arXiv subjects
Publications and source records attributed to Weizhe Zheng.
We prove that for any real number $ξ\neq 0$ and any coprime integers $p>q\ge1$ such that $ξ$ is irrational or $q>1$, the image in $\mathbb{R}/\mathbb{Z}$ of the sequence $(ξ(-p/q)^n)_{n\ge 0}$ is not contained in any interval of length less than $(1+q/p-q^2/p^2)/p$.
We study the distribution modulo one of linear recurrent sequences of real numbers. We prove criteria for the finiteness of the set of limit values of the fractional parts of such a sequence and give lower bounds for the maximal distance between two limit values. Our results generalize theorems of Flatto, Lagarias, Pollington, and Dubickas.
Astorg and Boc Thaler studied the dynamics of certain skew-products $f$ tangent to the identity on $\mathbb{C}^2$, with two real parameters $α>1$ and $β$ derived from its coefficients. They proved that if there exists a strictly increasing sequence of positive integers $(n_k)_{k\geqslant 1}$ such that $(σ_k)_{k\geqslant 1}:=(n_{k+1}-αn_k-β\ln n_k)_{k\geqslant 1}$ converges, then $f$ admits wandering domains of rank one. They also proved that for $α>1$ with the Pisot property, the condition that $θ:=\frac{β\lnα}{α-1}$ is rational is sufficient for the existence of $(n_k)_{k\geqslant 1}$ such that $(σ_k)_{k\geqslant 1}$ converges to a cycle. They asked if this condition is necessary. When $α$ is an algebraic number, we answer the question of Astorg and Boc Thaler in the affirmative. Furthermore, denoting by $P(x)\in\mathbb{Z}[x]$ the minimal polynomial of~$α$, we prove that $θ\in\frac{1}{P(1)}\mathbb{Z}$ is necessary and sufficient for the existence of $(n_k)_{k\geqslant 1}$ such that $(σ_k)_{k\geqslant 1}$ converges. Combined with the work of Astorg and Boc Thaler, our result provides explicit new examples of skew-products on $\mathbb{C}^2$ with wandering domains of rank one.
Let $b\ge 2$ be an integer. Using Sturmian words we describe all irrational real numbers $ξ$ such that the image in $\mathbb{R}/\mathbb{Z}$ of the sequence $(ξ(-b)^n)_{n\ge 0}$ is contained in an interval of length $b^{-1}+b^{-2}-b^{-3}$. In previous work (arXiv:2603.16794) we showed that the image cannot be contained in a shorter interval.
We prove that Frobenius eigenvalues of $\ell$-adic cohomology and $\ell$-adic intersection cohomology of rigid spaces over $p$-adic local fields are algebraic integers and we give bounds for their $p$-adic valuations. As an application, we deduce bounds for their weights, proving conjectures of Bhatt, Hansen, and Zavyalov. We also give examples of monodromy-pure perverse sheaves on projective curves with non monodromy-pure cohomology, answering a question of Hansen and Zavyalov.
We show that Schur classes of ample vector bundles on smooth projective varieties satisfy Hodge-Riemann relations on $H^{p,q}$ under the assumption that $H^{p-2,q-2}$ vanishes. More generally, we study Hodge-Riemann polynomials, which are partially symmetric polynomials that produce cohomology classes satisfying the Hodge-Riemann property when evaluated at Chern roots of ample vector bundles. In the case of line bundles and in bidegree $(1,1)$, these are precisely the nonzero dually Lorentzian polynomials. We prove various properties of Hodge-Riemann polynomials, confirming predictions and answering questions of Ross and Toma. As an application, we show that the derivative sequence of any product of Schur polynomials is Schur log-concave, confirming conjectures of Ross and Wu.
In this article, we develop a theory of Grothendieck's six operations for derived categories in étale cohomology of Artin stacks, for both torsion and adic coefficients. We prove several desired properties of the operations, including the base change theorem in derived categories. This extends many previous theories on this subject, including the one developed by Laszlo and Olsson, in which the operations are subject to more assumptions and the base change isomorphism is only constructed on the level of sheaves. Moreover, our theory works for higher Artin stacks as well. In addition, we define perverse t-structures on higher Artin stacks for general perversity, extending Gabber's work on schemes. Our method differs from previous approaches, as we exploit the theory of stable $\infty$-categories developed by Lurie. We enhance derived categories, functors, and natural isomorphisms to the level of $\infty$-categories and introduce $\infty$-categorical (co)homological descent. To handle the issue of ``homotopy coherence'', we develop a general technique for gluing subcategories of $\infty$-categories and several other $\infty$-categorical techniques. We obtain categorical equivalences between simplicial sets associated to certain multisimplicial sets. Such equivalences can be used to construct functors in different contexts. One of our category-theoretical results generalizes Deligne's gluing theory developed in the construction of the extraordinary pushforward operation in étale cohomology of schemes.
We prove a relative Lefschetz-Verdier theorem for locally acyclic objects over a Noetherian base scheme. This is done by studying duals and traces in the symmetric monoidal $2$-category of cohomological correspondences. We show that local acyclicity is equivalent to dualizability and deduce that duality preserves local acyclicity. As another application of the category of cohomological correspondences, we show that the nearby cycle functor over a Henselian valuation ring preserves duals, generalizing a theorem of Gabber.
Let $\mathbf{F}_q$ be a finite field of $q$ elements. We show that the normalized Jacobi sum $q^{-(m-1)/2}J(χ_1,\dots,χ_m)$ ($χ_1\dotsm χ_m$ nontrivial) is asymptotically equidistributed on the unit circle, when $χ_1\in \mathcal{A}_1,\dots, χ_m\in \mathcal{A}_m$ run through arbitrary sets of nontrivial multiplicative characters of $\mathbf{F}_q^\times$, if $\#\mathcal{A}_1\ge q^{\frac{1}{2}+ε}$, $\#\mathcal{A}_2 \ge (\log q)^{\frac{1}δ-1}$ for $ε>δ>0$ fixed and $q\to \infty$ or if $\#\mathcal{A}_1\#\mathcal{A}_2/q\to \infty$. This extends previous results of Xi, Z. Zheng, and the authors.
This paper studies the sliced nearby cycle functor and its commutation with duality. Over a Henselian discrete valuation ring, we show that this commutation holds, confirming a prediction of Deligne. As an application we give a new proof of Beilinson's theorem that the vanishing cycle functor commutes with duality up to twist. Over an excellent base scheme, we show that the sliced nearby cycle functor commutes with duality up to modification of the base. We deduce that duality preserves universal local acyclicity over an excellent regular base. We also present Gabber's theorem that local acyclicity implies universal local acyclicity over a Noetherian base.
We show that compatible systems of $\ell$-adic sheaves on a scheme of finite type over the ring of integers of a local field are compatible along the boundary up to stratification. This extends a theorem of Deligne on curves over a finite field. As an application, we deduce the equicharacteristic case of classical conjectures on $\ell$-independence for proper smooth varieties over complete discrete valuation fields. Moreover, we show that compatible systems have compatible ramification. We also prove an analogue for integrality along the boundary.
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.
In this paper, we show that the Galois representations provided by $\ell$-adic cohomology of proper smooth varieties, and more generally by $\ell$-adic intersection cohomology of proper varieties, over any field, are orthogonal or symplectic according to the degree. We deduce this from a preservation result of orthogonal and symplectic pure perverse sheaves by proper direct image. We show moreover that the subgroup of the Grothendieck group generated by orthogonal pure perverse sheaves of even weights and symplectic pure perverse sheaves of odd weights are preserved by Grothendieck's six operations. Over a finite field, we deduce parity and symmetry results for Jordan blocks appearing in the Frobenius action on intersection cohomology of proper varieties, and virtual parity results for the Frobenius action on ordinary cohomology of arbitrary varieties.
In this sequel of arXiv:1211.5294 and arXiv:1211.5948, we develop an adic formalism for étale cohomology of Artin stacks and prove several desired properties including the base change theorem. In addition, we define perverse t-structures on Artin stacks for general perversity, extending Gabber's work on schemes. Our results generalize results of Laszlo and Olsson on adic formalism and middle perversity. We continue to work in the world of $\infty$-categories in the sense of Lurie, by enhancing all the derived categories, functors, and natural transformations to the level of $\infty$-categories.
We present Gabber's theorem of independence of $l$ for the intersection cohomology of a proper equidimensional scheme over the spectrum of a finite field. We follow [Fuji] very closely. ----- On expose ici le théorème de Gabber d'indépendance de $l$ pour la cohomologie d'intersection d'un schéma propre équidimensionnel sur le spectre d'un corps fini. On suit [Fuji] à très peu près.
Gluing of two pseudo functors has been studied by Deligne, Ayoub, and others in the construction of extraordinary direct image functors in étale cohomology, stable homotopy, and mixed motives of schemes. In this article, we study more generally the gluing of finitely many pseudo functors. Given pseudo functors $F_i\colon \mathcal{A}_i\to \mathcal{D}$ defined on sub-$2$-categories $\mathcal{A}_i$ of a $2$-category $\mathcal{C}$, we are concerned with the problem of finding pseudo functors $\mathcal{C}\to \mathcal{D}$ extending $F_i$ up to pseudo natural equivalences. With the help of $n$-fold categories, we organize gluing data for $n$ pseudo functors into $2$-categories. We establish general criteria for equivalence between such $2$-categories for $n$ pseudo functors and for $n-1$ pseudo functors, which can be applied inductively to the gluing problem. Results of this article are used in arXiv:1006.3810 to construct extraordinary direct image functors in étale cohomology of Deligne-Mumford stacks.
Let $k$ be an algebraically closed field. Let $Λ$ be a noetherian commutative ring annihilated by an integer invertible in $k$ and let $\ell$ be a prime number different from the characteristic of $k$. We prove that if $X$ is a separated algebraic space of finite type over $k$ endowed with an action of a $k$-algebraic group $G$, the equivariant étale cohomology algebra $H^*([X/G],Λ)$, where $[X/G]$ is the quotient stack of $X$ by $G$, is finitely generated over $Λ$. Moreover, for coefficients $K \in D^+_c([X/G],\mathbb{F}_{\ell})$ endowed with a commutative multiplicative structure, we establish a structure theorem for $H^*([X/G],K)$, involving fixed points of elementary abelian $\ell$-subgroups of $G$, which is similar to Quillen's theorem in the case $K = \mathbb{F}_{\ell}$. One key ingredient in our proof of the structure theorem is an analysis of specialization of points of the quotient stack. We also discuss variants and generalizations for certain Artin stacks.
Let $\mathbf{F}_q$ be a finite field of $q$ elements. For multiplicative characters $χ_1,\dots, χ_m$ of $\mathbf{F}_q^\times$, we let $J(χ_1,\dots, χ_m)$ denote the Jacobi sum. Nicholas Katz and Zhiyong Zheng showed that for $m=2$, the normalized Jacobi sum $q^{-1/2}J(χ_1,χ_2)$ ($χ_1χ_2$ nontrivial) is asymptotically equidistributed on the unit circle as $q\to \infty$, when $χ_1$ and $χ_2$ run through all nontrivial multiplicative characters of $\mathbf{F}_q^\times$. In this paper, we show a similar property for $m\ge 2$. More generally, we show that the normalized Jacobi sum $q^{-(m-1)/2}J(χ_1,\dots,χ_m)$ ($χ_1\dotsm χ_m$ nontrivial) is asymptotically equidistributed on the unit circle, when $χ_1,\dots, χ_m$ run through arbitrary sets of nontrivial multiplicative characters of $\mathbf{F}_q^\times$ with two of the sets being sufficiently large. The case $m=2$ answers a question of Shparlinski.