Curved Space Research Articles

Counting meromorphic differentials on CP1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {C}\\mathbb {P}}^1$$\\end{document}

We give explicit formulas for the number of meromorphic differentials on CP1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb{C}\\mathbb{P}^1$$\\end{document} with two zeros and any number of residueless poles and for the number of meromorphic differentials on CP1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb{C}\\mathbb{P}^1$$\\end{document} with one zero, two poles with unconstrained residue and any number of residueless poles, in terms of the orders of their zeros and poles. These are the only two finite families of differentials on CP1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb{C}\\mathbb{P}^1$$\\end{document} with vanishing residue conditions at a subset of poles, up to the action of PGL(2,C)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{PGL}(2,\\mathbb {C})$$\\end{document}. The first family of numbers is related to triple Hurwitz numbers by simple integration and we show its connection with the representation theory of SL2(C)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{SL}_2(\\mathbb {C})$$\\end{document} and the equations of the dispersionless KP hierarchy. The second family has a very simple generating series, and we recover it through surprisingly involved computations using intersection theory of moduli spaces of curves and differentials.

Exponential Decay Estimates and Smoothness of the Moduli Space of Pseudoholomorphic Curves

The compacified moduli space of bordered stable maps carries a Kuranishi structure with boundary. Smoothness of Kuranishi structure along the boundary requires smoothness of coordinate changes along the boundary. The proof of smoothness is written by the authors in [AMS/IP Stud. Adv. Math., 46.1, 2009, xii+396 pp.], [AMS/IP Stud. Adv. math., 46.2, 2009, pp. i–xii and 397–805], and [Surv. Differ. Geom., vol. 22, pp. 133–190, 2018] based on some uniform exponential decay estimates of the stable maps with respect to a parameter T T , the length of the gluing cylindrical neck, near the boundary of the moduli space. In this paper we establish this exponential decay estimates in a precise manner by carefully examining the dependence on the parameter T T of the gluing construction in the setting of bordered Riemann surfaces with boundary punctures. We also show that the smoothness of the collar follows from the aforementioned exponential estimates by taking s = 1 / T s = 1/T as the radial coordinate of the collar.

Wall crossing for K‐moduli spaces of plane curves

AbstractWe construct proper good moduli spaces parametrizing K‐polystable ‐Gorenstein smoothable log Fano pairs , where is a Fano variety and is a rational multiple of the anticanonical divisor. We then establish a wall‐crossing framework of these K‐moduli spaces as varies. The main application in this paper is the case of plane curves of degree as boundary divisors of . In this case, we show that when the coefficient is small, the K‐moduli space of these pairs is isomorphic to the GIT moduli space. We then show that the first wall crossing of these K‐moduli spaces are weighted blow‐ups of Kirwan type. We also describe all wall crossings for degree 4,5,6 and relate the final K‐moduli spaces to Hacking's compactification and the moduli of K3 surfaces.

Inverse localization and global approximation for some Schrödinger operators on hyperbolic spaces

We consider the question of whether the high-energy eigenfunctions of certain Schrödinger operators on the d-dimensional hyperbolic space of constant curvature −κ2 are flexible enough to approximate an arbitrary solution of the Helmholtz equation Δh + h = 0 on Rd, over the natural length scale O(λ−1/2) determined by the eigenvalue λ ≫ 1. This problem is motivated by the fact that, by the asymptotics of the local Weyl law, approximate Laplace eigenfunctions do have this approximation property on any compact Riemannian manifold. In this paper we are specifically interested in the Coulomb and harmonic oscillator operators on the hyperbolic spaces Hd(κ). As the dimension of the space of bound states of these operators tends to infinity as κ ↘ 0, one can hope to approximate solutions to the Helmholtz equation by eigenfunctions for some κ > 0 that is not fixed a priori. Our main result shows that this is indeed the case, under suitable hypotheses. We also prove a global approximation theorem with decay for the Helmholtz equation on manifolds that are isometric to the hyperbolic space outside a compact set, and consider an application to the study of the heat equation on Hd(κ). Although global approximation and inverse approximation results are heuristically related in that both theorems explore flexibility properties of solutions to elliptic equations on hyperbolic spaces, we will see that the underlying ideas behind these theorems are very different.

DR Hierarchies: From the Moduli Spaces of Curves to Integrable Systems

DR Hierarchies: From the Moduli Spaces of Curves to Integrable Systems

A Generalization of Operads Based on Subgraph Contractions

Abstract We introduce a generalization of the notion of operad that we call a contractad, whose set of operations is indexed by connected graphs and whose composition rules are numbered by contractions of connected subgraphs. We show that many classical operads, such as the operad of commutative algebras, Lie algebras, associative algebras, pre-Lie algebras, the little disks operad, and the operad of moduli spaces of stable curves $\operatorname{\overline{{\mathcal{M}}}}_{0,n+1}$, admit generalizations to contractads. We explain that standard tools like Koszul duality and the machinery of Gröbner bases can be easily generalized to contractads. We verify the Koszul property of the commutative, Lie, associative, and Gerstenhaber contractads.

The characterization of (𝑛 − 1)-spheres with 𝑛 + 4 vertices having maximal Buchstaber number

Abstract We present a computationally efficient algorithm that is suitable for graphic processing unit implementation. This algorithm enables the identification of all weak pseudo-manifolds that meet specific facet conditions, drawn from a given input set. We employ this approach to enumerate toric colorable seeds. Consequently, we achieve a comprehensive characterization of ( n − 1 ) (n-1) -dimensional PL spheres with n + 4 n+4 vertices that possess a maximal Buchstaber number. A primary focus of this research is the fundamental categorization of non-singular complete toric varieties of Picard number 4. This classification serves as a valuable tool for addressing questions related to toric manifolds of Picard number 4. Notably, we have determined which of these manifolds satisfy equality within an inequality regarding the number of minimal components in their rational curve space. This addresses a question posed by Chen, Fu, and Hwang in 2014 for this specific case.

The expanding universe of the geometric mean

AbstractIn this paper the authors seek to trace in an accessible fashion the rapid recent development of the theory of the matrix geometric mean in the cone of positive definite matrices up through the closely related operator geometric mean in the positive cone of a unital $$C^*$$ C ∗ -algebra. The story begins with the two-variable matrix geometric mean, moves to the breakthrough developments in the multivariable matrix setting, the main focus of the paper, and then on to the extension to the positive cone of the $$C^*$$ C ∗ -algebra of operators on a Hilbert space, even to general unital $$C^*$$ C ∗ -algebras, and finally to the consideration of barycentric maps that grow out of the geometric mean on the space of integrable probability measures on the positive cone. Besides expected tools from linear algebra and operator theory, one observes a surprisingly substantial interplay with geometrical notions in metric spaces, particularly the notion of nonpositive curvature. Added features include a glance at the probabilistic theory of random variables with values in a metric space of nonpositive curvature, and the appearance of related means such as the inductive and power means. The authors also consider in a much briefer fashion the extension of the theory to the setting of Lie groups and briefer still to the positive symmetric cones of finite-dimensional Euclidean Jordan algebras.

Infinite (continuous) spin particle in constant curvature space

We present a new particle model that generalize for constant curvature space an infinite spin particle in flat space. The model is described by commuting Weyl spinor additional coordinates. It proved that such a model is consistent only in external gravitational field corresponding to the constant curvature spaces. Full set of the first-class constraints in the de Sitter and anti-de Sitter spaces is obtained.

Hypersurfaces in spaces of constant curvature satisfying a particular Roter type equation

Let M be a hypersurface isometrically immersed in an (n+1)-dimensional semi-Riemannian space of constant curvature, n>3, such that its shape operator A satisfies A3=ϕA2+ψA+ρId, where ϕ, ψ and ρ are some functions on M and Id is the identity operator. The main result of this paper states that on the set U of all points of M at which the square S2 of the Ricci operator S of M is not a linear combination of S and Id, the Riemann-Christoffel curvature tensor R of M is a linear combination of some Kulkarni-Nomizu products formed by the metric tensor g, the Ricci tensor S and the tensor S2 of M, i.e., the tensor R satisfies on U some Roter type equation. Moreover, the (0,4)-tensor R⋅S is on U a linear combination of some Tachibana tensors formed by the tensors g, S and S2. In particular, if M is a hypersurface isometrically immersed in the (n+1)-dimensional Riemannian space of constant curvature, n>3, with three distinct principal curvatures and the Ricci operator S with three distinct eigenvalues then the Riemann-Christoffel curvature tensor R of M also satisfies a Roter type equation of this kind.

Polynomial point counts and odd cohomology vanishing on moduli spaces of stable curves

Polynomial point counts and odd cohomology vanishing on moduli spaces of stable curves

Gaussian maps for singular curves on Enriques surfaces

AbstractA marked Prym curve is a triple $$(C,\alpha ,T_d)$$ ( C , α , T d ) where C is a smooth algebraic curve, $$\alpha $$ α is a $$2-$$ 2 - torsion line bundle on C, and $$T_d$$ T d is a divisor of degree d. We give obstructions—in terms of Gaussian maps—for a marked Prym curve $$(C,\alpha ,T_d)$$ ( C , α , T d ) to admit a singular model lying on an Enriques surface with only one ordinary singular point of multiplicity d, such that $$T_d$$ T d is the pull-back of the singular point by the normalization map. More precisely, let (S, H) be a polarized Enriques surface and let (C, f) be a smooth curve together with a morphism $$f:C \rightarrow S$$ f : C → S birational onto its image and such that $$f(C) \in |H|$$ f ( C ) ∈ | H | , f(C) has exactly one ordinary singular point of multiplicity d. Let $$\alpha =f^*\omega _S$$ α = f ∗ ω S and $$T_d$$ T d be the divisor over the singular point of f(C). We show that if H is sufficiently positive then certain natural Gaussian maps on C, associated with $$\omega _C$$ ω C , $$\alpha $$ α , and $$T_d$$ T d are not surjective. On the contrary, we show that for the general triple in the moduli space of marked Prym curves $$(C,\alpha ,T_d)$$ ( C , α , T d ) , the same Gaussian maps are surjective.

The Irreducibility of the Spaces of Rational Curves on del Pezzo Manifolds

Abstract We prove the irreducibility of the spaces of rational curves on del Pezzo manifolds of Picard rank $1$ and dimension $n \ge 4$ by analyzing the fibers of evaluation maps. As a corollary, we prove Geometric Manin’s Conjecture in these cases.

Multimatroids and Rational Curves with Cyclic Action

Abstract We study the connection between multimatroids and moduli spaces of rational curves with cyclic action. Multimatroids are generalizations of matroids and delta-matroids that naturally arise in topological graph theory. The perspective of moduli of curves provides a tropical framework for studying multimatroids, generalizing the previous connection between type-$A$ permutohedral varieties (Losev–Manin moduli spaces) and matroids, and the connection between type-$B$ permutohedral varieties and delta-matroids. Specifically, we equate a combinatorial nef cone of the moduli space with the space of ${\mathbb {R}}$-multimatroids, a generalization of multimatroids, and we introduce the independence polytopal complex of a multimatroid, whose volume is identified with an intersection number on the moduli space. As an application, we give a combinatorial formula for a natural class of intersection numbers on the moduli space by relating to the volumes of independence polytopal complexes of multimatroids.

On the symmetry algebra of a one-dimensional quantum-mechanical oscillator on a hyperbola

The quantum-mechanical problem of a harmonic oscillator on a hyperbola as a one-dimensional space of constant negative curvature is considered in this article. A generalization to the singular oscillator model in the context of one-dimensional Cayley – Klein geometries is given by the factorization method. The energy spectrum and wave functions of stationary states are found having the curvature of space as a parameter. For the energy levels of the singular oscillator, the effect of non-zero curvature is clearly manifested through a positive or negative term, depending on the sign of the curvature, which is quadratic in the level number. The results obtained are consistent with those previously published. The dynamical symmetry of the problem is shown explicitly as a quadratic Hahn algebra QH(3) or its isomorphic Higgs algebra.

Higher Airy Structures, 𝒲 Algebras and Topological Recursion

We define higher quantum Airy structures as generalizations of the Kontsevich–Soibelman quantum Airy structures by allowing differential operators of arbitrary order (instead of only quadratic). We construct many classes of examples of higher quantum Airy structures as modules of W ( g ) \mathcal {W}(\mathfrak {g}) algebras at self-dual level, with g = g l N + 1 \mathfrak {g}= \mathfrak {gl}_{N+1} , s o 2 N \mathfrak {so}_{2 N } or e N \mathfrak {e}_N . We discuss their enumerative geometric meaning in the context of (open and closed) intersection theory of the moduli space of curves and its variants. Some of these W \mathcal {W} constraints have already appeared in the literature, but we find many new ones. For g l N + 1 \mathfrak {gl}_{N+1} our result hinges on the description of previously unnoticed Lie subalgebras of the algebra of modes. As a consequence, we obtain a simple characterization of the spectral curves (with arbitrary ramification) for which the Bouchard–Eynard topological recursion gives symmetric ω g , n \omega _{g,n} s and is thus well defined. For all such cases, we show that the topological recursion is equivalent to W ( g l ) \mathcal {W}(\mathfrak {gl}) constraints realized as higher quantum Airy structures, and obtain a Givental-like decomposition for the corresponding partition functions.

Algebraic Conditions for Conformal Superintegrability in Arbitrary Dimension

We consider second order (maximally) conformally superintegrable systems and explain how the definition of such a system on a (pseudo-)Riemannian manifold gives rise to a conformally invariant interpretation of superintegrability. Conformal equivalence in this context is a natural extension of the classical (linear) Stäckel transform, originating from the Maupertuis-Jacobi principle. We extend our recently developed algebraic geometric approach for the classification of second order superintegrable systems in arbitrarily high dimension to conformally superintegrable systems, which are presented via conformal scale choices of second order superintegrable systems defined within a conformal geometry. For superintegrable systems on constant curvature spaces, we find that the conformal scales of Stäckel equivalent systems arise from eigenfunctions of the Laplacian and that their equivalence is characterised by a conformal density of weight two. Our approach yields an algebraic equation that governs the classification under conformal equivalence for a prolific class of second order conformally superintegrable systems. This class contains all non-degenerate examples known to date, and is given by a simple algebraic constraint of degree two on a general harmonic cubic form. In this way the yet unsolved classification problem is put into the reach of algebraic geometry and geometric invariant theory. In particular, no obstruction exists in dimension three, and thus the known classification of conformally superintegrable systems is reobtained in the guise of an unrestricted univariate sextic. In higher dimensions, the obstruction is new and has never been revealed by traditional approaches.

Umbilicity of constant mean curvature hypersurfaces into space forms

In this paper we establish conditions on the length of the traceless part of the second fundamental form of a complete constant mean curvature hypersurface immersed in a space of constant sectional curvature in order to show that it is totally umbilical.

Superintegrability and deformed oscillator realizations of quantum TTW Hamiltonians on constant-curvature manifolds and with reflections in a plane

We extend the method for constructing symmetry operators of higher order for two-dimensional quantum Hamiltonians by Kalnins et al (2010 J. Phys. A: Math. Theor. 43 265205). This expansion method expresses the integral in a finite power series in terms of lower degree integrals so as to exhibit it as a first-order differential operators. One advantage of this approach is that it does not require the a priori knowledge of the explicit eigenfunctions of the Hamiltonian nor the action of their raising and lowering operators as in their recurrence approach (Kalnins et al 2011 SIGMA 7 031). We obtain insight into the two-dimensional Hamiltonians of radial oscillator type with general second-order differential operators for the angular variable. We then re-examine the Hamiltonian of Tremblay et al (2009 J. Phys. A: Math. Theor. 42 242001) as well as a deformation discovered by Post et al (2011 J. Phys. A: Math. Theor. 44 505201) which possesses reflection operators. We will extend the analysis to spaces of constant curvature. We present explicit formulas for the integrals and the symmetry algebra, the Casimir invariant and oscillator realizations with finite-dimensional irreps which fill a gap in the literature.

The rank of syzygies of canonical curves

Abstract We prove that the linear syzygy spaces of a general canonical curve are spanned by syzygies of minimal rank.