Geometry & Topology Articles (Project Euclid)
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The latest articles from Geometry & Topology on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2017 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 19 Oct 2017 14:27 EDTThu, 19 Oct 2017 14:27 EDThttps://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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On representation varieties of $3$–manifold groups
https://projecteuclid.org/euclid.gt/1508437634
<strong>Michael Kapovich</strong>, <strong>John Millson</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 4, 1931--1968.</p><p><strong>Abstract:</strong><br/>
We prove universality theorems (“Murphy’s laws”) for representation varieties of fundamental groups of closed [math] –dimensional manifolds. We show that germs of [math] –representation schemes of such groups are essentially the same as germs of schemes over [math] of finite type.
</p>projecteuclid.org/euclid.gt/1508437634_20171019142734Thu, 19 Oct 2017 14:27 EDTMirror theorem for elliptic quasimap invariantshttps://projecteuclid.org/euclid.gt/1522461620<strong>Bumsig Kim</strong>, <strong>Hyenho Lho</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 3, 1459--1481.</p><p><strong>Abstract:</strong><br/>
We propose and prove a mirror theorem for the elliptic quasimap invariants of smooth Calabi–Yau complete intersections in projective spaces. This theorem, combined with the wall-crossing formula of Ciocan-Fontanine and Kim, implies mirror theorems of Zinger and Popa for the elliptic Gromov–Witten invariants of those varieties. This paper and the wall-crossing formula provide a unified framework for the mirror theory of rational and elliptic Gromov–Witten invariants.
</p>projecteuclid.org/euclid.gt/1522461620_20180330220033Fri, 30 Mar 2018 22:00 EDTCounting problem on wind-tree modelshttps://projecteuclid.org/euclid.gt/1522461621<strong>Angel Pardo</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 3, 1483--1536.</p><p><strong>Abstract:</strong><br/>
We study periodic wind-tree models, that is, billiards in the plane endowed with [math] –periodically located identical connected symmetric right-angled obstacles. We give asymptotic formulas for the number of (isotopy classes of) closed billiard trajectories (up to [math] –translations) on the wind-tree billiard. We also explicitly compute the associated Siegel–Veech constant for generic wind-tree billiards depending on the number of corners on the obstacle.
</p>projecteuclid.org/euclid.gt/1522461621_20180330220033Fri, 30 Mar 2018 22:00 EDTGroup trisections and smooth $4$–manifoldshttps://projecteuclid.org/euclid.gt/1522461622<strong>Aaron Abrams</strong>, <strong>David T Gay</strong>, <strong>Robion Kirby</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 3, 1537--1545.</p><p><strong>Abstract:</strong><br/>
A trisection of a smooth, closed, oriented [math] –manifold is a decomposition into three [math] –dimensional [math] –handlebodies meeting pairwise in [math] –dimensional [math] –handlebodies, with triple intersection a closed surface. The fundamental groups of the surface, the [math] –dimensional handlebodies, the [math] –dimensional handlebodies and the closed [math] –manifold, with homomorphisms between them induced by inclusion, form a commutative diagram of epimorphisms, which we call a trisection of the [math] –manifold group. A trisected [math] –manifold thus gives a trisected group; here we show that every trisected group uniquely determines a trisected [math] –manifold. Together with Gay and Kirby’s existence and uniqueness theorem for [math] –manifold trisections, this gives a bijection from group trisections modulo isomorphism and a certain stabilization operation to smooth, closed, connected, oriented [math] –manifolds modulo diffeomorphism. As a consequence, smooth [math] –manifold topology is, in principle, entirely group-theoretic. For example, the smooth [math] –dimensional Poincaré conjecture can be reformulated as a purely group-theoretic statement.
</p>projecteuclid.org/euclid.gt/1522461622_20180330220033Fri, 30 Mar 2018 22:00 EDTSymmetric products and subgroup latticeshttps://projecteuclid.org/euclid.gt/1522461623<strong>Markus Hausmann</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 3, 1547--1591.</p><p><strong>Abstract:</strong><br/>
Let [math] be a finite group. We show that the rational equivariant homotopy groups of symmetric products of the [math] –equivariant sphere spectrum are naturally isomorphic to the rational homology groups of certain subcomplexes of the subgroup lattice of [math] .
</p>projecteuclid.org/euclid.gt/1522461623_20180330220033Fri, 30 Mar 2018 22:00 EDTConvex projective structures on nonhyperbolic three-manifoldshttps://projecteuclid.org/euclid.gt/1522461624<strong>Samuel A Ballas</strong>, <strong>Jeffrey Danciger</strong>, <strong>Gye-Seon Lee</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 3, 1593--1646.</p><p><strong>Abstract:</strong><br/>
Y Benoist proved that if a closed three-manifold [math] admits an indecomposable convex real projective structure, then [math] is topologically the union along tori and Klein bottles of finitely many submanifolds each of which admits a complete finite volume hyperbolic structure on its interior. We describe some initial results in the direction of a potential converse to Benoist’s theorem. We show that a cusped hyperbolic three-manifold may, under certain assumptions, be deformed to convex projective structures with totally geodesic torus boundary. Such structures may be convexly glued together whenever the geometry at the boundary matches up. In particular, we prove that many doubles of cusped hyperbolic three-manifolds admit convex projective structures.
</p>projecteuclid.org/euclid.gt/1522461624_20180330220033Fri, 30 Mar 2018 22:00 EDTHyperbolic Dehn filling in dimension fourhttps://projecteuclid.org/euclid.gt/1522461625<strong>Bruno Martelli</strong>, <strong>Stefano Riolo</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 3, 1647--1716.</p><p><strong>Abstract:</strong><br/>
We introduce and study some deformations of complete finite-volume hyperbolic four-manifolds that may be interpreted as four-dimensional analogues of Thurston’s hyperbolic Dehn filling.
We construct in particular an analytic path of complete, finite-volume cone four-manifolds [math] that interpolates between two hyperbolic four-manifolds [math] and [math] with the same volume [math] . The deformation looks like the familiar hyperbolic Dehn filling paths that occur in dimension three, where the cone angle of a core simple closed geodesic varies monotonically from [math] to [math] . Here, the singularity of [math] is an immersed geodesic surface whose cone angles also vary monotonically from [math] to [math] . When a cone angle tends to [math] a small core surface (a torus or Klein bottle) is drilled, producing a new cusp.
We show that various instances of hyperbolic Dehn fillings may arise, including one case where a degeneration occurs when the cone angles tend to [math] , like in the famous figure-eight knot complement example.
The construction makes an essential use of a family of four-dimensional deforming hyperbolic polytopes recently discovered by Kerckhoff and Storm.
</p>projecteuclid.org/euclid.gt/1522461625_20180330220033Fri, 30 Mar 2018 22:00 EDTSemidualities from products of treeshttps://projecteuclid.org/euclid.gt/1522461626<strong>Daniel Studenmund</strong>, <strong>Kevin Wortman</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 3, 1717--1758.</p><p><strong>Abstract:</strong><br/>
Let [math] be a global function field of characteristic [math] , and let [math] be a finite-index subgroup of an arithmetic group defined with respect to [math] and such that any torsion element of [math] is a [math] –torsion element. We define semiduality groups, and we show that [math] is a [math] –semiduality group if [math] acts as a lattice on a product of trees. We also give other examples of semiduality groups, including lamplighter groups, Diestel–Leader groups, and countable sums of finite groups.
</p>projecteuclid.org/euclid.gt/1522461626_20180330220033Fri, 30 Mar 2018 22:00 EDTBrane actions, categorifications of Gromov–Witten theory and quantum K–theoryhttps://projecteuclid.org/euclid.gt/1522461627<strong>Etienne Mann</strong>, <strong>Marco Robalo</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 3, 1759--1836.</p><p><strong>Abstract:</strong><br/>
Let [math] be a smooth projective variety. Using the idea of brane actions discovered by Toën, we construct a lax associative action of the operad of stable curves of genus zero on the variety [math] seen as an object in correspondences in derived stacks. This action encodes the Gromov–Witten theory of [math] in purely geometrical terms and induces an action on the derived category [math] which allows us to recover the quantum K–theory of Givental and Lee.
</p>projecteuclid.org/euclid.gt/1522461627_20180330220033Fri, 30 Mar 2018 22:00 EDTRicci flow on asymptotically Euclidean manifoldshttps://projecteuclid.org/euclid.gt/1522461628<strong>Yu Li</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 3, 1837--1891.</p><p><strong>Abstract:</strong><br/>
In this paper, we prove that if an asymptotically Euclidean manifold with nonnegative scalar curvature has long-time existence of Ricci flow, the ADM mass is nonnegative. We also give an independent proof of the positive mass theorem in dimension three.
</p>projecteuclid.org/euclid.gt/1522461628_20180330220033Fri, 30 Mar 2018 22:00 EDTFrom operator categories to higher operadshttps://projecteuclid.org/euclid.gt/1523584815<strong>Clark Barwick</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 4, 1893--1959.</p><p><strong>Abstract:</strong><br/>
We introduce the notion of an operator category and two different models for homotopy theory of [math] –operads over an operator category — one of which extends Lurie’s theory of [math] –operads, the other of which is completely new, even in the commutative setting. We define perfect operator categories, and we describe a category [math] attached to a perfect operator category [math] that provides Segal maps. We define a wreath product of operator categories and a form of the Boardman–Vogt tensor product that lies over it. We then give examples of operator categories that provide universal properties for the operads [math] and [math] ( [math] ) and also a collection of new examples.
</p>projecteuclid.org/euclid.gt/1523584815_20180412220031Thu, 12 Apr 2018 22:00 EDTQuantitative bi-Lipschitz embeddings of bounded-curvature manifolds and orbifoldshttps://projecteuclid.org/euclid.gt/1523584816<strong>Sylvester Eriksson-Bique</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 4, 1961--2026.</p><p><strong>Abstract:</strong><br/>
We construct bi-Lipschitz embeddings into Euclidean space for bounded-diameter subsets of manifolds and orbifolds of bounded curvature. The distortion and dimension of such embeddings is bounded by diameter, curvature and dimension alone. We also construct global bi-Lipschitz embeddings for spaces of the form [math] , where [math] is a discrete group acting properly discontinuously and by isometries on [math] . This generalizes results of Naor and Khot. Our approach is based on analyzing the structure of a bounded-curvature manifold at various scales by specializing methods from collapsing theory to a certain class of model spaces. In the process, we develop tools to prove collapsing theory results using algebraic techniques.
</p>projecteuclid.org/euclid.gt/1523584816_20180412220031Thu, 12 Apr 2018 22:00 EDTUnfolded Seiberg–Witten Floer spectra, I: Definition and invariancehttps://projecteuclid.org/euclid.gt/1523584817<strong>Tirasan Khandhawit</strong>, <strong>Jianfeng Lin</strong>, <strong>Hirofumi Sasahira</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 4, 2027--2114.</p><p><strong>Abstract:</strong><br/>
Let [math] be a closed and oriented [math] –manifold. We define different versions of unfolded Seiberg–Witten Floer spectra for [math] . These invariants generalize Manolescu’s Seiberg–Witten Floer spectrum for rational homology [math] –spheres. We also compute some examples when [math] is a Seifert space.
</p>projecteuclid.org/euclid.gt/1523584817_20180412220031Thu, 12 Apr 2018 22:00 EDTA family of compact complex and symplectic Calabi–Yau manifolds that are non-Kählerhttps://projecteuclid.org/euclid.gt/1523584818<strong>Lizhen Qin</strong>, <strong>Botong Wang</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 4, 2115--2144.</p><p><strong>Abstract:</strong><br/>
We construct a family of [math] –dimensional compact manifolds [math] which are simultaneously diffeomorphic to complex Calabi–Yau manifolds and symplectic Calabi–Yau manifolds. They have fundamental groups [math] , their odd-degree Betti numbers are even, they satisfy the hard Lefschetz property, and their real homotopy types are formal. However, [math] is never homotopy equivalent to a compact Kähler manifold for any topological space [math] . The main ingredient to show the non-Kählerness is a structure theorem of cohomology jump loci due to the second author.
</p>projecteuclid.org/euclid.gt/1523584818_20180412220031Thu, 12 Apr 2018 22:00 EDTRotation intervals and entropy on attracting annular continuahttps://projecteuclid.org/euclid.gt/1523584819<strong>Alejandro Passeggi</strong>, <strong>Rafael Potrie</strong>, <strong>Martín Sambarino</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 4, 2145--2186.</p><p><strong>Abstract:</strong><br/>
We show that if [math] is an annular homeomorphism admitting an attractor which is an irreducible annular continua with two different rotation numbers, then the entropy of [math] is positive. Further, the entropy is shown to be associated to a [math] –robust rotational horseshoe . On the other hand, we construct examples of annular homeomorphisms with such attractors for which the rotation interval is uniformly large but the entropy approaches zero as much as desired.
The developed techniques allow us to obtain similar results in the context of Birkhoff attractors .
</p>projecteuclid.org/euclid.gt/1523584819_20180412220031Thu, 12 Apr 2018 22:00 EDTPrimes and fields in stable motivic homotopy theoryhttps://projecteuclid.org/euclid.gt/1523584820<strong>Jeremiah Heller</strong>, <strong>Kyle M Ormsby</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 4, 2187--2218.</p><p><strong>Abstract:</strong><br/>
Let [math] be a field of characteristic different from [math] . We establish surjectivity of Balmer’s comparison map
ρ
∙
:
Spc
(
SH
A
1
(
F
)
c
)
→
Spec
h
(
K
∗
M
W
(
F
)
)
from the tensor triangular spectrum of the homotopy category of compact motivic spectra to the homogeneous Zariski spectrum of Milnor–Witt [math] –theory. We also comment on the tensor triangular geometry of compact cellular motivic spectra, producing in particular novel field spectra in this category. We conclude with a list of questions about the structure of the tensor triangular spectrum of the stable motivic homotopy category.
</p>projecteuclid.org/euclid.gt/1523584820_20180412220031Thu, 12 Apr 2018 22:00 EDTSurgery for partially hyperbolic dynamical systems, I: Blow-ups of invariant submanifoldshttps://projecteuclid.org/euclid.gt/1523584821<strong>Andrey Gogolev</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 4, 2219--2252.</p><p><strong>Abstract:</strong><br/>
We suggest a method to construct new examples of partially hyperbolic diffeomorphisms. We begin with a partially hyperbolic diffeomorphism [math] which leaves invariant a submanifold [math] . We assume that [math] is an Anosov submanifold for [math] , that is, the restriction [math] is an Anosov diffeomorphism and the center distribution is transverse to [math] . By replacing each point in [math] with the projective space (real or complex) of lines normal to [math] , we obtain the blow-up [math] . Replacing [math] with [math] amounts to a surgery on the neighborhood of [math] which alters the topology of the manifold. The diffeomorphism [math] induces a canonical diffeomorphism [math] . We prove that under certain assumptions on the local dynamics of [math] at [math] the diffeomorphism [math] is also partially hyperbolic. We also present some modifications, such as the connected sum construction, which allows to “paste together” two partially hyperbolic diffeomorphisms to obtain a new one. Finally, we present several examples to which our results apply.
</p>projecteuclid.org/euclid.gt/1523584821_20180412220031Thu, 12 Apr 2018 22:00 EDTEigenvalues of curvature, Lyapunov exponents and Harder–Narasimhan filtrationshttps://projecteuclid.org/euclid.gt/1523584822<strong>Fei Yu</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 4, 2253--2298.</p><p><strong>Abstract:</strong><br/>
Inspired by the Katz–Mazur theorem on crystalline cohomology and by the numerical experiments of Eskin, Kontsevich and Zorich, we conjecture that the polygon of the Lyapunov spectrum lies above (or on) the Harder–Narasimhan polygon of the Hodge bundle over any Teichmüller curve. We also discuss the connections between the two polygons and the integral of eigenvalues of the curvature of the Hodge bundle by using the works of Atiyah and Bott, Forni, and Möller. We obtain several applications to Teichmüller dynamics conditional on the conjecture.
</p>projecteuclid.org/euclid.gt/1523584822_20180412220031Thu, 12 Apr 2018 22:00 EDTLower bounds for Lyapunov exponents of flat bundles on curveshttps://projecteuclid.org/euclid.gt/1523584823<strong>Alex Eskin</strong>, <strong>Maxim Kontsevich</strong>, <strong>Martin Möller</strong>, <strong>Anton Zorich</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 4, 2299--2338.</p><p><strong>Abstract:</strong><br/>
Consider a flat bundle over a complex curve. We prove a conjecture of Fei Yu that the sum of the top [math] Lyapunov exponents of the flat bundle is always greater than or equal to the degree of any rank- [math] holomorphic subbundle. We generalize the original context from Teichmüller curves to any local system over a curve with nonexpanding cusp monodromies. As an application we obtain the large-genus limits of individual Lyapunov exponents in hyperelliptic strata of abelian differentials, which Fei Yu proved conditionally on his conjecture.
Understanding the case of equality with the degrees of subbundle coming from the Hodge filtration seems challenging, eg for Calabi–Yau-type families. We conjecture that equality of the sum of Lyapunov exponents and the degree is related to the monodromy group being a thin subgroup of its Zariski closure.
</p>projecteuclid.org/euclid.gt/1523584823_20180412220031Thu, 12 Apr 2018 22:00 EDTHyperbolic jigsaws and families of pseudomodular groups, Ihttps://projecteuclid.org/euclid.gt/1523584824<strong>Beicheng Lou</strong>, <strong>Ser Peow Tan</strong>, <strong>Anh Duc Vo</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 4, 2339--2366.</p><p><strong>Abstract:</strong><br/>
We show that there are infinitely many commensurability classes of pseudomodular groups, thus answering a question raised by Long and Reid. These are Fuchsian groups whose cusp set is all of the rationals but which are not commensurable to the modular group. We do this by introducing a general construction for the fundamental domains of Fuchsian groups obtained by gluing together marked ideal triangular tiles, which we call hyperbolic jigsaw groups.
</p>projecteuclid.org/euclid.gt/1523584824_20180412220031Thu, 12 Apr 2018 22:00 EDTSubflexible symplectic manifoldshttps://projecteuclid.org/euclid.gt/1523584825<strong>Emmy Murphy</strong>, <strong>Kyler Siegel</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 4, 2367--2401.</p><p><strong>Abstract:</strong><br/>
We introduce a class of Weinstein domains which are sublevel sets of flexible Weinstein manifolds but are not themselves flexible. These manifolds exhibit rather subtle behavior with respect to both holomorphic curve invariants and symplectic flexibility. We construct a large class of examples and prove that every flexible Weinstein manifold can be Weinstein homotoped to have a nonflexible sublevel set.
</p>projecteuclid.org/euclid.gt/1523584825_20180412220031Thu, 12 Apr 2018 22:00 EDTNormalized entropy versus volume for pseudo-Anosovshttps://projecteuclid.org/euclid.gt/1523584826<strong>Sadayoshi Kojima</strong>, <strong>Greg McShane</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 4, 2403--2426.</p><p><strong>Abstract:</strong><br/>
Thanks to a recent result by Jean-Marc Schlenker, we establish an explicit linear inequality between the normalized entropies of pseudo-Anosov automorphisms and the hyperbolic volumes of their mapping tori. As corollaries, we give an improved lower bound for values of entropies of pseudo-Anosovs on a surface with fixed topology, and a proof of a slightly weaker version of the result by Farb, Leininger and Margalit first, and by Agol later, on finiteness of cusped manifolds generating surface automorphisms with small normalized entropies. Also, we present an analogous linear inequality between the Weil–Petersson translation distance of a pseudo-Anosov map (normalized by multiplying by the square root of the area of a surface) and the volume of its mapping torus, which leads to a better bound.
</p>projecteuclid.org/euclid.gt/1523584826_20180412220031Thu, 12 Apr 2018 22:00 EDTTowers of regular self-covers and linear endomorphisms of torihttps://projecteuclid.org/euclid.gt/1523584827<strong>Wouter van Limbeek</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 4, 2427--2464.</p><p><strong>Abstract:</strong><br/>
Let [math] be a closed manifold that admits a self-cover [math] of degree [math] . We say [math] is strongly regular if all iterates [math] are regular covers. In this case, we establish an algebraic structure theorem for the fundamental group of [math] : We prove that [math] surjects onto a nontrivial free abelian group [math] , and the self-cover is induced by a linear endomorphism of [math] . Under further hypotheses we show that a finite cover of [math] admits the structure of a principal torus bundle. We show that this applies when [math] is Kähler and [math] is a strongly regular, holomorphic self-cover, and prove that a finite cover of [math] splits as a product with a torus factor.
</p>projecteuclid.org/euclid.gt/1523584827_20180412220031Thu, 12 Apr 2018 22:00 EDTClassification and arithmeticity of toroidal compactifications with $3\bar{c}_2 = \bar{c}_1^{2} = 3$https://projecteuclid.org/euclid.gt/1523584828<strong>Luca F Di Cerbo</strong>, <strong>Matthew Stover</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 4, 2465--2510.</p><p><strong>Abstract:</strong><br/>
We classify the minimum-volume smooth complex hyperbolic surfaces that admit smooth toroidal compactifications, and we explicitly construct their compactifications. There are five such surfaces, and they are all arithmetic; ie they are associated with quotients of the ball by an arithmetic lattice. Moreover, the associated lattices are all commensurable. The first compactification, originally discovered by Hirzebruch, is the blowup of an abelian surface at one point. The others are bielliptic surfaces blown up at one point. The bielliptic examples are new and are the first known examples of smooth toroidal compactifications birational to bielliptic surfaces.
</p>projecteuclid.org/euclid.gt/1523584828_20180412220031Thu, 12 Apr 2018 22:00 EDTKähler–Ricci flow, Kähler–Einstein metric, and K–stabilityhttps://projecteuclid.org/euclid.gt/1538186735<strong>Xiuxiong Chen</strong>, <strong>Song Sun</strong>, <strong>Bing Wang</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 6, 3145--3173.</p><p><strong>Abstract:</strong><br/>
We prove the existence of a Kähler–Einstein metric on a K–stable Fano manifold using the recent compactness result on Kähler–Ricci flows. The key ingredient is an algebrogeometric description of the asymptotic behavior of Kähler–Ricci flow on Fano manifolds. This is in turn based on a general finite-dimensional discussion, which is interesting on its own and could potentially apply to other problems. As one application, we relate the asymptotics of the Calabi flow on a polarized Kähler manifold to K–stability, assuming bounds on geometry.
</p>projecteuclid.org/euclid.gt/1538186735_20180928220544Fri, 28 Sep 2018 22:05 EDTTropical refined curve counting via motivic integrationhttps://projecteuclid.org/euclid.gt/1538186736<strong>Johannes Nicaise</strong>, <strong>Sam Payne</strong>, <strong>Franziska Schroeter</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 6, 3175--3234.</p><p><strong>Abstract:</strong><br/>
We propose a geometric interpretation of Block and Göttsche’s refined tropical curve counting invariants in terms of virtual [math] specializations of motivic measures of semialgebraic sets in relative Hilbert schemes. We prove that this interpretation is correct for linear series of genus 1, and in arbitrary genus after specializing from [math] –genus to Euler characteristic.
</p>projecteuclid.org/euclid.gt/1538186736_20180928220544Fri, 28 Sep 2018 22:05 EDTAdditive invariants for knots, links and graphs in $3$–manifoldshttps://projecteuclid.org/euclid.gt/1538186737<strong>Scott A Taylor</strong>, <strong>Maggy Tomova</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 6, 3235--3286.</p><p><strong>Abstract:</strong><br/>
We define two new families of invariants for ( [math] –manifold, graph) pairs which detect the unknot and are additive under connected sum of pairs and ( [math] ) additive under trivalent vertex sum of pairs. The first of these families is closely related to both bridge number and tunnel number. The second of these families is a variation and generalization of Gabai’s width for knots in the [math] –sphere. We give applications to the tunnel number and higher-genus bridge number of connected sums of knots.
</p>projecteuclid.org/euclid.gt/1538186737_20180928220544Fri, 28 Sep 2018 22:05 EDTPhase tropical hypersurfaceshttps://projecteuclid.org/euclid.gt/1538186738<strong>Gabriel Kerr</strong>, <strong>Ilia Zharkov</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 6, 3287--3320.</p><p><strong>Abstract:</strong><br/>
We prove a conjecture of Viro (Tr. Mat. Inst. Steklova 273 (2011) 271–303) that a smooth complex hypersurface in [math] is homeomorphic to the corresponding phase tropical hypersurface.
</p>projecteuclid.org/euclid.gt/1538186738_20180928220544Fri, 28 Sep 2018 22:05 EDTOn the Farrell–Jones conjecture for Waldhausen's $A$–theoryhttps://projecteuclid.org/euclid.gt/1538186739<strong>Nils-Edvin Enkelmann</strong>, <strong>Wolfgang Lück</strong>, <strong>Malte Pieper</strong>, <strong>Mark Ullmann</strong>, <strong>Christoph Winges</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 6, 3321--3394.</p><p><strong>Abstract:</strong><br/>
We prove the Farrell–Jones conjecture for (nonconnective) [math] –theory with coefficients and finite wreath products for hyperbolic groups, [math] –groups, cocompact lattices in almost connected Lie groups and fundamental groups of manifolds of dimension less or equal to three. Moreover, we prove inheritance properties such as passing to subgroups, colimits of direct systems of groups, finite direct products and finite free products. These results hold also for Whitehead spectra and spectra of stable pseudoisotopies in the topological, piecewise linear and smooth categories.
</p>projecteuclid.org/euclid.gt/1538186739_20180928220544Fri, 28 Sep 2018 22:05 EDTFaithful actions from hyperplane arrangementshttps://projecteuclid.org/euclid.gt/1538186740<strong>Yuki Hirano</strong>, <strong>Michael Wemyss</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 6, 3395--3433.</p><p><strong>Abstract:</strong><br/>
We show that if [math] is a smooth quasiprojective [math] –fold admitting a flopping contraction, then the fundamental group of an associated simplicial hyperplane arrangement acts faithfully on the derived category of [math] . The main technical advance is to use torsion pairs as an efficient mechanism to track various objects under iterations of the flop functor (or mutation functor). This allows us to relate compositions of the flop functor (or mutation functor) to the theory of Deligne normal form, and to give a criterion for when a finite composition of [math] –fold flops can be understood as a tilt at a single torsion pair. We also use this technique to give a simplified proof of a result of Brav and Thomas (Math. Ann. 351 (2011) 1005–1017) for Kleinian singularities.
</p>projecteuclid.org/euclid.gt/1538186740_20180928220544Fri, 28 Sep 2018 22:05 EDTGenerators for a complex hyperbolic braid grouphttps://projecteuclid.org/euclid.gt/1538186741<strong>Daniel Allcock</strong>, <strong>Tathagata Basak</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 6, 3435--3500.</p><p><strong>Abstract:</strong><br/>
We give generators for a certain complex hyperbolic braid group. That is, we remove a hyperplane arrangement from complex hyperbolic [math] –space, take the quotient of the remaining space by a discrete group, and find generators for the orbifold fundamental group of the quotient space. These generators have the most natural form: loops corresponding to the hyperplanes which come nearest the basepoint. Our results support the conjecture that motivated this study, the “monstrous proposal”, which posits a relationship between this braid group and the monster finite simple group.
</p>projecteuclid.org/euclid.gt/1538186741_20180928220544Fri, 28 Sep 2018 22:05 EDTA formal Riemannian structure on conformal classes and uniqueness for the $\sigma_2$–Yamabe problemhttps://projecteuclid.org/euclid.gt/1538186742<strong>Matthew Gursky</strong>, <strong>Jeffrey Streets</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 6, 3501--3573.</p><p><strong>Abstract:</strong><br/>
We define a new formal Riemannian metric on a conformal classes of four-manifolds in the context of the [math] –Yamabe problem. Exploiting this new variational structure we show that solutions are unique unless the manifold is conformally equivalent to the round sphere.
</p>projecteuclid.org/euclid.gt/1538186742_20180928220544Fri, 28 Sep 2018 22:05 EDTChern–Schwartz–MacPherson classes of degeneracy locihttps://projecteuclid.org/euclid.gt/1538186743<strong>László M Fehér</strong>, <strong>Richárd Rimányi</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 6, 3575--3622.</p><p><strong>Abstract:</strong><br/>
The Chern–Schwartz–MacPherson class (CSM) and the Segre–Schwartz–MacPherson class (SSM) are deformations of the fundamental class of an algebraic variety. They encode finer enumerative invariants of the variety than its fundamental class. In this paper we offer three contributions to the theory of equivariant CSM/SSM classes. First, we prove an interpolation characterization for CSM classes of certain representations. This method — inspired by recent work of Maulik and Okounkov and of Gorbounov, Rimányi, Tarasov and Varchenko — does not require a resolution of singularities and often produces explicit (not sieve) formulas for CSM classes. Second, using the interpolation characterization we prove explicit formulas — including residue generating sequences — for the CSM and SSM classes of matrix Schubert varieties. Third, we suggest that a stable version of the SSM class of matrix Schubert varieties will serve as the building block of equivariant SSM theory, similarly to how the Schur functions are the building blocks of fundamental class theory. We illustrate these phenomena, and related stability and (two-step) positivity properties for some relevant representations.
</p>projecteuclid.org/euclid.gt/1538186743_20180928220544Fri, 28 Sep 2018 22:05 EDTComputational complexity and $3$–manifolds and zombieshttps://projecteuclid.org/euclid.gt/1538186744<strong>Greg Kuperberg</strong>, <strong>Eric Samperton</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 6, 3623--3670.</p><p><strong>Abstract:</strong><br/>
We show the problem of counting homomorphisms from the fundamental group of a homology [math] –sphere [math] to a finite, nonabelian simple group [math] is almost parsimoniously [math] –complete, when [math] is fixed and [math] is the computational input. In the reduction, we guarantee that every nontrivial homomorphism is a surjection. As a corollary, any nontrivial information about the number of nontrivial homomorphisms is computationally intractable assuming standard conjectures in computer science. In particular, deciding if there is a nontrivial homomorphism is [math] –complete. Another corollary is that for any fixed integer [math] , it is [math] –complete to decide whether [math] admits a connected [math] –sheeted covering.
Given a classical reversible circuit [math] , we construct [math] so that evaluations of [math] with certain initialization and finalization conditions correspond to homomorphisms [math] . An intermediate state of [math] likewise corresponds to homomorphism [math] , where [math] is a Heegaard surface of [math] of genus [math] . We analyze the action on these homomorphisms by the pointed mapping class group [math] and its Torelli subgroup [math] . Using refinements of results of Dunfield and Thurston, we show that the actions of these groups are as large as possible when [math] is large. Our results and our construction are inspired by universality results in topological quantum computation, even though the present work is nonquantum.
One tricky step in the construction is handling an inert “zombie” symbol in the computational alphabet, which corresponds to a trivial homomorphism from the fundamental group of a subsurface of the Heegaard surface.
</p>projecteuclid.org/euclid.gt/1538186744_20180928220544Fri, 28 Sep 2018 22:05 EDT$C^*$–algebraic higher signatures and an invariance theorem in codimension twohttps://projecteuclid.org/euclid.gt/1538186745<strong>Nigel Higson</strong>, <strong>Thomas Schick</strong>, <strong>Zhizhang Xie</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 6, 3671--3699.</p><p><strong>Abstract:</strong><br/>
We revisit the construction of signature classes in [math] –algebra [math] –theory, and develop a variation that allows us to prove equality of signature classes in some situations involving homotopy equivalences of noncompact manifolds that are only defined outside a compact set. As an application, we prove a counterpart for signature classes of a codimension-two vanishing theorem for the index of the Dirac operator on spin manifolds (the latter is due to Hanke, Pape and Schick, and is a development of well-known work of Gromov and Lawson).
</p>projecteuclid.org/euclid.gt/1538186745_20180928220544Fri, 28 Sep 2018 22:05 EDTContractible stability spaces and faithful braid group actionshttps://projecteuclid.org/euclid.gt/1538186746<strong>Yu Qiu</strong>, <strong>Jon Woolf</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 6, 3701--3760.</p><p><strong>Abstract:</strong><br/>
We prove that any “finite-type” component of a stability space of a triangulated category is contractible. The motivating example of such a component is the stability space of the Calabi–Yau– [math] category [math] associated to an ADE Dynkin quiver. In addition to showing that this is contractible we prove that the braid group [math] acts freely upon it by spherical twists, in particular that the spherical twist group [math] is isomorphic to [math] . This generalises the result of Brav–Thomas for the [math] case. Other classes of triangulated categories with finite-type components in their stability spaces include locally finite triangulated categories with finite-rank Grothendieck group and discrete derived categories of finite global dimension.
</p>projecteuclid.org/euclid.gt/1538186746_20180928220544Fri, 28 Sep 2018 22:05 EDTParametrized spectra, multiplicative Thom spectra and the twisted Umkehr maphttps://projecteuclid.org/euclid.gt/1544756689<strong>Matthew Ando</strong>, <strong>Andrew J Blumberg</strong>, <strong>David Gepner</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 7, 3761--3825.</p><p><strong>Abstract:</strong><br/>
We introduce a general theory of parametrized objects in the setting of [math] –categories. Although parametrised spaces and spectra are the most familiar examples, we establish our theory in the generality of families of objects of a presentable [math] –category parametrized over objects of an [math] –topos. We obtain a coherent functor formalism describing the relationship of the various adjoint functors associated to base-change and symmetric monoidal structures.
Our main applications are to the study of generalized Thom spectra. We obtain fiberwise constructions of twisted Umkehr maps for twisted generalized cohomology theories using a geometric fiberwise construction of Atiyah duality. In order to characterize the algebraic structures on generalized Thom spectra and twisted (co)homology, we express the generalized Thom spectrum as a categorification of the well-known adjunction between units and group rings.
</p>projecteuclid.org/euclid.gt/1544756689_20181213220504Thu, 13 Dec 2018 22:05 ESTA Morse lemma for quasigeodesics in symmetric spaces and euclidean buildingshttps://projecteuclid.org/euclid.gt/1544756690<strong>Michael Kapovich</strong>, <strong>Bernhard Leeb</strong>, <strong>Joan Porti</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 7, 3827--3923.</p><p><strong>Abstract:</strong><br/>
We prove a Morse lemma for regular quasigeodesics in nonpositively curved symmetric spaces and euclidean buildings. We apply it to give a new coarse geometric characterization of Anosov subgroups of the isometry groups of such spaces simply as undistorted subgroups which are uniformly regular.
</p>projecteuclid.org/euclid.gt/1544756690_20181213220504Thu, 13 Dec 2018 22:05 ESTRicci flow from spaces with isolated conical singularitieshttps://projecteuclid.org/euclid.gt/1544756691<strong>Panagiotis Gianniotis</strong>, <strong>Felix Schulze</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 7, 3925--3977.</p><p><strong>Abstract:</strong><br/>
Let [math] be a compact [math] –dimensional Riemannian manifold with a finite number of singular points, where the metric is asymptotic to a nonnegatively curved cone over [math] . We show that there exists a smooth Ricci flow starting from such a metric with curvature decaying like [math] . The initial metric is attained in Gromov–Hausdorff distance and smoothly away from the singular points. In the case that the initial manifold has isolated singularities asymptotic to a nonnegatively curved cone over [math] , where [math] acts freely and properly discontinuously, we extend the above result by showing that starting from such an initial condition there exists a smooth Ricci flow with isolated orbifold singularities.
</p>projecteuclid.org/euclid.gt/1544756691_20181213220504Thu, 13 Dec 2018 22:05 ESTQuasi-projectivity of even Artin groupshttps://projecteuclid.org/euclid.gt/1544756692<strong>Rubén Blasco-García</strong>, <strong>José Ignacio Cogolludo-Agustín</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 7, 3979--4011.</p><p><strong>Abstract:</strong><br/>
Even Artin groups generalize right-angled Artin groups by allowing the labels in the defining graph to be even. We give a complete characterization of quasi-projective even Artin groups in terms of their defining graphs. Also, we show that quasi-projective even Artin groups are realizable by [math] quasi-projective spaces.
</p>projecteuclid.org/euclid.gt/1544756692_20181213220504Thu, 13 Dec 2018 22:05 ESTHigher enveloping algebrashttps://projecteuclid.org/euclid.gt/1544756693<strong>Ben Knudsen</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 7, 4013--4066.</p><p><strong>Abstract:</strong><br/>
We provide spectral Lie algebras with enveloping algebras over the operad of little [math] –framed [math] –dimensional disks for any choice of dimension [math] and structure group [math] , and we describe these objects in two complementary ways. The first description is an abstract characterization by a universal mapping property, which witnesses the higher enveloping algebra as the value of a left adjoint in an adjunction. The second, a generalization of the Poincaré–Birkhoff–Witt theorem, provides a concrete formula in terms of Lie algebra homology. Our construction pairs the theories of Koszul duality and Day convolution in order to lift to the world of higher algebra the fundamental combinatorics of Beilinson–Drinfeld’s theory of chiral algebras. Like that theory, ours is intimately linked to the geometry of configuration spaces and has the study of these spaces among its applications. We use it here to show that the stable homotopy types of configuration spaces are proper homotopy invariants.
</p>projecteuclid.org/euclid.gt/1544756693_20181213220504Thu, 13 Dec 2018 22:05 ESTVolumes of $\mathrm{SL}_n(\mathbb{C})$–representations of hyperbolic $3$–manifoldshttps://projecteuclid.org/euclid.gt/1544756694<strong>Wolfgang Pitsch</strong>, <strong>Joan Porti</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 7, 4067--4112.</p><p><strong>Abstract:</strong><br/>
Let [math] be a compact oriented three-manifold whose interior is hyperbolic of finite volume. We prove a variation formula for the volume on the variety of representations of [math] in [math] . Our proof follows the strategy of Reznikov’s rigidity when [math] is closed; in particular, we use Fuks’s approach to variations by means of Lie algebra cohomology. When [math] , we get Hodgson’s formula for variation of volume on the space of hyperbolic Dehn fillings. Our formula also recovers the variation of volume on the space of decorated triangulations obtained by Bergeron, Falbel and Guilloux and Dimofte, Gabella and Goncharov.
</p>projecteuclid.org/euclid.gt/1544756694_20181213220504Thu, 13 Dec 2018 22:05 ESTThe normal closure of big Dehn twists and plate spinning with rotating familieshttps://projecteuclid.org/euclid.gt/1544756695<strong>François Dahmani</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 7, 4113--4144.</p><p><strong>Abstract:</strong><br/>
We study the normal closure of a big power of one or several Dehn twists in a mapping class group. We prove that it has a presentation whose relators consist only of commutators between twists of disjoint support, thus answering a question of Ivanov. Our method is to use the theory of projection complexes of Bestvina, Bromberg and Fujiwara, together with the theory of rotating families, simultaneously on several spaces.
</p>projecteuclid.org/euclid.gt/1544756695_20181213220504Thu, 13 Dec 2018 22:05 ESTEndotrivial representations of finite groups and equivariant line bundles on the Brown complexhttps://projecteuclid.org/euclid.gt/1544756696<strong>Paul Balmer</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 7, 4145--4161.</p><p><strong>Abstract:</strong><br/>
We relate endotrivial representations of a finite group in characteristic [math] to equivariant line bundles on the simplicial complex of nontrivial [math] –subgroups, by means of weak homomorphisms.
</p>projecteuclid.org/euclid.gt/1544756696_20181213220504Thu, 13 Dec 2018 22:05 ESTIndicability, residual finiteness, and simple subquotients of groups acting on treeshttps://projecteuclid.org/euclid.gt/1544756697<strong>Pierre-Emmanuel Caprace</strong>, <strong>Phillip Wesolek</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 7, 4163--4204.</p><p><strong>Abstract:</strong><br/>
We establish three independent results on groups acting on trees. The first implies that a compactly generated locally compact group which acts continuously on a locally finite tree with nilpotent local action and no global fixed point is virtually indicable; that is to say, it has a finite-index subgroup which surjects onto [math] . The second ensures that irreducible cocompact lattices in a product of nondiscrete locally compact groups such that one of the factors acts vertex-transitively on a tree with a nilpotent local action cannot be residually finite. This is derived from a general result, of independent interest, on irreducible lattices in product groups. The third implies that every nondiscrete Burger–Mozes universal group of automorphisms of a tree with an arbitrary prescribed local action admits a compactly generated closed subgroup with a nondiscrete simple quotient. As applications, we answer a question of D Wise by proving the nonresidual finiteness of a certain lattice in a product of two regular trees, and we obtain a negative answer to a question of C Reid, concerning the structure theory of locally compact groups.
</p>projecteuclid.org/euclid.gt/1544756697_20181213220504Thu, 13 Dec 2018 22:05 ESTAn application of the Duistermaat–Heckman theorem and its extensions in Sasaki geometryhttps://projecteuclid.org/euclid.gt/1544756698<strong>Charles P Boyer</strong>, <strong>Hongnian Huang</strong>, <strong>Eveline Legendre</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 7, 4205--4234.</p><p><strong>Abstract:</strong><br/>
Building on an idea laid out by Martelli, Sparks and Yau (2008), we use the Duistermaat–Heckman localization formula and an extension of it to give rational and explicit expressions of the volume, the total transversal scalar curvature and the Einstein–Hilbert functional, seen as functionals on the Sasaki cone (Reeb cone). Studying the leading terms, we prove they are all proper. Among consequences thereof we get that the Einstein–Hilbert functional attains its minimal value and each Sasaki cone possesses at least one Reeb vector field with vanishing transverse Futaki invariant.
</p>projecteuclid.org/euclid.gt/1544756698_20181213220504Thu, 13 Dec 2018 22:05 ESTThe resolution of paracanonical curves of odd genushttps://projecteuclid.org/euclid.gt/1544756699<strong>Gavril Farkas</strong>, <strong>Michael Kemeny</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 7, 4235--4257.</p><p><strong>Abstract:</strong><br/>
We prove the Prym–Green conjecture on minimal free resolutions of paracanonical curves of odd genus. The proof proceeds via curves lying on ruled surfaces over an elliptic curve.
</p>projecteuclid.org/euclid.gt/1544756699_20181213220504Thu, 13 Dec 2018 22:05 ESTRigidity of Teichmüller spacehttps://projecteuclid.org/euclid.gt/1544756700<strong>Alex Eskin</strong>, <strong>Howard Masur</strong>, <strong>Kasra Rafi</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 7, 4259--4306.</p><p><strong>Abstract:</strong><br/>
We prove that every quasi-isometry of Teichmüller space equipped with the Teichmüller metric is a bounded distance from an isometry of Teichmüller space. That is, Teichmüller space is quasi-isometrically rigid.
</p>projecteuclid.org/euclid.gt/1544756700_20181213220504Thu, 13 Dec 2018 22:05 ESTStein fillings and $\mathrm{SU}(2)$ representationshttps://projecteuclid.org/euclid.gt/1544756701<strong>John A Baldwin</strong>, <strong>Steven Sivek</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 7, 4307--4380.</p><p><strong>Abstract:</strong><br/>
We recently defined invariants of contact [math] –manifolds using a version of instanton Floer homology for sutured manifolds. In this paper, we prove that if several contact structures on a [math] –manifold are induced by Stein structures on a single [math] –manifold with distinct Chern classes modulo torsion then their contact invariants in sutured instanton homology are linearly independent. As a corollary, we show that if a [math] –manifold bounds a Stein domain that is not an integer homology ball then its fundamental group admits a nontrivial homomorphism to [math] . We give several new applications of these results, proving the existence of nontrivial and irreducible [math] representations for a variety of [math] –manifold groups.
</p>projecteuclid.org/euclid.gt/1544756701_20181213220504Thu, 13 Dec 2018 22:05 ESTClassifying matchbox manifoldshttps://projecteuclid.org/euclid.gt/1552356077<strong>Alex Clark</strong>, <strong>Steven Hurder</strong>, <strong>Olga Lukina</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 1, 1--27.</p><p><strong>Abstract:</strong><br/>
Matchbox manifolds are foliated spaces with totally disconnected transversals. Two matchbox manifolds which are homeomorphic have return equivalent dynamics, so that invariants of return equivalence can be applied to distinguish nonhomeomorphic matchbox manifolds. In this work we study the problem of showing the converse implication: when does return equivalence imply homeomorphism? For the class of weak solenoidal matchbox manifolds, we show that if the base manifolds satisfy a strong form of the Borel conjecture, then return equivalence for the dynamics of their foliations implies the total spaces are homeomorphic. In particular, we show that two equicontinuous [math] –like matchbox manifolds of the same dimension are homeomorphic if and only if their corresponding restricted pseudogroups are return equivalent. At the same time, we show that these results cannot be extended to include the “ adic surfaces”, which are a class of weak solenoids fibering over a closed surface of genus [math] .
</p>projecteuclid.org/euclid.gt/1552356077_20190311220143Mon, 11 Mar 2019 22:01 EDTQuasi-asymptotically conical Calabi–Yau manifoldshttps://projecteuclid.org/euclid.gt/1552356078<strong>Ronan J Conlon</strong>, <strong>Anda Degeratu</strong>, <strong>Frédéric Rochon</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 1, 29--100.</p><p><strong>Abstract:</strong><br/>
We construct new examples of quasi-asymptotically conical ( [math] ) Calabi–Yau manifolds that are not quasi-asymptotically locally Euclidean ( [math] ). We do so by first providing a natural compactification of [math] –spaces by manifolds with fibered corners and by giving a definition of [math] –metrics in terms of an associated Lie algebra of smooth vector fields on this compactification. Thanks to this compactification and the Fredholm theory for elliptic operators on [math] –spaces developed by the second author and Mazzeo, we can in many instances obtain Kähler [math] –metrics having Ricci potential decaying sufficiently fast at infinity. This allows us to obtain [math] Calabi–Yau metrics in the Kähler classes of these metrics by solving a corresponding complex Monge–Ampère equation.
</p>projecteuclid.org/euclid.gt/1552356078_20190311220143Mon, 11 Mar 2019 22:01 EDTThe homotopy groups of the algebraic $K$–theory of the sphere spectrumhttps://projecteuclid.org/euclid.gt/1552356079<strong>Andrew J Blumberg</strong>, <strong>Michael A Mandell</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 1, 101--134.</p><p><strong>Abstract:</strong><br/>
We calculate [math] , the homotopy groups of [math] away from [math] , in terms of the homotopy groups of [math] , the homotopy groups of [math] and the homotopy groups of [math] . This builds on work of Waldhausen, who computed the rational homotopy groups (building on work of Quillen and Borel) and Rognes, who calculated the groups at odd regular primes in terms of the homotopy groups of [math] and the homotopy groups of [math] .
</p>projecteuclid.org/euclid.gt/1552356079_20190311220143Mon, 11 Mar 2019 22:01 EDTTopology of automorphism groups of parabolic geometrieshttps://projecteuclid.org/euclid.gt/1552356080<strong>Charles Frances</strong>, <strong>Karin Melnick</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 1, 135--169.</p><p><strong>Abstract:</strong><br/>
We prove for the automorphism group of an arbitrary parabolic geometry that the [math] – and [math] –topologies coincide, and the group admits the structure of a Lie group in this topology. We further show that this automorphism group is closed in the homeomorphism group of the underlying manifold.
</p>projecteuclid.org/euclid.gt/1552356080_20190311220143Mon, 11 Mar 2019 22:01 EDTRigidity of convex divisible domains in flag manifoldshttps://projecteuclid.org/euclid.gt/1552356081<strong>Wouter Van Limbeek</strong>, <strong>Andrew Zimmer</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 1, 171--240.</p><p><strong>Abstract:</strong><br/>
In contrast to the many examples of convex divisible domains in real projective space, we prove that up to projective isomorphism there is only one convex divisible domain in the Grassmannian of [math] –planes in [math] when [math] . Moreover, this convex divisible domain is a model of the symmetric space associated to the simple Lie group [math] .
</p>projecteuclid.org/euclid.gt/1552356081_20190311220143Mon, 11 Mar 2019 22:01 EDTUbiquitous quasi-Fuchsian surfaces in cusped hyperbolic $3$–manifoldshttps://projecteuclid.org/euclid.gt/1552356082<strong>Daryl Cooper</strong>, <strong>David Futer</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 1, 241--298.</p><p><strong>Abstract:</strong><br/>
We prove that every finite-volume hyperbolic [math] –manifold [math] contains a ubiquitous collection of closed, immersed, quasi-Fuchsian surfaces. These surfaces are ubiquitous in the sense that their preimages in the universal cover separate any pair of disjoint, nonasymptotic geodesic planes. The proof relies in a crucial way on the corresponding theorem of Kahn and Markovic for closed [math] –manifolds. As a corollary of this result and a companion statement about surfaces with cusps, we recover Wise’s theorem that the fundamental group of [math] acts freely and cocompactly on a [math] cube complex.
</p>projecteuclid.org/euclid.gt/1552356082_20190311220143Mon, 11 Mar 2019 22:01 EDTOperads of genus zero curves and the Grothendieck–Teichmüller grouphttps://projecteuclid.org/euclid.gt/1552356083<strong>Pedro Boavida de Brito</strong>, <strong>Geoffroy Horel</strong>, <strong>Marcy Robertson</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 1, 299--346.</p><p><strong>Abstract:</strong><br/>
We show that the group of homotopy automorphisms of the profinite completion of the genus zero surface operad is isomorphic to the (profinite) Grothendieck–Teichmüller group. Using a result of Drummond-Cole, we deduce that the Grothendieck–Teichmüller group acts nontrivially on [math] , the operad of stable curves of genus zero. As a second application, we give an alternative proof that the framed little [math] –disks operad is formal.
</p>projecteuclid.org/euclid.gt/1552356083_20190311220143Mon, 11 Mar 2019 22:01 EDTBirational models of moduli spaces of coherent sheaves on the projective planehttps://projecteuclid.org/euclid.gt/1552356084<strong>Chunyi Li</strong>, <strong>Xiaolei Zhao</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 1, 347--426.</p><p><strong>Abstract:</strong><br/>
We study the birational geometry of moduli spaces of semistable sheaves on the projective plane via Bridgeland stability conditions. We show that the entire MMP of their moduli spaces can be run via wall-crossing. Via a description of the walls, we give a numerical description of their movable cones, along with its chamber decomposition corresponding to minimal models. As an application, we show that for primitive vectors, all birational models corresponding to open chambers in the movable cone are smooth and irreducible.
</p>projecteuclid.org/euclid.gt/1552356084_20190311220143Mon, 11 Mar 2019 22:01 EDTMotivic hyper-Kähler resolution conjecture, I: Generalized Kummer varietieshttps://projecteuclid.org/euclid.gt/1552356085<strong>Lie Fu</strong>, <strong>Zhiyu Tian</strong>, <strong>Charles Vial</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 1, 427--492.</p><p><strong>Abstract:</strong><br/>
Given a smooth projective variety [math] endowed with a faithful action of a finite group [math] , following Jarvis–Kaufmann–Kimura (Invent. Math. 168 (2007) 23–81), and Fantechi–Göttsche (Duke Math. J. 117 (2003) 197–227), we define the orbifold motive (or Chen–Ruan motive) of the quotient stack [math] as an algebra object in the category of Chow motives. Inspired by Ruan (Contemp. Math. 312 (2002) 187–233), one can formulate a motivic version of his cohomological hyper-Kähler resolution conjecture (CHRC). We prove this motivic version, as well as its K–theoretic analogue conjectured by Jarvis–Kaufmann–Kimura in loc. cit., in two situations related to an abelian surface [math] and a positive integer [math] . Case (A) concerns Hilbert schemes of points of [math] : the Chow motive of [math] is isomorphic as algebra objects, up to a suitable sign change, to the orbifold motive of the quotient stack [math] . Case (B) concerns generalized Kummer varieties: the Chow motive of the generalized Kummer variety [math] is isomorphic as algebra objects, up to a suitable sign change, to the orbifold motive of the quotient stack [math] , where [math] is the kernel abelian variety of the summation map [math] . As a by-product, we prove the original cohomological hyper-Kähler resolution conjecture for generalized Kummer varieties. As an application, we provide multiplicative Chow–Künneth decompositions for Hilbert schemes of abelian surfaces and for generalized Kummer varieties. In particular, we have a multiplicative direct sum decomposition of their Chow rings with rational coefficients, which is expected to be the splitting of the conjectural Bloch–Beilinson–Murre filtration. The existence of such a splitting for holomorphic symplectic varieties is conjectured by Beauville (London Math. Soc. Lecture Note Ser. 344 (2007) 38–53). Finally, as another application, we prove that over a nonempty Zariski open subset of the base, there exists a decomposition isomorphism [math] in [math] which is compatible with the cup products on both sides, where [math] is the relative generalized Kummer variety associated to a (smooth) family of abelian surfaces [math] .
</p>projecteuclid.org/euclid.gt/1552356085_20190311220143Mon, 11 Mar 2019 22:01 EDTTowards a quantum Lefschetz hyperplane theorem in all generahttps://projecteuclid.org/euclid.gt/1552356086<strong>Honglu Fan</strong>, <strong>Yuan-Pin Lee</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 1, 493--512.</p><p><strong>Abstract:</strong><br/>
An effective algorithm of determining Gromov–Witten invariants of smooth hypersurfaces in any genus (subject to a degree bound) from Gromov–Witten invariants of the ambient space is proposed.
</p>projecteuclid.org/euclid.gt/1552356086_20190311220143Mon, 11 Mar 2019 22:01 EDT(Log-)epiperimetric inequality and regularity over smooth cones for almost area-minimizing currentshttps://projecteuclid.org/euclid.gt/1552356087<strong>Max Engelstein</strong>, <strong>Luca Spolaor</strong>, <strong>Bozhidar Velichkov</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 1, 513--540.</p><p><strong>Abstract:</strong><br/>
We prove a new logarithmic epiperimetric inequality for multiplicity-one stationary cones with isolated singularity by flowing any given trace in the radial direction along appropriately chosen directions. In contrast to previous epiperimetric inequalities for minimal surfaces (eg work of Reifenberg, Taylor and White), we need no a priori assumptions on the structure of the cone (eg integrability). If the cone is integrable (not only through rotations), we recover the classical epiperimetric inequality. As a consequence we deduce a new regularity result for almost area-minimizing currents at singular points where at least one blowup is a multiplicity-one cone with isolated singularity. This result is similar to the one for stationary varifolds of Leon Simon (1983), but independent from it since almost-minimizers do not satisfy any equation.
</p>projecteuclid.org/euclid.gt/1552356087_20190311220143Mon, 11 Mar 2019 22:01 EDT