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We prove that for a fixed braid index there are only finitely many possible shapes of the annular Rasmussen invariant of braid closures. Focusing on the case of 3-braids, we compute the Rasmussen s-invariant and the annular Rasmussen invariant of all 3-braid closures. As a corollary, we show that the vanishing/non-vanishing of the ψ invariant is entirely determined by the s-invariant and the self-linking number for 3-braid closures.
We show that noetherian splinters ascend under essentially étale homomorphisms. Along the way, we also prove that the henselization of a noetherian local splinter is always a splinter and that the completion of a local splinter with geometrically regular formal fibers is a splinter. Finally, we give an example of a (nonexcellent) Gorenstein local splinter with mild singularities whose completion is not a splinter. Our results provide evidence for a strengthening of the direct summand theorem, namely that regular maps preserve the splinter property.
We prove that for each positive integer n, there exists a positive number such that n-dimensional toric quotient singularities satisfy the ACC for MLDs on the interval . In the course of the proof, we show a geometric Jordan property for finite automorphism groups of affine toric varieties.
Rees-like algebras have played a major role in settling the Eisenbud–Goto conjecture. This paper concerns the structure of the canonical module of the Rees-like algebra and its class groups. Via an explicit computation based on linkage, we provide an explicit and surprisingly well-structured resolution of the canonical module in terms of a type of double-Koszul complex. Additionally, we give descriptions of both the divisor class group and the Picard group of a Rees-like algebra.
We introduce the notion of pseudo-Néron model and give new examples of varieties admitting pseudo-Néron models other than Abelian varieties. As an application of pseudo-Néron models, given a scheme admitting a finite morphism to an Abelian scheme over a positive-dimensional base, we prove that for a very general genus-0 degree-d curve in the base with d sufficiently large, every section of the scheme over the curve is contained in a unique section over the entire base.
We study the monodromies and the limit mixed Hodge structures of families of complete intersection varieties over a punctured disk in the complex plane. For this purpose, we express their motivic nearby fibers in terms of the geometric data of some Newton polyhedra. In particular, the limit mixed Hodge numbers and some part of the Jordan normal forms of the monodromies of such a family will be described very explicitly.
We show the existence of several new infinite families of polynomially-growing automorphisms of free groups whose mapping tori are CAT(0) free-by-cyclic groups. Such mapping tori are thick, and thus not relatively hyperbolic. These are the first families comprising infinitely many examples for each rank of the nonabelian free group; they contrast strongly with Gersten’s example of a thick free-by-cyclic group which cannot be a subgroup of a CAT(0) group.
For rational points on algebraic varieties defined over a number field K, we study the behavior of the property of weak approximation with Brauer–Manin obstruction under extension of the ground field. We construct K-varieties accompanied with a quadratic extension such that the property holds over K (conditionally on a conjecture) whereas fails over L. The result is unconditional when K equals or certain quadratic number fields. We give an explicit example when .
In this paper, we are concerned with the well-posedness of Vlasov–Poisson equation near vacuum in weighted fractional Sobolev space , that is, Bessel potential space. The difficulty lies in the estimates of the electronic term . To overcome it, we establish the - estimate and take advantage of commutator estimates. The -version energy method and pseudo-differential operator techniques are also introduced.
Regular semisimple Hessenberg varieties are smooth subvarieties of the flag variety, and their examples contain the flag variety itself and the permutohedral variety, which is a toric variety. We give a complete classification of Fano and weak Fano regular semisimple Hessenberg varieties in type A in terms of combinatorics of Hessenberg functions. In particular, we show that if the anticanonical bundle of a regular semisimple Hessenberg variety is nef, then it is in fact nef and big.
Given a dominant rational self-map on a projective variety over a number field, we can define the arithmetic degree at a rational point. It is known that the arithmetic degree at any point is less than or equal to the first dynamical degree. In this paper, we show that there are densely many -rational points with maximal arithmetic degree (i.e., whose arithmetic degree is equal to the first dynamical degree) for self-morphisms on projective varieties. For unirational varieties and Abelian varieties, we show that there are densely many rational points with maximal arithmetic degree over a sufficiently large number field. We also give a generalization of a result of Kawaguchi and Silverman in the Appendix.
In this paper we revisit Ribenboim’s notion of higher derivations of modules and relate it to the recent work of de Fernex and Docampo on the sheaf of differentials of the arc space. In particular, we derive their formula for the Kähler differentials of the Hasse–Schmidt algebra as a consequence of the fact that the Hasse–Schmidt algebra functors commute.
The main goal of this paper is to study the different definitions of generating sequences appearing in the literature. We present these definitions and show that under certain situations they are equivalent. We also present an example that shows that they are not, in general, equivalent. We also present the relation of generating sequences and key polynomials.
A ring is said to be rigid if it admits no nonzero locally nilpotent derivations, and an affine variety is rigid if its coordinate ring is rigid. In this paper, we improve some techniques for determining the rigidity of k-domains (affine varieties) over a field k of characteristic zero. First, we generalize the ABC theorem. Then we study locally nilpotent derivations of a simple algebraic extension of a k-domain R, where for some nonzero and some positive integer n. Subsequently, we study locally nilpotent derivations and rigidity of an extension of R such that or for some nonzero and some positive integers . Finally, as applications of these general results, we prove the rigidity of some quadrinomial varieties and Pham–Brieskorn hypersurfaces.
We generalize the classical semicontinuity theorem for GIT (semi)stable loci under variations of linearizations to a relative situation of an equivariant projective morphism over an affine base S. As an application to moduli problems, we consider degenerations of Hilbert schemes and give a conceptual interpretation of the (semi)stable loci of the degeneration families constructed in [GHH19].
Given a natural number k and an orientable surface S of finite type, define the k-curve graph to be the graph with vertices corresponding to isotopy classes of essential simple closed curves on S and with edges corresponding to pairs of such curves admitting representatives that intersect at most k times. We prove that the automorphism group of the k-curve graph of a surface S is isomorphic to the extended mapping class group for all k satisfying . We prove the same result for the so-called systolic complex, a variant of the curve graph with many complete subgraphs coming from interesting collections of systoles with respect to a hyperbolic metric. This resolves a conjecture of Schmutz Schaller.
We obtain a new bound on Weyl sums with degree polynomials of the form , , with fixed and , which holds for almost all and all . We improve and generalize some recent results of Erdoǧan and Shakan (2019), whose work also shows links between this question and some classical partial differential equations. We extend this to more general settings of families of polynomials for all with for a set of of full Lebesgue measure, provided that f is a Hölder function.
We use bordered Floer theory to study properties of twisted Mazur pattern satellite knots . We prove that is not Floer homologically thin, with two exceptions. We calculate the 3-genus of in terms of the twisting parameter n and the 3-genus of the companion K, and we determine when is fibered. As an application to our results on Floer thickness and 3-genus, we verify the Cosmetic Surgery Conjecture for many of these satellite knots.
Let L be a nonnegative self-adjoint operator acting on , where X is a space of homogeneous type of dimension n. Suppose that the heat kernel of L satisfies a Gaussian upper bound. It is known that the operator is bounded on for and (see, e.g., [7, 22, 33]). The index was only obtained recently in [9, 10], and this range of s is sharp since it is precisely the range known in the case where L is the Laplace operator Δ on . In this paper, we establish that for , the operator is of weak type , that is, there is a constant C, independent of t and f, such that
(for when and when ). Moreover, we also show that the index is sharp when L is the Laplacian on by providing an example.
Our results are applicable to Schrödinger groups for large classes of operators including elliptic operators on compact manifolds, Schrödinger operators with nonnegative potentials and Laplace operators acting on Lie groups of polynomial growth or irregular nondoubling domains of Euclidean spaces.
We study the local equivalence problem for real-analytic () hypersurfaces that, in some holomorphic coordinates with , are rigid in the sense that their graphing functions
are independent of v. Specifically, we study the group of rigid local biholomorphic transformations of the form
where and , which preserve the rigidity of hypersurfaces.
After performing a Cartan-type reduction to an appropriate -structure, we find exactly two primary invariants and , which we express explicitly in terms of the 5-jet of the graphing function F of M. The identical vanishing then provides a necessary and sufficient condition for M to be locally rigidly biholomorphic to the known model hypersurface
We establish that always .
If one of these two primary invariants or does not vanish identically, then on either of the two Zariski-open sets or , we show that this rigid equivalence problem between rigid hypersurfaces reduces to an equivalence problem for a certain five-dimensional -structure on M, that is, we get an invariant absolute parallelism on . Hence drops from 7 to 5, illustrating the gap phenomenon.
For a classical link, Milnor defined a family of isotopy invariants, called Milnor -invariants. Recently, Chrisman extended Milnor -invariants to welded links by a topological approach. The aim of this paper is showing that Milnor -invariants can be extended to welded links by a combinatorial approach. The proof contains an alternative proof of the invariance of the original -invariants for classical links.
We study truncated point schemes of connected graded algebras as families over the parameter space of varying relations for the algebras, proving that the families are flat over the open dense locus where the point schemes achieve the expected (i.e., minimal) dimension.
When the truncated point scheme is zero-dimensional, we obtain its number of points counted with multiplicity via a Chow ring computation. This latter application in particular confirms a conjecture of Brazfield that a generic two-generator two-relation algebra has seventeen truncated point modules of length six.
In 1979, Pisier proved remarkably that a sequence of independent and identically distributed standard Gaussian random variables determines, via random Fourier series, a homogeneous Banach algebra strictly contained in , the class of continuous functions on the unit circle and strictly containing the classical Wiener algebra , that is, . This improved some previous results obtained by Zafran in solving a long-standing problem raised by Katznelson. In this paper, we extend Pisier’s result by showing that any probability measure on the unit circle defines a homogeneous Banach algebra contained in . Thus Pisier algebra is not an isolated object but rather an element in a large class of Pisier-type algebras. We consider the case of spectral measures of stationary sequences of Gaussian random variables and obtain a sufficient condition for the boundedness of the random Fourier series in the general setting of dependent random variables .
We prove that integral projective curves with Cohen–Macaulay singularities over an algebraically closed field of characteristic zero are determined by their triangulated category of perfect complexes. This partially extends a theorem of Bondal and Orlov to the case of singular projective varieties and in particular shows that all integral complex projective curves are determined by their category of perfect complexes.
Every L-space knot is fibered and strongly quasi-positive, but this does not hold for L-space links. In this paper, we use the so-called H-function, which is a concordance link invariant, to introduce a subfamily of fibered strongly quasi-positive L-space links. Furthermore, we present an infinite family of L-space links that are not quasi-positive.
For a link in a thickened annulus , we define a filtration on Sarkar–Seed–Szabó’s perturbation of the geometric spectral sequence. The filtered chain homotopy type is an invariant of the isotopy class of the annular link. From this, we define a two-dimensional family of annular link invariants and study their behavior under cobordisms. In the case of annular links obtained from braid closures, we obtain a necessary condition for braid quasi-positivity and a sufficient condition for right-veeringness, as well as Bennequin-type inequalities.
Let be a -dimensional compact CR manifold with codimension , , and let G be a d-dimensional compact Lie group with CR action on X and T be a globally defined vector field on X such that , where is the space of vector fields on X induced by the Lie algebra of G. In this work, we show that if X is strongly pseudoconvex in the direction of T and , then there exists a G-equivariant CR embedding of X into for some . We also establish a CR orbifold version of Boutet de Monvel’s embedding theorem.
and let be the group generated by a and . In this paper, we study the problem of determining when the group is not free for rational. We give a robust computational criterion, which allows us to prove that if for , then is non-free with the possible exception of . In this latter case, we prove that the set of denominators for which is non-free has natural density 1. For a general numerator , we prove that the lower density of denominators for which is non-free has a lower bound
Finally, we show that for a fixed s, there are arbitrarily long sequences of consecutive denominators r such that is non-free. The proofs of some of the results are computer assisted, and Mathematica code has been provided together with suitable documentation.
We determine explicit generators for a cohomology group constructed from a solution of a Fuchsian linear differential equation and describe its relation with cohomology groups with coefficients in a local system. In the parametrized case, this yields into an algorithm which computes new Fuchsian differential equations from those depending on multi-parameters. This generalizes the classical convolution of solutions of Fuchsian differential equations.
As a generalization of the ideals of star configurations of hypersurfaces, we consider the a-fold product ideal when is a sequence of n-generic forms and . Firstly, we show that this ideal has complete intersection quotients when these forms are of the same degree and essentially linear. Then, we study its symbolic powers while focusing on the uniform case with . For large a, we describe its resurgence and symbolic defect. And for general a, we also investigate the corresponding invariants for meeting-at-the-minimal-components version of symbolic powers.
We prove local uniformization of Abhyankar valuations of an algebraic function field K over a ground field k. Our result generalizes the proof of this result, with the additional assumption that the residue field of the valuation ring is separable over k, by Hagen Knaf and Franz-Viktor Kuhlmann. The proof in this paper uses different methods, being inspired by the approach of Zariski and Abhyankar.
For families of surfaces, we establish a sufficient criterion for real or complex multiplication. Our criterion is arithmetic in nature. It may show, at first, that the generic fibre of the family has a nontrivial endomorphism field. Moreover, the endomorphism field does not shrink under specialization. As an application, we present two explicit families of surfaces having real multiplication by and , respectively.
We study behavior of solutions to the nonlinear generalized Hartree equation, where the nonlinearity is of nonlocal type and is expressed as a convolution
Our main goal is to understand global behavior of solutions of this equation in various settings. In this work we make an initial attempt towards this goal and study (finite energy) solutions. We first investigate the local well-posedness and small data theory. We then, in the intercritical regime (), classify the behavior of solutions under the mass-energy assumption , identifying the sharp threshold for global versus finite time solutions via the sharp constant of the corresponding convolution type Gagliardo–Nirenberg interpolation inequality (note that the uniqueness of a ground state is not known in the general case). In particular, depending on the size of the initial mass and gradient, solutions will either exist for all time and scatter in , or blow up in finite time, or diverge along an infinite time sequence. To obtain scattering or divergence to infinity, in this paper we employ the well-known concentration compactness and rigidity method of Kenig and Merle  with the novelty of studying the nonlocal, convolution nonlinearity.
In this paper, we associate with isometries of cube complexes specific subspaces, referred to as median sets, which play a similar role as minimizing sets of semisimple isometries in spaces. Various applications are deduced, including a cubulation of centralizers, a splitting theorem, a proof that Dehn twists in mapping class groups must be elliptic for every action on a cube complex, a cubical version of the flat torus theorem, and a structural theorem about polycyclic groups acting on cube complexes.
In , Grahl and Nevo obtained a significant improvement for the well-known normality criterion of Montel. They proved that for a family of meromorphic functions in a domain , and for a positive constant ε, if for each there exist meromorphic functions such that f omits in D and
for all , then is normal in D. Here, ρ is the spherical metric in . In this paper, we establish the high-dimensional versions for the above result and for the following well-known result of Lappan: A meromorphic function f in the unit disc is normal if there are five distinct values such that
In this paper, via the contraction mapping principle, we give a proof of a Bloch-type theorem for normalized harmonic Bochner–Takahashi K-mappings and for solutions to equations of the form , where P is a homogeneous differential operator with an analytic fundamental solution, that is, homogeneous elliptic operators with constant coefficients.
In this paper, we give complex geometric descriptions of the notions of algebraic geometric positivity of vector bundles and torsion-free coherent sheaves, such as nef, big, pseudo-effective, and weakly positive, by using singular hermitian metrics. As an application, we obtain a generalization of Mori’s result. We also give a characterization of the augmented base locus by using singular hermitian metrics on vector bundles and the Lelong numbers.
Let X be a simply connected rational elliptic space of formal dimension m, and let denote the group of homotopy classes of self-equivalences of X. If Y is the space obtained by attaching rational cells of dimension q to X, where , then we prove that and , where . Here denotes the subgroup of of the elements inducing the identity on the homology groups. Consequently, we show that, for any finite group G and for any , there exists a simply connected space X such that .
We extend the notion of Frobenius Betti numbers to large classes of finitely generated modules over rings of prime characteristic, which are not assumed to be local. To do so, we introduce new invariants, which we call Frobenius Euler characteristics. We prove uniform convergence and upper semicontinuity results for Frobenius Betti numbers and Euler characteristics. These invariants detect the singularities of a ring, extending two results from the local to the global setting.
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