Open Dense Subset Research Articles

Rank 2 local systems and abelian varieties II

Let $X/\mathbb {F}_{q}$ be a smooth, geometrically connected, quasi-projective scheme. Let $\mathcal {E}$ be a semi-simple overconvergent $F$-isocrystal on $X$. Suppose that irreducible summands $\mathcal {E}_i$ of $\mathcal {E}$ have rank 2, determinant $\bar {\mathbb {Q}}_p(-1)$, and infinite monodromy at $\infty$. Suppose further that for each closed point $x$ of $X$, the characteristic polynomial of $\mathcal {E}$ at $x$ is in $\mathbb {Q}[t]\subset \mathbb {Q}_p[t]$. Then there exists a dense open subset $U\subset X$ such that $\mathcal {E}|_U$ comes from a family of abelian varieties on $U$. As an application, let $L_1$ be an irreducible lisse $\bar {\mathbb {Q}}_l$ sheaf on $X$ that has rank 2, determinant $\bar {\mathbb {Q}}_l(-1)$, and infinite monodromy at $\infty$. Then all crystalline companions to $L_1$ exist (as predicted by Deligne's crystalline companions conjecture) if and only if there exist a dense open subset $U\subset X$ and an abelian scheme $\pi _U\colon A_U\rightarrow U$ such that $L_1|_U$ is a summand of $R^{1}(\pi _U)_*\bar {\mathbb {Q}}_l$.

The monodromy of unit-root F-isocrystals with geometric origin

Let $C$ be a smooth curve over a finite field of characteristic $p$ and let $M$ be an overconvergent $\mathbf {F}$-isocrystal over $C$. After replacing $C$ with a dense open subset, $M$ obtains a slope filtration. This is a purely $p$-adic phenomenon; there is no counterpart in the theory of lisse $\ell$-adic sheaves. The graded pieces of this slope filtration correspond to lisse $p$-adic sheaves, which we call geometric. Geometric lisse $p$-adic sheaves are mysterious, as there is no $\ell$-adic analogue. In this article, we study the monodromy of geometric lisse $p$-adic sheaves with rank one. More precisely, we prove exponential bounds on their ramification breaks. When the generic slopes of $M$ are integers, we show that the local ramification breaks satisfy a certain type of periodicity. The crux of the proof is the theory of $\mathbf {F}$-isocrystals with log-decay. We prove a monodromy theorem for these $\mathbf {F}$-isocrystals, as well as a theorem relating the slopes of $M$ to the rate of log-decay of the slope filtration. As a consequence of these methods, we provide a new proof of the Drinfeld–Kedlaya theorem for irreducible $\mathbf {F}$-isocrystals on curves.

Complete symplectic quadrics and Kontsevich spaces of conics in Lagrangian Grassmannians

A wonderful compactification of an orbit under the action of a semi-simple and simply connected group is a smooth projective variety containing the orbit as a dense open subset, and where the added boundary divisor is simple normal crossing. We construct the wonderful compactification of the space of symmetric and symplectic matrices, and investigate its geometry. As an application, we describe the birational geometry of the Kontsevich spaces parametrizing conics in Lagrangian Grassmannians.

Orlicz affine isoperimetric inequalities for star bodies

The original goal of this paper is to extend the affine isoperimetric inequality and Steiner type inequality of Orlicz projection bodies (which originated to Lutwak, Yang, and Zhang [32]), from convex bodies to Lipschitz star bodies (whose radial functions are locally Lipschitz).In order to achieve it, we investigate the graph functions of the given Lipschitz star body K: Along almost all directions u, we can define the graph functions on an open dense subset of the orthogonal projection of K onto u⊥.

Approximation and Gâteaux differentiability of convex function in Banach spaces

AbstractThis paper shows that if f is a convex continuous function on a Gâteaux differentiability space X, then for any separable closed subspace E of X, there exists a sequence of continuous convex functions such that (1) on X; (2) is Gâteaux differentiable at all points of a dense open subset of X; (3) on E. Moreover, if X is separable, then there exists a continuous convex function sequence such that (1) and (2) are true and on X.

Strict pure strategy Nash equilibrium in large finite-player games when the action set is a manifold

We present results on the relationship between non-atomic games (in distributional form) and approximating games with a large but finite number of players. Specifically, in a setting with differentiable payoff functions, we show that: (1) The set of all non-atomic games has an open dense subset such that any finite-player game that is sufficiently close (in terms of distributions of players’ characteristics) to a game in this subset and has sufficiently many players has a strict pure strategy Nash equilibrium (Theorem 1), and (2) any equilibrium distribution of any non-atomic game is the limit of equilibrium distributions defined from strict pure strategy Nash equilibria of finite-player games (Theorem 2). This supplements our paper Carmona and Podczeck (2020b), where analogous results are established for the case where the action set of players is a subset of some Euclidean space, with non-empty interior, and payoff functions are such that equilibrium actions are in the interior of the action set. The goal of the present paper is to remove these assumptions.

Effective Bertini theorem and formulas for multiplicity and the local Łojasiewicz exponent

The classical Bertini theorem on generic intersection of an algebraic set with hyperplanes states the following: Let X be a nonsingular closed subvariety ofPkn, where k is an algebraically closed field. Then there exists a hyperplaneH⊂Pknnot containing X and such that the schemeH∩Xis regular at every point. Furthermore, the set of hyperplanes with this property forms an open dense subset of the complete linear system|H|considered as a projective space. We show that one can effectively indicate a finite family of hyperplanes H such that at least one of them satisfies the assertion of the Bertini theorem, provided the characteristic of the field k is equal to zero. As an application of the method used in the proof we give effective formulas for the multiplicity and the Łojasiewicz exponent of polynomial mappings.

Equivariant realizations of Hermitian symmetric space of noncompact type

Let \(M=G/K\) be a Hermitian symmetric space of noncompact type. We provide a way of constructing K-equivariant embeddings from M to its tangent space \(T_oM\) at the origin by using the polarity of the K-action. As an application, we reconstruct the K-equivariant holomorphic embedding so called the Harish-Chandra realization and the K-equivariant symplectomorphism constructed by Di Scala-Loi and Roos under appropriate identifications of spaces. Moreover, we characterize the holomorphic/symplectic embedding of M by means of the polarity of the K-action. Furthermore, we show a special class of totally geodesic submanifolds in M is realized as either linear subspaces or bounded domains of linear subspaces in \(T_oM\) by the K-equivariant embeddings. We also construct a K-equivariant holomorphic/symplectic embedding of an open dense subset of the compact dual \(M^*\) into its tangent space at the origin as a dual of the holomorphic/symplectic embedding of M.

Extended Hamilton–Jacobi Theory, Symmetries and Integrability by Quadratures

In this paper, we study the extended Hamilton–Jacobi Theory in the context of dynamical systems with symmetries. Given an action of a Lie group G on a manifold M and a G-invariant vector field X on M, we construct complete solutions of the Hamilton–Jacobi equation (HJE) related to X (and a given fibration on M). We do that along each open subset U⊆M, such that πU has a manifold structure and πU:U→πU, the restriction to U of the canonical projection π:M→M/G, is a surjective submersion. If XU is not vertical with respect to πU, we show that such complete solutions solve the reconstruction equations related to XU and G, i.e., the equations that enable us to write the integral curves of XU in terms of those of its projection on πU. On the other hand, if XU is vertical, we show that such complete solutions can be used to construct (around some points of U) the integral curves of XU up to quadratures. To do that, we give, for some elements ξ of the Lie algebra g of G, an explicit expression up to quadratures of the exponential curve expξt, different to that appearing in the literature for matrix Lie groups. In the case of compact and of semisimple Lie groups, we show that such expression of expξt is valid for all ξ inside an open dense subset of g.

On the μ-invariants of abelian varieties over function fields of positive characteristic

Let $A$ be an abelian variety over a global function field $K$ of characteristic $p$. We study the $\mu$-invariant appearing in the Iwasawa theory of $A$ over the unramified $\mathbb{Z}_p$-extension of $K$. Ulmer suggests that this invariant is equal to what he calls the dimension of the Tate-Shafarevich group of $A$ and that it is indeed the dimension of some canonically defined group scheme. Our first result is to verify his suggestions. He also gives a formula for the dimension of the Tate-Shafarevich group (which is now the $\mu$-invariant) in terms of other quantities including the Faltings height of $A$ and Frobenius slopes of the numerator of the Hasse-Weil $L$-function of $A / K$ assuming the conjectural Birch-Swinnerton-Dyer formula. Our next result is to prove this $\mu$-invariant formula unconditionally for Jacobians and for semistable abelian varieties. Finally, we show that the "$\mu=0$" locus of the moduli of isomorphism classes of minimal elliptic surfaces endowed with a section and with fixed large enough Euler characteristic is a dense open subset.

Weierstrass n-semigroups with even n and curves on toric surfaces

First, we give some Weierstrass semigroups which cannot be attained by any smooth curve on a smooth compact toric surface. Next, for any integer l ≥ 2 we describe the Weierstrass semigroup of a total ramification point of a cyclic covering of the projective line with degree l using l − 1 non-negative integers. And finally, we will study smooth curves C lying on a smooth compact toric surface S acted by the torus T which is a dense open subset of S . For an even integer n ≤ 10 we characterize the Weierstrass semigroup of a total ramification point of a cyclic covering of degree n which is the restriction to C of a toric fibration of S such that the ramification point lies on some T -invariant divisor.

Local Flexibility for Open Partial Differential Relations

AbstractWe show that local deformations, near closed subsets, of solutions to open partial differential relations can be extended to global deformations, provided all but the highest derivatives stay constant along the subset. The applicability of this general result is illustrated by a number of examples, dealing with convex embeddings of hypersurfaces, differential forms, and lapse functions in Lorentzian geometry.The main application is a general approximation result by sections that have very restrictive local properties on open dense subsets. This shows, for instance, that given any K ∈ ℝ every manifold of dimension at least 2 carries a complete C1, 1‐metric which, on a dense open subset, is smooth with constant sectional curvature K. Of course, this is impossible for C2‐metrics in general. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

Examples and applications of the density of strongly norm attaining Lipschitz maps

We study the density of the set SNA (M,Y) of those Lipschitz maps from a (complete pointed) metric space M to a Banach space Y which strongly attain their norm (i.e., the supremum defining the Lipschitz norm is actually a maximum). We present new and somehow counterintuitive examples, and we give some applications. First, we show that SNA (\mathbb T,Y) is not dense in Lip _0(\mathbb T,Y) for any Banach space Y , where \mathbb T denotes the unit circle in the Euclidean plane. This provides the first example of a Gromov concave metric space (i.e., every molecule is a strongly exposed point of the unit ball of the Lipschitz-free space) for which the density does not hold. Next, we construct metric spaces M satisfying that SNA (M,Y) is dense in Lip _0(M,Y) regardless Y but which contain isometric copies of [0,1] and so the Lipschitz-free space \mathcal F(M) fails the Radon–Nikodym property, answering in the negative a question posed by Cascales et al. in 2019 and by Godefroy in 2015. Furthermore, an example of such M can be produced failing all the previously known sufficient conditions for the density of strongly norm attaining Lipschitz maps. Finally, among other applications, we show that if M is a boundedly compact metric space for which SNA (M,\mathbb R) is dense in Lip _0(M,\mathbb R) , then the unit ball of the Lipschitz-free space on M is the closed convex hull of its strongly exposed points. Further, we prove that given a compact metric space M which does not contain any isometric copy of [0,1] and a Banach space Y , if SNA (M,Y) is dense, then SNA (M,Y) actually contains an open dense subset.

Geometric Regularity Results on B k α , β -Manifolds, I: Affine Connections

In this paper we consider existence and multiplicity results concerning affine connections on $C^{k}$-manifolds $M$ whose coefficients are as regular as one needs, following the regularity theory introduced in [10]. We show that if $M$ admits a $B_{\alpha,\beta}^{k}$-structure, then the existence of such regular connections can be established in terms of properties of the structural presheaf $B$. In other words, we propose a solution to the existence problem in this setting. With regard to the multiplicity problem, we show that the space of regular affine connections is an affine space of the space of regular $\operatorname{End}(TM)$-valued 1-forms, and that if two regular connections are locally additively different, then they are actually locally different. The existence of a topology in which the space of regular connections is a nonempty open dense subset of the space of all regular $\operatorname{End}(TM)$-valued 1-forms is suggested. <br><br> У цій роботі розглядаються результати існування та кратності для афінних зв’язностей на $C^{k}$-многовидах $M$, коефіцієнти яких настільки регулярні, наскільки це необхідно, відповідно до теорії регулярності, введеної в [10]. Показано, що якщо $M$ допускає $B_{\alpha,\beta}^{k}$-структуру, то існування таких регулярних зв’язностей можна встановити з точки зору властивостей структурного передпучка $B$. Іншими словами, ми пропонуємо розв'язання проблеми існування в цій постановці. Стосовно проблеми кратності, ми показуємо, що простір регулярних зв’язностей є афінним простором простору регулярних 1-форм зі значеннями в $\operatorname{End}(TM)$, і що якщо дві регулярні зв’язності локально адитивно різні, то вони насправді є локально різні. Запропоновано топологію, в якій простір регулярних зв’язностей є непорожньою відкритою щільною підмножиною простору всіх регулярних $\operatorname{End}(TM)$-значних 1-форм.

Moduli space of logarithmic connections singular over a finite subset of a compact Riemann surface

Let $S$ be a finite subset of a compact connected Riemann surface $X$ of genus $g \geq 2$. Let $\cat{M}_{lc}(n,d)$ denote the moduli space of pairs $(E,D)$, where $E$ is a holomorphic vector bundle over $X$ and $D$ is a logarithmic connection on $E$ singular over $S$, with fixed residues in the centre of $\mathfrak{gl}(n,\C)$, where $n$ and $d$ are mutually corpime. Let $L$ denote a fixed line bundle with a logarithmic connection $D_L$ singular over $S$. Let $\cat{M}'_{lc}(n,d)$ and $\cat{M}_{lc}(n,L)$ be the moduli spaces parametrising all pairs $(E,D)$ such that underlying vector bundle $E$ is stable and $(\bigwedge^nE, \tilde{D}) \cong (L,D_L)$ respectively. Let $\cat{M}'_{lc}(n,L) \subset \cat{M}_{lc}(n,L)$ be the Zariski open dense subset such that the underlying vector bundle is stable. We show that there is a natural compactification of $\cat{M}'_{lc}(n,d)$ and $\cat{M}'_{lc}(n,L)$ and compute their Picard groups. We also show that $\cat{M}'_{lc}(n,L)$ and hence $\cat{M}_{lc}(n,L)$ do not have any non-constant algebraic functions but they admit non-constant holomorhic functions. We also study the Picard group and algebraic functions on the moduli space of logarithmic connections singular over $S$, with arbitrary residues.

Strong submeasures and applications to non-compact dynamical systems

AbstractA strong submeasure on a compact metric space X is a sub-linear and bounded operator on the space of continuous functions on X. A strong submeasure is positive if it is non-decreasing. By the Hahn–Banach theorem, a positive strong submeasure is the supremum of a non-empty collection of measures whose masses are uniformly bounded from above. There are many natural examples of continuous maps of the form $f:U\rightarrow X$ , where X is a compact metric space and $U\subset X$ is an open-dense subset, where f cannot extend to a reasonable function on X. We can mention cases such as transcendental maps of $\mathbb {C}$ , meromorphic maps on compact complex varieties, or continuous self-maps $f:U\rightarrow U$ of a dense open subset $U\subset X$ where X is a compact metric space. For the aforementioned mentioned the use of measures is not sufficient to establish the basic properties of ergodic theory, such as the existence of invariant measures or a reasonable definition of measure-theoretic entropy and topological entropy. In this paper we show that strong submeasures can be used to completely resolve the issue and establish these basic properties. In another paper we apply strong submeasures to the intersection of positive closed $(1,1)$ currents on compact Kähler manifolds.

The smooth locus in infinite-level Rapoport–Zink spaces

Rapoport–Zink spaces are deformation spaces for $p$-divisible groups with additional structure. At infinite level, they become preperfectoid spaces. Let ${{\mathscr M}}_{\infty }$ be an infinite-level Rapoport–Zink space of EL type, and let ${{\mathscr M}}_{\infty }^{\circ }$ be one connected component of its geometric fiber. We show that ${{\mathscr M}}_{\infty }^{\circ }$ contains a dense open subset which is cohomologically smooth in the sense of Scholze. This is the locus of $p$-divisible groups which do not have any extra endomorphisms. As a corollary, we find that the cohomologically smooth locus in the infinite-level modular curve $X(p^{\infty })^{\circ }$ is exactly the locus of elliptic curves $E$ with supersingular reduction, such that the formal group of $E$ has no extra endomorphisms.

On orthogonal projections of Nöbeling spaces

Suppose that . We prove that there is a dense open subset of the Grassmann space such that the orthogonal projection of the standard Nöbeling space (which lies in for sufficiently large ) to every -dimensional plane in this subset is -soft and possesses the strong -universal property with respect to Polish spaces. Every such orthogonal projection is a natural counterpart of the standard Nöbeling space for the category of maps.

A quasi-locally constant function with given cluster sets

We continue to study cluster sets of a function defined on an open subset of a topological space. In our previous papers we characterized the cluster multifunction of any function and of a continuous function. The next aim was to study cluster sets of a quasi-continuous function. But here we solve a more specific problem on cluster sets of a quasi-locally constant function (i.e., a function that is quasi-continuous with respect to the discrete topology on the range space). We prove that for any open dense subset D of a metrizable space X and a dense subspace Y of a metrizable compact space $${\overline{Y}}$$, every upper continuous compact-valued multifunction $$\Phi :X\setminus D\multimap {\overline{Y}}$$ is the cluster multifunction $${\overline{f}}$$ of some quasi-locally constant function $$f:D\rightarrow Y$$.

Kostant–Toda lattices and the universal centralizer

To each complex semisimple Lie algebra g decorated with appropriate data, one may associate two completely integrable systems. One is the well-studied Kostant–Toda lattice, while the second is an integrable system defined on the universal centralizer Zg of g. These systems are similar in that each exploits and closely reflects the invariant theory of g, as developed by Chevalley, Kostant, and others. One also has Kostant’s description of level sets in the Kostant–Toda lattice, which turns out to suggest deeper similarities between the two integrable systems in question. Some more recent works allude to a strong connection between the two systems, e.g. Bezrukavnikov et al. (2005), Teleman (2014, 2018).We study relationships between the two aforementioned integrable systems, partly to understand and contextualize the similarities mentioned above. Our main result is a canonical open embedding of a flow-invariant open dense subset of the Kostant–Toda lattice into Zg. Secondary results include some qualitative features of the integrable system on Zg.