Strictly Nef Vector Bundles and Characterizations of P
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REMARKS on the ABUNDANCE CONJECTURE 3 Log flips for Log Canonical Pairs Is Known for All Dimensions (Cf
REMARKS ON THE ABUNDANCE CONJECTURE KENTA HASHIZUME Abstract. We prove the abundance theorem for log canonical n- folds such that the boundary divisor is big assuming the abundance conjecture for log canonical (n − 1)-folds. We also discuss the log minimal model program for log canonical 4-folds. Contents 1. Introduction 1 2. Notations and definitions 3 3. Proof of the main theorem 5 4. Minimal model program in dimension four 9 References 10 1. Introduction One of the most important open problems in the minimal model theory for higher-dimensional algebraic varieties is the abundance con- jecture. The three-dimensional case of the above conjecture was com- pletely solved (cf. [KeMM] for log canonical threefolds and [F1] for semi log canonical threefolds). However, Conjecture 1.1 is still open in dimension ≥ 4. In this paper, we deal with the abundance conjecture in relative setting. arXiv:1509.04626v2 [math.AG] 3 Nov 2015 Conjecture 1.1 (Relative abundance). Let π : X → U be a projective morphism of varieties and (X, ∆) be a (semi) log canonical pair. If KX + ∆ is π-nef, then it is π-semi-ample. Hacon and Xu [HX1] proved that Conjecture 1.1 for log canonical pairs and Conjecture 1.1 for semi log canonical pairs are equivalent (see also [FG]). If (X, ∆) is Kawamata log terminal and ∆ is big, then Date: 2015/10/30, version 0.21. 2010 Mathematics Subject Classification. Primary 14E30; Secondary 14J35. Key words and phrases. abundance theorem, big boundary divisor, good minimal model, finite generation of adjoint ring. 1 2 KENTAHASHIZUME Conjecture 1.1 follows from the usual Kawamata–Shokurov base point free theorem in any dimension. -
Positivity of Hodge Bundles of Abelian Varieties Over Some Function Fields
Positivity of Hodge bundles of abelian varieties over some function fields Xinyi Yuan August 31, 2018 Contents 1 Introduction2 1.1 Positivity of Hodge bundle....................2 1.2 Purely inseparable points.....................4 1.3 Partial finiteness of Tate{Shafarevich group..........5 1.4 Reduction of Tate conjecture...................6 1.5 Idea of proofs...........................7 1.6 Notation and terminology.................... 10 2 Positivity of Hodge bundle 13 2.1 Group schemes of constant type................. 13 2.2 The quotient process....................... 18 2.3 Control by heights........................ 23 2.4 Lifting p-divisible groups..................... 27 3 Purely inseparable points on torsors 31 3.1 Preliminary results on torsors.................. 31 3.2 Purely inseparable points..................... 34 4 Reduction of the Tate conjecture 40 4.1 Preliminary results........................ 41 4.2 Reduction of the Tate conjecture................ 44 1 1 Introduction Given an abelian variety A over the rational function field K = k(t) of a finite field k, we prove the following results: (1) A is isogenous to the product of a constant abelian variety over K and 1 an abelian variety over K whose N´eronmodel over Pk has an ample Hodge bundle. (2) finite generation of the abelian group A(Kper) if A has semi-abelian 1 reduction over Pk, as part of the \full" Mordell{Lang conjecture for A over K; (3) finiteness of the abelian group X(A)[F 1], the subgroup of elements of the Tate{Shafarevich group X(A) annihilated by iterations of the relative Frobenius homomorphisms, if A has semi-abelian reduction 1 over Pk; (4) the Tate conjecture for all projective and smooth surfaces X over finite 1 fields with H (X; OX ) = 0 implies the Tate conjecture for all projective and smooth surfaces over finite fields. -
Arxiv:1307.5568V2 [Math.AG]
PARTIAL POSITIVITY: GEOMETRY AND COHOMOLOGY OF q-AMPLE LINE BUNDLES DANIEL GREB AND ALEX KURONYA¨ To Rob Lazarsfeld on the occasion of his 60th birthday Abstract. We give an overview of partial positivity conditions for line bundles, mostly from a cohomological point of view. Although the current work is to a large extent of expository nature, we present some minor improvements over the existing literature and a new result: a Kodaira-type vanishing theorem for effective q-ample Du Bois divisors and log canonical pairs. Contents 1. Introduction 1 2. Overview of the theory of q-ample line bundles 4 2.1. Vanishing of cohomology groups and partial ampleness 4 2.2. Basic properties of q-ampleness 7 2.3. Sommese’s geometric q-ampleness 15 2.4. Ample subschemes, and a Lefschetz hyperplane theorem for q-ample divisors 17 3. q-Kodaira vanishing for Du Bois divisors and log canonical pairs 19 References 23 1. Introduction Ampleness is one of the central notions of algebraic geometry, possessing the extremely useful feature that it has geometric, numerical, and cohomological characterizations. Here we will concentrate on its cohomological side. The fundamental result in this direction is the theorem of Cartan–Serre–Grothendieck (see [Laz04, Theorem 1.2.6]): for a complete arXiv:1307.5568v2 [math.AG] 23 Jan 2014 projective scheme X, and a line bundle L on X, the following are equivalent to L being ample: ⊗m (1) There exists a positive integer m0 = m0(X, L) such that L is very ample for all m ≥ m0. (2) For every coherent sheaf F on X, there exists a positive integer m1 = m1(X, F, L) ⊗m for which F ⊗ L is globally generated for all m ≥ m1. -
Ample Line Bundles, Global Generation and $ K 0 $ on Quasi-Projective
Ample line bundles, global generation and K0 on quasi-projective derived schemes Toni Annala Abstract The purpose of this article is to extend some classical results on quasi-projective schemes to the setting of derived algebraic geometry. Namely, we want to show that any vector bundle on a derived scheme admitting an ample line bundle can be twisted to be globally generated. Moreover, we provide a presentation of K0(X) as the Grothendieck group of vector bundles modulo exact sequences on any quasi- projective derived scheme X. Contents 1 Introduction 2 2 Background 2 2.1 ∞-Categories .................................. 2 2.2 DerivedSchemes ................................ 4 2.3 Quasi-CoherentSheaves . 5 3 Strong Sheaves 6 4 Ample Line Bundles 9 arXiv:1902.08331v2 [math.AG] 26 Nov 2019 4.1 The Descent Spectral Sequence . .. 9 4.2 TwistingSheaves ................................ 10 4.3 RelativeAmpleness ............................... 12 5 K0 of Derived Schemes 14 1 1 Introduction The purpose of this paper is to give proofs for some general facts concerning quasi- projective derived schemes stated and used in [An]. In that paper, the author constructed a new extension of algebraic cobordism from smooth quasi-projective schemes to all quasi- projective (derived) schemes. The main result of the paper is that the newly obtained extension specializes to K0(X) – the 0th algebraic K-theory of the scheme X, giving strong evidence that the new extension is the correct one. All previous extensions (operational, A1-homotopic) fail to have this relation with the Grothendieck ring K0. A crucial role in the proof is played by the construction of Chern classes for Ω∗. -
Euler Sequence and Koszul Complex of a Module
Ark. Mat. DOI: 10.1007/s11512-016-0236-4 c 2016 by Institut Mittag-Leffler. All rights reserved Euler sequence and Koszul complex of a module Bj¨orn Andreas, Dar´ıo S´anchez G´omez and Fernando Sancho de Salas Abstract. We construct relative and global Euler sequences of a module. We apply it to prove some acyclicity results of the Koszul complex of a module and to compute the cohomology of the sheaves of (relative and absolute) differential p-forms of a projective bundle. In particular we generalize Bott’s formula for the projective space to a projective bundle over a scheme of characteristic zero. Introduction This paper deals with two related questions: the acyclicity of the Koszul com- plex of a module and the cohomology of the sheaves of (relative and absolute) differential p-forms of a projective bundle over a scheme. Let M be a module over a commutative ring A. One has the Koszul complex · ⊗ · · · Kos(M)=Λ M A S M,whereΛM and S M stand for the exterior and symmetric algebras of M. It is a graded complex Kos(M)= n≥0 Kos(M)n,whosen-th graded component Kos(M)n is the complex: 0 −→ ΛnM −→ Λn−1M ⊗M −→ Λn−2M ⊗S2M −→ ... −→ SnM −→ 0 It has been known for many years that Kos(M)n is acyclic for n>0, provided that M is a flat A-module or n is invertible in A (see [3]or[10]). It was conjectured in [11]thatKos(M) is always acyclic. A counterexample in characteristic 2 was given in [5], but it is also proved there that Hμ(Kos(M)μ)=0 for any M,whereμ is the minimal number of generators of M. -
Compact Complex Manifolds with Numerically Effective Tangent Bundles
COMPACT COMPLEX MANIFOLDS WITH NUMERICALLY EFFECTIVE TANGENT BUNDLES Jean-Pierre Demailly⋆, Thomas Peternell⋆⋆, Michael Schneider⋆⋆ ⋆ Universit´ede Grenoble I ⋆⋆ Universit¨at Bayreuth Institut Fourier, BP 74 Mathematisches Institut U.R.A.188duC.N.R.S. Postfach101251 38402 Saint-Martin d’H`eres, France D-8580 Bayreuth, Deutschland Contents 0. Introduction................................................... ........................p. 2 1. Basic properties of nef line bundles ................................................... p. 5 1.A. Nef line bundles ................................................... ............... p. 5 1.B. Nef vector bundles................................................... ............p. 11 2. Inequalities for Chern classes ................................................... ...... p. 18 3. Compact manifolds with nef tangent bundles. Structure of the Albanese map..........p. 22 3.A. Some examples ................................................... ............... p. 22 3.B. General properties ................................................... ............ p. 23 3.C. Structure of the Albanese map (K¨ahler case) .................................... p. 25 3.D. Numerical flatness of the Albanese map ......................................... p. 32 4. Moishezon manifolds with nef tangent bundles ........................................ p. 36 5. Two structure theorems ................................................... ........... p. 37 6. Surfaces with nef tangent bundles .................................................. -
Koszul Cohomology and Algebraic Geometry Marian Aprodu Jan Nagel
Koszul Cohomology and Algebraic Geometry Marian Aprodu Jan Nagel Institute of Mathematics “Simion Stoilow” of the Romanian Acad- emy, P.O. Box 1-764, 014700 Bucharest, ROMANIA & S¸coala Normala˘ Superioara˘ Bucures¸ti, 21, Calea Grivit¸ei, 010702 Bucharest, Sector 1 ROMANIA E-mail address: [email protected] Universite´ Lille 1, Mathematiques´ – Bat.ˆ M2, 59655 Villeneuve dAscq Cedex, FRANCE E-mail address: [email protected] 2000 Mathematics Subject Classification. Primary 14H51, 14C20, 14F99, 13D02 Contents Introduction vii Chapter 1. Basic definitions 1 1.1. The Koszul complex 1 1.2. Definitions in the algebraic context 2 1.3. Minimal resolutions 3 1.4. Definitions in the geometric context 5 1.5. Functorial properties 6 1.6. Notes and comments 10 Chapter 2. Basic results 11 2.1. Kernel bundles 11 2.2. Projections and linear sections 12 2.3. Duality 17 2.4. Koszul cohomology versus usual cohomology 19 2.5. Sheaf regularity. 21 2.6. Vanishing theorems 22 Chapter 3. Syzygy schemes 25 3.1. Basic definitions 25 3.2. Koszul classes of low rank 32 3.3. The Kp,1 theorem 34 3.4. Rank-2 bundles and Koszul classes 38 3.5. The curve case 41 3.6. Notes and comments 45 Chapter 4. The conjectures of Green and Green–Lazarsfeld 47 4.1. Brill-Noether theory 47 4.2. Numerical invariants of curves 49 4.3. Statement of the conjectures 51 4.4. Generalizations of the Green conjecture. 54 4.5. Notes and comments 57 Chapter 5. Koszul cohomology and the Hilbert scheme 59 5.1. -
Positivity in Algebraic Geometry I
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 48 Positivity in Algebraic Geometry I Classical Setting: Line Bundles and Linear Series Bearbeitet von R.K. Lazarsfeld 1. Auflage 2004. Buch. xviii, 387 S. Hardcover ISBN 978 3 540 22533 1 Format (B x L): 15,5 x 23,5 cm Gewicht: 1650 g Weitere Fachgebiete > Mathematik > Geometrie > Elementare Geometrie: Allgemeines Zu Inhaltsverzeichnis schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, eBooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte. Introduction to Part One Linear series have long stood at the center of algebraic geometry. Systems of divisors were employed classically to study and define invariants of pro- jective varieties, and it was recognized that varieties share many properties with their hyperplane sections. The classical picture was greatly clarified by the revolutionary new ideas that entered the field starting in the 1950s. To begin with, Serre’s great paper [530], along with the work of Kodaira (e.g. [353]), brought into focus the importance of amplitude for line bundles. By the mid 1960s a very beautiful theory was in place, showing that one could recognize positivity geometrically, cohomologically, or numerically. During the same years, Zariski and others began to investigate the more complicated be- havior of linear series defined by line bundles that may not be ample. -
Big Q-Ample Line Bundles
COMPOSITIO MATHEMATICA Big q-ample line bundles Morgan V. Brown Compositio Math. 148 (2012), 790{798. doi:10.1112/S0010437X11007457 FOUNDATION COMPOSITIO MATHEMATICA Downloaded from https://www.cambridge.org/core. IP address: 170.106.202.126, on 27 Sep 2021 at 16:49:00, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1112/S0010437X11007457 Compositio Math. 148 (2012) 790{798 doi:10.1112/S0010437X11007457 Big q-ample line bundles Morgan V. Brown Abstract A recent paper of Totaro developed a theory of q-ample bundles in characteristic 0. Specifically, a line bundle L on X is q-ample if for every coherent sheaf F on X, there i exists an integer m0 such that m > m0 implies H (X; F ⊗ O(mL)) = 0 for i > q. We show that a line bundle L on a complex projective scheme X is q-ample if and only if the restriction of L to its augmented base locus is q-ample. In particular, when X is a variety and L is big but fails to be q-ample, then there exists a codimension-one subscheme D of X such that the restriction of L to D is not q-ample. 1. Introduction A recent paper of Totaro [Tot10] generalized the notion of an ample line bundle, with the object of relating cohomological, numerical, and geometric properties of these line bundles. Let q be a natural number. Totaro called a line bundle L on X q-ample if for every coherent sheaf F on X, i there exists an integer m0 such that m > m0 implies H (X; F ⊗ O(mL)) = 0 for i > q. -
Moving Codimension-One Subvarieties Over Finite Fields
Moving codimension-one subvarieties over finite fields Burt Totaro In topology, the normal bundle of a submanifold determines a neighborhood of the submanifold up to isomorphism. In particular, the normal bundle of a codimension-one submanifold is trivial if and only if the submanifold can be moved in a family of disjoint submanifolds. In algebraic geometry, however, there are higher-order obstructions to moving a given subvariety. In this paper, we develop an obstruction theory, in the spirit of homotopy theory, which gives some control over when a codimension-one subvariety moves in a family of disjoint subvarieties. Even if a subvariety does not move in a family, some positive multiple of it may. We find a pattern linking the infinitely many obstructions to moving higher and higher multiples of a given subvariety. As an application, we find the first examples of line bundles L on smooth projective varieties over finite fields which are nef (L has nonnegative degree on every curve) but not semi-ample (no positive power of L is spanned by its global sections). This answers questions by Keel and Mumford. Determining which line bundles are spanned by their global sections, or more generally are semi-ample, is a fundamental issue in algebraic geometry. If a line bundle L is semi-ample, then the powers of L determine a morphism from the given variety onto some projective variety. One of the main problems of the minimal model program, the abundance conjecture, predicts that a variety with nef canonical bundle should have semi-ample canonical bundle [15, Conjecture 3.12]. -
Tensor Categorical Foundations of Algebraic Geometry
Tensor categorical foundations of algebraic geometry Martin Brandenburg { 2014 { Abstract Tannaka duality and its extensions by Lurie, Sch¨appi et al. reveal that many schemes as well as algebraic stacks may be identified with their tensor categories of quasi-coherent sheaves. In this thesis we study constructions of cocomplete tensor categories (resp. cocontinuous tensor functors) which usually correspond to constructions of schemes (resp. their morphisms) in the case of quasi-coherent sheaves. This means to globalize the usual local-global algebraic geometry. For this we first have to develop basic commutative algebra in an arbitrary cocom- plete tensor category. We then discuss tensor categorical globalizations of affine morphisms, projective morphisms, immersions, classical projective embeddings (Segre, Pl¨ucker, Veronese), blow-ups, fiber products, classifying stacks and finally tangent bundles. It turns out that the universal properties of several moduli spaces or stacks translate to the corresponding tensor categories. arXiv:1410.1716v1 [math.AG] 7 Oct 2014 This is a slightly expanded version of the author's PhD thesis. Contents 1 Introduction 1 1.1 Background . .1 1.2 Results . .3 1.3 Acknowledgements . 13 2 Preliminaries 14 2.1 Category theory . 14 2.2 Algebraic geometry . 17 2.3 Local Presentability . 21 2.4 Density and Adams stacks . 22 2.5 Extension result . 27 3 Introduction to cocomplete tensor categories 36 3.1 Definitions and examples . 36 3.2 Categorification . 43 3.3 Element notation . 46 3.4 Adjunction between stacks and cocomplete tensor categories . 49 4 Commutative algebra in a cocomplete tensor category 53 4.1 Algebras and modules . 53 4.2 Ideals and affine schemes . -
Algebraic Surfaces and Hyperbolic Geometry
Current Developments in Algebraic Geometry MSRI Publications Volume 59, 2011 Algebraic surfaces and hyperbolic geometry BURT TOTARO We describe the Kawamata–Morrison cone conjecture on the structure of Calabi–Yau varieties and more generally klt Calabi–Yau pairs. The conjecture is true in dimension 2. We also show that the automorphism group of a K3 surface need not be commensurable with an arithmetic group, which answers a question by Mazur. 1. Introduction Many properties of a projective algebraic variety can be encoded by convex cones, such as the ample cone and the cone of curves. This is especially useful when these cones have only finitely many edges, as happens for Fano varieties. For a broader class of varieties which includes Calabi–Yau varieties and many rationally connected varieties, the Kawamata–Morrison cone conjecture predicts the structure of these cones. I like to think of this conjecture as what comes after the abundance conjecture. Roughly speaking, the cone theorem of Mori, Kawamata, Shokurov, Kollár, and Reid describes the structure of the curves on a projective variety X on which the canonical bundle K X has negative degree; the abundance conjecture would give strong information about the curves on which K X has degree zero; and the cone conjecture fully describes the structure of the curves on which K X has degree zero. We give a gentle summary of the proof of the cone conjecture for algebraic surfaces, with plenty of examples [Totaro 2010]. For algebraic surfaces, these cones are naturally described using hyperbolic geometry, and the proof can also be formulated in those terms.