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On Some Problems of Kobayashi and Lang
On some problems of Kobayashi and Lang Claire Voisin ii Contents 0 Introduction 3 1 The analytic setting 9 1.1 Basic results and definitions . 9 1.1.1 Definition of the pseudo-metrics/volume forms . 9 1.1.2 First properties . 11 1.1.3 Brody’s theorem, applications . 14 1.1.4 Relation between volume and metric . 18 1.2 Curvature arguments . 18 1.2.1 Hyperbolic geometry and the Ahlfors-Schwarz lemma 19 1.2.2 Another definition of the Kobayashi infinitesimal pseudo- metric . 22 1.2.3 Jet differentials . 24 1.3 Conjectures, examples . 26 1.3.1 Kobayashi’s conjecture on volumes . 26 1.3.2 Conjectures on metrics . 30 1.3.3 Campana’s construction and conjectures . 32 2 Algebraic versions, variants, Lang’s conjectures 33 2.1 Subvarieties and hyperbolicity . 33 2.1.1 Algebraic hyperbolicity . 33 2.1.2 Lang’s conjectures . 37 2.1.3 Algebraic measure-hyperbolicity . 40 2.2 K-correspondences and a variant of Kobayashi’s conjecture . 42 2.2.1 K-correspondences . 42 2.2.2 Existence theorem for K-correspondences . 45 2.2.3 A variant of the Kobayashi-Eisenman pseudo-volume form . 48 3 General hypersurfaces in projective space 51 3.1 Hyperbolicity of hypersurfaces and their complements . 51 3.1.1 Algebraic hyperbolicity . 51 iii CONTENTS 1 3.1.2 Geometry of the complement . 56 3.1.3 Kobayashi’s hyperbolicity . 58 3.2 Calabi-Yau hypersurfaces . 61 3.2.1 Rational curves on Calabi-Yau hypersurfaces . 61 3.2.2 Sweeping out a general hypersurface by subvarieties . -
Arxiv:1308.5667V3 [Math.AG] 1 Apr 2021 R-S,R Oenetgat G 11.G34.31.0023
L.Kamenova, S.Lu, M.Verbitsky Kobayashi metric on hyperk¨ahler manifolds Kobayashi pseudometric on hyperk¨ahler manifolds Ljudmila Kamenova, Steven Lu1, Misha Verbitsky2 Abstract The Kobayashi pseudometric on a complex manifold is the maximal pseudometric such that any holomorphic map from the Poincar´edisk to the manifold is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi-Yau manifolds. Using ergodicity of complex structures, we prove this conjecture for any hyperk¨ahler manifold that admits a deformation with two Lagrangian fibrations and whose Picard rank is not maximal. The Strominger-Yau-Zaslow (SYZ) conjecture claims that parabolic nef line bundles on hyperk¨ahler manifolds are semi-ample. We prove that the Kobayashi pseudomet- ric vanishes for any hyperk¨ahler manifold with b2 > 13 if the SYZ conjecture holds for all its deformations. This proves the Kobayashi conjecture for all K3 surfaces and their Hilbert schemes. Contents 1 Introduction 2 1.1 Teichm¨uller spaces and hyperk¨ahler geometry . 3 1.2 Ergodiccomplexstructures . 5 1.3 Kobayashi pseudometric/pseudodistance . 5 1.4 Upper semi-continuity . 7 2 Vanishing of the Kobayashi pseudometric 8 2.1 Kobayashi pseudometric and ergodicity . 8 2.2 Lagrangian fibrations in hyperk¨ahler geometry . 9 2.3 Kobayashi pseudometrics and Lagrangian fibrations . 11 arXiv:1308.5667v3 [math.AG] 1 Apr 2021 3 Vanishing of the infinitesimal pseudometric 13 4 Appendix on abelian fibrations 14 1Partially supported by an NSERC discovery grant 2Partially supported by RFBR grants 12-01-00944-a, NRI-HSE Academic Fund Pro- gram in 2013-2014, research grant 12-01-0179, Simons-IUM fellowship, and AG Laboratory NRI-HSE, RF government grant, ag. -
On Asymmetric Distances
Analysis and Geometry in Metric Spaces Research Article • DOI: 10.2478/agms-2013-0004 • AGMS • 2013 • 200-231 On Asymmetric Distances Abstract ∗ In this paper we discuss asymmetric length structures and Andrea C. G. Mennucci asymmetric metric spaces. A length structure induces a (semi)distance function; by using Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy the total variation formula, a (semi)distance function induces a length. In the first part we identify a topology in the set of paths that best describes when the above operations are idempo- tent. As a typical application, we consider the length of paths Received 24 March 2013 defined by a Finslerian functional in Calculus of Variations. Accepted 15 May 2013 In the second part we generalize the setting of General metric spaces of Busemann, and discuss the newly found aspects of the theory: we identify three interesting classes of paths, and compare them; we note that a geodesic segment (as defined by Busemann) is not necessarily continuous in our setting; hence we present three different notions of intrinsic metric space. Keywords Asymmetric metric • general metric • quasi metric • ostensible metric • Finsler metric • run–continuity • intrinsic metric • path metric • length structure MSC: 54C99, 54E25, 26A45 © Versita sp. z o.o. 1. Introduction Besides, one insists that the distance function be symmetric, that is, d(x; y) = d(y; x) (This unpleasantly limits many applications [...]) M. Gromov ([12], Intr.) The main purpose of this paper is to study an asymmetric metric theory; this theory naturally generalizes the metric part of Finsler Geometry, much as symmetric metric theory generalizes the metric part of Riemannian Geometry. -
Abstract Commensurability and Quasi-Isometry Classification of Hyperbolic Surface Group Amalgams
ABSTRACT COMMENSURABILITY AND QUASI-ISOMETRY CLASSIFICATION OF HYPERBOLIC SURFACE GROUP AMALGAMS EMILY STARK Abstract. Let XS denote the class of spaces homeomorphic to two closed orientable surfaces of genus greater than one identified to each other along an essential simple closed curve in each surface. Let CS denote the set of fundamental groups of spaces in XS . In this paper, we characterize the abstract commensurability classes within CS in terms of the ratio of the Euler characteristic of the surfaces identified and the topological type of the curves identified. We prove that all groups in CS are quasi-isometric by exhibiting a bilipschitz map between the universal covers of two spaces in XS . In particular, we prove that the universal covers of any two such spaces may be realized as isomorphic cell complexes with finitely many isometry types of hyperbolic polygons as cells. We analyze the abstract commensurability classes within CS : we characterize which classes contain a maximal element within CS ; we prove each abstract commensurability class contains a right-angled Coxeter group; and, we construct a common CAT(0) cubical model geometry for each abstract commensurability class. 1. Introduction Finitely generated infinite groups carry both an algebraic and a geometric structure, and to study such groups, one may study both algebraic and geometric classifications. Abstract commensurability defines an algebraic equivalence relation on the class of groups, where two groups are said to be abstractly commensurable if they contain isomorphic subgroups of finite-index. Finitely generated groups may also be viewed as geometric objects, since a finitely generated group has a natural word metric which is well-defined up to quasi-isometric equivalence. -
Abstract FINSLER METRICS with PROPERTIES of the KOBAYASHI METRIC on CONVEX DOMAINS
Publicacions Matemátiques, Vol 36 (1992), 131-155. FINSLER METRICS WITH PROPERTIES OF THE KOBAYASHI METRIC ON CONVEX DOMAINS MYUNG-YULL PANG Abstract The structure of complex Finsler manifolds is studied when the Finsler metric has the property of the Kobayashi metric on con- vex domains: (real) geodesics locally extend to complex curves (extremal disks) . lt is shown that this property of the Finsler metric induces a complex foliation of the cotangent space closely related to geodesics. Each geodesic of the metric is then shown to have a unique extension to a maximal totally geodesic complex curve E which has,properties of extremal disks. Under the addi- tional conditions that the metric is complete and the holomorphic sectional curvature is -4, E coincides with an extrema¡ disk and a theorem of Faran is recovered: the Finsler metric coincides with the Kobayashi metric. 1. Introduction The Riemann mapping theorem says that all simply connected do- mains in C, different from C are biholomorphically equivalent . It is a well known fact that this theorem does not hold for domains in T' for n > 1, and the classification of bounded domains up to biholomorphism has been an important problem in several complex variables. One approach to understanding the structure of bounded domains is to study biholo- morphically invariant métrics such as the Kobayashi or Carathéodory metrics [K]] [BD] [L3] [Pa] . In [L1] and [L2], Lempert showed that these metrics are extremely well-behaved in the special case when the domain is strictly linearly convex and has smooth boundary: In this case, the two metrics coincide, and the infinitesimal form FK of the Kobayashi metric falls into a special class of smooth Finsler metrics with constant holomorphic sectional curvature K = -4. -
Isometric Actions on Spheres with an Orbifold Quotient
ISOMETRIC ACTIONS ON SPHERES WITH AN ORBIFOLD QUOTIENT CLAUDIO GORODSKI AND ALEXANDER LYTCHAK Abstract. We classify representations of compact connected Lie groups whose induced action on the unit sphere has an orbit space isometric to a Riemannian orbifold. 1. Introduction In this paper we classify all orthogonal representations of compact connected groups G on Euclidean unit spheres Sn for which the quotient Sn/G is a Riemannian orbifold. We are going to call such representations infinitesimally polar, since they can be equivalently defined by the condition that slice representations at all non-zero vectors are polar. The interest in such representations is two-fold. On the one hand, one can hope to isolate an interesting class of representations in this way. This class contains all polar representations, which can be defined by the property that the corresponding quotient space Sn/G is an orbifold of constant curvature 1. Polar representations very often play an important and special role in geometric questions concern- ing representations (cf. [PT88, BCO03]), and the class investigated in this paper consists of closest relatives of polar representations. On the other hand, any such quotient has positive curvature and all geodesics in the quotient are closed. Any of these two properties is extremely interesting and in both classes the number of known manifold examples is very limited (cf. [Zil12, Bes78]). We have hoped to obtain new examples of positively curved manifolds and of manifolds with closed geodesics as universal orbi-covering of such spaces. Should the orbifolds be bad one could still hope to obtain new examples of interesting orbifolds. -
Colimit Theorems for Coarse Coherence with Applications
COLIMIT THEOREMS FOR COARSE COHERENCE WITH APPLICATIONS BORIS GOLDFARB AND JONATHAN L. GROSSMAN Abstract. We establish two versions of a central theorem, the Family Colimit Theorem, for the coarse coherence property of metric spaces. This is a coarse geometric property and so is well-defined for finitely generated groups with word metrics. It is known that coarse coherence of the fundamental group has important implications for classification of high-dimensional manifolds. The Family Colimit Theorem is one of the permanence theorems that give struc- ture to the class of coarsely coherent groups. In fact, all known permanence theorems follow from the Family Colimit Theorem. We also use this theorem to construct new groups from this class. 1. Introduction Let Γ be a finitely generated group. Coarse coherence is a property of the group viewed as a metric space with a word metric, defined in terms of algebraic proper- ties invariant under quasi-isometries. It has emerged in [1, 6] as a condition that guarantees an equivalence between the K-theory of a group ring K(R[Γ]) and its G-theory G(R[Γ]). The K-theory is the central home for obstructions to a num- ber of constructions in geometric topology and number theory. We view K(R[Γ]) as a non-connective spectrum associated to a commutative ring R and the group Γ whose stable homotopy groups are the Quillen K-groups in non-negative dimen- sions and the negative K-groups of Bass in negative dimensions. The corresponding G-theory, introduced in this generality in [2], is well-known as a theory with better computational tools available compared to K-theory. -
ON BI-INVARIANT WORD METRICS 1. Introduction 1.1. the Result Let OV
June 28, 2011 11:19 WSPC/243-JTA S1793525311000556 Journal of Topology and Analysis Vol. 3, No. 2 (2011) 161–175 c World Scientific Publishing Company DOI: 10.1142/S1793525311000556 ON BI-INVARIANT WORD METRICS SWIATOS´ LAW R. GAL∗ and JAREK KEDRA† ∗University of Wroclaw, Poland and Universit¨at Wien, Austria †University of Aberdeen, UK and University of Szczecin ∗[email protected] †[email protected] Received 7 February 2011 We prove that bi-invariant word metrics are bounded on certain Chevalley groups. As an application we provide restrictions on Hamiltonian actions of such groups. Keywords: Bi-invariant metric; Hamiltonian actions; Chevalley group. 1. Introduction 1.1. The result Let OV ⊂ K be a ring of V-integers in a number field K,whereV is a set of valuations containing all Archimedean ones. Let Gπ(Φ, OV) be the Chevalley group associated with a faithful representation π : g → gl(V ) of a simple complex Lie algebra of rank at least two (see Sec. 4.1 for details). Theorem 1.1. Let Γ be a finite extension or a supergroup of finite index of the Chevalley group Gπ(Φ, OV). Then any bi-invariant metric on Γ is bounded. The examples to which Theorem 1.1 applies include the following groups. (1) SL(n; Z);√ it is a non-uniform lattice in SL(n; R). (2) SL(n; Z[ 2]); the image of its diagonal embedding into the product SL(n; R) × SL(n; R) is a non-uniform lattice. (3) SO(n; Z):={A ∈ SL(n, Z) | AJAT = J}, where J is the matrix with ones on the anti-diagonal and zeros elsewhere. -
Integrating We’Re Actually So ) Z ( = D 0 R = ) Integrating − Dt T Log + Ihe Mcquillan Michael Z ∆( Esnsformula
ICM 2002 · Vol. I · 547–554 Integrating ∂∂ Michael McQuillan∗ Abstract We consider the algebro-geometric consequences of integration by parts. 2000 Mathematics Subject Classification: 32, 14. Keywords and Phrases: Jensen’s formula. 1. Jensen’s formula Recall that for a suitably regular function ϕ on the unit disc ∆ we can apply integration by parts/Stoke’s formula twice to obtain for r< 1, r dt ddcϕ = ϕ − ϕ(0) (1.1) t Z0 Z∆(t) Z∂∆(r) c 1 1 where d = 4πi (∂ − ∂) so actually we’re integrating 2πi ∂∂. In the presence of singularities things continue to work. For example suppose f : ∆ → X is a holo- morphic map of complex spaces and D a metricised effective Cartier divisor on X, ∗ with f(0) ∈/ D, and ϕ = − log f k1IDk, where 1ID ∈ OX (D) is the tautological section, then we obtain, r dt f ∗c (D) = − log kf ∗1I k + log kf ∗1I k(0) t 1 D D Z0 Z∆(t) Z∂∆(r) (1.2) ∗ r arXiv:math/0212409v1 [math.AG] 1 Dec 2002 + ord (f D) log . z z 0<|z|<r X Obviously it’s not difficult to write down similar formulae for not necessarily effective Cartier divisors, meromorphic functions, drop the condition that f(0) ∈/ D provided f(∆) 6⊂ D, extend to ramified covers p : Y → ∆, etc., but in all cases what is clear is, ∗Institut des Hautes Etudes Scientifiques, 35 route de Chartres, 91440 Bures-sur-Yvette, France and Department of Mathematics, University of Glasgow G128QW, Scotland. E-mail: mquil- [email protected] 548 Michael McQuillan r dt ∗ Facts 1.3. -
Math 395: Category Theory Northwestern University, Lecture Notes
Math 395: Category Theory Northwestern University, Lecture Notes Written by Santiago Can˜ez These are lecture notes for an undergraduate seminar covering Category Theory, taught by the author at Northwestern University. The book we roughly follow is “Category Theory in Context” by Emily Riehl. These notes outline the specific approach we’re taking in terms the order in which topics are presented and what from the book we actually emphasize. We also include things we look at in class which aren’t in the book, but otherwise various standard definitions and examples are left to the book. Watch out for typos! Comments and suggestions are welcome. Contents Introduction to Categories 1 Special Morphisms, Products 3 Coproducts, Opposite Categories 7 Functors, Fullness and Faithfulness 9 Coproduct Examples, Concreteness 12 Natural Isomorphisms, Representability 14 More Representable Examples 17 Equivalences between Categories 19 Yoneda Lemma, Functors as Objects 21 Equalizers and Coequalizers 25 Some Functor Properties, An Equivalence Example 28 Segal’s Category, Coequalizer Examples 29 Limits and Colimits 29 More on Limits/Colimits 29 More Limit/Colimit Examples 30 Continuous Functors, Adjoints 30 Limits as Equalizers, Sheaves 30 Fun with Squares, Pullback Examples 30 More Adjoint Examples 30 Stone-Cech 30 Group and Monoid Objects 30 Monads 30 Algebras 30 Ultrafilters 30 Introduction to Categories Category theory provides a framework through which we can relate a construction/fact in one area of mathematics to a construction/fact in another. The goal is an ultimate form of abstraction, where we can truly single out what about a given problem is specific to that problem, and what is a reflection of a more general phenomenom which appears elsewhere. -
Moduli Spaces of Riemann Surfaces
MODULI SPACES OF RIEMANN SURFACES ALEX WRIGHT Contents 1. Different ways to build and deform Riemann surfaces 2 2. Teichm¨uller space and moduli space 4 3. The Deligne-Mumford compactification of moduli space 11 4. Jacobians, Hodge structures, and the period mapping 13 5. Quasiconformal maps 16 6. The Teichm¨ullermetric 18 7. Extremal length 24 8. Nielsen-Thurston classification of mapping classes 26 9. Teichm¨uller space is a bounded domain 29 10. The Weil-Petersson K¨ahlerstructure 40 11. Kobayashi hyperbolicity 46 12. Geometric Shafarevich and Mordell 48 References 57 These are course notes for a one quarter long second year graduate class at Stanford University taught in Winter 2017. All of the material presented is classical. For the softer aspects of Teichm¨ullertheory, our reference was the book of Farb-Margalit [FM12], and for Teichm¨ullertheory as a whole our comprehensive ref- erence was the book of Hubbard [Hub06]. We also used Gardiner and Lakic's book [GL00] and McMullen's course notes [McM] as supple- mental references. For period mappings our reference was the book [CMSP03]. For Geometric Shafarevich our reference was McMullen's survey [McM00]. The author thanks Steve Kerckhoff, Scott Wolpert, and especially Jeremy Kahn for many helpful conversations. These are rough notes, hastily compiled, and may contain errors. Corrections are especially welcome, since these notes will be re-used in the future. 2 WRIGHT 1. Different ways to build and deform Riemann surfaces 1.1. Uniformization. A Riemann surface is a topological space with an atlas of charts to the complex plane C whose translation functions are biholomorphisms. -
Geometrization of Three-Dimensional Orbifolds Via Ricci Flow
GEOMETRIZATION OF THREE-DIMENSIONAL ORBIFOLDS VIA RICCI FLOW BRUCE KLEINER AND JOHN LOTT Abstract. A three-dimensional closed orientable orbifold (with no bad suborbifolds) is known to have a geometric decomposition from work of Perelman [50, 51] in the manifold case, along with earlier work of Boileau-Leeb-Porti [4], Boileau-Maillot-Porti [5], Boileau- Porti [6], Cooper-Hodgson-Kerckhoff [19] and Thurston [59]. We give a new, logically independent, unified proof of the geometrization of orbifolds, using Ricci flow. Along the way we develop some tools for the geometry of orbifolds that may be of independent interest. Contents 1. Introduction 3 1.1. Orbifolds and geometrization 3 1.2. Discussion of the proof 3 1.3. Organization of the paper 4 1.4. Acknowledgements 5 2. Orbifold topology and geometry 5 2.1. Differential topology of orbifolds 5 2.2. Universal cover and fundamental group 10 2.3. Low-dimensional orbifolds 12 2.4. Seifert 3-orbifolds 17 2.5. Riemannian geometry of orbifolds 17 2.6. Critical point theory for distance functions 20 2.7. Smoothing functions 21 3. Noncompact nonnegatively curved orbifolds 22 3.1. Splitting theorem 23 3.2. Cheeger-Gromoll-type theorem 23 3.3. Ruling out tight necks in nonnegatively curved orbifolds 25 3.4. Nonnegatively curved 2-orbifolds 27 Date: May 28, 2014. Research supported by NSF grants DMS-0903076 and DMS-1007508. 1 2 BRUCE KLEINER AND JOHN LOTT 3.5. Noncompact nonnegatively curved 3-orbifolds 27 3.6. 2-dimensional nonnegatively curved orbifolds that are pointed Gromov- Hausdorff close to an interval 28 4.