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Central limit theorem for toric Kähler manifolds Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210601
Steve Zelditch, Peng ZhouAssociated to the Bergman kernels of a polarized toric Kähler manifold $(M,\omega,L,h)$ are sequences of measures ${\lbrace \mu^z_k \rbrace}^\infty_{k=1}$ parametrized by the points $z \in M$. For each $z$ in the open orbit, we prove a central limit theorem for $\mu^z_k$. The center of mass of $\mu^z_k$ is the image of $z$ under the moment map up to $\mathcal{O}(1/k)$; after recentering at $0$ and

A class of curvature type equation Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210601
Pengfei Guan, Xiangwen ZhangIn this paper, we study the solvability of a general class of fully nonlinear curvature equations, which can be viewed as generalizations of the equations for Christoffel–Minkowski problem in convex geometry. We will also study the Dirichlet problem of the corresponding degenerate equations as an extension of the equations studied by Krylov.

Continuity of the Yang–Mills flow on the set of semistable bundles Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210601
Benjamin Sibley, Richard WentworthA recent paper [16] studied properties of a compactification of the moduli space of irreducible Hermitian–Yang–Mills connections on a hermitian bundle over a projective algebraic manifold. In this followup note, we show that the Yang–Mills flow at infinity on the space of semistable integrable connections defines a continuous map to the set of ideal connections used to define this compactification

Products of random matrices: a dynamical point of view Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210601
TienCuong Dinh, Lucas Kaufmann, Hao WuWe study products of random matrices in $\mathrm{SL}^2 (\mathbb{C})$ from the point of view of holomorphic dynamics. For nonelementary measures with finite first moment we obtain the exponential convergence towards the stationary measure in Sobolev norm. As a consequence we obtain the exponentially fast equidistribution of forward images of points towards the stationary measure. We also give a new

Pluripotential solutions versus viscosity solutions to complex Monge–Ampère flows Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210601
Vincent Guedj, Chinh H. Lu, Ahmed ZeriahiWe compare various notions of weak subsolutions to degenerate complex Monge–Ampère flows, showing that they all coincide. This allows us to show that the viscosity solution coincides with the envelope of pluripotential subsolutions.

Morsetype integrals on nonKähler manifolds Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210601
Sławomir Kołodziej, Valentino TosattiWe pose a conjecture about Morsetype integrals in nef $(1,1)$ classes on compact Hermitian manifolds, and we show that it holds for semipositive classes, or when the manifold admits certain special Hermitian metrics.

The complex Monge–Ampère equation with a gradient term Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210601
Valentino Tosatti, Ben WeinkoveWe consider the complex Monge–Ampère equation with an additional linear gradient term inside the determinant. We prove existence and uniqueness of solutions to this equation on compact Hermitian manifolds.

Twisted Kähler–Einstein metrics Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210601
Julius Ross, Gábor SzékelyhidiWe prove an existence result for twisted Kähler–Einstein metrics, assuming an appropriate twisted K‑stability condition. An improvement over earlier results is that certain nonnegative twisting forms are allowed.

Positivity of Weil–Petersson currents on canonical models Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210601
Bin Guo, Jian SongWe show that the Weil–Petersson current is a global nonnegative closed $(1,1)$current in the twisted Kähler–Einstein equation on nongeneral type canonical models.

Concave elliptic equations and generalized Khovanskii–Teissier inequalities Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210601
Tristan C. CollinsWe explain a general construction through which concave elliptic operators on complex manifolds give rise to concave functions on cohomology. In particular, this leads to generalized versions of the Khovanskii–Teissier inequalities.

Anomaly flow and Tduality Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210601
Teng Fei, Sebastien PicardIn this paper, we study the dual Anomaly flow, which is a dual version of the Anomaly flow under T‑duality. A family of monotone functionals is introduced and used to estimate the dilaton function along the flow. Many examples and reductions of the dual Anomaly flow are worked out in detail.

Weak geodesics for the deformed Hermitian–Yang–Mills equation Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210601
Adam JacobWe study weak geodesics in the space of potentials for the deformed Hermitian–Yang–Mills equation. The geodesic equation can be formulated as a degenerate elliptic equation, allowing us to employ nonlinear Dirichlet duality theory, as developed by Harvey–Lawson. By exploiting the convexity of the level sets of the Lagrangian angle operator in the highest branch, we are able to construct $C^0$ solutions

Positive projectively flat manifolds are locally conformally flatKähler Hopf manifolds Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210601
Simone CalamaiWe define a partition of the space of projectively flat metrics in three classes according to the sign of the Chern scalar curvature; we prove that the class of negative projectively flat metrics is empty, and that the class of positive projectively flat metrics consists precisely of locally conformally flatKähler metrics on Hopf manifolds, explicitly characterized by Vaisman [23]. Finally, we review

Cartan–Iwahori–Matsumoto decompositions for reductive groups Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210401
Jarod Alper, Jochen Heinloth, Daniel HalpernLeistnerWe provide a short and selfcontained argument for the existence of Cartan–Iwahori–Matsumoto decompositions for reductive groups.

From algebraic geometry to machine learning Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210401
Michael R. DouglasDavid Mumford made groundbreaking contributions in many fields, including the pure mathematics of algebraic geometry and the applied mathematics of machine learning and artificial intelligence. His work in both fields influenced my career at several key moments.

Mumford’s influence on the moduli theory of algebraic varieties Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210401
János KollárWe give a short appreciation of Mumford’s work on the moduli of varieties by putting it into historical context. By reviewing earlier works we highlight the innovations introduced by Mumford. Then we discuss recent developments whose origins can be traced back to Mumford’s ideas.

Brauer groups of involution surface bundles Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210401
Andrew Kresch, Yuri TschinkelWe present an algorithm to compute the Brauer group of involution surface bundles over rational surfaces.

Stability and applications Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210401
Emanuele Macrì, Benjamin SchmidtWe give a brief overview of Bridgeland’s theory of stability conditions, focusing on applications to algebraic geometry. We sketch the basic ideas in Bayer’s proof of the Brill–Noether Theorem and in the authors’ proof of a theorem by Gruson–Peskine and Harris on the genus of space curves. This note originated from the lecture of the first author at the conference From Algebraic Geometry to Vision

A method in deformation theory Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210401
Frans OortWe describe a method in deformation theory that David Mumford and the present author developed in 1966.

Relations in the tautological ring of the moduli space of curves Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210401
R. Pandharipande, A. PixtonThe virtual geometry of the moduli space of stable quotients is used to obtain Chow relations among the $\kappa$ classes on the moduli space of nonsingular genus $g$ curves. In a series of steps, the stable quotient relations are rewritten in successively simpler forms. The final result is the proof of the Faber–Zagier relations (conjectured in 2000).

On Shimura varieties for unitary groups Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210401
M. Rapoport, B. Smithling, W. ZhangThis is a largely expository article based on our paper [31] on arithmetic diagonal cycles on unitary Shimura varieties. We define a class of Shimura varieties closely related to unitary groups which represent a moduli problem of abelian varieties with additional structure, and which admit interesting algebraic cycles. We generalize to arbitrary signature type the results of loc. cit. valid under special

Areas of totally geodesic surfaces of hyperbolic $3$orbifolds Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210101
Benjamin Linowitz, D. B. McReynolds, Nicholas MillerThe geodesic length spectrum of a complete, finite volume, hyperbolic $3$‑orbifold $M$ is a fundamental invariant of the topology of $M$ via Mostow–Prasad Rigidity. Motivated by this, the second author and Reid defined a twodimensional analogue of the geodesic length spectrum given by the multiset of isometry types of totally geodesic, immersed, finitearea surfaces of $M$ called the geometric genus

Irregular Eguchi–Hanson type metrics and their soliton analogues Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210101
Akito FutakiWe verify the extension to the zero section of momentum construction of Kähler–Einstein metrics and Kähler–Ricci solitons on the total space $Y$ of positive rational powers of the canonical line bundle of toric Fano manifolds with possibly irregular Sasaki–Einstein metrics. More precisely, we show that the extended metric along the zero section has an expression which can be extended to $Y$, restricts

Complex structures on jet spaces and bosonic Fock space dynamics for causal variational principles Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210101
Felix Finster, Niky KamranBased on conservation laws for surface layer integrals for critical points of causal variational principles, it is shown how jet spaces can be endowed with an almostcomplex structure. We analyze under which conditions the almostcomplex structure can be integrated to a canonical complex structure. Combined with the scalar product expressed by a surface layer integral, we obtain a complex Hilbert space

A diffeomorphisminvariant metric on the space of vectorvalued oneforms Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210101
Martin Bauer, Eric Klassen, Stephen C. Preston, Zhe SuIn this article we introduce a diffeomorphisminvariant Riemannian metric on the space of vector valued oneforms. The particular choice of metric is motivated by potential future applications in the field of functional data and shape analysis and by connections to the Ebin metric on the space of all Riemannian metrics. In the present work we calculate the geodesic equations and obtain an explicit

Graded tilting for gauged Landau–Ginzburg models and geometric applications Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210101
Christian Okonek, Andrei TelemanIn this paper we develop a graded tilting theory for gauged Landau–Ginzburg models of regular sections in vector bundles over projective varieties. Our main theoretical result describes—under certain conditions—the bounded derived category of the zero locus $Z(s)$ of such a section s as a graded singularity category of a noncommutative quotient algebra $\Lambda / {\langle s \rangle} : D^b (\operatorname{coh}Z(s))

Round handle problem Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210101
Min Hoon Kim, Mark Powell, Peter TeichnerWe present the Round Handle Problem (RHP), proposed by Freedman and Krushkal. It asks whether a collection of links, which contains the Generalised Borromean Rings (GBRs), are slice in a $4$‑manifold $R$ constructed from adding round handles to the four ball. A negative answer would contradict the union of the surgery conjecture and the $s$cobordism conjecture for $4$‑manifolds with free fundamental

Rational representation of real functions Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210101
Wojciech Kucharz, Krzysztof KurdykaLet $X$ be an irreducible smooth real algebraic variety of dimension at least $2$ and let $f : U \to \mathbb{R}$ be a function defined on a connected open subset $U \subset X(\mathbb{R})$. Assume that for every irreducible smooth real algebraic curve $C \subset X$, for which $C(\mathbb{R})$ is the boundary of a disc embedded in $U$, the restriction ${f \vert}_{C(\mathbb{R})}$ is continuous and has

CMC foliations of open spacetimes asymptotic to open Robertson–Walker spacetimes Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210101
Claus GerhardtWe consider open globally hyperbolic spacetimes $N$ of dimension $n + 1, n \geq 3$, which are spatially asymptotic to a Robertson–Walker spacetime or an open Friedmann universe with spatial curvature $\tilde{\kappa} = 0, 1$ and prove, under reasonable assumptions, that there exists a unique foliation by spacelike hypersurfaces of constant mean curvature and that the mean curvature function $\tau$

On the Euler characteristics of certain moduli spaces of $1$dimensional closed subschemes Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210101
Mazen M. AlhwaimelGeneralizing the ideas in [W. Li and Z. Qin, “On The Euler Number of Moduli Spaces of Curves and Points”, Commu. in Anal. and Geom. 14 (2006), 387–410] and using virtual Hodge polynomials as well as tours actions, we compute the Euler characteristics of certain moduli spaces of $1$‑dimensional closed subschemes when the ambient smooth projective variety admits a Zariskilocally trivial fibration to

Evolution and monotonicity of a geometric constant under the Ricci flow Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210101
Shouwen Fang, Junwei Yu, Peng ZhuLet $(M,g(t))$ be a compact Riemannian manifold and the metric $g(t)$ evolve by the Ricci flow. In the paper we derive the evolution equation for a geometric constant $\lambda$ under the Ricci flow and the normalized Ricci flow, such that there exist positive solutions to the nonlinear equation\[\Delta_{\phi} f + af \: \ln \, f + bRf = \lambda f \: \textrm{,}\]where $\Delta \phi$ is the Witten–Laplacian

GulbrandsenHalleHulek degeneration and HilbertChow morphism Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210101
Yasunari NagaiFor a semistable degeneration of surfaces without a triple point, we show that two models of degeneration of Hilbert scheme of points of the family, Gulbrandsen–Halle–Hulek degeneration given in [M. G. Gulbrandsen, L. H. Halle, and K. Hulek, “A GIT construction of degenerations of Hilbert schemes of points”, Doc. Math. 24 (2019), 421–472] and the one given by the author in [Y. Nagai, “Symmetric products

Hilbert manifold structure for weakly asymptotically hyperbolic relativistic initial data Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210101
Erwann Delay, Jérémie FougeirolWe construct a Hilbert manifold structure à la Bartnik for the space of weakly asymptotically hyperbolic initial data for the vacuum constraint equations. The proofs requires new weighted Poincaré and Korntype inequalities for asymptotically hyperbolic manifolds with inner boundary.

The Tanaka–Thomas’s Vafa–Witten invariants via surface Deligne–Mumford stacks Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210101
Yunfeng Jiang, Promit KunduWe provide a definition of Vafa–Witten invariants for projective surface DeligneMumford stacks, generalizing the construction of Tanaka–Thomas on the Vafa–Witten invariants for projective surfaces inspired by the $S$duality conjecture. We give calculations for a root stack over a general type quintic surface, and quintic surfaces with ADE singularities. The relationship between the Vafa–Witten invariants

Entropy rigidity for foliations by strictly convex projective manifolds Pure Appl. Math. Q. (IF 0.659) Pub Date : 20210101
Alessio SaviniLet $N$ be a compact manifold with a foliation $\mathscr{F}_N$ whose leaves are compact strictly convex projective manifolds. Let $M$ be a compact manifold with a foliation $\mathscr{F}_M$ whose leaves are compact hyperbolic manifolds of dimension bigger than or equal to $3$. Suppose we have a foliationpreserving homeomorphism $f : (N,\mathscr{F}_N) \to (M, \mathscr{F}_M)$ which is $C^1$regular when

Symplectic coordinates on $\operatorname{PSL}_3(\mathbb{R})$Hitchin components Pure Appl. Math. Q. (IF 0.659) Pub Date : 20201201
Suhyoung Choi, Hongtaek Jung, Hong Chan KimGoldman parametrizes the $\operatorname{PSL}_3(\mathbb{R})$Hitchin component of a closed oriented hyperbolic surface of genus $g$ by $16g  16$ parameters. Among them, $10g  10$ coordinates are canonical. We prove that the $\operatorname{PSL}_3(\mathbb{R})$Hitchin component equipped with the Atiyah–Bott–Goldman symplectic form admits a global Darboux coordinate system such that the half of its coordinates

Frobenius’ theta function and Arakelov invariants in genus three Pure Appl. Math. Q. (IF 0.659) Pub Date : 20201201
Robin de JongWe give explicit formulas for the Kawazumi–Zhang invariant and Faltings deltainvariant of a compact and connected Riemann surface of genus three. The formulas are in terms of two integrals over the associated jacobian, one integral involving the standard Riemann theta function, and another involving a theta function particular to genus three that was discovered by Frobenius. We review part of Frobenius’

Admissible restrictions of irreducible representations of reductive Lie groups: symplectic geometry and discrete decomposability Pure Appl. Math. Q. (IF 0.659) Pub Date : 20201201
Toshiyuki KobayashiLet $G$ be a real reductive Lie group, $L$ a compact subgroup, and $\pi$ an irreducible admissible representation of $G$. In this article we prove a necessary and sufficient condition for the finiteness of the multiplicities of $L$types occurring in $\pi$ based on symplectic techniques. This leads us to a simple proof of the criterion for discrete decomposability of the restriction of unitary representations

On the image of MRC fibrations of projective manifolds with semipositive holomorphic sectional curvature Pure Appl. Math. Q. (IF 0.659) Pub Date : 20201201
ShinIchi MatsumuraIn this paper, we pose several conjectures on structures and images of maximal rationally connected fibrations of smooth projective varieties admitting semipositive holomorphic sectional curvature. Toward these conjectures, we prove that the canonical bundle of images of such fibrations is not big. Our proof gives a generalization of Yang’s solution using RC positivity for Yau’s conjecture. As an

The perverse filtration for the Hitchin fibration is locally constant Pure Appl. Math. Q. (IF 0.659) Pub Date : 20201201
Mark Andrea A. de Cataldo, Davesh MaulikWe prove that the perverse Leray filtration for the Hitchin morphism is locally constant in families, thus providing some evidence towards the validity of the $P = W$ conjecture due to de Cataldo, Hausel and Migliorini in non Abelian Hodge theory.

Berkovich log discrepancies in positive characteristic Pure Appl. Math. Q. (IF 0.659) Pub Date : 20201201
Eric CantonWe introduce and study a log discrepancy function on the space of semivaluations centered on an integral noetherian scheme of positive characteristic. Our definition shares many properties with the analogue in characteristic zero; we prove that if log resolutions exist in positive characteristic, then our definition agrees with previous approaches to log discrepancies of semivaluations that use these

On exchange spectra of valued quivers and cluster algebras Pure Appl. Math. Q. (IF 0.659) Pub Date : 20201201
Fang Li, Siyang LiuInspirited by the importance of the spectral theory of graphs, we introduce the spectral theory of the valued quiver of a cluster algebra. Our aim is to characterize a cluster algebra via its spectrum so as to use the spectral theory as a tool. First, we give relations between exchange spectrum of valued quivers and adjacency spectrum of their underlying valued graphs, and between exchange spectra

Rank of ordinary webs in codimension one an effective method Pure Appl. Math. Q. (IF 0.659) Pub Date : 20201201
JeanPaul Dufour, Daniel LehmannWe are interested by holomorphic $d$webs $W$ of codimension one in a complex $n$dimensional manifold $M$. If they are ordinary, i.e. if they satisfy to some condition of genericity (whose precise definition is recalled below), we proved in [CL] that their rank $\rho (W)$ is upperbounded by a certain number $\pi^\prime (n, d)$ (which, for $n \geq 3$, is strictly smaller than the Castelnuovo–Chern’s

Regularity of fully nonlinear elliptic equations on Kähler cones Pure Appl. Math. Q. (IF 0.659) Pub Date : 20201201
Rirong YuanWe derive quantitative boundary estimates, and then solve the Dirichlet problem for a general class of fully nonlinear elliptic equations on annuli of Kähler cones over closed Sasakian manifolds. This extends extensively a result concerning the geodesic equations in the space of Sasakian metrics due to Guan–Zhang. Our results show that the solvability is deeply affected by the transverse Kähler structures

Singular mappings and their zeroforms Pure Appl. Math. Q. (IF 0.659) Pub Date : 20201201
Goo Ishikawa, Stanisław JaneczkoWe study the quotient complexes of the de Rham complex on singular mappings; the complex of algebraic restrictions, the complex of geometric restrictions and the residual complex. Vanishing theorem for algebraic, geometric and residual cohomologies on quasihomogeneous mapgerms was proved. The finite order and symplectic zeroforms were characterized on parametric singularities. In this context the

Categorical representation of superschemes Pure Appl. Math. Q. (IF 0.659) Pub Date : 20201201
Yasuhiro WakabayashiIn the present paper, we prove that a locally noetherian superscheme $X^{\circledS}$ may be reconstructed (up to certain equivalence) categorytheoretically from the category of noetherian superschemes over $X^{\circledS}$. This result is a supergeometric generalization of the result proved by Shinichi Mochizuki concerning categorical reconstruction of schemes.

Hilbert schemes of points and quasimodularity Pure Appl. Math. Q. (IF 0.659) Pub Date : 20201201
Zhongyan Shen, Zhenbo QinWe study further connections between Hilbert schemes of points on a smooth projective (complex) surface and quasimodular forms. We prove that the leading terms of certain generating series (with variable $q$) involving intersections with the total Chern classes of the tangent bundles of these Hilbert schemes are quasimodular forms. The main idea is to link these leading terms with those coming from

Infinitesimal deformation of Deligne cycle class map Pure Appl. Math. Q. (IF 0.659) Pub Date : 20201201
Sen YangIn this note, we study the infinitesimal forms of Deligne cycle class maps. As an application, we prove that the infinitesimal form of a conjecture by Beilinson [1] is true.

Vanishing viscosity limit to the 3D Burgers equation in Gevrey class Pure Appl. Math. Q. (IF 0.659) Pub Date : 20201201
Ridha Selmi, Abdelkerim ChaabaniWe consider the Cauhcy problem to the 3D diffusive periodic Burgers equation. We prove that a unique solution exists on time interval independent of the viscosity and tends, as the viscosity vanishes, to the solution of the limiting equation, the inviscid periodic threedimensional Burgers equation, in Gevrey–Sobolev spaces. Compared to Navier–Stokes equations, the main difficulties come from the lack

A Nekhoroshev type theorem for the nonlinear wave equation on the torus Pure Appl. Math. Q. (IF 0.659) Pub Date : 20201201
Lufang Mi, Chunyong Liu, Guanghua Shi, Rong ZhaoIn this paper, we prove a Nekhoroshev type theorem for the nonlinear wave equation\[u_{tt} = u_{xx}  mu  f (u)\]on the finite $x$interval $[0, \pi]$. The parameter m is real and positive, and the nonlinearity $f$ is assumed to be real analytic in $u$. More precisely, we prove that if the initial datum is analytic in a district of width $2 \rho \gt 0$ whose norm on this district is equal to $\varepsilon$

Symmetrization of convex plane curves Pure Appl. Math. Q. (IF 0.659) Pub Date : 20201201
Peter Giblin, Stanisław JaneczkoSeveral point symmetrizations of a convex curve $\Gamma$ are introduced and one, the affinely invariant ‘central symmetric transform’ (CST) with respect to a given basepoint inside $\Gamma$, is investigated in detail. Examples for $\Gamma$ include triangles, rounded triangles, ellipses, curves defined by support functions and piecewise smooth curves. Of particular interest is the region of basepoints

A view on elliptic integrals from primitive forms (period integrals of type $\mathrm{A}_2$, $\mathrm{B}_2$ and $\mathrm{G}_2$) Pure Appl. Math. Q. (IF 0.659) Pub Date : 20201001
Kyoji SaitoElliptic integrals, since Euler’s finding of addition theorem 1751, has been studied extensively from various view points. The present paper gives a view point from primitive integrals of types $\mathrm{A}_2$, $\mathrm{B}_2$ and $\mathrm{G}_2$. We solve Jacobi inversion problem for the period maps by introducing generalized Eisenstein series of types $\mathrm{A}_2$, $\mathrm{B}_2$ and $\mathrm{G}_2$

Connectedness of Milnor ﬁbres and Stein factorization of compactiﬁable holomorphic functions Pure Appl. Math. Q. (IF 0.659) Pub Date : 20201001
Helmut A. HammWe start with conditions under which the Milnor fibre of a holomorphic function on a singular space is connected. In this case the special fibre is contractible, hence connected. So we pass to a more general question: compare the number of connected components of the fibres of a holomorphic function. Useful ingredients are local Lefschetz theorems and some kind of a Stein factorization.

Critical points and mKdV hierarchy of type $C^{(1)}_n$ Pure Appl. Math. Q. (IF 0.659) Pub Date : 20201001
Alexander Varchenko, Tyler WoodruffWe consider the population of critical points, generated from the critical point of the master function with no variables, which is associated with the trivial representation of the twisted affine Lie algebra ${C_{n}^{(1)}}$. The population is naturally partitioned into an infinite collection of complex cells ${\mathbb{C}^{m}}$, where m are positive integers. For each cell we define an injective rational

Generalization of the Weierstrass ℘ function and Maass lifts of weak Jacobi forms Pure Appl. Math. Q. (IF 0.659) Pub Date : 20200601
Hiroki AokiTypically, a Maass lift is a map from (holomorphic) Jacobi forms of index $1$ to Siegel modular forms of degree $2$ or other kinds of modular forms. In this paper, we construct Maass lifts from weak Jacobi forms to (nonholomorphic) Siegel modular forms of degree $2$ with or without levels and characters, as formal series. By the Koecher principle, the images of our lifts are not holomorphic at cusps

Categoriﬁcation of Legendrian knots Pure Appl. Math. Q. (IF 0.659) Pub Date : 20200601
Tatsuki KuwagakiThe concept of a perverse schober defined by Kapranov–Schechtman is a categorification of the notion of a perverse sheaf. In their definition, a key ingredient is a certain purity property of perverse sheaves. In this short note, we attempt to describe a real analogue of the above story, as categorification of Legendrian points/knots. The notion turns out to include various notions such as semiorthogonal

On the area formulas of inscribed polygons in classical geometry Pure Appl. Math. Q. (IF 0.659) Pub Date : 20200601
Yohei Komori, Runa Umezawa, Takuro YasuiWe show that there is no area formula of the general inscribed $n$‑gon for $n \geq 5$ only by using arithmetic operations and $k$‑th roots of its side lengths in classical geometry.

Curve counting on $\mathcal{A}_n \times \mathbb{C}^2$ Pure Appl. Math. Q. (IF 0.659) Pub Date : 20200601
Yalong CaoLet $\mathcal{A}_n \to \mathbb{C}^2 / \mathbb{Z}_{n+1}$ be the minimal resolution of $\mathcal{A}_n$singularity and $X = \mathcal{A}_n \times \mathbb{C}^2$ be the associated toric Calabi–Yau $4$fold. In this note, we study curve counting on $X$ from both Donaldson–Thomas and Gromov–Witten perspectives. In particular, we verify conjectural formulae relating them proposed by the author, Maulik and

On the $\operatorname{BV}$ structure on the cohomology of moduli space Pure Appl. Math. Q. (IF 0.659) Pub Date : 20200601
Sümeyra Sakallı, Alexander A. VoronovThe question of vanishing of the $\operatorname{BV}$ operator on the cohomology of the moduli space of Riemann surfaces is investigated. The $\operatorname{BV}$ structure, which comprises a $\operatorname{BV}$ operator and an antibracket, is identified, vanishing theorems are proven, and a counterexample is provided.

Quantizing Deformation Theory II Pure Appl. Math. Q. (IF 0.659) Pub Date : 20200101
Alexander A. VoronovA quantization of classical deformation theory, based on the MaurerCartan Equation $dS + \frac{1}{2}[S,S] = 0$ in dgLie algebras, a theory based on the Quantum Master Equation $dS + \hbar \Delta S + \frac{1}{2} \{S,S\} = 0$ in dgBValgebras, is proposed. Representability theorems for solutions of the Quantum Master Equation are proven. Examples of "quantum" deformations are presented.