Hélène Esnault
If $X$ is an abelian variety over a field and $L$ is an invertible sheaf, we know that the degree of the 0-cycle $L^g$ is divisible by $g!$. As a 0-cycle, it is not, even over a field of cohomological dimension 1. But we show that over a finite field there is perhaps some hope.
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The notion of modulus is a striking feature of Rosenlicht-Serre's theory of generalized Jacobian varieties of curves. It was carried over to algebraic cycles on general varieties by Bloch-Esnault, Park, Rülling, Krishna-Levine. Recently, Kerz-Saito introduced a notion of Chow group of $0$-cycles with modulus in connection with geometric class field theory with wild ramification for varieties over finite fields. We study the non-homotopy invariant part of the Chow group of $0$-cycles with modulus and show their torsion and divisibility properties. Modulus is being brought to sheaf theory by Kah…
Effective good divisibility of rational homogeneous varieties
本文计算了有理齐性簇的有效好整除性，推广了Naldi和Occhetta关于复Grassmann流形的结果，并由此证明了到经典型Lie群有理齐性簇的非常值态射不存在。
For several Hodge-type Shimura varieties of good reduction in characteristic $p$, we show that the cone of weights of automorphic forms is encoded by the stack of $G$-zips of Pink-Wedhorn-Ziegler. This establishes several instances of a general conjecture formulated in previous papers by the authors. Furthermore, we prove in these cases that any mod $p$ automorphic form whose weight lies in a specific region of the weight space is divisible by a partial Hasse invariant. This generalizes to other Shimura varieties previous results of Diamond--Kassaei on Hilbert modular forms.
We apply Vojta's conjecture to blowups and deduce a number of deep statements regarding (generalized) greatest common divisors on varieties, in particular on projective space and on abelian varieties. Special cases of these statements generalize earlier results and conjectures. We also discuss the relationship between generalized greatest common divisors and the divisibility sequences attached to algebraic groups, and we apply Vojta's conjecture to obtain a strong bound on the divisibility sequences attached to abelian varieties of dimension at least two.
We extend results of Colliot-Thélène and Raskind on the $\mathcal{K}_2$-cohomology of smooth projective varieties over a separably closed field $k$ to the étale motivic cohomology of smooth, not necessarily projective, varieties over $k$. Some consequences are drawn, such as the degeneration of the Bloch-Lichtenbaum spectral sequence for any field containing $k$.
We prove several statements about arithmetic hyperbolicity of certain blow-up varieties. As a corollary we obtain multiple examples of simply connected quasi-projective varieties that are pseudo-arithmetically hyperbolic. This generalizes results of Corvaja and Zannier obtained in dimension 2 to arbitrary dimension. The key input is an application of the Ru-Vojta's strategy. We also obtain the analogue results for function fields and Nevanlinna theory with the goal to apply them in a future paper in the context of Campana's conjectures.
Let $X$ be a product of smooth projective curves over a finite unramified extension $k$ of $\mathbb{Q}_p$. Suppose that the Albanese variety of $X$ has good reduction and that $X$ has a $k$-rational point. We propose the following conjecture. The kernel of the Albanese map $CH_0(X)^0\rightarrow\text{Alb}_X(k)$ is $p$-divisible. When $p$ is an odd prime, we prove this conjecture for a large family of products of elliptic curves and certain principal homogeneous spaces of abelian varieties. Using this, we provide some evidence for a local-to-global conjecture for zero-cycles of Colliot-Thélène a…
Maximal disjoint Schubert cycles in rational homogeneous varieties
本文通过研究经典类型有理齐次簇的 Chow 环中低余维数有效零因子及有效良好可除性，探讨了这些簇之间非常数映射的存在性，推广了射影空间和 Grassmann 簇的已知结果。
An old conjecture of Voisin describes how zero-cycles on a variety $X$ should behave when pulled-back to the self-product $X^m$ for $m$ larger than the geometric genus of $X$. Using complete intersections of quadrics, we give examples of varieties in any dimension and with arbitrarily high geometric genus that verify Voisin's conjecture.
A divisibility sequence is a sequence of integers $\{d_n\}$ such that $d_m$ divides $d_n$ if $m$ divides $n$. Results of Bugeaud, Corvaja, Zannier, among others, have shown that the gcd of two divisibility sequences corresponding to subgroups of the multiplicative group grows in a controlled way. Silverman conjectured that a similar behaviour should appear in many algebraic groups. We extend results by Ghioca-Hsia-Tucker and Silverman for elliptic curves and prove an analogue of Silverman's conjecture over function fields for abelian and split semiabelian varieties and some generalizations of …
The present paper provides a geometric characterization of complete flag varieties for semisimple algebraic groups. Namely, if $X$ is a Fano manifold whose all elementary contractions are $\mathbb P^1$-fibrations then $X$ is isomorphic to the complete flag manifold $G/B$ where $G$ is a semi-simple Lie algebraic group and $B$ is a Borel subgroup of $G$.
We give a positive answer to the questions 4.4 on Hodge type and 4.2 on divisibility of the eigenvalues of Frobenius of the article by N. Fakhruddin and C. S. Rajan (NT/0402230).
We give a characteristic-free proof that general codimension-1 Schubert varieties meet transversally in a Grassmannian and in some related varieties. Thus the corresponding intersection numbers computed in the Chow (and quantum Chow) rings of these varieties are enumerative in all characteristics. We show that known transversality results do not apply to these enumerative problems, emphasizing the need for additional theoretical work on transversality. The method of proof also strengthens some results in real enumerative geometry.
Given an arithmetic surface and a positive hermitian line bundle over it, we bound the successive minima of the lattice of global sections of this line bundle. Our method combines a result of C.Voisin on secant varieties of projective curves with previous work by the author on the arithmetic analog of the Kodaira vanishing theorem. The paper also includes a result of A.Granville on the divisibility properties of binomial coefficients in a given line of Pascal's triangle.
Elementary anabelian varieties are anabelian
本文研究初等可解亏减变种的基本群同构与变种同构之间的双射关系，并证明占优映射与稳定同调注入映射之间的双射，验证了Grothendieck对Faltings的猜想。
Revisiting Dwork cohomology: Visibility and divisibility of Frobenius eigenvalues in rigid cohomology
该研究通过Dwork方法探究有限域上代数簇紧支集刚性上同调的Frobenius特征值，得到两个算术推论：仿射簇zeta函数可见证相关特征值，且刚性上同调的q-可除性下界优于ℓ进上同调的已知结果。
We use topological K-theory to study non-singular varieties with quadratic entry locus. We thus obtain a new proof of Russo's Divisibility Property for locally quadratic entry locus manifolds. In particular we obtain a K-theoretic proof of Zak's theorem that the dimension of a Severi variety must be 2, 4, 8 or 16 and so resolve a conjecture of Atiyah and Berndt. We also show how the same methods applied to dual varieties recover the Landman parity theorem.
Motivated by questions in modular representation theory, Carlson, Friedlander, and the first author introduced the varieties E(r, g) of r-dimensional abelian p-nilpotent subalgebras of a p-restricted Lie algebra g. In this paper, we identify the varieties E(r, g) for a reductive restricted Lie algebra g and r the maximal dimension of an abelian p-nilpotent subalgebra of g.
We study the functional codes of order $h$ defined by G. Lachaud on $\mathcal{X} \subset {\mathbb{P}}^n(\mathbb{F}_q)$ a non-degenerate Hermitian variety. We give a condition of divisibility of the weights of the codewords. For $\mathcal{X}$ a non-degenerate Hermitian surface, we list the first five weights and the corresponding codewords and give a positive answer on a conjecture formulated on this question. The paper ends with a conjecture on the minimum distance and the distribution of the codewords of the first $2h+1$ weights of the functional codes for the functional codes of order $h$ on…
Let X be a smooth proper variety over the quotient field of a Henselian discrete valuation ring with algebraically closed residue field of characteristic p. We show that for any coherent sheaf E on X, the index of X divides the Euler-Poincaré characteristic χ(X,E) if p=0 or p>dim(X)+1. If 0<p\leq dim(X)+1, the prime-to-p part of the index of X divides χ(X,E). Combining this with the Hattori-Stong theorem yields an analogous result concerning the divisibility of the cobordism class of X by the index of X. As a corollary, rationally connected varieties over the maximal unramified extension of a …
We use multiplication maps to give a characteristic-free approach to vanishing theorems on toric varieties. Our approach is very elementary but is enough powerful to prove vanishing theorems.
How does the first order language of fields encode birational invariants of varieties?... This question is related to rational points on varieties and effectiveness in algebraic/arithmetic geometry.
Reflective hyperbolic 2-elementary lattices, K3 surfaces and hyperkahler varieties
本文计算大反射偶2-初等双曲格的Coxeter图及其极大抛物子图，并将其应用于K3曲面和超凯勒簇理论。
The Sarkisov Program studies birational maps between varieties that are end products of the Minimal Model Program (MMP) on nonsingular uniruled varieties. If X and Y are terminal Q-factorial projective varieties endowed with a structure of Mori fibre space, any birational map between them can be decomposed into a finite number of elementary Sarkisov links. This decomposition is not unique in general, and any two distinct decompositions define a relation in the Sarkisov Program. This paper shows that relations in the Sarkisov Program are generated by some elementary relations. Roughly speaking,…
We prove an elementary but somewhat unexpected result about projective embeddings of smooth varieties X whose cotangent bundles are numerically effective. Specifically, we show that the degree of X in any projective embedding must grow (almost) exponentially in the dimension of X.
We discuss a non-computational elementary approach to a well-known criterion of divisibility by 2 in the group of rational points on an elliptic curve.
Ribet Points, geometric divisibility sequence and order of reductions on semiabelian varieties
本文在数域几何非分裂半阿贝尔簇上构造了Silverman猜想所述现象的无条件例子，推导了相关整数整除关系，证明存在无平方因子整数满足分母理想不变的条件。
We study a particular kind of fiber type contractions between complex, projective, smooth varieties f:X->Y, called Fano conic bundles. This means that X is a Fano variety, and every fiber of f is isomorphic to a plane conic. Denoting by rho_{X} the Picard number of X, we investigate such contractions when rho_{X}-rho_{Y} is greater than 1, called non-elementary. We prove that rho_{X}-rho_{Y} is at most 8, and we deduce new geometric information about our varieties, depending on rho_{X}-rho_{Y}. Moreover, when X is locally factorial with canonical singularities and with at most finitely many no…
This note studies, and partially solves, 3 elementary questions about continuous rational functions on real (and p-adic) algebraic varieties: Can one restrict such a function to a subvariety? Can one extend such a function from a subvariety? Which linear equations can be solved using such functions?
This is a small note meant to be published in a Conference Proceedings. We discuss elementary rationality questions in the Grothendieck ring of varieties for the quotient of a finite dimensional vector space over a characteristic 0 field by a finite group. Part of it reproduces the content of a letter dated September 27, 2008 addressed to Johannes Nicaise
We construct explicit examples of elementary extremal contractions, both birational and of fiber type, from smooth projective n-dimensional varieties, n\geq 4, onto smooth projective varieties, arising from classical projective geometry and defined over sufficiently small fields, not necessarily algebraically closed. The examples considered come from particular special homaloidal and subhomaloidal linear systems, which usually are degenerations of general phenomena classically investigated by Bordiga, Severi, Todd, Room, Fano, Semple and Tyrrell and more recently by Ein and Shepherd-Barron. Th…
Let ${\mathcal O}$ be the ring of $S$-integers in a number field $K$. For $A\in\rm{SL}_{2}(\mathcal{O})$ and $k\geq 1$, we define matrix-factorization varieties $V_k(A)$ over ${\mathcal O}$ which parametrize factoring $A$ into a product of $k$ elementary matrices; the equations defining $V_k(A)$ are written in terms of Euler's continuant polynomials. We show that the $V_k(A)$ are rational $(k-3)$-folds with an inductive fibration structure. We combine this geometric structure with arithmetic results to study the Zariski closure of the ${\mathcal O}$-points of $V_k(A)$. We prove that for $k\geq…
We give a sharp divisibility bound, in terms of g, for the degree of the field extension required to realize the endomorphisms of an abelian variety of dimension g over an arbitrary number field; this refines a result of Silverberg. This follows from a stronger result giving the same bound for the order of the component group of the Sato-Tate group of the abelian variety, which had been proved for abelian surfaces by Fite-Kedlaya-Rotger-Sutherland. The proof uses Minkowski's reduction method, but with some care required in the extremal cases when p equals 2 or a Fermat prime.
We present some simple examples of smooth projective varieties in positive characteristic, arising from linear algebra, which do not admit a lifting neither to characteristic zero, nor to the ring of second Witt vectors. Our first construction is the blow-up of the graph of the Frobenius morphism of a homogeneous space. The second example is a blow-up of $\mathbb{P}^3$ in a 'purely characteristic-$p$' configuration of points and lines.
The multiplicity of a Mori Dream Space: applications to polar duality and the classification of toric $\mathbb{Q}$-Fano varieties
本文将伪加权射影空间的重数概念推广到Mori梦想空间，引入完备toric簇的权群$G_Q$和权模$|Q|$，为Fano类toric簇分类提供框架，并给出Batyrev极对偶性的代数解释。
We discuss an elementary approach to the existence of non-uniruled ramified covers of varieties.
Let X be a normal affine algebraic variety with regular action of a torus \TT and T\subset\TT be a subtorus. We prove that each root of X with respect to T can be obtained by restriction of some root of X with respect to \TT. This allows to get an elementary proof of the description of roots of the affine Cremona group. Several results on restriction of roots in the case of subtorus action on an affine toric variety are obtained.
We prove that the function field of a variety which possesses a special correspondence in the sense of M. Rost preserves the rationality of cycles of small codimensions. This fact was proven by Vishik in the case of quadrics and played the crucial role in his construction of fields with $u$-invariant $2^r+1$. The main technical tools are algebraic cobordism of Levine-Morel, generalized Rost degree formula and divisibility of Chow traces of certain Landweber-Novikov operations. As a direct application of our methods we prove the Vishik's Theorem for all $F_4$-varieties.
Singular intersections in split semiabelian varieties
本文证明了分裂半阿贝尔簇中不可约曲线与代数子群相切交点的有限性，推广了二维环面情形，并应用于几何可整除序列的无平方性。
Let X be a projective hypersurface in P_k^n of degree d <= n. In this paper we study the relation between the class [X] in K_0(Var_k) and the existence of k-rational points. Using elementary geometric methods we show, for some particular X, that X(k) is nonempty if and only if [X] is equivalent to 1 modulo the class of the affine line in K_0(Var_k). More precisely we consider the following cases: a union of hyperplanes, a quadric, a cubic hypersurface with a singular k-rational point, and a quartic which is a union of two quadrics one of which being smooth.
These lectures give a short introduction to the study of curves on algebraic varieties. After an elementary proof of the dimension formula for the space of curves, we summarize the basic properties of uniruled and of rationally connected varieties. At the end, some connections with symplectic geometry are considered.
We prove that, if $A$ is a positively graded, graded commutative, local, finite Hopf algebra, its cohomology is finitely generated, thus unifying classical results of Wilkerson and Hopkins-Smith, and of Friedlander-Suslin. We do this by showing the existence of conormal elementary quotients.
We study the Selmer variety associated to a canonical quotient of the $\Q_p$-pro-unipotent fundamental group of a smooth projective curve of genus at least two defined over $\Q$ whose Jacobian decomposes into a product of abelian varieties with complex multiplication. Elementary multi-variable Iwasawa theory is used to prove dimension bounds, which, in turn, lead to a new proof of Diophantine finiteness over $\Q$ for such curves.
We construct higher-dimensional Calabi-Yau varieties defined over a given number field with Zariski dense sets of rational points. We give two elementary constructions in arbitrary dimensions as well as another construction in dimension three which involves certain Calabi-Yau threefolds containing an Enriques surface. The constructions also show that potential density holds for (sufficiently) general members of the families.
An explicit form of the standard conjecture of Hodge type for Hermitian varieties, with $p$-adic applications
本文给出 Hermitian 簇上 Hodge 型标准猜想的显式初等证明，利用对偶极图控制相交形式，并证明 $p$-adic 初等因子整除 $q^m$，从而得到代数环的数值等价性及猜想对有限覆盖簇的推广。
We show that the presentation of affine $\mathbb{T}$-varieties of complexity one in terms of polyhedral divisors holds over an arbitrary field. We also describe a class of multigraded algebras over Dedekind domains. We study how the algebra associated to a polyhedral divisor changes when we extend the scalars. As another application, we provide a combinatorial description of affine $\mathbf{G}$-varieties of complexity one over a field, where $\mathbf{G}$ is a (not-nescessary split) torus, by using elementary facts on Galois descent. This class of affine $\mathbf{G}$-varieties is described via …
Classical toric varieties are among the simplest objects in algebraic geometry. They arise in an elementary fashion as varieties parametrized by monomials whose exponents are a finite subset $\mathcal{A}$ of $\mathbb{Z}^n$. They may also be constructed from a rational fan $Σ$ in $\mathbb{R}^n$. The combinatorics of the set $\mathcal{A}$ or fan $Σ$ control the geometry of the associated toric variety. These toric varieties have an action of an algebraic torus with a dense orbit. Applications of algebraic geometry in geometric modeling and algebraic statistics have long studied the nonnegative r…
We present the elementary properties of log canonical centers of log varieties.
Let V be a finite dimensional vector space over a local field. Let us say that a complex function on V is elementary if it is a product of the additive character of a rational function Q on V and multiplicative characters of polynomials on V. In this paper we study when the Fourier transform of an elementary function is elementary. If Q has a nonzero Hessian, a necessary condition for this is that the Legendre transform Q_* of Q is rational. The basic example is a nondegenerate quadratic form. We study such functions Q, give examples, and find all of them such that both Q and Q_* are of the fo…
The purpose of this note is to give a refinement of the product formula proved in [1] for the Poincare polynomial of a smooth Schubert variety in the flag variety of an algebraic group G over C. This yields a factorization of the number of elements in a Bruhat interval [e,w] in the Weyl group W of G provided the Schubert variety associated to w is smooth. This gives an elementary necessary condition for a Schubert variety in the flag variety to be smooth.
A Frobenius difference field is an algebraically closed field of characteristic $p>0$, enriched with a symbol for $x \mapsto x^{p^m}$. We study a sentence or formula in the language of fields with a distinguished automorphism, interpreted in Frobenius difference fields with $p$ or $m$ tending to infinity. In particular, a decision procedure is found to determine when a sentence is true in almost every Frobenius difference field. This generalizes Cebotarev's density theorem and Weil's Riemann hypothesis for curves (both in qualitative versions), but hinges on a result going slightly beyond the …
We prove several results on symplectic varieties with a Hamiltonian action of a reductive group having invariant Lagrangian subvarieties. Our main result states that the images of the moment maps of a Hamiltonian variety and of the cotangent bundle over an invariant Lagrangian subvariety coincide. This implies that the complexity and rank of the Lagrangian subvariety are equal to the half of the corank and to the defect of the Hamiltonian variety, respectively. This result generalizes a theorem of Panyushev on the complexity and rank of a conormal bundle. A simple elementary proof of this theo…
We describe an algorithm which verifies whether linear algebraic cycles of the Fermat variety generate the lattice of Hodge cycles. A computer implementation of this confirms the integral Hodge conjecture for quartic and quintic Fermat fourfolds. Our algorithm is based on computation of the list of elementary divisors of both the lattice of linear algebraic cycles, and the lattice of Hodge cycles written in terms of vanishing cycles, and observing that these two lists are the same.
Let $A_g$ be an abelian variety of dimension $g$ and $p$-rank $λ\leq 1$ over an algebraically closed field of characteristic $p>0$. We compute the number of homomorphisms from $π_1^{\text{ét}}(A_g,a)$ to $GL_n(\mathbb F_q)$, where $q$ is any power of $p$. We show that for fixed $g$, $λ$, and $n$, the number of such representations is polynomial in $q$, and give an explicit formula for this polynomial. We show that the set of such homomorphisms forms a constructible set, and use the geometry of this space to deduce information about the coefficients and degree of the polynomial. In the last sec…
The purpose of this note is to point out an elementary but somewhat surprising connection between the work of Buser and Sarnak on lengths of periods of abelian varieties and the Seshadri constants measuring the local positivity of theta divisors. The link is established via symplectic blowing up, in the spirit of McDuff and Polterovich. As an application of the main inequality, we get a simple new proof of a statement of Buser-Sarnak type to the effect that the Jacobian of a curve has a period of unusually small length. We also deduce a lower bound on the Seshadri constant of a very general p.…
This paper studies the geometry and combinatorics of three interrelated varieties: Springer fibers, Steinberg varieties, and parabolic Hessenberg varieties. We prove that each parabolic Hessenberg variety is the pullback of a Steinberg variety under the projection of the flag variety to an appropriate partial flag variety and we give three applications of this result. The first application constructs an explicit paving of all Steinberg varieties in Lie type $A$ in terms of semistandard tableaux. As a result, we obtain an elementary proof of a theorem of Steinberg and Shimomura that the well-kn…
Semidefinite representation of convex hulls of rational varieties
利用正半定矩矩阵和多项式平方和分解的对偶性质，证明了有理参数化代数簇的凸包在曲线、二次参数化超曲面和双变量四次参数化超曲面情形下具有半定表示。
A complex projective manifold is rationally connected, resp. rationally simply connected, if finite subsets are connected by a rational curve, resp. the spaces parameterizing these connecting rational curves are themselves rationally connected. We prove that a projective scheme over a global function field with vanishing "elementary obstruction" has a rational point if it deforms to a rationally simply connected variety in characteristic 0. This gives new, uniform proofs over these fields of the Period-Index Theorem, the quasi-split case of Serre's "Conjecture II", and Lang's $C_2$ property.
We give a description of the tropical variety of univariate polynomials of degree n having two double roots. As a set, it is given as the union of three types of maximal cones of dimension n-1, where only cones of two of these types are cones of the secondary fan of {0,...,n}. Through Kapranov's theorem, this goal is achieved by a careful study of the possible valuations of the elementary symmetric functions of the roots of a polynomial with two double root. Despite its apparent simplicity, the computation of the tropical Severi variety has both combinatorial and arithmetic ingredients.
We introduce a new class of algebraic varieties which we call frieze varieties. Each frieze variety is determined by an acyclic quiver. The frieze variety is defined in an elementary recursive way by constructing a set of points in affine space. From a more conceptual viewpoint, the coordinates of these points are specializations of cluster variables in the cluster algebra associated to the quiver. We give a new characterization of the finite--tame--wild trichotomy for acyclic quivers in terms of their frieze varieties. We show that an acyclic quiver is representation finite, tame, or wild, re…
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