Abstract In this article, we consider a generalization of the conjugate Hardy H 2 $H^2$ upper H squared spaces and give some properties of the minimal norm of the generalization and some relations between the norm of the generalization and the minimal L 2 $L^2$ upper L squared integrals. As applications, we give some monotonicity results for the conjugate Hardy H 2 $H^2$ upper H squared kernels and the Bergman kernels on planar regions, and some relations between the conjugate Hardy H 2 $H^2$ upper H squared kernels and the Bergman kernels on planar regions.
We give a new definition of matrix Schwarzian derivative, which is simpler than the Lagrange Schwarzian derivative and also provides an alternative to other definitions which appear in the literature. Some basic properties are discussed, in particular, analogs of M & ouml;bius invariance and the result of a change of independent variable, these being the two properties of the scalar Schwarzian derivative often considered to account for its universality. We then use our new definition of matrix Schwarzian derivative to construct new Schwarzian matrix ordinary and partial differential equation hierarchies: a Schwarzian matrix second Painlev & eacute; hierarchy and a Schwarzian matrix Korteweg-de Vries hierarchy, respectively. In addition, we define a new matrix second Painlev & eacute; hierarchy.
Abstract In this text, we will consider schemes, which are smooth and proper over a field, and whose tangent sheaf is free. We will call such schemes T-trivial varieties . Over the complex numbers, the T -trivial varieties are precisely the abelian varieties. Igusa observed however that in characteristic p ≤ 3 $p\leq 3$ p less than or equals 3 there are T -trivial bielliptic surfaces, which are not isomorphic to abelian varieties. In this context, we show that T -trivial varieties X separably dominated by abelian varieties A can exist only for p ≤ 3 $p\leq 3$ p less than or equals 3 . Furthermore, we prove that every T -trivial variety, after passing to a finite étale covering, can be fibered in T -trivial varieties with Betti number b 1 = 0 $b_1=0$ b 1 equals 0 . We also show that if some n -dimensional T -trivial variety X lifts to characteristic zero and p ≥ 2 n + 1 $p\geq 2n+1$ p greater than or equals 2 n plus 1 , then X admits a finite étale covering by an abelian variety. Along the way, we establish several results about the automorphism group of abelian varieties, and the existence of relative Albanese maps.
Let G be a finite group acting on an ice quiver with potential ( Q , F , W ) $(Q, F, W)$ left parenthesis upper Q comma upper F comma upper W right parenthesis . We construct the corresponding G-equivariant relative cluster category and G-equivariant Higgs category, extending the work of Demonet. Using the orbit mutations on the set of G-stable cluster-tilting objects of the Higgs category and an appropriate cluster character, we can link these data to a skew-symmetrizable cluster algebra with coefficients. As a specific example, this provides an additive categorification for cluster algebras with principal coefficients in the non-simply laced case.
Okounkov (2003, Progr. Math. 213, 329-347) conjectured the log-concavity about the structure constants for many interesting basis from representation theory. For the cluster algebra, Gross et al. (2018, J. Amer. Math. Soc. 31, 497-608) introduced the atomic theta basis. We prove that the coefficients of the exponents of any cluster variable of type $A_n$ are log-concave. We show that the structure constants for the theta basis of type $A_2$ are log-concave. As for larger generality, we conjecture the log-concavity of the structure constants for the theta basis of the cluster algebra.
The Askey-Wilson algebras illustrate the bispectral property of orthogonal polynomials in the Askey scheme. The universal Askey-Wilson algebra $\triangle _q$ is a central extension of the Askey-Wilson algebras associated with the most general orthogonal polynomials in the Askey scheme. The Verma $\triangle _q$ -modules are a family of infinite-dimensional $\triangle _q$ -modules with marginal weights. Under the condition that q is not a root of unity, it was shown that every finite-dimensional irreducible $\triangle _q$ -module has a marginal weight and is isomorphic to a quotient of a Verma $\triangle _q$ -module. Assume that q is a root of unity. We prove that every finite-dimensional irreducible $\triangle _q$ -module with a marginal weight is isomorphic to a quotient of a Verma $\triangle _q$ -module. More precisely, two natural families of finite-dimensional quotients of Verma $\triangle _q$ -modules contain all finite-dimensional irreducible $\triangle _q$ -modules with marginal weights up to isomorphism. Furthermore, we classify the finite-dimensional irreducible $\triangle _q$ -modules with marginal weights up to isomorphism.
We discuss the relative log minimal model theory for log surfaces in the analytic setting. More precisely, we show that the minimal model program, the abundance theorem, and the finite generation of log canonical rings hold for log pairs of complex surfaces which are projective over complex analytic varieties.
Using metric techniques introduced by Berndtsson, we show a result on constancy of families dominated by a constant variety and, on the opposite side, a result on the strong non isotriviality of certain families of surfaces with positive index. We also give metric interpretations of liftability of relative volume forms and of strong non isotriviality in terms of the complex conjugate of a suitable representative of the Kodaira-Spencer class.
We show that Miyaoka's bound for the number of conics on a degree-2h K3 surface is attained for high h, and analogously for higher even degree (smooth) rational curves.
In this paper we investigate the p-rank stratification of the moduli space of curves of genus g that admit a double cover to a fixed elliptic curve E in characteristic p>2. We show that the closed p-rank strata of this moduli space are equidimensional of the expected dimension. We also show the existence of a smooth double cover of E of all the possible values of the p-rank on this moduli space.
We show that a divisor in a rational homogenous variety with split normal sequence is the preimage of a hyperplane section in either the projective space or a quadric.
In this article, we classify irregular threefolds with numerically trivial canonical divisors in positive characteristic. For a threefold, if its Albanese dimension is not maximal, then the Albanese morphism will induce a fibration which either maps to a curve or is fibered by curves. In practice, we treat arbitrary dimensional irregular varieties with either one-dimensional Albanese fiber or one-dimensional Albanese image. We prove that such a variety carries another fibration transversal to its Albanese morphism (a "bi-fibration" structure), which is an analog structure of bielliptic or quasi-bielliptic surfaces. In turn, we give an explicit description of irregular threefolds with trivial canonical divisors.
We present a unified construction of perfectoid towers from specific prisms which covers all the previous constructions of (p-torsion-free) perfectoid towers. By virtue of the construction, perfectoid towers can be systematically constructed for a large class of rings with Frobenius lift. Especially, any Frobenius lifting of a reduced $\mathbb{F}_p$-algebra has a perfectoid tower.
The notion of faithful flatness of a module over a commutative ring is studied for two R-modules M arising in functional analysis, where R is a Banach algebra and M is a Hilbert space. The following results are shown: If X is a locally compact Hausdorff topological space, and & micro; is a positive Radon measure on X, then L-2(X, & micro;) is a flat L-infinity(X, & micro;)-module. Moreover: center dot If & micro; is o--finite, then for every finitely generated, nonzero, proper ideal n of L infinity(X, & micro;), there holds n L-2(X, & micro;) c L-2(X,& micro;). center dot If X is the union of an increasing family of Borel sets U-n, n is an element of N, such that for each n is an element of N, U-n is compact and & micro; (Un+1 \U-n) >0, then L-2(X, & micro;) is not a faithfully flat L infinity(X, & micro;)-module. In addition, it is shown that the classical Hardy space H-2 is a flat, but not a faithfully flat H infinity module, which answers a 2005 question of Alban Quadrat.
In this article, we obtain uniform effective upper bounds for the projective dimension and the Castelnuovo-Mumford regularity of homogeneous ideals inside a standard graded polynomial ring S over a field. Such bounds are independent of the number of variables of S, in the spirit of Stillman's conjecture and of the Ananyan-Hochster's theorem, and depend on partial data extracted from the beginning or the end of the resolution. The main result is an extension of a theorem due to McCullough from 2012. Namely, we bound the projective dimension and the regularity of an ideal in terms of the regularity of a fraction of the syzygies.
We describe the behavior of a free reduced plane projective curve with respect to the addition, respectively, deletion, of a smooth conic. These results apply in particular to conic-line arrangements. We present some obstructions to the geometry and combinatorics of a free reduced curve, generalizing results known a priori only for free projective line arrangements.
For the module category of an Artin algebra, we generalize the notion of torsion pairs to ideal torsion pairs. Instead of full subcategories of modules, ideals of morphisms of the ambient category are considered. We characterize the functorially finite ideal torsion pairs, which are those fulfilling some nice approximation conditions, first through corresponding functors and then through the notion of ideals determined by objects introduced in this work. As an application of this theory, we generalize preprojective modules, introduce a new homological dimension, the torsion dimension, and establish its connection with the Krull-Gabriel dimension. In particular, it is shown that both dimensions coincide for hereditary Artin algebras.
In this article, by utilizing the properties of elliptic functions, we characterize the meromorphic solutions of Fermat-type functional equations $f(z)<^>{n}+f(L(z))<^>{m}=1$ over the complex plane $\mathbb {C}$ , where $L(z)$ is a nonconstant entire function, and m and n are two positive integers. As applications, we also investigate the meromorphic solutions of Fermat-type difference and q-difference equations.
For any power q of the positive ground field characteristic, a smooth q-bic threefold-the Fermat threefold of degree $q+1$ , for example-has a smooth surface S of lines which behaves like the Fano surface of a smooth cubic threefold. I develop projective, moduli-theoretic, and degeneration techniques to study the geometry of S. Using, in addition, the modular representation theory of the finite unitary group and the geometric theory of filtrations, I compute the cohomology of the structure sheaf of S when q is prime.
