• Math. Ann. (IF 1.356) Pub Date : 2020-06-03
Peng Chen, Xuan Thinh Duong, Ji Li, Lixin Yan

Let L be a non-negative self-adjoint operator acting on $$L^2(X)$$ where X is a space of homogeneous type with a dimension n. Suppose that the heat operator $$e^{-tL}$$ satisfies the generalized Gaussian $$(p_0, p'_0)$$-estimates of order m for some $$1\le p_0 < 2$$. In this paper we prove sharp endpoint $$L^p$$-Sobolev bound for the Schrödinger group $$e^{itL}$$, that is for every $$p\in (p_0, p'_0)$$

更新日期：2020-06-03
• Math. Ann. (IF 1.356) Pub Date : 2020-06-03
Van Hoang Nguyen

This paper is addressed to study the existence of maximizers for the singular Moser–Trudinger supremum under constraints in the critical case \begin{aligned} MT_{N}(a,\beta ) = \sup _{u\in W^{1,N}({\mathbb {R}}^N),\, \Vert \nabla u\Vert _N^a + \Vert u\Vert _N^N =1} \int _{{\mathbb {R}}^N}\Phi _N\left( (1-\beta /N)\alpha _N |u|^{\frac{N}{N-1}}\right) |x|^{-\beta } dx, \end{aligned} where $$a>0$$

更新日期：2020-06-03
• Math. Ann. (IF 1.356) Pub Date : 2020-06-03
Jaehyun Hong, Sui-Chung Ng

Let G/P be a rational homogeneous space (not necessarily irreducible) and $$x_0\in G/P$$ be the point at which the isotropy group is P. The G-translates of the orbit $$Qx_0$$ of a parabolic subgroup $$Q\subsetneq G$$ such that $$P\cap Q$$ is parabolic are called Q-cycles. We established an extension theorem for local biholomorphisms on G/P that map local pieces of Q-cycles into Q-cycles. We showed

更新日期：2020-06-03
• Math. Ann. (IF 1.356) Pub Date : 2020-06-02
Ki-Ahm Lee, Jinwan Park

In this paper, we study the regularity of the free boundaries of the parabolic double obstacle problem for the heat operator and fully nonlinear operator. The result in this paper are generalizations of the theory for the elliptic problem in Lee et al. (Calc Var Partial Differ Equ 58(3):104, 2019) and Lee and Park (The regularity theory for the double obstacle problem for fully nonlinear operator,

更新日期：2020-06-02
• Math. Ann. (IF 1.356) Pub Date : 2020-05-14
Emma Brakkee

Motivated by the relation between (twisted) K3 surfaces and special cubic fourfolds, we construct moduli spaces of polarized twisted K3 surfaces of any fixed degree and order. We do this by mimicking the construction of the moduli space of untwisted polarized K3 surfaces as a quotient of a bounded symmetric domain.

更新日期：2020-05-14
• Math. Ann. (IF 1.356) Pub Date : 2020-05-08
Damien Roy

We show that Hermite’s approximations to values of the exponential function at given algebraic numbers are nearly optimal when considered from an adelic perspective. We achieve this by taking into account the ratio of these values whenever they make sense in the various completions (Archimedean or p-adic) of a number field containing these algebraic numbers.

更新日期：2020-05-08
• Math. Ann. (IF 1.356) Pub Date : 2020-05-07
Raphael S. Steiner

We prove sub-convex bounds on the fourth moment of Hecke–Laplace eigenforms on $$S^3$$. As a corollary, we get a Burgess-type sub-convex bound on the sup-norm of an individual eigenform. This constitutes an improvement over what is achievable through employing the Iwaniec–Sarnak amplifier.

更新日期：2020-05-07
• Math. Ann. (IF 1.356) Pub Date : 2020-05-02
Man-Chun Lee

In this work, we show that along a particular choice of Hermitian curvature flow, the non-positivity of the first Ricci curvature will be preserved if the initial metric has Griffiths non-positive Chern curvature. If in addition, the first Ricci curvature is negative at a point, then the canonical line bundle is ample.

更新日期：2020-05-02
• Math. Ann. (IF 1.356) Pub Date : 2020-04-28
Antonio Alfieri, Sungkyung Kang, András I. Stipsicz

Using the covering involution on the double branched cover of $$S^3$$ branched along a knot, and adapting ideas of Hendricks–Manolescu and Hendricks–Hom–Lidman, we define new knot (concordance) invariants and apply them to deduce novel linear independence results in the smooth concordance group of knots.

更新日期：2020-04-28
• Math. Ann. (IF 1.356) Pub Date : 2020-04-28
Octave Curmi

We prove that the boundaries of the Milnor fibers of smoothings of non-isolated reduced complex surface singularities are graph manifolds. Moreover, we give a method, inspired by previous work of Némethi and Szilard, to compute associated plumbing graphs.

更新日期：2020-04-28
• Math. Ann. (IF 1.356) Pub Date : 2020-04-25
Benjamin Ambrosio, Arnaud Ducrot, Shigui Ruan

Traveling wave solutions in general time-dependent (including time-periodic) reaction–diffusion equations and systems of equations have attracted great attention in the last two decades. The aim of this paper is to study the propagation phenomenon in a general time-heterogeneous environment. More specifically, we investigate generalized traveling wave solutions for a two-component time-dependent non-cooperative

更新日期：2020-04-25
• Math. Ann. (IF 1.356) Pub Date : 2020-04-23
Fabien Priziac

Using the geometric quotient of a real algebraic set by the action of a finite group G, we construct invariants of G-affine real algebraic varieties with respect to equivariant homeomorphisms with algebraic graph, including additive invariants with values in $${\mathbb {Z}}$$. The construction requires to consider the wider category of $$\mathcal {AS}$$-sets.

更新日期：2020-04-23
• Math. Ann. (IF 1.356) Pub Date : 2020-04-22
Kazuki Tokimoto

Following Weinstein, Boyarchenko–Weinstein and Imai–Tsushima, we construct a family of affinoids in the Lubin–Tate perfectoid space and their formal models such that the cohomology of the reduction of each formal model realizes the local Langlands correspondence and the local Jacquet–Langlands correspondence for certain representations. In the terminology of the essentially tame local Langlands correspondence

更新日期：2020-04-23
• Math. Ann. (IF 1.356) Pub Date : 2020-04-22
Mohan Ravichandran, Jonathan Leake

We adapt the arguments of Marcus, Spielman and Srivastava in their proof of the Kadison–Singer problem to prove improved paving estimates. Working with Anderson’s paving formulation of Kadison–Singer instead of Weaver’s vector balancing version, we show that the machinery of interlacing polynomials due to Marcus, Spielman and Srivastava works in this setting as well. The relevant expected characteristic

更新日期：2020-04-23
• Math. Ann. (IF 1.356) Pub Date : 2020-04-22
Natalia Garcia-Fritz, Hector Pasten

For a positive proportion of primes p and q, we prove that $${\mathbb {Z}}$$ is Diophantine in the ring of integers of $${\mathbb {Q}}(\root 3 \of {p},\sqrt{-q})$$. This provides a new and explicit infinite family of number fields K such that Hilbert’s tenth problem for $$O_K$$ is unsolvable. Our methods use Iwasawa theory and congruences of Heegner points in order to obtain suitable rank stability

更新日期：2020-04-23
• Math. Ann. (IF 1.356) Pub Date : 2020-04-18
Elia Brué, Quoc-Hung Nguyen

It is known, after Jabin (J Differ Equ 260(5):4739–4757, 2016) and Alberti et al. (Ann PDE 5(1):9, 2019), that ODE flows and solutions of the transport equation associated to Sobolev vector fields do not propagate Sobolev regularity, even of fractional order. In this paper, we improve the result at Clop and Jylha (J Differ Equ 266(8):4544–4567, 2019) and show that some kind of propagation of Sobolev

更新日期：2020-04-23
• Math. Ann. (IF 1.356) Pub Date : 2020-04-16
Kathryn Mann, Sam Nariman

Motivated by a question of Ghys, we study obstructions to extending group actions on the boundary $$\partial M$$ of a 3-manifold to a $$C^0$$-action on M. Among other results, we show that for a 3-manifold M, the $$S^1 \times S^1$$ action on the boundary does not extend to a $$C^0$$-action of $$S^1 \times S^1$$ via homeomorphisms that are isotopic to the identity as a discrete group on M, except in

更新日期：2020-04-23
• Math. Ann. (IF 1.356) Pub Date : 2020-04-13
Chris Connell, Shi Wang

We show that any closed manifold with a metric of nonpositive curvature that admits either a single point rank condition or a single point curvature condition has positive simplicial volume. We use this to provide a differential geometric proof of a conjecture of Gromov in dimension three.

更新日期：2020-04-23
• Math. Ann. (IF 1.356) Pub Date : 2020-04-02
Cristian Minoccheri

A variety is rationally connected if two general points can be joined by a rational curve. A higher version of this notion is rational simple connectedness, which requires suitable spaces of rational curves through two points to be rationally connected themselves. We prove that smooth, complex, weighted complete intersections of low enough degree are rationally simply connected. This result has strong

更新日期：2020-04-23
• Math. Ann. (IF 1.356) Pub Date : 2020-03-16
Jacopo Bellazzini, Vladimir Georgiev, Nicola Visciglia

The assumptions of Theorem are not correct in case

更新日期：2020-04-23
• Math. Ann. (IF 1.356) Pub Date : 2019-09-13
Erik Carlsson, Eugene Gorsky, Anton Mellit

The earlier work of the first and the third name authors introduced the algebra $${\mathbb {A}}_{q,t}$$ and its polynomial representation. In this paper we construct an action of this algebra on the equivariant K-theory of certain smooth strata in the flag Hilbert scheme of points on the plane. In this presentation, the fixed points of the torus action correspond to generalized Macdonald polynomials

更新日期：2020-04-23
• Math. Ann. (IF 1.356) Pub Date : 2020-02-17
Takashi Taniguchi, Frank Thorne

In our companion paper [28], we developed an efficient algebraic method for computing the Fourier transforms of certain functions defined on prehomogeneous vector spaces over finite fields, and we carried out these computations in a variety of cases. Here we develop a method, based on Fourier analysis and algebraic geometry, which exploits these Fourier transform formulas to yield level of distribution

更新日期：2020-04-23
• Math. Ann. (IF 1.356) Pub Date : 2018-10-08
Wen-yuan Yang

We establish that, for statistically convex-cocompact actions, contracting elements are exponentially generic in counting measure. We obtain as corollaries results on the exponential genericity for the set of hyperbolic elements in relatively hyperbolic groups, the set of rank-1 elements in CAT(0) groups, and the set of pseudo-Anosov elements in mapping class groups. For a proper action with purely

更新日期：2020-04-23
• Math. Ann. (IF 1.356) Pub Date : 2019-09-18
Long Jin, Ruixiang Zhang

We prove an explicit formula for the dependence of the exponent $$\beta$$ in the fractal uncertainty principle of Bourgain–Dyatlov (Ann Math 187:1–43, 2018) on the dimension $$\delta$$ and on the regularity constant $$C_R$$ for the regular set. In particular, this implies an explicit essential spectral gap for convex co-compact hyperbolic surfaces when the Hausdorff dimension of the limit set is

更新日期：2020-04-23
• Math. Ann. (IF 1.356) Pub Date : 2019-11-27
Haining Wang

In this article we study the special fiber of the Rapoport–Zink space attached to a quaternionic unitary group. The special fiber is described using the so called Bruhat–Tits stratification and is intimately related to the Bruhat–Tits building of a split symplectic group. As an application we describe the supersingular locus of the related Shimura variety.

更新日期：2020-04-23
• Math. Ann. (IF 1.356) Pub Date : 2019-12-02
Quy Thuong Lê, Hong Duc Nguyen

We develop Denef–Loeser’s motivic integration to an equivariant version and use it to prove the full integral identity conjecture for regular functions. In comparison with Hartmann’s work, the equivariant Grothendieck ring defined in this article is more elementary and it yields the application to the conjecture.

更新日期：2020-04-23
• Math. Ann. (IF 1.356) Pub Date : 2019-10-10
Eckhard Meinrenken, María Amelia Salazar

The classical Van Est theory relates the smooth cohomology of Lie groups with the cohomology of the associated Lie algebra, or its relative versions. Some aspects of this theory generalize to Lie groupoids and their Lie algebroids. In this paper, continuing an idea from Li-Bland and Meinrenken (Enseign Math 61(1–2):93–137, 2015), we revisit the van Est theory using the Perturbation Lemma from homological

更新日期：2020-04-23
• Math. Ann. (IF 1.356) Pub Date : 2019-11-07
Ofir Gorodetsky, Will Sawin

For a fixed polynomial $$\varDelta$$, we study the number of polynomials f of degree n over $${\mathbb {F}}_q$$ such that f and $$f+\varDelta$$ are both irreducible, an $${\mathbb {F}}_q[T]$$-analogue of the twin primes problem. In the large-q limit, we obtain a lower-order term for this count if we consider non-monic polynomials, which depends on $$\varDelta$$ in a manner which is consistent with

更新日期：2020-04-23
• Math. Ann. (IF 1.356) Pub Date : 2019-09-25
David Damanik, Jake Fillman, Selim Sukhtaiev

We establish spectral and dynamical localization for several Anderson models on metric and discrete radial trees. The localization results are obtained on compact intervals contained in the complement of discrete sets of exceptional energies. All results are proved under the minimal hypothesis on the type of disorder: the random variables generating the trees assume at least two distinct values. This

更新日期：2020-04-23
• Math. Ann. (IF 1.356) Pub Date : 2019-02-15
Federico Cacciafesta, Anne-Sophie de Suzzoni, Diego Noja

The system describing a single Dirac electron field coupled with classically moving point nuclei is presented and studied. The model is a semi-relativistic extension of corresponding time-dependent one-body Hartree-Fock equation coupled with classical nuclear dynamics, already known and studied both in quantum chemistry and in rigorous mathematical literature. We prove local existence of solutions

更新日期：2020-04-23
• Math. Ann. (IF 1.356) Pub Date : 2019-08-21
Yûsuke Okuyama

We introduce the f-crucial function $${\text {Crucial}}_f$$ associated to a rational function $$f\in K(z)$$ of degree $$>1$$ over an algebraically closed field K of possibly positive characteristic that is complete with respect to a non-trivial and non-archimedean absolute value, and give a global and explicit expression of Rumely’s (resultant) function $${\text {ordRes}}_f$$ in terms of the hyperbolic

更新日期：2020-04-23
• Math. Ann. (IF 1.356) Pub Date : 2020-03-30
Andrzej Wiśnicki

Suppose that Q is a $$\hbox {weak}^{*}$$ compact convex subset of a dual Banach space with the Radon–Nikodým property. We show that if (S, Q) is a nonexpansive and norm-distal dynamical system, then there is a fixed point of S in Q and the set of fixed points is a nonexpansive retract of Q. As a consequence we obtain a nonlinear extension of the Bader–Gelander–Monod theorem concerning isometries in

更新日期：2020-03-30
• Math. Ann. (IF 1.356) Pub Date : 2020-03-25
P. Bengoechea, Ö. Imamoglu

In 2008, Kaneko made several interesting observations about the values of the modular j invariant at real quadratic irrationalities. The values of modular functions at real quadratics are defined in terms of their cycle integrals along the associated geodesics. In this paper we prove some of the conjectures of Kaneko for a general modular function.

更新日期：2020-03-25
• Math. Ann. (IF 1.356) Pub Date : 2020-03-19
Jitendra Bajpai, Günter Harder, Ivan Horozov, Matias Victor Moya Giusti

In this article, several cohomology spaces associated to the arithmetic groups $$\mathrm {SL}_3({\mathbb {Z}})$$ and $$\mathrm {GL}_3({\mathbb {Z}})$$ with coefficients in any highest weight representation $${\mathcal {M}}_\lambda$$ have been computed, where $$\lambda$$ denotes their highest weight. Consequently, we obtain detailed information of their Eisenstein cohomology with coefficients in $${\mathcal 更新日期：2020-03-19 • Math. Ann. (IF 1.356) Pub Date : 2020-03-16 Simon Brandhorst We characterize Salem numbers which have some power arising as dynamical degree of an automorphism on a complex (projective) 2-Torus, K3 or Enriques surface. 更新日期：2020-03-16 • Math. Ann. (IF 1.356) Pub Date : 2020-03-04 Lionel Lang Harmonic amoebas are generalisations of amoebas of algebraic curves immersed in complex tori. Introduced by Krichever in 2014, the consideration of such objects suggests to enlarge the scope of tropical geometry. In the present paper, we introduce the notion of harmonic morphisms from tropical curves to affine spaces and show how these morphisms can be systematically described as limits of families 更新日期：2020-03-04 • Math. Ann. (IF 1.356) Pub Date : 2020-02-29 John Christian Ottem, Fumiaki Suzuki We prove that there exists a pencil of Enriques surfaces defined over \({\mathbb {Q}}$$ with non-algebraic integral Hodge classes of non-torsion type. This gives the first example of a threefold with the trivial Chow group of zero-cycles on which the integral Hodge conjecture fails. As an application, we construct a fourfold which gives the negative answer to a classical question of Murre on the universality

更新日期：2020-02-29
• Math. Ann. (IF 1.356) Pub Date : 2020-02-05
Richard Hepworth

We prove a general homological stability theorem for certain families of groups equipped with product maps, followed by two theorems of a new kind that give information about the last two homology groups outside the stable range. (These last two unstable groups are the ‘edge’ in our title.) Applying our results to automorphism groups of free groups yields a new proof of homological stability with an

更新日期：2020-02-05
• Math. Ann. (IF 1.356) Pub Date : 2020-01-29
Lev Birbrair, Alexandre Fernandes, J. Edson Sampaio, Misha Verbitsky

It was conjectured that multiplicity of a singularity is bi-Lipschitz invariant. We disprove this conjecture constructing examples of bi-Lipschitz equivalent complex algebraic singularities with different values of multiplicity.

更新日期：2020-01-29
• Math. Ann. (IF 1.356) Pub Date : 2019-11-07
Tommaso Bruno, Marco M. Peloso, Maria Vallarino

In this paper we develop a theory of Besov and Triebel–Lizorkin spaces on general noncompact connected Lie groups endowed with a sub-Riemannian structure. Such spaces are defined by means of hypoelliptic sub-Laplacians with drift, and endowed with a measure whose density with respect to a right Haar measure is a continuous positive character of the group. We prove several equivalent characterizations

更新日期：2019-11-07
• Math. Ann. Pub Date : 2019-07-02
Matthew C H Tointon

We show that a K-approximate subgroup A of a residually nilpotent group G is contained in boundedly many cosets of a finite-by-nilpotent subgroup, the nilpotent factor of which is of bounded step. Combined with an earlier result of the author, this implies that A is contained in boundedly many translates of a coset nilprogression of bounded rank and step. The bounds are effective and depend only on

更新日期：2019-11-01
• Math. Ann. Pub Date : 2018-01-27
John E McCarthy,James E Pascoe

We prove a Julia inequality for bounded non-commutative functions on polynomial polyhedra. We use this to deduce a Julia inequality for holomorphic functions on classical domains in ℂ d . We look at differentiability at a boundary point for functions that have a certain regularity there.

更新日期：2019-11-01
• Math. Ann. Pub Date : null
Marek Kaluba,Piotr W Nowak,Narutaka Ozawa

We give a constructive, computer-assisted proof that Aut ( F 5 ) , the automorphism group of the free group on 5 generators, has Kazhdan's property (T).

更新日期：2019-11-01
• Math. Ann. Pub Date : null
Jürgen Jost,Lei Liu,Miaomiao Zhu

Let { u n } be a sequence of maps from a compact Riemann surface M with smooth boundary to a general compact Riemannian manifold N with free boundary on a smooth submanifold K ⊂ N satisfying sup n ‖ ∇ u n ‖ L 2 ( M ) + ‖ τ ( u n ) ‖ L 2 ( M ) ≤ Λ , where τ ( u n ) is the tension field of the map u n . We show that the energy identity and the no neck property hold during a blow-up process. The assumptions

更新日期：2019-11-01
• Math. Ann. Pub Date : null
Jason D Lotay,Goncalo Oliveira

We initiate the systematic study of G 2 -instantons with SU(2) 2 -symmetry. As well as developing foundational theory, we give existence, non-existence and classification results for these instantons. We particularly focus on R 4 × S 3 with its two explicitly known distinct holonomy G 2 metrics, which have different volume growths at infinity, exhibiting the different behaviour of instantons in these

更新日期：2019-11-01
• Math. Ann. Pub Date : null
Claudia Garetto,Christian Jäh,Michael Ruzhansky

In this paper we analyse the well-posedness of the Cauchy problem for a rather general class of hyperbolic systems with space-time dependent coefficients and with multiple characteristics of variable multiplicity. First, we establish a well-posedness result in anisotropic Sobolev spaces for systems with upper triangular principal part under interesting natural conditions on the orders of lower order

更新日期：2019-11-01
• Math. Ann. Pub Date : null
Kevin Henriot,Kevin Hughes

We obtain restriction estimates of ε -removal type for the set of k-th powers of integers, and for discrete d-dimensional surfaces of the form { ( n 1 , ⋯ , n d , n 1 k + ⋯ + n d k ) : | n 1 | , ⋯ , | n d | ⩽ N } , which we term 'k-paraboloids'. For these surfaces, we obtain a satisfying range of exponents for large values of d, k. We also obtain estimates of ε -removal type in the full supercritical

更新日期：2019-11-01
• Math. Ann. Pub Date : null
Jussi Behrndt,Fritz Gesztesy,Shu Nakamura

The spectral shift function of a pair of self-adjoint operators is expressed via an abstract operator-valued Titchmarsh-Weyl m-function. This general result is applied to different self-adjoint realizations of second-order elliptic partial differential operators on smooth domains with compact boundaries and Schrödinger operators with compactly supported potentials. In these applications the spectral

更新日期：2019-11-01
• Math. Ann. (IF 1.356) Pub Date : 2019-10-17
Shaoming Guo, Joris Roos, Andreas Seeger, Po-Lam Yung

Let $$H^{(u)}$$ be the Hilbert transform along the parabola $$(t, ut^2)$$ where $$u\in \mathbb {R}$$. For a set U of positive numbers consider the maximal function $${\mathcal {H}}^U \,f= \sup \{|H^{(u)}\, f|: u\in U\}$$. We obtain an (essentially) optimal result for the $$L^p$$ operator norm of $${\mathcal {H}}^U$$ when $$2 更新日期：2019-10-17 • Math. Ann. (IF 1.356) Pub Date : 2019-06-08 Marta Lewicka, Juan Manfredi, Diego Ricciotti We study mean value properties of \(\mathbf{p }$$-harmonic functions on the first Heisenberg group $${\mathbb {H}}$$, in connection to the dynamic programming principles of certain stochastic processes. We implement the approach of Peres and Sheffield (Duke Math J 145(1):91–120, 2008) to provide a game-theoretical interpretation of the sub-elliptic $$\mathbf{p }$$-Laplacian; and of Manfredi et al.

更新日期：2019-06-08
• Math. Ann. (IF 1.356) Pub Date : 2019-05-22
Evgeny Korotyaev, Natalia Saburova

We consider a Laplacian on periodic discrete graphs. Its spectrum consists of a finite number of bands. In a class of periodic 1-forms, i.e., functions defined on edges of the periodic graph, we introduce a subclass of minimal forms with a minimal number $${{\mathcal {I}}}$$ of edges in their supports on the period. We obtain a specific decomposition of the Laplacian into a direct integral in terms

更新日期：2019-05-22
• Math. Ann. (IF 1.356) Pub Date : 2019-03-21
Matti Lassas, Tony Liimatainen, Mikko Salo

We introduce a new approach to the anisotropic Calderón problem, based on a map called Poisson embedding that identifies the points of a Riemannian manifold with distributions on its boundary. We give a new uniqueness result for a large class of Calderón type inverse problems for quasilinear equations in the real analytic case. The approach also leads to a new proof of the result of Lassas et al. (Annales

更新日期：2019-03-21
• Math. Ann. (IF 1.356) Pub Date : 2019-01-30
Yuxing Deng, Xiaohua Zhu

In this paper, we show that any n-dimensional $$\kappa$$-noncolla-psed steady (gradient) Kähler–Ricci soliton with nonnegative bisectional curvature must be flat. The result is an improvement to our former work in Deng and Zhu (Trans Am Math Soc 370(4):2855–2877, 2018).

更新日期：2019-01-30
• Math. Ann. (IF 1.356) Pub Date : 2019-01-18
Min Ru, Nessim Sibony

We develop Nevanlinna’s theory for a class of holomorphic maps when the source is a disc. Such maps appear in the theory of foliations by Riemann Surfaces.

更新日期：2019-01-18
• Math. Ann. (IF 1.356) Pub Date : 2018-08-28
Laurentiu Maxim, Morihiko Saito, Jörg Schürmann

We introduce spectral Hirzebruch–Milnor classes for singular hypersurfaces. These can be identified with Steenbrink spectra in the isolated singularity case, and may be viewed as their global analogues in general. Their definition uses vanishing cycles of mixed Hodge modules and the Todd class transformation. These are compatible with the pushforward by proper morphisms, and the classes can be calculated

更新日期：2018-08-28
• Math. Ann. (IF 1.356) Pub Date : 2018-08-01
Laura G. DeMarco, Sarah C. Koch, Curtis T. McMullen

The postcritical set P(f) of a rational map $$f:{\mathbb P}^1\rightarrow {\mathbb P}^1$$ is the smallest forward invariant subset of $${\mathbb P}^1$$ that contains the critical values of f. In this paper we show that every finite set $$X\subset {\mathbb P}^1({\overline{{\mathbb Q}}})$$ can be realized as the postcritical set of a rational map. We also show that every map $$F:X\rightarrow X$$ defined

更新日期：2018-08-01
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