The Grothendieck--Katz $p$-curvature conjecture predicts that an arithmetic differential equation whose reduction modulo $p$ has vanishing $p$-curvatures for {\em almost all} $p,$ has finite monodromy. It is known that it suffices to prove the conjecture for differential equations on $\mathbb{P}^{1}-\{0,1,\infty\}.$ We prove a variant of this conjecture for $\mathbb{P}^{1}-\{0,1,\infty\},$ which asserts that if the equation satisfies a certain convergence condition for {\em all} $p,$ then its monodromy is trivial. For those $p$ for which the $p$-curvature makes sense, its vanishing implies our condition. We deduce from this a description of the differential Galois group of the equation in terms of $p$-curvatures and certain local monodromy groups. We also prove similar variants of the $p$-curvature conjecture for the elliptic curve with $j$-invariant $1728$ minus its identity and for $\mathbb{P}^1-\{\pm 1,\pm i,\infty\}$.
In this paper, we establish a link between the structure theory of the pro-unipotent motivic fundamental group of the projective line minus three points and Diophantine geometry. In particular, we give a p-adic proof of Siegel's theorem.
We look at natural foliations on the Painlevé VI moduli space of regular connections of rank 2 on $\pp ^1 -{t_1,t_2,t_3,t_4}$. These foliations are fibrations, and are interpreted in terms of the nonabelian Hodge filtration, giving a proof of the nonabelian Hodge foliation conjecture in this case. Two basic kinds of fibrations arise: from apparent singularities, and from quasiparabolic bundles. We show that these are transverse. Okamoto's additional symmetry, which may be seen as Katz's middle convolution, exchanges the quasiparabolic and apparent-singularity foliations.
In this paper, the author constructs a family of algebraic cycles in Bloch's cubical cycle complex over the projective line minus three points which are expected to correspond to multiple polylogarithms in one variable. Elements in this family are in particular equidimensional over the projective line minus three points. In weight greater or equal to $2$, they are naturaly extended as equidimensional cycle over the affine line. This allows to consider their fibers at the point 1 and this is one of the main differences with Gangl, Goncharov and Levin work where generic arguments are imposed for…
This article deals with the tamely ramified geometric Langlands correspondence for GL_2 on $\mathbf{P}_{\mathbf{F}_q}^1$, where $q$ is a prime power, with tame ramification at four distinct points $D = \{\infty, 0,1, t\} \subset \mathbf{P}^1(\mathbf{F}_q)$. We describe in an explicit way (1) the action of the Hecke operators on a basis of the cusp forms, which consists of $q$ elements; and (2) the correspondence that assigns to a pure irreducible rank 2 local system $E$ on $\mathbf{P}^1 \setminus D$ with unipotent monodromy its Hecke eigensheaf on the moduli space of rank 2 parabolic vector bu…
In a previous work, the author have built two families of distinguished algebraic cycles in Bloch-Kriz cubical cycle complex over the projective line minus three points. The goal of this paper is to show how these cycles induce well-defined elements in the $\HH^0$ of the bar construction of the cycle complex and thus generated comodules over this $\HH^0$, that is a mixed Tate motives as in Bloch and Kriz construction. In addition, it is shown that out of the two families only ones is needed at the bar construction level. As a consequence, the author obtains that one of the family gives a basis…
The affine line minus one point is the underlying space of the algebraic torus of dimension one. However the fibration of an affine algebraic threefold by the affine line minus one point is not always the quotient morphism of the threefold by the algebraic torus. We investigate the structure of affine algebraic threefolds having fibrations by the affine line minus one point. Our results cover not only algebro-geometric but also topological properties of the threefolds. We also consider when the fibration by the affine line minus one point becomes the quotient morphism by an algebraic torus of …
We describe a construction of an object associated to the fundamental group of the projective line minus three points in the Bloch-Kriz category of mixed Tate motives. This description involves Massey products of Steinberg symbols in the motivic cohomology of the ground field.
For two properly intersecting effective cycles in projective space X,Y, and their intersection product Z, the metric Bezout Theorem relates the degrees, heights of X,Y, and Z, as well as their distances and algebraic distances to a given point theta. Applications of this Theorem are in the area of Diophantine Approximation, giving estimates for approximation properties of Z with respect to $θ$ against the ones of X, and Y.
A set of multi-homogeneous equations for the Jacobian of a genus two curve is given. The approach used is to write down affine equations for the Jacobian minus various tranlations of the Theta-divisor by [2]-division points, and then to write down affine glue equations for the overlaps. Taking multi-projective completions for all of these then yields a complete multi-projective two-dimensional variety whose points are in one-to-one correspondence with degree zero divisor classes on the curve (i.e. the Picard group). This multi-projective variety then becomes a complete projective two-dimension…
We study the moduli space of logarithmic connections of rank 2 on the Riemann sphere minus n points with fixed spectral data. There are two natural Lagrangian maps: one towards apparent singularities of the associated fuchsian scalar equation, and another one towards moduli of parabolic bundles. We show that these are transversal and dual to each other. In case n=5, we recover the beautiful geometry of Del Pezzo surfaces of degree 4.
In this paper, we study the geometric invariant theory on algebraic spaces, and construct te moduli spaces of $\mathcal{H}$-semistable sheaves on projective Deligne-Mumford stacks over algebraic spaces $S$. We prove that this moduli space is projective over $S$ as an algebraic space.
We prove that a generic holomorphic foliation on a weighted projective plane has no algebraic solutions when the degree is big enough. We also prove an analogous result for foliations on Hirzebruch surfaces.
Let $X$ be a smooth projective connected curve of genus $g\ge 2$ defined over an algebraically closed field $k$ of characteristic $p>0$. Let $G$ be a finite group, $P$ a Sylow $p$-subgroup of $G$ and $N_G(P)$ its normalizer in $G$. We show that if there exists an étale Galois cover $Y\to X$ with group $N_G(P)$, then $G$ is the Galois group wan étale Galois cover $\mathcal{Y}\to\mathcal{X}$, where the genus of $\mathcal{X}$ depends on the order of $G$, the number of Sylow $p$-subgroups of $G$ and $g$. Suppose that $G$ is an extension of a group $H$ of order prime to $p$ by a $p$-group $P$ and $…
For $θ$ a non-algebraic point on a quasi projective variety over a number field, I prove that $θ$ has an approximation by a series of algebraic points of bounded height and degree which is essentially best possible. Applications of this result will include a proof of a slightly strengthened version of the Philippon criterion, some new algebraic independence criteria, statements concerning metric transcendence theory on varieties of arbitrary dimension, and a rather accurate estimate for the number of algebraic points of bounded height and degree on quasi projective varieties over number fields…
We prove that the universal cover of a normal, projective variety X is quasi-projective if and only if a finite, étale cover of X is a fiber bundle over an Abelian variety with simply connected fiber.
The Deligne-Lusztig varieties associated to the Coxeter classes of the algebraic groups 2A2, 2B2 and 2G2 are affine algebraic curves. We produce explicit projective models of the closures of these curves. Furthermore for $d$ the Coxeter number of these groups, we find polynomials for each of these models that cut out the $\F_q$-points, the $\F_{q^d}$-points and the $\F_{q^{d+1}}$-points, and demonstrate a relation satisfied by these polynomials.
Constructing families of abelian varieties of $\text{GL}_2$-type over $4$-punctured complex projective line via $p$-adic Hodge theory and Langlands correspondence and application to algebraic solutions of Painleve VI equation
To understand the structure of an algebraic variety we often embed it in various projective spaces. This develops the notion of projective geometry which has been an invaluable tool in algebraic geometry. We develop a perfectoid analog of projective geometry, and explore how equipping a perfectoid space with a map to a certain analog of projective space can be a powerful tool to understand its geometric and arithmetic structure. In particular, we show that maps from a perfectoid space $X$ to the perfectoid analog of projective space correspond to line bundles on $X$ together with some extra da…
A criterion for the existence of a birational embedding of an algebraic curve into a projective plane with two Galois points is presented. Several novel examples of plane curves with two inner Galois points as an application are described.
A general method of computing cohomology groups of the space of nonsingular algebraic hypersurfaces of degree $d$ in $CP^n$ is described. Using this method, rational cohomology groups of such spaces with $n=2, d \le 4$ and $n=3=d$ are calculated.
In this article, we address the classification of smooth projective algebraic surfaces over complex numbers admitting algebraic semigroup structures. We give a full description of those surfaces $S$, which has at least one non-trivial algebraic semigroup structure, when the Kodaira dimension of $S$ is $ -\infty$ and $ 0$. For the case "$ κ(S)=1$", we give a description of one special type of elliptic surfaces which admit non-trivial algebraic semigroup laws. \\ For a given surface $S$, it is an interesting problem to describe all algebraic semigroup structures on it and determine the dimension…
We provide a Hodge theoretical characterization of the set of algebraic numbers which arises from the complete list, due to A. Beauville, of semistable families of elliptic curves over $\mathbb{P}^1$ with four singular fibers. Our technical innovation is the analysis of the periodicity of the uniformizing Higgs bundle attached to $\mathbb{P}^1$ minus four points over the field of complex numbers.
Let $\mathbf{x}=(x,y)$. A projective 2-dimensional flow is a solution to a 2-dimensional projective translation equation (PrTE) $(1-z)ϕ(\mathbf{x})=ϕ(ϕ(\mathbf{x}z)(1-z)/z)$, $ϕ:\mathbb{C}^{2}\mapsto\mathbb{C}^{2}$. Previously we have found all solutions of the PrTE which are rational functions. The rational flow gives rise to a vector field $\varpi(x,y)\bullet\varrho(x,y)$ which is a pair of 2-homogenic rational functions. On the other hand, only very special pairs of 2-homogenic rational functions, as vector fields, give rise to rational flows. The main ingredient in the proof of the classif…
Let C be an ACM (projectively normal) nondegenerate smooth curve in projective 3-space, and suppose C is general in its Hilbert scheme - this is irreducible once the postulation is fixed. Answering a question posed by Peskine, we show the gonality of C is d-l, where d is the degree of the curve, and l is the maximum order of a multisecant line of C. Furthermore l=4 except for two series of cases, in which the postulation of C forces every surface of minimum degree containing C to contain a line as well. We compute the value of l in terms of the postulation of C in these exceptional cases. We a…
Given a complex projective surface with an ADE singularity and p_{g}=0, we construct ADE bundles over it and its minimal resolution. Furthermore, we descibe their minuscule representation bundles in terms of configurations of (reducible) (-1)-curves.
This paper is devoted to the investigation of selected situations when the computation of projective (and other) equivalences of algebraic varieties can be efficiently solved with the help of finding projective equivalences of finite sets on the projective line. In particular, we design a unifying approach that finds for two algebraic varieties $X,Y$ from special classes an associated set of automorphisms of the projective line (the so called good candidate set) consisting of candidates for the construction of possible mappings $X\rightarrow Y$. The functionality of the designed method is pres…
We study the homotopy types of spaces of algebraic (rational) maps from real projective spaces into complex projective spaces. In a previous paper we have shown that the inclusion of the first space into the second one is a homotopy equivalence. In this paper we prove that the homotopy types of the terms of the natural "degree" filtration approximate closer and closer the homotopy type of the space of continuous maps and obtain bounds that describe the closeness of the approximation in terms of the degree. Moreover, we compute the homotopy groups of the spaces in low dimensions.
We look at rank two parabolic Higgs bundles over the projective line minus five points which are semistable with respect to a weight vector $μ\in[0,1]^5$. The moduli space corresponding to the central weight $μ_c=(\frac{1}{2}, \dots, \frac{1}{2})$ is studied in details and all singular fibers of the Hitchin map are described, including the nilpotent cone. After giving a description of fixed points of the $\mathbb C^*$-action we obtain a proof of Simpson's foliation conjecture in this case. For each $n\ge 5$, we remark that there is a weight vector so that the foliation conjecture in the moduli…
Non-dicritical codimension one foliations on projective spaces of dimension four or higher always have an invariant algebraic hypersurface. The proof relies on a strengthening of a result by Rossi on the algebraization/continuation of analytic subvarieties of projective spaces.
Regular algebraic surfaces isogenous to a higher product of curves can be obtained from finite groups with ramification structures. We find unmixed ramification structures for finite groups constructed as p-quotients of particular infinite groups with special presentations related to finite projective planes.
A. Girand has constructed an explicit two-parameter family of flat connections over the complex projective plane $\mathbb{P}^2$. These connections have dihedral monodromy and their polar locus is a prescribed quintic composed of a conic and three tangent lines. In this paper, we give a generalization of this construction. That is, we construct an explicit $n$-parameter family of flat connections over the complex projective space $\mathbb{P}^n$. Moreover, we discuss the relation between these connections and the Garnier system.
We discover a class of projective self-dual algebraic varieties. Namely, we consider actions of isotropy groups of complex symmetric spaces on the projectivized nilpotent varieties of isotropy modules. For them, we classify all orbit closures $X$ such that $X=\check X$ where $\check X$ is the projective dual of $X$. We give algebraic criteria of projective self-duality for the considered orbit closures.
Nagoya University（名古屋大学）；Korea Institute for Advanced Study（韩国高等研究院）；The University of Tokyo（东京大学）
We discuss the geometry of rational maps from a projective space of an arbitrary dimension to the product of projective spaces of lower dimensions induced by linear projections. In particular, we give an algebro-geometric variant of the projective reconstruction theorem by Hartley and Schaffalitzky [HS09].
We show an analogue of Jordan's theorem for algebraic groups defined over a field $\mathbb k$ of arbitrary characteristic. As a consequence, a Jordan-type property holds for the automorphism group of any projective variety over $\mathbb k$.
Atiyah and Hirzebruch gave examples ofeven degree torsion classes in the singularcohomology of a smooth complex projective manifold, which arenot Poincaré dual to an algebraiccycle. We notice that the order ofthese classes are small comparedto the dimension of the manifold.However, building upon a construction ofKollàr, one can provide such examples witharbitrary high prime order, the dimension being fixed. This method alsoprovides examples of torsion algebraiccycles, which are non trivial in the Griffiths' groups, and lie in a arbitrary high level of the H.Saito filtration onChow groups.
We provide a natural smooth projective compactification of the space of algebraic maps from the projective line to the projective space of dimension n by adding a divisor with simple normal crossings.
The set of common roots of a finite set $I$ (it is an ideal) of homogeneous polynomials is known as projective algebraic set $V$. In this article I show how to dualize such projective algebraic sets $V$ by elimination of variables from a system of polynomials with the Gröbner bases method. A dualization algorithm is implemented in the computer algebra system {\sc Singular}. Some examples are given. The main diagram shows the relationship between the ideal $I$, its radical $\sqrt{I}$ and their dual ideals.
In the present paper, we study the relationship between deformation quantizations and Frobenius-projective structures defined on an algebraic curve in positive characteristic. A Frobenius-projective structure is an analogue of a complex projective structure on a Riemann surface, which was introduced by Y. Hoshi. Such an additional structure has some equivalent objects, e.g., a dormant $\mathrm{PGL}_2$-oper and a projective connection having a full set of solutions. The main result of the present paper provides a canonical construction of a Frobenius-constant quantization on the cotangent space…
It is well-known that a Riemann surface can be decomposed into the so-called pairs-of-pants. Each pair-of-pants is diffeomorphic to a Riemann sphere minus 3 points. We show that a smooth complex projective hypersurface of arbitrary dimension admits a similar decomposition. The n-dimensional pair-of-pants is diffeomorphic to the complex projective n-space minus n+2 hyperplanes. Alternatively, these decompositions can be treated as certain fibrations on the hypersurfaces. We show that there exists a singular fibration on the hypersurface with an n-dimensional polyhedral complex as its base and a…
In our previous work, we provided an algebraic proof of the Zinger's comparison formula between genus one Gromov-Witten invariants and reduced invariants when the target space is a complete intersection of dimension two or three in a projective space. In this paper, we extend the result in any dimensions and for descendant invariants.
We show that the spectral radius for the action of a self map $f$ of a smooth projective variety (over an arbitrary base field) on its $\ell$-adic cohomology is achieved on the $f^*$-stable sub-algebra generated by any ample class. This generalizes a result of Esnault-Srinivas who had obtained an analogous result for automorphisms of surfaces. Over $\mathbb{C}$ we also show that this sub-algebra is naturally an irreducible representation of a Looijenga-Lunts-Verbitsky type Lie algebra acting on the cohomology of a smooth projective variety.
The purpose of this paper is to give a linear and effective height inequality for algebraic points on curves over functional fields. Our height inequality can be viewed as the logarithmic canonical class inequality of a punctured curve over a functional field (a fibered surface minus a section).
We introduce a notion of Homological Projective Duality for smooth algebraic varieties in dual projective spaces, a homological extension of the classical projective duality. If algebraic varieties $X$ and $Y$ in dual projective spaces are Homologically Projectively Dual, then we prove that the orthogonal linear sections of $X$ and $Y$ admit semiorthogonal decompositions with an equivalent nontrivial component. In particular, it follows that triangulated categories of singularities of these sections are equivalent. We also investigate Homological Projective Duality for projectivizations of vec…
In this article we prove a result comparing rationality of algebraic cycles over the function field of a projective homogeneous variety under a linear algebraic group of type $F_4$ or $E_8$ and over the base field, which can be of any characteristic.
We classify minimal-degree curves in the Hilbert schemes of points on algebraic surfaces. When the algebraic surface is the projective plane, the nef cone and a flip structure of these Hilbert schemes are determined.
