DOCSLIB.ORG

On Generalised Abundance, II

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

ON GENERALISED ABUNDANCE, II VLADIMIR LAZIC´ AND THOMAS PETERNELL Abstract. In our previous work, we introduced the Generalised Non- vanishing Conjecture, which generalises several central conjectures in algebraic geometry. In this paper, we derive some surprising nonvanish- ing results for pluricanonical bundles which were not predicted by the Minimal Model Program, by making progress towards the Generalised Nonvanishing Conjecture in every dimension. The main step is to estab- lish that a somewhat stronger version of the Generalised Nonvanishing Conjecture holds almost always in the presence of metrics with gener- alised algebraic singularities, assuming the Minimal Model Program in lower dimensions. Contents 1. Introduction 1 2. Preliminaries 5 3. Reduction to the Weak Generalised Nonvanishing Conjecture 12 4. A criterion for Nonvanishing 22 5. Metrics with generalised algebraic singularities, I 28 6. Metrics with generalised algebraic singularities, II 31 7. Numerical dimension 1 39 8. Maps to abelian varieties 40 References 41 1. Introduction arXiv:1809.02500v4 [math.AG] 7 May 2019 In this paper we study the Generalised Nonvanishing Conjecture in every dimension, and derive some surprising nonvanishing results for pluricanoni- cal bundles which were not predicted by the Minimal Model Program. Generalised Nonvanishing Conjecture. Let (X, ∆) be a klt pair such that KX +∆ is pseudoeffective. Let L be a nef Q-divisor on X. Then for Lazi´cwas supported by the DFG-Emmy-Noether-Nachwuchsgruppe “Gute Strukturen in der h¨oherdimensionalen birationalen Geometrie”. Peternell was supported by the DFG grant “Zur Positivit¨at in der komplexen Geometrie”. We would like to thank S. Schreieder for asking a question about the paper [LP18a] at a seminar in Munich in December 2017 which motivated this paper, and to J.-P. Demailly for useful discussions. 2010 Mathematics Subject Classification: 14E30. Keywords: abundance conjecture, Minimal Model Program. 1 2 VLADIMIR LAZIC´ AND THOMAS PETERNELL every t ≥ 0 the numerical class of the divisor KX +∆+ tL belongs to the effective cone. If the numerical class of a divisor belongs to the effective cone, then we call the divisor num-effective, see Section 2. The Generalised Nonvanishing Conjecture as well as the Generalised Abundance Conjecture were intro- duced in our paper [LP18b] as generalisations of several central conjectures in algebraic geometry. This conjecture contains as special cases the Nonvanishing Conjecture – often viewed as part of the Abundance Conjecture – and the nonvanishing part of the Semiampleness Conjecture (for nef line bundle on varieties with numerically trivial canonical class). 1.1. Numerical versus linear equivalence. Constructing many non-tri- vial effective or basepoint free divisors on a projective variety is one of the main goals of the Minimal Model Program. Assume, for instance, that (X, ∆) is a projective klt pair such that KX + ∆ is semiample. It is a basic question to determine which Q-divisors D in the numerical class of KX +∆ are effective (that is, κ(X, D) ≥ 0) or semiample. If the numerical and Q- 1 linear equivalence on X coincide, i.e. if H (X, OX ) = 0, then all such D are trivially effective, respectively semiample, and one would expect that this is the only case where such behaviour happens. Our first main result is that, surprisingly, such behaviour happens al- most always. More precisely, whether each such D is effective, respectively semiample, depends not (only) on the vanishing of the cohomology group 1 H (X, OX ), but on the behaviour of the Euler-Poincar´echaracteristic of the structure sheaf of X. We obtain the following unconditional results in dimensions at most 3, which were previously not even conjectured. Theorem A. Let (X, ∆) be a projective klt pair of dimension at most 3 such that KX +∆ is pseudoeffective. Assume that χ(X, OX ) 6= 0. (i) Then for every Q-divisor G with KX +∆ ≡ G we have κ(X, G) ≥ 0. (ii) If KX +∆ is semiample, then every Q-divisor G with KX +∆ ≡ G is semiample. Theorem A is a special case of Corollaries 6.7 and 6.8 below. Another instance where our results hold unconditionally in every dimen- sion is the case where the numerical dimension is 1; this is the first highly nontrivial case beyond those when the numerical dimension is 0 or maximal. Theorem B. Let (X, ∆) be a projective terminal pair of dimension n such that KX +∆ is nef and let L be a nef Q-Cartier divisor on X. Assume that ν(X, KX +∆+ L) = 1 and χ(X, OX ) 6= 0. Then KX +∆+ tL is num-effective for all t ≥ 0. This is proved in Section 7. ONGENERALISEDABUNDANCE,II 3 As an illustration, consider a terminal Calabi-Yau pair (X, ∆), i.e. a ter- minal pair such that KX +∆ ≡ 0. Let L be a nef divisor on X of numerical dimension 1. If χ(X, OX ) 6= 0, then L is num-effective. 1.2. Metrics with algebraic singularities. Theorem A above is a special instance of more general results which work in all dimensions but require inductive hypotheses. Let (X, ∆) be a klt pair such that KX +∆ is nef and X is not uniruled, and let L be a nef Q-divisor on X. Our main technical result, Theorem 4.1 below, gives a general criterion for the existence of sections of some multiple of KX +∆+ L. The criterion generalises the main results of [LP18a]: the proof uses the birational stability of the cotangent bundle from [CP15] (this is where the non-uniruledness of X is necessary) together with the very carefully chosen MMP techniques in Theorem 4.2 (this is where the nefness of KX + ∆ is used). We then apply Theorem 4.1 to line bundles possessing metrics with gen- eralised algebraic singularities; the importance of this class of metrics is explained in Section 2. We expect these methods to be crucial in the reso- lution of the Generalised Nonvanishing Conjecture. In Section 6 we prove our central inductive statement. Theorem C. Assume the termination of flips in dimensions at most n − 1, and the Abundance Conjecture in dimensions at most n − 1. Let (X, ∆) be a projective klt pair of dimension n such that KX +∆ is pseudoeffective, and let L be a nef Q-divisor on X. Suppose that KX +∆ and KX +∆+ L have singular metrics with generalised algebraic singularities and semipositive curvature currents, and that χ(X, OX ) 6= 0. (i) Then for every Q-divisor L′ with L ≡ L′ there exists a positive ra- tional number t0 such that ′ κ(X, KX +∆+ tL ) ≥ 0 for every 0 ≤ t ≤ t0. (ii) Assume additionally the semiampleness part of the Abundance Con- jecture in dimension n. Then KX +∆+ tL is num-effective for every t ≥ 0. With notation from Theorem C, we note that by [LP18b, Theorem A], all divisors of the form KX +∆+ L should possess singular metrics with generalised algebraic singularities and semipositive curvature currents. We mention a special case, where the conclusion of Theorem C is much stronger: when KX +∆ and KX +∆+ L additionally have smooth metrics with semipositive curvature, or, more generally, singular metrics with semi- positive curvature currents and vanishing Lelong numbers, then KX +∆+L is actually semiample. This is Theorem 5.2, and the proof uses crucially the main result of [GM17]. As a corollary of Theorem C, we obtain the following: 4 VLADIMIR LAZIC´ AND THOMAS PETERNELL Corollary D. Let (X, ∆) be a projective klt pair of dimension 4 such that KX +∆ is pseudoeffective and let L be a nef Q-divisor on X. Suppose that KX +∆ and KX +∆+ L have singular metrics with generalised algebraic singularities and semipositive curvature currents, and that χ(X, OX ) 6= 0. Then there exists a positive rational number t0 such that κ(X, KX +∆+ tL) ≥ 0 for every 0 ≤ t ≤ t0. Theorem C has an unexpected consequence for the Minimal Model Pro- gram in terms of the existence of sections of divisors numerically equivalent to adjoint divisors, generalising Theorem A to every dimension. The follow- ing result is a special case of Corollary 6.5 and Theorem 6.6. Theorem E. Assume the termination of flips in dimensions at most n − 1, and the existence of good models in dimensions at most n. Let (X, ∆) be a projective klt pair of dimension n such that KX +∆ is pseudoeffective. Assume that χ(X, OX ) 6= 0. (i) Then for every Q-divisor G with KX +∆ ≡ G we have κ(X, G) ≥ 0. (ii) If KX +∆ is semiample, then every Q-divisor G with KX +∆ ≡ G is semiample. 1.3. The Weak Generalised Nonvanishing Conjecture. As an impor- tant part of this paper on which the previous results depend, we show that the assumptions of the Generalised Nonvanishing Conjecture can be weak- ened considerably. In fact, we reduce the Generalised Nonvanishing Conjec- ture to the following. Weak Generalised Nonvanishing Conjecture. Let (X, ∆) be a Q-facto- rial projective klt pair of dimension n such that X is not uniruled and KX +∆ is nef. Let L be a nef Q-divisor on X such that n(X, KX +∆+ L) = n. Then for every t ≥ 0 the numerical class of the divisor KX +∆+ tL belongs to the effective cone. Here, for any Q-Cartier divisor D on X, we denote by n(X, D) the nef dimension of D, see Section 2. The condition n(X, D) = dim X is equivalent to saying that X cannot be covered by curves C such that D · C = 0. Conjecturally, when n(X, KX +∆+ L) = dim X as above, then KX +∆+ L is big. This conjecture is weaker than the Generalised Nonvanishing Conjecture in three different ways: the variety X is not uniruled; the pair (X, ∆) is a minimal model; and the nef dimension of KX +∆+ L is maximal; see Section 2 for details.
Recommended publications
  • Recent Results in Higher-Dimensional Birational Geometry

    Recent Results in Higher-Dimensional Birational Geometry

    Complex Algebraic Geometry MSRI Publications Volume 28, 1995 Recent Results in Higher-Dimensional Birational Geometry ALESSIO CORTI ct. Abstra This note surveys some recent results on higher-dimensional birational geometry, summarising the views expressed at the conference held at MSRI in November 1992. The topics reviewed include semistable flips, birational theory of Mori fiber spaces, the logarithmic abundance theorem, and effective base point freeness. Contents 1. Introduction 2. Notation, Minimal Models, etc. 3. Semistable Flips 4. Birational theory of Mori fibrations 5. Log abundance 6. Effective base point freeness References 1. Introduction The purpose of this note is to survey some recent results in higher-dimensional birational geometry. A glance to the table of contents may give the reader some idea of the topics that will be treated. I have attempted to give an informal presentation of the main ideas, emphasizing the common grounds, addressing a general audience. In 3, I could not resist discussing some details that perhaps § only the expert will care about, but hopefully will also introduce the non-expert reader to a subtle subject. Perhaps the most significant trend in Mori theory today is the increasing use, more or less explicit, of the logarithmic theory. Let me take this opportunity This work at the Mathematical Sciences Research Institute was supported in part by NSF grant DMS 9022140. 35 36 ALESSIO CORTI to advertise the Utah book [Ko], which contains all the recent software on log minimal models. Our notation is taken from there. I have kept the bibliography to a minimum and made no attempt to give proper credit for many results.
  • REMARKS on the ABUNDANCE CONJECTURE 3 Log flips for Log Canonical Pairs Is Known for All Dimensions (Cf

    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.
  • Introduction to Birational Geometry of Surfaces (Preliminary Version)

    Introduction to Birational Geometry of Surfaces (Preliminary Version)

    INTRODUCTION TO BIRATIONAL GEOMETRY OF SURFACES (PRELIMINARY VERSION) JER´ EMY´ BLANC (Very) quick introduction Let us recall some classical notions of algebraic geometry that we will need. We recommend to the reader to read the first chapter of Hartshorne [Har77] or another book of introduction to algebraic geometry, like [Rei88] or [Sha94]. We fix a ground field k. All results of the first 4 sections work over any alge- braically closed field, and a few also on non-closed fields. In the last section, the case of all perfect fields will be discussed. For our purpose, the characteristic is not important. n n 0.1. Affine varieties. The affine n-space Ak, or simply A , is the set of n-tuples n of elements of k. Any point x 2 A can be written as x = (x1; : : : ; xn), where x1; : : : ; xn 2 k are the coordinates of x. An algebraic set X ⊂ An is the locus of points satisfying a set of polynomial equations: n X = (x1; : : : ; xn) 2 A f1(x1; : : : ; xn) = ··· = fk(x1; : : : ; xn) = 0 where each fi 2 k[x1; : : : ; xn]. We denote by I(X) ⊂ k[x1; : : : ; xn] the set of polynomials vanishing along X, it is an ideal of k[x1; : : : ; xn], generated by the fi. We denote by k[X] or O(X) the set of algebraic functions X ! k, which is equal to k[x1; : : : ; xn]=I(X). An algebraic set X is said to be irreducible if any writing X = Y [ Z where Y; Z are two algebraic sets implies that Y = X or Z = X.
  • UC Berkeley UC Berkeley Electronic Theses and Dissertations

    UC Berkeley UC Berkeley Electronic Theses and Dissertations

    UC Berkeley UC Berkeley Electronic Theses and Dissertations Title Cox Rings and Partial Amplitude Permalink https://escholarship.org/uc/item/7bs989g2 Author Brown, Morgan Veljko Publication Date 2012 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California Cox Rings and Partial Amplitude by Morgan Veljko Brown A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, BERKELEY Committee in charge: Professor David Eisenbud, Chair Professor Martin Olsson Professor Alistair Sinclair Spring 2012 Cox Rings and Partial Amplitude Copyright 2012 by Morgan Veljko Brown 1 Abstract Cox Rings and Partial Amplitude by Morgan Veljko Brown Doctor of Philosophy in Mathematics University of California, BERKELEY Professor David Eisenbud, Chair In algebraic geometry, we often study algebraic varieties by looking at their codimension one subvarieties, or divisors. In this thesis we explore the relationship between the global geometry of a variety X over C and the algebraic, geometric, and cohomological properties of divisors on X. Chapter 1 provides background for the results proved later in this thesis. There we give an introduction to divisors and their role in modern birational geometry, culminating in a brief overview of the minimal model program. In chapter 2 we explore criteria for Totaro's notion of q-amplitude. A line bundle L on X is q-ample if for every coherent sheaf F on X, there exists an integer m0 such that m ≥ m0 implies Hi(X; F ⊗ O(mL)) = 0 for i > q.
  • Birational Geometry of the Moduli Spaces of Curves with One Marked Point

    Birational Geometry of the Moduli Spaces of Curves with One Marked Point

    The Dissertation Committee for David Hay Jensen Certi¯es that this is the approved version of the following dissertation: BIRATIONAL GEOMETRY OF THE MODULI SPACES OF CURVES WITH ONE MARKED POINT Committee: Sean Keel, Supervisor Daniel Allcock David Ben-Zvi Brendan Hassett David Helm BIRATIONAL GEOMETRY OF THE MODULI SPACES OF CURVES WITH ONE MARKED POINT by David Hay Jensen, B.A. DISSERTATION Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Ful¯llment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY THE UNIVERSITY OF TEXAS AT AUSTIN May 2010 To Mom, Dad, and Mike Acknowledgments First and foremost, I would like to thank my advisor, Sean Keel. His sug- gestions, perspective, and ideas have served as a constant source of support during my years in Texas. I would also like to thank Gavril Farkas, Joe Harris, Brendan Hassett, David Helm and Eric Katz for several helpful conversations. iv BIRATIONAL GEOMETRY OF THE MODULI SPACES OF CURVES WITH ONE MARKED POINT Publication No. David Hay Jensen, Ph.D. The University of Texas at Austin, 2010 Supervisor: Sean Keel We construct several rational maps from M g;1 to other varieties for 3 · g · 6. These can be thought of as pointed analogues of known maps admitted by M g. In particular, they contract pointed versions of the much- studied Brill-Noether divisors. As a consequence, we show that a pointed 1 Brill-Noether divisor generates an extremal ray of the cone NE (M g;1) for these speci¯c values of g.
  • Positivity in Algebraic Geometry I

    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.
  • Moving Codimension-One Subvarieties Over Finite Fields

    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].
  • Arxiv:1609.05543V2 [Math.AG] 1 Dec 2020 Ewrs Aovreis One Aiis Iersystem Linear Families, Bounded Varieties, Program

    Arxiv:1609.05543V2 [Math.AG] 1 Dec 2020 Ewrs Aovreis One Aiis Iersystem Linear Families, Bounded Varieties, Program

    Singularities of linear systems and boundedness of Fano varieties Caucher Birkar Abstract. We study log canonical thresholds (also called global log canonical threshold or α-invariant) of R-linear systems. We prove existence of positive lower bounds in different settings, in particular, proving a conjecture of Ambro. We then show that the Borisov- Alexeev-Borisov conjecture holds, that is, given a natural number d and a positive real number ǫ, the set of Fano varieties of dimension d with ǫ-log canonical singularities forms a bounded family. This implies that birational automorphism groups of rationally connected varieties are Jordan which in particular answers a question of Serre. Next we show that if the log canonical threshold of the anti-canonical system of a Fano variety is at most one, then it is computed by some divisor, answering a question of Tian in this case. Contents 1. Introduction 2 2. Preliminaries 9 2.1. Divisors 9 2.2. Pairs and singularities 10 2.4. Fano pairs 10 2.5. Minimal models, Mori fibre spaces, and MMP 10 2.6. Plt pairs 11 2.8. Bounded families of pairs 12 2.9. Effective birationality and birational boundedness 12 2.12. Complements 12 2.14. From bounds on lc thresholds to boundedness of varieties 13 2.16. Sequences of blowups 13 2.18. Analytic pairs and analytic neighbourhoods of algebraic singularities 14 2.19. Etale´ morphisms 15 2.22. Toric varieties and toric MMP 15 arXiv:1609.05543v2 [math.AG] 1 Dec 2020 2.23. Bounded small modifications 15 2.25. Semi-ample divisors 16 3.
  • Automorphisms of Cubic Surfaces Without Points

    Automorphisms of Cubic Surfaces Without Points

    AUTOMORPHISMS OF CUBIC SURFACES WITHOUT POINTS CONSTANTIN SHRAMOV Abstract. We classify finite groups acting by birational transformations of a non- trivial Severi–Brauer surface over a field of characteristc zero that are not conjugate to subgroups of the automorphism group. Also, we show that the automorphism group of a smooth cubic surface over a field K of characteristic zero that has no K-points is abelian, and find a sharp bound for the Jordan constants of birational automorphism groups of such cubic surfaces. 1. Introduction Given a variety X, it is natural to try to describe finite subgroups of its birational automorphism group Bir(X) in terms of the birational models of X on which these finite subgroups are regularized. In dimension 2, this is sometimes possible due to the well developed Minimal Model Program. In the case of the group of birational automorphisms of the projective plane over an algebraically closed field of characteristic zero, this was done in the work of I. Dolgachev and V. Iskovskikh [DI09]. Over algebraically non-closed fields, some partial results are also known for non-trivial Severi–Brauer surfaces, i.e. del Pezzo surfaces of degree 9 not isomorphic to P2 over the base field. Note that the latter surfaces are exactly the del Pezzo surfaces of degree 9 without points over the base field, see e.g. [Ko16, Theorem 53(2)]. Recall that a birational map Y 99K X defines an embedding of a group Aut(Y ) into Bir(X). We will denote by rk Pic(X)Γ the invariant part of the Picard group Pic(X) with respect to the action of a group Γ, and by µn the cyclic group of order n.
  • Birational Geometry of Algebraic Varieties

    Birational Geometry of Algebraic Varieties

    Birational geometry of algebraic varieties Caucher Birkar Cambridge University Rome, 2019 Algebraic geometry is the study of solutions of systems of polynomial equations and associated geometric structures. Algebraic geometry is an amazingly complex but beautiful subject. It is deeply related to many branches of mathematics but also to mathematical physics, computer science, etc. Algebraic geometry and associated geometric structures. Algebraic geometry is an amazingly complex but beautiful subject. It is deeply related to many branches of mathematics but also to mathematical physics, computer science, etc. Algebraic geometry Algebraic geometry is the study of solutions of systems of polynomial equations Algebraic geometry is an amazingly complex but beautiful subject. It is deeply related to many branches of mathematics but also to mathematical physics, computer science, etc. Algebraic geometry Algebraic geometry is the study of solutions of systems of polynomial equations and associated geometric structures. It is deeply related to many branches of mathematics but also to mathematical physics, computer science, etc. Algebraic geometry Algebraic geometry is the study of solutions of systems of polynomial equations and associated geometric structures. Algebraic geometry is an amazingly complex but beautiful subject. Algebraic geometry Algebraic geometry is the study of solutions of systems of polynomial equations and associated geometric structures. Algebraic geometry is an amazingly complex but beautiful subject. It is deeply related to many branches of mathematics but also to mathematical physics, computer science, etc. Let k = Q, or R, or C. Let k[t] = polynomials in variable t with coefficients in k. Given f 2 k[t], we want to find its solutions.
  • Strictly Nef Vector Bundles and Characterizations of P

    Strictly Nef Vector Bundles and Characterizations of P

    Complex Manifolds 2021; 8:148–159 Research Article Open Access Jie Liu, Wenhao Ou, and Xiaokui Yang* Strictly nef vector bundles and characterizations of Pn https://doi.org/10.1515/coma-2020-0109 Received September 8, 2020; accepted February 10, 2021 Abstract: In this note, we give a brief exposition on the dierences and similarities between strictly nef and ample vector bundles, with particular focus on the circle of problems surrounding the geometry of projective manifolds with strictly nef bundles. Keywords: strictly nef, ample, hyperbolicity MSC: 14H30, 14J40, 14J60, 32Q57 1 Introduction Let X be a complex projective manifold. A line bundle L over X is said to be strictly nef if L · C > 0 for each irreducible curve C ⊂ X. This notion is also called "numerically positive" in literatures (e.g. [25]). The Nakai-Moishezon-Kleiman criterion asserts that L is ample if and only if Ldim Y · Y > 0 for every positive-dimensional irreducible subvariety Y in X. Hence, ample line bundles are strictly nef. In 1960s, Mumford constructed a number of strictly nef but non-ample line bundles over ruled surfaces (e.g. [25]), and they are tautological line bundles of stable vector bundles of degree zero over smooth curves of genus g ≥ 2. By using the terminology of Hartshorne ([24]), a vector bundle E ! X is called strictly nef (resp. ample) if its tautological line bundle OE(1) is strictly nef (resp. ample). One can see immediately that the strictly nef vector bundles constructed by Mumford are actually Hermitian-at. Therefore, some functorial properties for ample bundles ([24]) are not valid for strictly nef bundles.
  • Bimeromorphic Geometry of K'́ahler Threefolds

    Bimeromorphic Geometry of K'́ahler Threefolds

    Bimeromorphic geometry of K’́ahler threefolds Andreas Höring, Thomas Peternell To cite this version: Andreas Höring, Thomas Peternell. Bimeromorphic geometry of K’́ahler threefolds. 2015 Summer Institute on Algebraic Geometry, Jul 2015, Salt Lake City, United States. hal-01428621 HAL Id: hal-01428621 https://hal.archives-ouvertes.fr/hal-01428621 Submitted on 6 Jan 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Bimeromorphic geometry of K¨ahler threefolds Andreas H¨oring and Thomas Peternell Abstract. We describe the recently established minimal model program for (non-algebraic) K¨ahler threefolds as well as the abundance theorem for these spaces. 1. Introduction Given a complex projective manifold X, the Minimal Model Program (MMP) predicts that either X is covered by rational curves (X is uniruled) or X has a ′ - slightly singular - birational minimal model X whose canonical divisor KX′ is nef; and then the abundance conjecture says that some multiple mKX′ is spanned by global sections (so X′ is a good minimal model). The MMP also predicts how to achieve the birational model, namely by a sequence of divisorial contractions and flips. In dimension three, the MMP is completely established (cf.