Kähler-Einstein Metrics and Algebraic Geometry
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Arxiv:1705.02740V4 [Math.AG] 18 Dec 2018 Iease Oaqeto Se Yyce I Uigteamwo AIM the During 2017
BOUNDEDNESS OF Q-FANO VARIETIES WITH DEGREES AND ALPHA-INVARIANTS BOUNDED FROM BELOW CHEN JIANG Abstract. We show that Q-Fano varieties of fixed dimension with anti-canonical degrees and alpha-invariants bounded from below form a bounded family. As a corollary, K-semistable Q-Fano varieties of fixed dimension with anti-canonical degrees bounded from below form a bounded family. 1. Introduction Throughout the article, we work over an algebraically closed field of char- acteristic zero. A Q-Fano variety is defined to be a normal projective variety X with at most klt singularities such that the anti-canonical divisor KX is an ample Q-Cartier divisor. − When the base field is the complex number field, an interesting prob- lem for Q-Fano varieties is the existence of K¨ahler–Einstein metrics which is related to K-(semi)stability of Q-Fano varieties. It has been known that a Fano manifold X (i.e., a smooth Q-Fano variety over C) admits K¨ahler–Einstein metrics if and only if X is K-polystable by the works [DT92, Tia97, Don02, Don05, CT08, Sto09, Mab08, Mab09, Ber16] and [CDS15a, CDS15b, CDS15c, Tia15]. K-stability is stronger than K-polystability, and K-polystability is stronger than K-semistability. Hence K-semistable Q- Fano varieties are interesting for both differential geometers and algebraic geometers. It also turned out that K¨ahler–Einstein metrics and K-stability play cru- cial roles for construction of nice moduli spaces of certain Q-Fano varieties. For example, compact moduli spaces of smoothable K¨ahler–Einstein Q-Fano varieties have been constructed (see [OSS16] for dimension two case and [LWX14, SSY16, Oda15] for higher dimensional case). -
The Role of Partial Differential Equations in Differential Geometry
Proceedings of the International Congress of Mathematicians Helsinki, 1978 The Role of Partial Differential Equations in Differential Geometry Shing-Tung Yau In the study of geometric objects that arise naturally, the main tools are either groups or equations. In the first case, powerful algebraic methods are available and enable one to solve many deep problems. While algebraic methods are still important in the second case, analytic methods play a dominant role, especially when the defining equations are transcendental. Indeed, even in the situation where the geometric abject is homogeneous or algebraic, analytic methods often lead to important contributions. In this talk, we shall discuss a class of problems in dif ferential geometry and the analytic methods that are involved in solving such problems. One of the main purposes of differential geometry is to understand how a surface (or a generalization of it) is curved, either intrinsically or extrinsically. Naturally, the problems that are involved in studying such an object cannot be linear. Since curvature is defined by differentiating certain quantities, the equations that arise are nonlinear differential equations. In studying curved space, one of the most important tools is the space of tangent vectors to the curved space. In the language of partial differential equations, the main tool to study nonlinear equations is the use of the linearized operators. Hence, even when we are facing nonlinear objects, the theory of linear operators is unavoidable. Needless to say, we are then left with the difficult problem of how precisely a linear operator approximates a non linear operator. To illustrate the situation, we mention five important differential operators in differential geometry. -
Vanishing Theorems and Syzygies for K3 Surfaces and Fano Varieties
VANISHING THEOREMS AND SYZYGIES FOR K3 SURFACES AND FANO VARIETIES F. J. Gallego and B. P. Purnaprajna May 26, 1996 Abstract. In this article we prove some strong vanishing theorems on K3 surfaces. As an application of them, we obtain higher syzygy results for K3 surfaces and Fano varieties. 1. Introduction In this article we prove some vanishing theorems on K3 surfaces. An application of the vanishing theorems is a result on higher syzygies for K3 surfaces and Fano varieties. One part of our results fits a meta-principle stating that if L is a line bundle that is a product of (p+1) ample and base point free line bundles satisfying certain conditions, then L satisfies the condition Np ( a condition on the free resolution of the homogeneous coordinate ring of X embedded by L). Other illustrations of this meta-principle have been given in [GP1], [GP2] and [GP3]. The condition Np may be interpreted, through Koszul cohomology, as a vanishing condition on a certain vector bundle. arXiv:alg-geom/9608008v1 7 Aug 1996 The other part of our results provides strong vanishing theorems that imply, in particular, the vanishing needed for Np. We also prove stronger variants of the principle stated above for K3 surfaces and Fano varieties. Before stating our results in detail, we recall some key results in this area, namely the normal generation and normal presentation on K3 surfaces due to Mayer and St.Donat. Mayer and St. Donat proved that if L is a globally generated line bundle on a K3 surface X such that the general member in the linear system is a non hyperelliptic curve of genus g ≥ 3, then L is normally generated (in other words, the homogeneous coordinate ring of X in projective space P(H0(L)) is projectively normal). -
About the Calabi-Yau Theorem and Its Applications
About the Calabi-Yau theorem and its applications { Krageomp program { Julien Keller L.A.T.P., Marseille University { Universit´ede Provence October 21, 2009 1 Any comments, remarks and suggestions are welcome. These notes are summing up the series of lectures given by the author during the Krageomp program (2009). Please feel free to contact the author at [email protected] 2 1 About complex and K¨ahlermanifolds In this lecture, starting with the background of Riemannian geometry, we introduce the notion of complex manifolds: • Definition using charts and holomorphic transition functions • Almost complex structure, complex structure and Newlander-Nirenberg theorem • Examples: the projective space , S2, hypersurfaces, weighted projective spaces • Uniformization theorem for Riemann surfaces, Chow's theorem (com- n plex compact analytic submanifolds of CP are actually algebraic vari- eties). We introduce the language of tensors on complex manifolds, and some natural invariants of the complex structure: • k-forms, exterior diffentiation operator, d = @ +@¯ decomposition, (p; q) forms • De Rham cohomology, Dolbeault cohomology, Hodge numbers, Euler characteristic invariant Using all the previous material, we can explain what is a K¨ahlermanifold. In particular we will formulate the Calabi conjecture on that space. • K¨ahlerforms, K¨ahlerclasses, K¨ahlercone n • The Fubini-Study metric on CP is a K¨ahlermetric Finally, we sketched the notion of holomorphic line bundle, the correspon- dence with divisor and Chern classes. This very classical material can be read in different books: - F. Zheng, Complex Differential Geometry, Studies in Adv. Maths, AMS (2000) 3 - W. Ballmann, Lectures on K¨ahlermanifolds, Lectures in Math and Physics, EMS (2006). -
Lines, Conics, and All That Ciro Ciliberto, M Zaidenberg
Lines, conics, and all that Ciro Ciliberto, M Zaidenberg To cite this version: Ciro Ciliberto, M Zaidenberg. Lines, conics, and all that. 2020. hal-02318018v3 HAL Id: hal-02318018 https://hal.archives-ouvertes.fr/hal-02318018v3 Preprint submitted on 5 Jul 2020 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. LINES, CONICS, AND ALL THAT C. CILIBERTO, M. ZAIDENBERG To Bernard Shiffman on occasion of his seventy fifths birthday Abstract. This is a survey on the Fano schemes of linear spaces, conics, rational curves, and curves of higher genera in smooth projective hypersurfaces, complete intersections, Fano threefolds, on the related Abel-Jacobi mappings, etc. Contents Introduction 1 1. Counting lines on surfaces 2 2. The numerology of Fano schemes 3 3. Geometry of the Fano scheme 6 4. Counting conics in complete intersection 8 5. Lines and conics on Fano threefolds and the Abel-Jacobi mapping 11 5.1. The Fano-Iskovskikh classification 11 5.2. Lines and conics on Fano threefolds 12 5.3. The Abel-Jacobi mapping 14 5.4. The cylinder homomorphism 16 6. Counting rational curves 17 6.1. Varieties of rational curves in hypersurfaces 17 6.2. -
All Complete Intersection Varieties Are Fano Visitors 3
ALL COMPLETE INTERSECTION VARIETIES ARE FANO VISITORS YOUNG-HOON KIEM, IN-KYUN KIM, HWAYOUNG LEE, AND KYOUNG-SEOG LEE Abstract. We prove that the derived category of a smooth complete inter- section variety is equivalent to a full subcategory of the derived category of a smooth projective Fano variety. This enables us to define some new invariants of smooth projective varieties and raise many interesting questions. 1. Introduction In recent years, derived categories of projective varieties attracted tremendous interest among algebraic geometers as well as physicists. Fano varieties in particular have been most intensively studied because their derived categories (1) determine the varieties completely due to the reconstruction theorem of Bondal and Orlov and (2) have canonical semiorthogonal decompositions by canonical exceptional collections of line bundles by the Kodaira vanishing theorem. Moreover, the de- rived categories of many Fano varieties of low dimension have been calculated quite explicitly. Often the derived categories of Fano varieties are big enough to contain interest- ing subcategories. For example, the derived categories of hyperelliptic curves are full subcategories of the derived categories of intersections of two quadrics (cf. [6]) and the derived categories of some special cubic 4-folds contain the derived cate- gories of K3 surfaces (cf. [17]). These results provide derived category theoretic explanations for the corresponding geometric results of [10, 27] and [3]. Analysis of derived categories may tell us where to dig (or where not to dig) when we are searching for a specific type of varieties. In 2011, Bondal raised the following question (cf. [5]). Question 1.1. (Fano visitor problem) arXiv:1503.00125v2 [math.AG] 29 Apr 2015 Let Y be a smooth projective variety. -
Chapter V. Fano Varieties
Chapter V. Fano Varieties A variety X is called Fano if the anticanonical bundle of X is ample. Thus Fano surfaces are the same as Del pezzo surfaces. The importance of Fano varieties in the theory of higher dimensional varieties is similar to the sig nificance of Del Pezzo surfaces in the two dimensional theory. The interest in Fano varieties increased recently since Mori's program predicts that every uniruled variety is birational to a fiberspace whose general fiber is a Fano variety (with terminal singularities). From this point of view it is more important to study the general prop erties of Fano varieties with terminal singularities than to understand the properties of smooth Fano varieties. At the moment, however, we know much more about smooth Fano varieties, and their theory should serve as a guide to the more subtle questions of singular Fano varieties. Fano varieties also appear naturally as important examples of varieties. In characteristic zero every projective variety which is homogeneous under a linear algebraic group is Fano (1.4), and their study is indispensable for the theory of algebraic groups. Also, Fano varieties have a very rich internal geometry, which makes their study very rewarding. This is one of the reasons for the success of the theory of Fano threefolds. This is a beautiful subject, about which I say essentially nothing. Section 1 is devoted to presenting the basic examples of Fano varieties and to the study of low degree rational curves on them. The largest class of examples are weighted complete intersections (1.2-3); these are probably the most accessible by elementary methods. -
C Omp Act Riemann
Jü rgen Jost C omp actRiemann S urfaces An Introduction to Contemporary Mathematics Third Edition With 23 Figures 123 Jü rgen Jost Max Planck Institute for Mathematics in the Sciences Inselstr. 22 04103 Leipzig Germany e-mail: [email protected] Mathematics Subject Classification (2000): 30F10, 30F45, 30F60, 58E20, 14H55 Library of Congress Control Number: 2006924561 ISBN-10 3-540-33065-8 Springer Berlin Heidelberg New York ISBN-13 978-3-540-33065-3 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Dupli- cation of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2006 PrintedinGermany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Erich Kirchner, Heidelberg Typesetting by the author and SPI Publisher Services using a Springer LATEX macro package Printed on acid-free paper 11689881 41/sz - 5 4 3 2 1 0 Dedicated to the memory of my father Preface The present book started from a set of lecture notes for a course taught to stu- dents at an intermediate level in the German system (roughly corresponding to the beginning graduate student level in the US) in the winter term 86/87 in Bochum. -
Fano Varieties; Iskovskih's Classification
Fano varieties; Iskovskih’s classification Ekaterina Amerik For details and extensive bibliography, we refer to [2], chapter V, and [1]. A Fano variety is a projective manifold X such that the anticanonical line bundle −1 p,0 0,p KX is ample. By Kodaira vanishing, the Hodge numbers h (X) = h (X) are zero for p 6= 0. Furthermore, Fano manifolds are simply connected (this is implied for example by their property to be rationally connected; see the main article on rational curves and uniruled varieties). Simplest examples are obtained by taking smooth complete intersections of type n (m1, m2, . , mk) in P . By adjunction formula, such a complete intersection is Fano P if and only if i mi ≤ n. A larger class of examples is that of complete intersections n+1 ∗ ∗ in a weighted projective space P(a0, a1, . , an) (this is (C − 0)/C , where C acts with weights a0, a1, . , an; it is singular when not isomorphic to a usual projective space, but we consider complete intersections avoiding the singularities) : the Fano P P condition amounts then to i mi < i ai. Rational homogeneous varieties G/H (G semisimple, H parabolic) are Fano, too. A Fano curve is, obviously, P1. If n = dim(X) = 2 and X is Fano, then X is called a Del Pezzo surface. Such surfaces have been classically studied, and it is well-known that any such X is isomorphic either to P2, or to P1 ×P1, or to P2 blown up in d points (1 ≤ d ≤ 8) in general position, ”general position” meaning here that no three points are on a line and no six on a conic. -
Arxiv:1709.09238V3 [Math.AG] 5 Mar 2021 Xmlso Ahlgclvreisvoaigte N Osuytheir [Dcf15])
KAWAMATA-VIEHWEG VANISHING FAILS FOR LOG DEL PEZZO SURFACES IN CHARACTERISTIC 3 FABIO BERNASCONI Abstract. We construct a klt del Pezzo surface in characteristic three violating the Kawamata-Viehweg vanishing theorem. As a consequence we show that there exists a Kawamata log terminal threefold singularity which is not Cohen-Macaulay in characteristic three. Contents 1. Introduction 1 2. Preliminaries 3 2.1. Notation 3 2.2. Frobenius splitting 5 2.3. A cone construction for Weil Q-Cartier divisors 6 3. A Keel-McKernan surface in characteristic three 9 3.1. Construction 9 3.2. Failure of the Kawamata-Viehweg vanishing theorem 12 4. A klt threefold singularity not CM in characteristic three 13 5. Kodaira-type vanishing for klt del Pezzo surfaces 15 References 19 arXiv:1709.09238v3 [math.AG] 5 Mar 2021 1. Introduction In characteristic zero one of the main technical tool used to estab- lish the Minimal Model Program (MMP for short) is the Kawamata- Viehweg vanishing theorem. Unfortunately, vanishing theorems are known to fail in general for varieties defined over fields of positive characteristic and a great amount of work has been done to construct examples of pathological varieties violating them and to study their geography (see for example [Ray78], [Eke88], [Muk13] and [dCF15]). 2020 Mathematics Subject Classification. 14E30, 14G17, 14J17, 14J45. Key words and phrases. Log del Pezzo surfaces, vanishing theorems, Kawamata log terminal singularities, positive characteristic. 1 2 FABIOBERNASCONI In this context, varieties of Fano type over perfect fields of charac- teristic p > 0 violating Kodaira-type vanishing theorems seem rather rare and in fact are conjectured to exist only for small primes in each dimension. -
On the Birational Geometry of Singular Fano Varieties
On the birational geometry of singular Fano varieties Thesis submitted in accordance with the requirements of the University of Liverpool for the degree of Doctor in Philosophy by Ewan Johnstone under the supervision of Professor Aleksandr Pukhlikov Department of Mathematical Sciences University of Liverpool United Kingdom April 2017 2 Contents Abstract . .5 Acknowledgement . .6 Introduction . .7 1 Background 13 1 Varieties in projective space . 14 2 Rational maps on quasi-projective varieties . 17 3 Singularities of algebraic varieties . 23 4 Divisors and linear systems . 27 5 Fano varieties . 33 6 Algebraic cycles and intersection theory . 35 2 Methods 41 1 Birational Rigidity . 42 2 The Noether-Fano inequality . 47 3 The technique of counting multiplicities . 52 4 Inversion of adjunction . 61 5 The cone technique . 67 3 Birationally rigid singular double quadrics and cubics 71 1 Introduction . 72 2 Fano double quadrics . 74 3 Fano double cubics . 79 Conclusion and Further Work 85 3 4 Bibliography 89 On the birational geometry of singular Fano varieties Ewan Johnstone Abstract This thesis investigates the birational geometry of a class of higher dimen- sional Fano varieties of index 1 with quadratic hypersurface singularities. The main investigating question is, what structures of a rationally connected fibre space can these varieties have? Two cases are considered: double cov- ers over a hypersurface of degree two, known as double quadrics and double covers over a hypersurface of degree three, known as double cubics. This thesis extends the study of double quadrics and cubics, first studied in the non-singular case by Iskovskikh and Pukhlikov, by showing that these vari- eties have the property of birational superrigidity, under certain conditions on the singularities of the branch divisor. -
The Degree of Irrationality of Hypersurfaces in Various Fano Varieties
manuscripta math. 161, 377–408 (2020) © Springer-Verlag GmbH Germany, part of Springer Nature 2019 David Stapleton · Brooke Ullery The degree of irrationality of hypersurfaces in various Fano varieties Received: 3 May 2018 / Accepted: 21 December 2018 / Published online: 9 January 2019 Abstract. The purpose of this paper is to compute the degree of irrationality of hypersurfaces of sufficiently high degree in various Fano varieties: quadrics, Grassmannians, products of projective space, cubic threefolds, cubic fourfolds, and complete intersection threefolds of type (2,2). This extends the techniques of Bastianelli, De Poi, Ein, Lazarsfeld, and the second author who computed the degree of irrationality of hypersurfaces of sufficiently high degree in projective space. A theme in the paper is that the fibers of low degree rational maps from the hypersurfaces to projective space tend to lie on curves of low degree contained in the Fano varieties. This allows us to study these maps by studying the geometry of curves in these Fano varieties. Introduction The degree of irrationality of an n-dimensional algebraic variety X, denoted irr(X), is the minimal degree of a dominant rational map φ : XPn. The aim of this paper is to compute the degree of irrationality of hypersurfaces in various Fano varieties: quadrics, cubic threefolds, cubic fourfolds, complete intersection threefolds of type (2,2), Grassmannians, and products of projective spaces. Throughout we work with varieties over C. Recently there has been a great deal of interest in understanding different mea- sures of irrationality of higher dimensional varieties. Bastianelli, Cortini, and De Poi conjectured ([1, Conj. 1.5]) that if X is a very general d hypersurface n+1 X = Xd ⊂ P with d ≥ 2n + 1, then irr(X) = d − 1.