is difficult to compute in practice. Using the techniques of tropical geometry we present a new formula for the motivic nearby fiber. Moreover, since there is a range of available software implementing the main algorithms in tropical geometry, our formula can be…
We introduce motivic analogues of p-adic exponential integrals. We prove a basic multiplicativity property from which we deduce a motivic analogue of the Thom-Sebastiani Theorem. In particular, we obtain a new proof of the Thom-Sebastiani Theorem for the Hodge spectrum of (non isolated) singularities of functions.
The goal of this article is to try understand where Hodge cycles on a singular complex projective variety X come from. As a first step we consider Hodge cycles on the maximal pure quotient $H^{2p}(X)/W_{2p-1}$, and introduce a class of algebraic cycles that we call homologically Cartier, that should conjecturally describe all such Hodge cycles. Secondly, given a singular complex projective variety $X$, we show that there is a cycle map from motivic cohomology group $H^{2p}_M(X,Q(p))$ to the space of weight 2p Hodge cycles in $H^{2p}(X,Q)$. We conjecture that this is surjective when X is define…
We perform a refined count of BPS states in the compactification of M-theory on $K3 \times T^2$, keeping track of the information provided by both the $SU(2)_L$ and $SU(2)_R$ angular momenta in the $SO(4)$ little group. Mathematically, this four variable counting function may be expressed via the motivic Donaldson-Thomas counts of $K3 \times T^2$, simultaneously refining Katz, Klemm, and Pandharipande's motivic Donaldson-Thomas counts on $K3$ and Oberdieck-Pandharipande's Gromov-Witten counts on $K3 \times T^2$. This provides the first full answer for motivic curve counts of a compact Calabi-Y…
We find a recursive formula for the motivic Milnor fiber of an irreducible plane curve, using the notions of a truncation and derived curve. We then apply natural transformations to obtain a similar recursion for the Hodge-theoretic spectrum.
We compute the motivic Milnor fiber of a complex plane curve singularity in an inductive and combinatoric way using the extended simplified resolution graph. The method introduced in this article has a consequence that one can study the Hodge-Steenbrink spectrum of such a singularity in terms of that of a quasi-homogeneous singularity.
By calculating the equivariant mixed Hodge numbers of motivic Milnor fibers introduced by Denef-Loeser, we obtain explicit formulas for the Jordan normal forms of Milnor monodromies. The numbers of the Jordan blocks will be described by the Newton polyhedron of the polynomial.
Let $\mathcal{X} \to Y$ be a birational map from a smooth Artin stack to a (possibly singular) variety. We prove a change of variables formula that relates motivic integrals over arcs of $Y$ to motivic integrals over arcs of $\mathcal{X}$. With a view toward the study of stringy Hodge numbers, this change of variables formula leads to a new notion of crepantness for the map $\mathcal{X} \to Y$ that coincides with the usual notion in the special case that $\mathcal{X}$ is a scheme.
We prove a formula expressing the motivic integral (\cite{ls}) of a K3 surface over $\bC((t))$ with semi-stable reduction in terms of the associated limit Hodge structure. Secondly, for every smooth variety over a non-archimedean 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 degenerated K3 surfaces over an arbitrary non-archimedean field and prove this conjecture for K…
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 $E_\infty$-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 $\mathbb{R}$. 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 repres…
We study the geometry and Hodge theory of the cubic hypersurfaces attached to two-loop Feynman integrals for generic physical parameters. We show that the Hodge structure attached to planar two-loop Feynman graphs decomposes into mixed Tate pieces and the Hodge structures of families of hyperelliptic, elliptic, or rational curves depending on the space-time dimension. For two-loop graphs with a small number of edges, we give more precise results. In particular, we recover a result of Bloch arXiv:2105.06132 that in the well-known double box example, there is an underlying family of elliptic cur…
An abelian category of relative pure motives is constructed along the lines of André (over a field of characteristic 0). An algebraic stack is shown to possess a motive in this sense. This motive is studied for the moduli stack of G-bundles over the universal curve for appropriate G, and a motivic Atiyah-Bott formula is obtained. By taking Hodge realization, a description of the corresponding variations of Hodge structures over M_g is given. In particular, it follows that these lie in the tensor category generated by the variation of Hodge structure associated to the curve itself. Similar resu…
By associating a `motivic integral' to every complex projective variety X with at worst canonical, Gorenstein singularities, Kontsevich proved that, when there exists a crepant resolution of singularities Y of X, the Hodge numbers of Y do not depend upon the choice of the resolution. In this article we provide an elementary introduction to the theory of motivic integration, leading to a proof of the result described above. We calculate the motivic integral of several quotient singularities and discuss these calculations in the context of the cohomological McKay correspondence.
This paper is an exposition of the completion of a modular group with respect to its inclusion into SL_2(Q) and the connection with the theory of modular forms and variations of mixed Hodge structure over modular curves. Among the goals of this paper are to give a context to Manin's iterated Shimura integrals (iterated integrals of modular forms) and to study relations in the "maximal Eisenstein quotient" of the completion of SL_2(Z). In particular, we use a computation of Terasoma to prove that Pollack's quadratic relations in the derivation Lie algebra of a rank two free Lie algebra lift to …
We develop the theory of multiple polylogarithms from analytic, Hodge and motivic point of view. Define the category of mixed Tate motives over a ring of integers in a number field. Describe explicitly the multiple polylogarithm Hopf algebra.
With representation-theoretic applications in mind, we construct a formalism of reduced motives with integral coefficients. These are motivic sheaves from which the higher motivic cohomology of the base scheme has been removed. We show that reduced stratified Tate motives satisfy favorable properties including weight and t-structures. We also prove that reduced motives on cellular (ind-)schemes unify various approaches to mixed sheaves in representation theory, such as Soergel-Wendt's semisimplified Hodge motives, Achar-Riche's complexes of parity sheaves, as well as Ho-Li's recent category of…
We show that for a hypersurface Batyrev's stringy E-function can be seen as a residue of the Hodge zeta function, a specialization of the motivic zeta function of Denef and Loeser. This is a nice application of inversion of adjunction. If an affine hypersurface is given by a polynomial that is non-degenerate with respect to its Newton polyhedron, then the motivic zeta function and thus the stringy E-function can be computed from this Newton polyhedron (by work of Artal, Cassou-Nogues, Luengo and Melle based on an algorithm of Denef and Hoornaert). We use this procedure to obtain an easy way to…
In this paper, we prove a motivic enhancement of the theorem of the fixed part in Hodge theory due to Deligne. In the pure motivic case, this was done for the first time by André in [And96]. Our main result is an extension to the mixed case, which strengthens a result by Arapura [Ara13] and also provides an alternative and simpler proof, in the framework of Nori motives, of a result by Ayoub [Ayo14].
We consider $M$-theory compactified on $T^4 \times T^2$ and describe the count of spinning $1/8$-BPS states. This refines the classic count of Maldacena-Moore-Strominger in the physics literature and the recent mathematical work of Bryan-Oberdieck-Pandharipande-Yin, which studied reduced Donaldson-Thomas invariants of abelian surfaces and threefolds. As in previous work on $K3 \times T^2$ compactification, we track angular momenta under both the $SU(2)_L$ and $SU(2)_R$ factors in the 5d little group, providing predictions for the relevant motivic curve counts.
In this paper we construct extensions of the Mixed Hodge structure on the fundamental group of a pointed algebraic curve. These extensions correspond to the regulator of certain explicit motivic cohomology cycles in the self product of the curve which were first constructed by Bloch and Beilinson. This leads to a new iterated integral expression for the regulator. Our result is a generalization of a result of Colombo's where she constructs the extension corresponding to a motivic cycle class in the Jacobian of a hyperelliptic curve constructed by Collino. This is to appear in the Mathematical …
We started from computer experiments with simple one-dimensional ergodic dynamical systems called interval exchange transformations. Correlators in these systems decay as a power of time. In the simplest non-trivial case the exponent is equal to 1/3. We found a formula connecting characteristic exponents with explicit integrals over moduli spaces of algebraic curves with additional structures. Moreover, these integrals can be interpreted as correlators in a topological string theory. Also a new analogy arose between ergodic theory and complex algebraic geometry.
In mixed characteristic and in equal characteristic $p$ we define a filtration on topological Hochschild homology and its variants. This filtration is an analogue of the filtration of algebraic $K$-theory by motivic cohomology. Its graded pieces are related in mixed characteristic to the complex $AΩ$ constructed in our previous work, and in equal characteristic $p$ to crystalline cohomology. Our construction of the filtration on $\mathrm{THH}$ is via flat descent to semiperfectoid rings. As one application, we refine the construction of the $AΩ$-complex by giving a cohomological construction o…
We prove the conjectural relationship recently proposed in [9] between certain special cubic Hodge integrals of the Gopakumar--Mariño--Vafa type [17, 28] and GUE correlators, and the conjecture proposed in [7] that the partition function of these Hodge integrals is a tau function of the discrete KdV hierarchy.
We consider parametric Feynman integrals and their dimensional regularization from the point of view of differential forms on hypersurface complements and the approach to mixed Hodge structures via oscillatory integrals. We consider restrictions to linear subspaces that slice the singular locus, to handle the presence of non-isolated singularities. In order to account for all possible choices of slicing, we encode this extra datum as an enrichment of the Hopf algebra of Feynman graphs. We introduce a new regularization method for parametric Feynman integrals, which is based on Leray coboundari…
In this paper we construct extensions of mixed Hodge structures coming from the mixed Hodge structure on the graded quotients of the group ring of the fundamental group of a smooth, projective, pointed curve. These extensions correspond to the regulators of certain motivic cycles in the Jacobian of the curve which were constructed by Beilinson and Bloch. This leads to a new iterated integral expression for the regulator. This is a generalisation of a theorem of Colombo where she constructed the extension corresponding to Collino's cycles in the Jacobian of a hyperelliptic curve.
