Yakov Varshavsky
The goal of this paper is to give a simple proof of Deligne's conjecture (proven by Fujiwara) and to generalize it to the situation appearing in our joint project with David Kazhdan on the global Langlands correspondence over function fields. Our proof holds in the realm of ordinary algebraic geometry and does not use rigid geometry.
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Proof of the Deligne-Milnor conjecture
本文运用导出与非交换代数几何方法研究算术概形上的分歧现象，证明了Deligne-Milnor猜想，并在纯特征情形下推广了Bloch导体猜想。
We reformulate a conjecture of Deligne on 1-motives by using the integral weight filtration of Gillet and Soulé on cohomology, and prove it. This implies the original conjecture up to isogeny. If the degree of cohomology is at most two, we can prove the conjecture for the Hodge realization without isogeny, and even for 1-motives with torsion.
We construct full strong exceptional collections of line bundles on smooth toric Fano Deligne-Mumford stacks of Picard number at most two and of any Picard number in dimension two. It is hoped that the approach of this paper will eventually lead to the proof of the existence of such collections on all smooth toric nef-Fano Deligne-Mumford stacks.
I attempted to write the full translation of this article to make the remarkable proof of Pierre Deligne available to a greater number of people. Overviews of the proofs can be found elsewhere. I especially recommend the notes of James Milne on Etale Cohomology that also contain a justification for the theory underlying this article and proofs of the results used by Deligne. The footnotes are mostly claims that some details appear in Milne, clarifications of some of the terminology or my personal struggles. I have also made a thorough overview of the proof together with more detailed explanati…
Deligne's conjecture is the Lefschetz trace formula for correspondences defined over a finite field. In this paper, we prove an analogous statement of Deligne's conjecture with respect to $p^n$-torsion étale cohomology under certain conditions, where $p$ is the characteristic of the base field.
O'Grady's generalized Franchetta conjecture (GFC) is concerned with codimension 2 algebraic cycles on universal polarized K3 surfaces. In \cite{BL17}, this conjecture has been studied in the Betti cohomology groups. Following a suggestion of Voisin, we investigate this problem in the Deligne-Beilinson (DB) cohomology groups. In this paper, we develop the theory of Deligne-Beilinson cohomology groups on separated (smooth) Deligne-Mumford stacks. Using the automorphic cohomology group and Noether-Lefschetz theory, we compute the 4-th DB-cohomology group of universal oriented polarized K3 surface…
We study Deligne's conjecture on the monodromy weight filtration on the nearby cycles in the mixed characteristic case, and reduce it to the nondegeneracy of certain pairings in the semistable case. We also prove a related conjecture of Rapoport and Zink which uses only the image of the Cech restriction morphism, if Deligne's conjecture holds for a general hyperplane section. In general we show that Deligne's conjecture is true if the standard conjectures hold.
We formulate a strong positivity conjecture on characters afforded by the Alvis-Curtis dual of the intersection cohomology of Deligne-Lusztig varieties. This conjecture provides a powerful tool to determine decomposition numbers of unipotent $\ell$-blocks of finite reductive groups.
In this paper, we prove that the conductor formula of Bloch implies the conjecture of Deligne on Milnor numbers of isolated singularities. In particular, thanks to the work of Bloch on his conjecture, our result implies that this so-called Deligne-Milnor's conjecture is equally true in the unequal characteristic case.
The aim of this note, which raises more questions than it answers, is to study natural operations acting on the cohomology of various types of algebras. It contains a lot of very surprising partial results and examples.
We express the kernel of Griffiths' Abel-Jacobi map by using the inductive limit of Deligne cohomology in the generalized sense (i.e. the absolute Hodge cohomology of A. Beilinson). This generalizes a result of L. Barbieri-Viale and V. Srinivas in the surface case. We then show that the Abel-Jacobi map for codimension 2 cycles and the Albanese map are bijective if a general hyperplane section is a surface for which Bloch's conjecture is proved. In certain cases we verify Nori's conjecture on the Griffiths group. We also prove a weak Lefschetz-type theorem for (higher) Chow groups, generalize a…
We establish the Airy curve case of a conjecture of Gukov and Sułkowski by reducing to Dijkgraaf-Verlinde-Verlinde Virasoro constraints satisfied by the intersection numbers on moduli spaces of algebraic curves.
We introduce the notion of extension of 1-motives. Using the dictionary between strictly commutative Picard stacks and complexes of abelian sheaves concentrated in degrees -1 and 0, we check that an extension of 1-motives induces an extension of the corresponding strictly commutative Picard stacks. We compute the Hodge, the de Rham and the l-adic realizations of an extention of 1-motives. Using these results we can prove Deligne's conjecture on extensions of 1-motives.
Deligne has conjectured that certain mixed Hodge theoretic invariants of complex algebraic invariants are motivic. This conjecture specializes to an algebraic construction of the Jacobian for smooth projective curves, which was done by A. Weil. The conjecture (and one-motives) are motivated by means of Jacobians, generalized Jacobians of Rosenlicht, and Serre's generalized Albanese varieties. We discuss the connections with the Hodge and the generalized Hodge conjecture. We end with some applications to number theory by providing partial answers to questions of Serre, Katz and Jannsen.
We introduce new motivic invariants of arbitrary varieties over a perfect field. These cohomological invariants take values in the category of one-motives (considered up to isogeny in positive characteristic). The algebraic definition of these invariants presented here proves a conjecture of Deligne. Applications include some cases of conjectures of Serre, Katz and Jannsen on the independence of $\ell$ of parts of the étale cohomology of arbitrary varieties over number fields and finite fields.
We prove Rapoport's dimension conjecture for affine Deligne-Lusztig varieties for GL_h and superbasic b. From this case the general dimension formula for affine Deligne-Lusztig varieties for special maximal compact subgroups of split groups follows, as was shown in a recent paper by Goertz, Haines, Kottwitz, and Reuman.
In this paper we discuss the geometry of affine Deligne Lusztig varieties with very special level structure, determining their dimension and connected and irreducible components. As application, we prove the Grothendieck conjecture for Shimura varieties with very special level at $p$ and a function field analogue.
In this paper, we will study the connections between the mirror symmetry of Calabi-Yau threefolds and Deligne's conjecture on the special values of the $L$-functions of critical motives. Using the theory of mirror symmetry, we will develop a method to compute the Deligne's period for a Calabi-Yau threefold in the mirror family of a one-parameter mirror pair. We will give two examples to show how this method works, and we will express the Deligne's period in terms of the classical periods of the threeform. Using this method, we will compute the Deligne's period of a Calabi-Yau threefold studied…
In this paper we provide a full account of the Weil conjectures including Deligne's proof of the conjecture about the eigenvalues of the Frobenius endomorphism. Section 1 is an introduction into the subject. Our exposition heavily relies on the Etale Cohomology theory of Grothendieck so I included an overview in Section 2. Once one verifies (or takes for granted) the results therein, proofs of most of the Weil conjectures are straightforward as we show in Section 3. Sections 4-8 constitute the proof of the remaining conjecture. The exposition is mostly similar to that of Deligne in [7] though …
Motivated by the S-duality conjecture of Vafa-Witten, Tanaka-Thomas have developed a theory of Vafa-Witten invariants for projective surfaces using the moduli space of Higgs sheaves. Their definition and calculation prove the S-duality prediction of Vafa-Witten in many cases in the side of gauge group $SU(r)$. In this survey paper for ICCM-2019 we review the S-duality conjecture in physics by Vafa-Witten and the definition of Vafa-Witten invariants for smooth projective surfaces and surface Deligne-Mumford stacks. We make a prediction that the Vafa-Witten invariants for Deligne-Mumford surface…
In this paper, we prove a conjecture of Kottwitz and Rapoport on a union of (generalized) affine Deligne-Lusztig varieties $X(μ, b)_J$ for any tamely ramified group $G$ and its parahoric subgroup $P_J$. We show that $X(μ, b)_J \neq \emptyset$ if and only if the group-theoretic version of Mazur's inequality is satisfied. In the process, we obtain a generalization of Grothendieck's conjecture on the closure relation of $\s$-conjugacy classes of a twisted loop group.
We prove that the existence of an automorphism of finite order on a (defined over a number field) variety X implies the existence of algebraic linear relations between the logarithm of certain periods of X and the logarithm of special values of the Gamma-function. This implies that a slight variation of results by Anderson, Colmez and Gross on the periods of CM abelian varieties is valid for a larger class of CM motives. In particular, we prove a weak form of the period conjecture of Gross-Deligne. Our proof relies on the arithmetic fixed point formula (equivariant arithmetic Riemann-Roch theo…
This is the write-up of a talk given in honour of Prof. Ihara's 80th Birthday conference in Kyoto in 2018. After briefly reviewing the work of Ihara on the projective line minus 3 points, I outline the main ideas in the proof of the Deligne-Ihara conjecture and provide an update on recent progress in this area and raise some new questions. The second part of the talk outlines the main features of the corresponding theory in genus one, i.e., on the moduli stack of elliptic curves. It contains many concrete and elementary examples to illustrate the main features of the theory and to make it more…
We provide stacky generalizations of classical gauge-theoretic results inspired by Donaldson, the Uhlenbeck-Yau theorem and variants due to Bando and his collaborators. Moreover, we show an application of this machinery in the study of ALE spaces.
This paper concerns the dimensions of certain affine Deligne-Lusztig varieties, both in the affine Grassmannian and in the affine flag manifold. Rapoport conjectured a formula for the dimensions of the varieties X_mu(b) in the affine Grassmannian. We prove his conjecture for b in the split torus; we find that these varieties are equidimensional; and we reduce the general conjecture to the case of superbasic b. In the affine flag manifold, we prove a formula that reduces the dimension question for X_x(b) with b in the split torus to computations of dimensions of intersections of Iwahori orbits …
We combine Deligne's global invariant cycle theorem, and the algebraicity theorem of Cattani, Deligne and Kaplan, for the connected components of the locus of Hodge classes, to conclude that under simple assumptions these components are defined over number fields (assuming the initial family is), as expected from the Hodge conjecture. We also show that the Hodge conjecture for (weakly) absolute Hodge classes reduces to the Hodge conjecture for (weakly) absolute Hodge classes on varieties defined over number fields.
In this paper, we will study the arithmetic geometry of rank-2 attractors, which are Calabi-Yau threefolds whose Hodge structures admit interesting splits. We will develop methods to analyze the algebraic de Rham cohomologies of rank-2 attractors, and we will illustrate how our methods work by focusing on an example in a recent paper by Candelas, de la Ossa, Elmi and van Straten. We will look at the interesting connections between rank-2 attractors in string theory and Deligne's conjecture on the special values of $L$-functions. We will also formulate several open questions concerning the pote…
Let $Bun_G(X)$ be the moduli stack of $G$-torsors on a smooth projective curve $X$ for a reductive group $G$. We prove a conjecture made by Drinfeld-Wang and Gaitsgory on the Deligne-Lusztig duality for D-modules on $Bun_G(X)$. This conjecture relates Drinfeld-Gaitsgory's pseudo-identity functors to the enhanced Eisenstein series and geometric constant term functors on $DMod(Bun_G(X))$. We also prove a "second adjointness" result for these enhanced functors.
The cyclic Deligne conjecture and Calabi-Yau structures
本文证明了循环Deligne猜想，指出配备Calabi-Yau结构的线性范畴的Hochschild上链具有framed $E_2$-代数结构，并应用于相对环路同调上的弦拓扑运算。
Non-commutative intersection theory and unipotent Deligne-Milnor formula
本文利用导出与非交换代数几何范畴化 Bloch 相交数，并结合非交换 Chern 特征，在新情形下推广 Bloch 导子猜想，建立幂幺 Deligne–Milnor 公式。
We prove a 1979 conjecture of Lusztig on the cohomology of semi-infinite Deligne--Lusztig varieties attached to division algebras over local fields. We also prove the two conjectures of Boyarchenko on these varieties. It is known that in this setting, the semi-infinite Deligne--Lusztig varieties are ind-schemes comprised of limits of certain finite-type schemes $X_h$. Boyarchenko's two conjectures are on the maximality of $X_h$ and on the behavior of the torus-eigenspaces of their cohomology. Both of these conjectures were known in full generality only for division algebras with Hasse invarian…
Let X be a smooth variety over $F_p$. Let E be a number field. For each nonarchimedean place $λ$ of E prime to p consider the set of isomorphism classes of irreducible lisse $\bar{E}_λ$-sheaves on X with determinant of finite order such that for every closed point x in X the characteristic polynomial of the Frobenius $F_x$ has coefficents in E. We prove that this set does not depend on $λ$. The idea is to use a method developed by G.Wiesend to reduce the problem to the case where X is a curve. This case was treated by L. Lafforgue.
We propose a non-commutative generalization of Beilinson's Conjecture on the regulator map from algebraic K-theory to Deligne cohomology of algebraic varieties over Q. We also check a baby case of the generalized conjecture, namely, the case of finite-dimensional associative algebras.
This is the final version of the 2007 preprint titled "On the derived category of 1-motives, I". It has been substantially expanded to contain a motivic proof of (two thirds of) Deligne's conjecture on 1-motives with rational coefficients, hence the new title. Compared to the 2007 preprint, the additions mainly concern an abstract theory of realisations with weight filtrations; Deligne's conjecture is tackled though them by an adjunction game.
Deligne's weight-monodromy conjecture gives control over the poles of local factors of L-functions of varieties at places of bad reduction. His proof in characteristic p was a step in his proof of the generalized Weil conjectures. Scholze developed the theory of perfectoid spaces to transfer Deligne's proof to characteristic 0, proving the conjecture for complete intersections in toric varieties. Building on Scholze's techniques, we prove the weight-monodromy conjecture for complete intersections in abelian varieties.
We prove that the Deligne-Lusztig varieties associated to elements of the Weyl group which are of minimal length in their twisted class are affine. Our proof differs from the proof of He and Orlik-Rapoport and it is inspired by the case of regular elements, which correspond to the varieties involved in Broué's conjectures.
In present paper we develop the deformation theory of operads and algebras over operads. Free resolutions (constructed via Boardman-Vogt approach) are used in order to describe formal moduli spaces of deformations. We apply the general theory to the proof of Deligne's conjecture. The latter says that the Hochschild complex of an associative algebra carries a canonical structure of a dg-algebra over the chain operad of the little discs operad. In the course of the proof we construct an operad of geometric nature which acts on the Hochschild complex. It seems to be different from the brace opera…
In this paper, we will study the connections between the mirror symmetry of K3 surfaces and the geometry of the Legendre family of elliptic curves. We will prove that the mirror map of the Dwork family is equal to the period map of the Legendre family. This result provides an interesting explanation to the modularities of counting functions for K3 surfaces from the mirror symmetry point of view. We will also discuss the relations between the arithmetic geometry of smooth fibers of the Fermat pencil (Dwork family) and that of the smooth fibers of the Legendre family, e.g. Shioda-Inose structure…
If $X$ is a smooth projective variety over ${\mathbb R}$, the Hodge ${\mathcal D}$-conjecture of Beilinson asserts the surjectivity of the regulator map to Deligne cohomology with real coefficients. It is known to be false in general but is true in some special cases like Abelian surfaces and $K3$-surfaces - and still expected to be true when the variety is defined over a number field. We prove an analogue of this for Abelian surfaces at a non-Archimedean place where the surface has bad reduction. Here the Deligne cohomology is replaced by a certain Chow group of the special fibre. The case of…
Given a family of smooth complex projective varieties, the Hodge conjecture predicts the algebraicity of the locus of Hodge classes. This was proven unconditionnally by Cattani, Deligne and Kaplan in 1995. In a similar way, conjectures on algebraic cycles have led Green and Griffiths to conjecture the algebraicity of the zero locus of normal functions. This corresponds to a mixed version of the theorem of Cattani, Deligne and Kaplan. This result has been proven recently by Brosnan-Pearlstein, Kato-Nakayama-Usui, and Schnell building on work of M. Saito. We will present some of the ideas around…
In this paper, we prove that, if Deligne's "petites camarades conjecture" holds, then a Langlands type correspondence holds also for $p$-adic coefficients on a smooth curve over a finite field. We also prove that any overconvergent $F$-isocrystal of rank less than or equal to 2 on a smooth variety is $ι$-mixed.
In this note, we show that the normalized Hochschild co--chains of an associative algebra with a non--degenerate, symmetric, invariant inner product are an algebra over a chain model of the framed little discs operad which is given by cacti. In particular, in this sense they are a BV algebra up to homotopy and the Hochschild cohomology of such an algebra is a BV algebra whose induced bracket coincides with Gerstenhaber's bracket. To show this, we use a cellular chain model for the framed little disc operad in terms of normalized cacti. This model is given by tensoring our chain model for the l…
The scalar product formula for parahoric Deligne--Lusztig induction
本文针对p进群仿射抛物子群正深度表示的仿射抛物型德利涅-卢斯蒂格归纳，解决了其是否满足标量积公式的基础猜想，证明所有豪可分解分裂泛型对及特定椭圆型子群与非挠素数情形下该公式成立。
Tempered currents and Deligne cohomology of Shimura varieties, with an application to $\mathrm{GSp}_6$
本文利用缓增电流给出任意Shimura簇Deligne-Beilinson上同调的新描述，并应用于计算Siegel六维簇motivic上同调类的Beilinson高阶调节子像，将其与Spin $L$-函数非临界特殊值联系，验证Beilinson猜想。
Let X and Y be K-equivalent toric Deligne-Mumford stacks related by a single toric wall-crossing. We prove the Crepant Transformation Conjecture in this case, fully-equivariantly and in genus zero. That is, we show that the equivariant quantum connections for X and Y become gauge-equivalent after analytic continuation in quantum parameters. Furthermore we identify the gauge transformation involved, which can be thought of as a linear symplectomorphism between the Givental spaces for X and Y, with a Fourier-Mukai transformation between the K-groups of X and Y, via an equivariant version of the …
In despair, as Deligne (2000) put it, of proving the Hodge and Tate conjectures, we can try to find substitutes. For abelian varieties in characteristic zero, Deligne (1982) constructed a theory of Hodge classes having many of the properties that the algebraic classes would have if the Hodge conjecture were known. In this article I investigate whether there exists a theory of "rational Tate classes" on varieties over finite fields having the properties that the algebraic classes would have if the Hodge and Tate conjectures were known. v3. Submitted version.
A Deligne conjecture for prestacks
本文证明预叠的 Deligne 猜想类比，通过证实 Hawkins 关于 $\mathsf{Quilt}$ operad 正次同调消失的猜想，赋予 Gerstenhaber–Schack 复形自然的 $E_2$-代数结构。
Deligne's conjecture that $\ell$-adic sheaves on normal schemes over a finite field admit $\ell'$-companions was proved by L. Lafforgue in the case of curves and by Drinfeld in the case of smooth schemes. In this paper, we extend Drinfeld's theorem to smooth Artin stacks and deduce Deligne's conjecture for coarse moduli spaces of smooth Artin stacks. We also extend related theorems on Frobenius eigenvalues and traces to Artin stacks.
We analyze the geometry of the tower of Lubin-Tate deformation spaces, which parametrize deformations of a one-dimensional formal module of height h together with level structure. According to the conjecture of Deligne-Carayol, these spaces realize the local Langlands correspondence in their l-adic cohomology. This conjecture is now a theorem, but currently there is no purely local proof. Working in the equal characteristic case, we find a family of affinoids in the Lubin-Tate tower with good reduction equal to a rather curious nonsingular hypersurface, whose equation we present explicitly. Gr…
Convex elements and cohomology of deep level Deligne-Lusztig varieties
本文完整描述了深层 Deligne-Lusztig 簇的上同调，证明其 φ-权部分集中于单一次数且由 Yu 型子群诱导，并应用于表示分解、Fargues-Scholze 参数及验证 Chan-Oi 猜想。
In this paper, we shall generalize the theory of mixed Hodge structures due to Deligne and obtain a subcategory GMHS in the category of mixed Hodge structures such that we have Ext_{GMHS}^2(Q,-)\not=0 in general.
Affine Deligne-Lusztig varieties beyond the minute case
针对过去仅能完整描述的极小情形仿射德利涅-卢斯提格簇，本文通过推广其等价条件并引入志村数据的深度概念（研究深度小于2的情形），得到了一类几何性质仍良好且可计算的更大类ADLVs。
This is a revision of the paper that was previously entitled "Weighted Completion of Galois Groups and Some Conjectures of Deligne". Fix a prime number $ł$. We prove a conjecture stated by Ihara, which he attributes to Deligne, about the action of the absolute Galois group on the pro-$ł$ completion of the fundamental group of the thrice punctured projective line. Similar techniques are also used to prove part of a conjecture of Goncharov, also about the action of the absolute Galois group on the fundamental group of the thrice punctured projective line. The main technical tool is the weighted …
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