J. Wildeshaus
This paper has been withdrawn by the author; its content is properly cantained in the paper arXiv:0706.4447, entitled "Pure motives, mixed motives and extensions of motives associated to singular surfaces", and submitted on June 29, 2007.
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In this article we further the study of non-commutative motives. We prove that bivariant cyclic cohomology (and its variants) becomes representable in the category of non-commutative motives. Furthermore, Connes' bilinear pairings correspond to the composition operation. As an application, we obtain a simple model, given in terms of infinite matrices, for the (de)suspension of these bivariant cohomology theories.
The purpose of this work is to generalize, in the context of 1-motives, the $p$-adic height pairings constructed by B. Mazur and J. Tate on abelian varieties. Following their approach, we define a global pairing between the rational points of a 1-motive and its dual. We also provide local pairings between zero-cycles and divisors on a curve, which is done by considering its Picard and Albanese 1-motives.
The purpose of this article is to define and study the notion of absolute intersection motive.
In this article, we construct a Hecke-equivariant Chow motive whose realizations equal intersection cohomology of Siegel threefolds with regular algebraic coefficients. As a consequence, we are able to define Grothendieck motives for Siegel modular forms.
For a split reductive group G over a finite field, we show that the intersection (cohomology) motive of the moduli stack of iterated G-shtukas with bounded modification and level structure is defined independently of the standard conjectures on motivic t-structures on triangulated categories of motives. This is in accordance with general expectations on the independence of l in the Langlands correspondence for function fields.
Using intersections of two Grassmannians in ${\mathbb{P}}^9$, Ottem-Rennemo and Borisov-Căldăraru-Perry have exhibited pairs of Calabi-Yau threefolds $X$ and $Y$ that are deformation equivalent, L-equivalent and derived equivalent, but not birational. To complete the picture, we show that $X$ and $Y$ have isomorphic Chow motives.
Intersection complex of any threefold as a Chow motive
本文提出了一种在三重方案上构造的Chow动机，该动机通过重量轮廓和motivic sheaves的设定，满足Wildeshaus对交错复形的刻画。
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 $A_k(X) \to H_{2k}(X)$ are injective for $k<c$. We give variants involving group actions which le…
The Beilinson-Bloch type conjectures predict that the low degree rational Chow groups of intersections of quadrics are one dimensional. This conjecture was proved by Otwinowska. Making use of homological projective duality and the recent theory of (Jacobians of) noncommutative Chow motives, we give an alternative proof of this conjecture in the case of a complete intersection of either two quadrics or three odd-dimensional quadrics. Moreover, without the use of the powerful Lefschetz theorem, we prove that in these cases the unique non-trivial algebraic Jacobian is the middle one. As an applic…
Pairing Anderson motives via formal residues in the Frobenius endomorphism
本工作通过取 Frobenius 自同态中形式幂级数的留数，构建了 Anderson 模块动机与对偶动机间的完美配对，并以此回答了 Hartl-Juschka 问题，将“abelian 等于 A-finite”定理推广至更广泛的代数类。
We first recall the construction of the Chow motive modelling intersection cohomology of a proper surface and study its fundamental properties. Using Voevodsky's category of effective geometrical motives, we then study the motive of the exceptional divisor in a non-singular blow-up. If all geometric irreducible components of the divisor are of genus zero, then Voevodsky's formalism allows us to construct certain one-extensions of Chow motives, as canonical sub-quotients of the motive with compact support of the smooth part of the surface. Specializing to Hilbert--Blumenthal surfaces, we recove…
The purpose of this article is to construct a Hecke-equivariant Chow motive whose realizations equal interior (or intersection) cohomology of Picard surfaces with regular algebraic coefficients. As a consequence, we are able to define Grothendieck motives for Picard modular forms.
We study genus 2 Hilbert-Siegel varieties, i.e. Shimura varieties $S_K$ corresponding to the group $\mbox{GSp}_{4,F}$ over a totally real field $F$, along with the relative Chow motives $^λ\mathcal{V}$ of abelian type over $S_K$ obtained from irreducible representations $V_λ$ of $\mbox{GSp}_{4,F}$. We analyse the weight filtration on the degeneration of such motives at the boundary of the Baily-Borel compactification and we find a criterion on the highest weight $λ$ which characterises the absence of the middle weights 0 and 1 in the corresponding degeneration. Thanks to Wildeshaus' theory, th…
In this note we prove the geometrical origin of pairings of abelian schemes. According to Deligne's philosophy of motives, this means that these pairings are motivic. We make also explicit the link between pairings and linear morphisms. We give a geometrical explanation of the main properties of the Weil pairings and of the Riemann forms of line bundles. We generalize our remarks to pairings of 1-motives.
Mumford-Tate groups of 1-motives and Weil pairing
本文研究1-动机的几何如何决定其动机Galois群的维数，通过周期矩阵描述Mumford-Tate群的作用，并在半椭圆情形下分类周期关系、计算群表示，为Grothendieck周期猜想提供新视角。
For a large class of good moduli spaces $X$ of symmetric stacks $\mathcal{X}$, we define noncommutative motives $\mathbb{D}^{\text{nc}}(X)$ which can be regarded as categorifications of the intersection cohomology of $X$. These motives are summands of noncommutative resolutions of singularities $\mathbb{D}(X)\subset D^b(\mathcal{X})$ of $X$. The category $\mathbb{D}(X)$ is a global analogue of the noncommutative resolutions of singularities of $V//G$ for $V$ a symmetric representation of a reductive group $G$ constructed by Špenko-Van den Bergh.
We show that the pairing on de Rham realizations of 1-motives in "Theorie di Hodge III", IHES 44, can be defined over any base scheme and we prove that it gives rise to a perfect duality if one is working with a 1-motive and its Cartier dual. Furthermore we study universal extensions of 1-motives and their relation with $\natural$-extensions.
G-functions, motives, and unlikely intersections -- old and new
本综述梳理G-函数在算术几何中的作用、其特殊值多项式关系与动机对称性的关联及Bombieri原理的应用，介绍该方法在非典型交与Zilber-Pink猜想研究中的新发展。
These notes are an account of a series of lectures I gave at the LMS-CMI Research School `Homotopy Theory and Arithmetic Geometry: Motivic and Diophantine Aspects', in July 2018, at the Imperial College London. The goal of these notes is to see how motives may be used to enhance cohomological methods, giving natural ways to prove independence of $\ell$ results for traces and zeta-functions, and constructions of characteristic classes (as $0$-cycles). This leads to the Grothendieck-Lefschetz formula, of which we give a new motivic proof. There are also a few additions to what have been told in …
In this note, we prove a criteria for supersingularity when the variety has a large automorphism group and a perfect bilinear pairing. This criteria unifies and extends many known results on the supersingularity of curves and varieties and in particular, applies to a large family of Artin-Schreier curves.
Generalized intersection pairings on moduli spaces of vector bundles over a curve
本文针对带紧合好模空间且稳定部分非空的Artin栈引入广义相交配对概念，通过建立壁交叉公式比较了光滑射影曲线上半稳定丛模栈上四种已知的该类配对构造，并给出低秩情形的显式计算。
In this note, we would like to propose a suitable extension of the arithmetic Chow group of codimension one, in which the Hodge index theorem holds. We also prove an arithmetic analogue of Bogomolov's instability theorem for rank 2 vector bundles on arbitrary regular projective arithmetic varieties.
We give a new definition of higher arithmetic Chow groups for smooth projective varieties defined over a number field, which is similar to Gillet and Soulé's definition of arithmetic Chow groups. We also give a compact description of the intersection theory of such groups. A consequence of this theory is the definition of a height pairing between two higher algebraic cycles, of complementary dimensions, whose real regulator class is zero. This description agrees with Beilinson's height pairing for the classical arithmetic Chow groups. We also give examples of the higher arithmetic intersection…
Real intersection homology II: A local duality obstruction
该研究属于代数几何领域，证明了研究者定义的实相交同调群上的相交配对一般不具备对偶性，为实相交同调的对偶性研究提供了关键结论。
One can realize higher laminations as positive configurations of points in the affine building. The duality pairings of Fock and Goncharov give pairings between higher laminations for two Langlands dual groups $G$ and $G^{\vee}$. These pairings are a generalization of the intersection pairing between measured laminations on a topological surface. We give a geometric interpretation of these intersection pairings in the case that $G=SL_n$. In particular, we show that they can be computed as the length of minimal weighted networks in the building. Thus we relate the intersection pairings to the m…
The four operations on perverse motives
本文利用 étale motives 三角范畴构造了反常混合动机的阿贝尔范畴，证明其导出范畴具备 Grothendieck 四运算及相关函子，并证明反常层的经典构造具有动机性且与 Hodge 理论相容。
This paper studies intersection theory on the compactified moduli space M(n,d) of holomorphic bundles of rank n and degree d over a fixed compact Riemann surface of genus g > 1 where n and d may have common factors. Because of the presence of singularities we work with the intersection cohomology groups defined by Goresky and MacPherson and the ordinary cohomology groups of a certain partial resolution of singularities of M(n,d). Based on our earlier work, we give a precise formula for the intersection cohomology pairing and provide a method to calculate pairings on the partial resolution of s…
Intersection theory on singular moduli spaces of vector bundles: a parabolic approach
本文通过抛物束方法，给出了奇异向量束模空间$M_0(r)$上交配对的显式公式，提供了更简洁的计算方式。
In this paper we shall give formulas for the pairings of intersection cohomology classes of complementary dimensions in the intersection cohomology of geometric invariant theoretic quotients for which semistability is not necessarily the same as stability (although we make some weaker assumptions on the action). We also give formulas for intersection pairings on resolutions of singularities (or more precisely partial resolutions, since orbifold singularities are allowed) of the quotients.
Properties of the Beilinson Height Pairing
本文研究了Beilinson高度配对的高维推广，通过构造新高度配对将其与相交积相联系，并证明其满足与相交积一致的投影公式。
For an arithmetic surface $X\to B=\operatorname{Spec} O_K$ the Deligne pairing $\left <\,,\,\right > \colon \operatorname{Pic}(X) \times \operatorname{Pic}(X) \to \operatorname{Pic}(B)$ gives the "schematic contribution" to the Arakelov intersection number. We present an idelic and adelic interpretation of the Deligne pairing; this is the first crucial step for a full idelic and adelic interpretation of the Arakelov intersection number. For the idelic approach we show that the Deligne pairing can be lifted to a pairing $\left<\,,\,\right>_i:\ker(d^1_\times)\times \ker(d^1_\times)\to\operatorna…
The purpose of this paper is to provide a way to compute the intersection cohomology of the GIT quotient of a nonsingular projective variety. We show that the middle perversity intersection cohomology of the GIT quotient $M//G$ is naturally embedded into the $G$ equivariant cohomology of $M^{ss}$ under an assumption satisfied by many interesting spaces. This embedding enables us to compute the intersection pairing in terms of the cup product structure of the equivariant cohomology ring. We also show that the image is a subspace defined by truncation. Overall, we can compute the intersection co…
This note proves combinatorially that the intersection pairing on the middle dimensional compactly supported cohomology of a smooth toric hyperkaehler variety is always definite, providing a large number of non-trivial L^2 harmonic forms for toric hyperkaehler metrics on these varieties. This is motivated by a result of Hitchin about the definiteness of the pairing of L^2 harmonic forms on complete hyperkaehler manifolds of linear growth.
The goal of this article is to provide an explicit algorithmic construction of formal $F$-manifold structures, formal Frobenius manifold structures, and higher residue pairings on the primitive middle-dimensional cohomology $\mathbb{H}$ of a smooth projective Calabi-Yau complete intersection variety $X$ defined by homogeneous polynomials $G_1(\underline x), \dots, G_k(\underline x)$. Our main method is to analyze a certain dGBV (differential Gerstenhaber-Batalin-Vilkovisky) algebra $\mathcal{A}$ obtained from the twisted de Rham complex which computes $\mathbb{H}$. More explicitly, we introduc…
We introduce a pairing on local intersection cohomology groups of variations of pure Hodge structure, which we call the asymptotic height pairing. Our original application of this pairing was to answer a question on the Ceresa cycle posed by R. Hain and D. Reed. (This question has since been answered independently by Hain.) Here we apply the pairing to show that a certain analytic line bundle, called the biextension line bundle, defined in terms of normal functions, always extends to any smooth partial compactification of the base. We show that the the pairing on intersection cohomology govern…
Matroids and amplitudes
该研究给出超平面排列补扭曲同调相交配对的组合公式，关联拟阵Bergman扇子扇拉普拉斯变换，定义拟阵散射振幅并验证其性质，推导散射振幅新老公式。
This paper has two aims. The former is to give an introduction to our earlier work on the Hodge theory of algebraic maps and more generally to some of the main themes of the theory of perverse sheaves and to some of its geometric applications. Particular emphasis is put on the topological properties of algebraic maps. The latter is to prove a motivic version of the decomposition theorem for the resolution of a threefold Y. This result allows to define a pure motive whose Betti realization is the intersection cohomology of Y. We provide a few explicit computations of perverse truncations and in…
Let R be a complete discrete valuation ring with algebraically closed residue field k and fraction field K. Let X_K be a projective smooth and geometrically connected scheme over K. Néron defined a canonical pairing on X_K between 0-cycles of degree zero and divisors which are algebraically equivalent to zero. When X_K is an abelian variety, and if one restricts to those 0-cycles supported by K-rational points, Néron gave an expression of his pairing involving intersection multiplicities on the Néron model A of A_K over R. When X_K is a curve, Gross and Hriljac gave independantly an analogous …
We show by studying the symplectic geometry of the extended moduli space that the intersection cohomology of the representation space $Hom(π_1(Σ),G)/G$ for a simply connected compact Lie group $G$ is naturally embedded into the $G$ equivariant cohomology of $Hom(π_1(Σ),G)$ where $Σ$ is a closed Riemann surface. This enables us to compute the intersection cohomology as a graded vector space with intersection pairing, in terms of the equivariant cohomology ring. The case where $G=SU(2)$ -- the moduli space of rank 2 holomorphic vector bundles of even degree -- is discussed in detail.
We introduce the p-adic analogue of Arakelov intersection theory on arithmetic surfaces. The intersection pairing in an extension of the p-adic height pairing for divisors of degree 0 in the form described by Coleman and Gross. It also uses Coleman integration and is related to work of Colmez on p-adic Green functions. We introduce the p-adic version of a metrized line bundle and define the metric on the determinant of its cohomology in the style of Faltings. It is possible to prove in this theory analogues of the Adjunction formula and the Riemann-Roch formula.
In two articles by Barthel, Brasselet, Fieseler and Kaup, and, Bressler and Lunts, a combinatorial theory of intersection cohomology and perverse sheaves has been developed on fans. In the first one, one tried to present everything on an elementary level,using only some commutative algebra and no derived categories. There remained two major gaps: First of all the Hard Lefschetz Theorem was only conjectured and secondly the intersection product seemed to depend on some non-canonical choices. Meanwhile the Hard Lefschetz theorem has been proved by Karu. The proof relies heavily on the intersecti…
Let $U$ be a smooth scheme over an algebraically closed field $\mathbb K$ of characteristic zero and $f:U\to{\mathbb A}^1$ a regular function, and write $X=$Crit$(f)$, as a closed subscheme of $U$. The motivic vanishing cycle $MF_{U,f}^ϕ$ is an element of the $\hatμ$-equivariant motivic Grothendieck ring ${\mathcal M}^{\hatμ}_X$ defined by Denef and Loeser math.AG/0006050 and Looijenga math.AG/0006220, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants, arXiv:0811.2435. We prove three main results: (a) $MF_{U,f}^ϕ$ depends only on the third-order thickenings $…
We prove formulas (found by Witten in 1992 using physical methods) for intersection pairings in the cohomology of the moduli space M(n,d) of stable holomorphic vector bundles of rank n and degree d (assumed coprime) on a Riemann surface of genus g greater than or equal to 2. We also use these formulas for intersection numbers to obtain a proof of the Verlinde formula for the dimension of the space of holomorphic sections of a line bundle over M(n,d).
The tropical (p, q)-homology groups of Itenberg, Katzarkov, Mikhalkin and Zharkov are the tropical analogues of the Hodge decomposition of the cohomology of complex algebraic varieties. There is a well-defined intersection pairing on tropical (1, 1)-classes of a compact non-singular tropical surface. Here we compute directly the (1, 1)-homology of a non-singular floor decomposed tropical surface in tropical projective space, along with the intersection form.
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