DOCSLIB.ORG

Enriques–Kodaira Classification of Compact Complex Surfaces – an Analysis *Ramesha

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Enriques–Kodaira Classification of Compact Complex Surfaces – an Analysis *Ramesha © 2018 JETIR December 2018, Volume 5, Issue 12 www.jetir.org (ISSN-2349-5162) Enriques–Kodaira classification of compact complex surfaces – An Analysis *Ramesha. H.G. Asst Professor of Mathematics. Govt First Grade College, Tiptur. Abstract This paper attempts to study the Enriques–Kodaira classification a classification of compact complex surfaces into ten classes. the Enriques-Kodaira classification is a classification of compact complex surfaces. For complex projective surfaces it was done by Federigo Enriques, and Kunihiko Kodaira later extended it to non- algebraic compact surfaces. The classification of a collection of objects generally means that a list has been constructed with exactly one member from each isomorphism type among the objects, and that tools and techniques can effectively be used to identify any combinatorially given object with its unique representative in the list. Examples of mathematical objects which have been classified include the finite simple groups and 2- manifolds but not, for example, knots. A compact manifold is a manifold that is compact as a topological space. Examples are the circle (the only one-dimensional compact manifold) and the -dimensional sphere and torus. Compact manifolds in two dimensions are completely classified by their orientation and the number of holes (genus). It should be noted that the term "compact manifold" often implies "manifold without boundary," which is the sense in which it is used here. When there is need for a separate term, a compact boundaryless manifold is called a closed manifold. It has been established that for given n the n-dimensional compact, connected complex manifolds X can be classified according to their Kodaira dimension kod(X), which can assume the values -00,0,1, ... , n. In the case n = 2 the surfaces in the classes kod(X) = -00 or kod(X) = 0, and to a lesser extent those with kod(X) = 1 can be classified in much more detail. Thus, starting from the rough classification by Kodaira dimension, surfaces are divided into ten classes. This classification is called the Enriques-Kodaira classification and is embodied in the following central result. The word "fibre space" will be used only for smooth surfaces fibred over smooth curves. Thus a fibre space is a triple (X , f , Y) , where X is a smooth surface, Y a smooth curve and f : X - Y a surjective morphism. The fibre space is called (relatively) minimal if no fibre contains an exceptional curve. It is called an elliptic fibre space if almost all fibres are elliptic curves. Sometimes a surface admitting at least one such elliptic fibration is called an elliptic surface. Algebraic surfaces X into four classes, mentioned in section ~, is the division according to the value of Kod(X) , which can be - - , 0 , 1 or 2 (we don’t need the interpretation given for the last two classes in section 1). Key words: Elliptic Curve, Elliptic Surface, Effective Divisor, Kodaira Dimension. JETIR1812D26 Journal of Emerging Technologies and Innovative Research (JETIR) www.jetir.org 874 © 2018 JETIR December 2018, Volume 5, Issue 12 www.jetir.org (ISSN-2349-5162) Introduction An Enriques surface is a smooth compact complex surface having irregularity and nontrivial canonical sheaf such that (Endraß). Such surfaces cannot be embedded in projective three-space, but there nonetheless exist transformations onto singular surfaces in projective three-space. There exists a family of such transformed surfaces of degree six which passes through each edge of a tetrahedron twice. A subfamily with tetrahedral symmetry is given by the two-parameter ( ) family of surfaces. For many problems in topology and geometry, it is convenient to study compact manifolds because of their "nice" behavior. Among the properties making compact manifolds "nice" are the fact that they can be covered by finitely many coordinate charts, and that any continuous real-valued function is bounded on a compact manifold. For any positive integer , a distinct nonorientable surface can be produced by replacing disks with Möbius strips. In particular, replacing one disk with a Möbius strip produces a cross surface and replacing two disks produces the Klein bottle. The sphere, the -holed tori, and this sequence of nonorientable surfaces form a complete list of compact, boundaryless two-dimensional manifolds. and the polynomial is a sphere with radius , (Endraß). There are a large number of invariants that are linear combinations of the Hodge numbers, as follows: * "b"0,"b"1,"b"2,"b"3,"b"4 are the Betti numbers: "b"i = dim("H"i("S")). "b"0 = "b"4 = 1 and "b"1 = "b"3 = "h""1","0" + "h""0","1" = "h""2","1" + "h""1","2" and "b"2 = "h""2","0" + "h""1","1" + "h""0","2" of the trivial bundle. (It should not be confused with the Euler number. Add minus signs to the rows for dimensions 1 and 3. This is the sum along any side of the diamond, while the topological Euler-Poincaré characteristic is the sum over the whole diamond.) By Noether's formula it is also equal to the Todd genus ("c"12 +"c"2)/12 j "i","j" *τ is the signature (of the second cohomology group) and is equal to 4χ−"e", which is Σi,j(−1) "h" . *"b"+ and "b"− are the dimensions of the maximal positive and negative definite subspaces of "H"2, so "b"+ + − "b" ="b"2 and "b"+ −"b"− =τ. 2 2 *"c"2 = "e" and "c"1 = "K" = 12χ − "e" are JETIR1812D26 Journal of Emerging Technologies and Innovative Research (JETIR) www.jetir.org 875 © 2018 JETIR December 2018, Volume 5, Issue 12 www.jetir.org (ISSN-2349-5162) the Chern numbers, defined as the integrals of various polynomials in the Chern classes over the manifold. Friedman and Morgan proved that the invariants above depend only on the underlying smooth 4-manifold. Objective: This paper intends to explore classification that goes under the name of Enriques- Kodaira birational classification of compact complex surfaces. For each of these classes, the surfaces in the class can be parametrized by a moduli space Blow-ups and birational maps Let be a surface. The structure of rational maps between and any projective variety is simple by the following Theorem 1 (Elimination of indeterminacy) Let be a rational map. Then there exists a composite of finite many blow-ups and a morphism such that . Proof We have a bijection between nondegenerate rational maps and linear systems on of dimension which has no fixed components. We use this correspondence and induct on the dimension of the linear systems. Each blow-up drops the dimension and this process terminates using the intersection pairing. □ The structure of birational morphisms between surfaces is also rather simple. Suppose is a birational morphism of surfaces. Then can be decomposed as a sequence of blow-ups , and an isomorphism . Combining this fact with the elimination of indeterminacy, we obtain that Theorem 2 Let be a birational map between surfaces. Then there exists morphisms and such that , where each of can be decomposed as a sequence of blow-ups and an isomorphism. Neron-Severi groups and minimal models The exponential exact sequence induces a long exact sequence Hodge theory shows that is a lattice in , so the Picard variety is a complex torus (and indeed an abelian variety). Since , we obtain an exact sequence where is the image of . Definition 1 The group is called the Neron-Severi group of . JETIR1812D26 Journal of Emerging Technologies and Innovative Research (JETIR) www.jetir.org 876 © 2018 JETIR December 2018, Volume 5, Issue 12 www.jetir.org (ISSN-2349-5162) Since is finitely generated, the Neron-Severi group is also finitely generated. The rank of is called the Picard number of . The Neron-Severi group can also be described as the group of divisors on modulo algebraic equivalence ([3, V 1.7]). The behavior of the Neron-Severi group under a blow-up can be easily seen. Theorem 3 Let be a blow-up at a point with exceptional divisor . Then there is a canonical isomorphism . Proof The canonical isomorphism given by descends to the Neron-Severi groups. □ In other words, a blow-up increases the Picard number by 1. Thus the Neron-Severi group gives us a canonical order on the set of isomorphism classes of surfaces birationally equivalent to a given surface . Definition 2 Let . We say that dominates if there exists a birationally morphism (in particular, ). We say that is minimal if every birational morphism is an isomorphism. In particular, every surface dominates a minimal surface since the Picard number is finite. Conversely, every surface is obtained by a sequence of blow-ups of a minimal surface. So the problem of birational classification boils down to classifying minimal surfaces. We can characterize minimal surfaces as those without exceptional curves. By the very definition of blow-ups, an exceptional curve is isomorphic to and has self-intersection number . This is actually a useful numerical criterion of minimality of surfaces ([1, II. 17]). Theorem 4 (Castelnuovo's contractibility criterion) Suppose a curve is isomorphic to with . Then is an exceptional curve on . Ruled surfaces In most cases, has a unique minimal element. However, the situation is not that simple for a large class of surfaces — ruled surfaces. Definition 3 A surface is called ruled if it is birational to for a smooth projective curve. As a little digression, we can calculate several birational invariants for ruled surfaces easily. Definition 4 Let be any surface. We define JETIR1812D26 Journal of Emerging Technologies and Innovative Research (JETIR) www.jetir.org 877 © 2018 JETIR December 2018, Volume 5, Issue 12 www.jetir.org (ISSN-2349-5162) the irregularity . It is equal to by Hodge theory.
Load more

Recommended publications
  • Threefolds of Kodaira Dimension One Such That a Small Pluricanonical System Do Not Define the Iitaka fibration
    THREEFOLDS OF KODAIRA DIMENSION ONE HSIN-KU CHEN Abstract. We prove that for any smooth complex projective threefold of Kodaira di- mension one, the m-th pluricanonical map is birational to the Iitaka fibration for every m ≥ 5868 and divisible by 12. 1. Introduction By the result of Hacon-McKernan [14], Takayama [25] and Tsuji [26], it is known that for any positive integer n there exists an integer rn such that if X is an n-dimensional smooth complex projective variety of general type, then |rKX| defines a birational morphism for all r ≥ rn. It is conjectured in [14] that a similar phenomenon occurs for any projective variety of non-negative Kodaira dimension. That is, for any positive integer n there exists a constant sn such that, if X is an n-dimensional smooth projective variety of non-negative Kodaira dimension and s ≥ sn is sufficiently divisible, then the s-th pluricanonical map of X is birational to the Iitaka fibration. We list some known results related to this problem. In 1986, Kawamata [17] proved that there is an integer m0 such that for any terminal threefold X with Kodaira dimension zero, the m0-th plurigenera of X is non-zero. Later on, Morrison proved that one can 5 3 2 take m0 =2 × 3 × 5 × 7 × 11 × 13 × 17 × 19. See [21] for details. In 2000, Fujino and Mori [12] proved that if X is a smooth projective variety with Kodaira dimension one and F is a general fiber of the Iitaka fibration of X, then there exists a integer M, which depends on the dimension of X, the middle Betti number of some finite covering of F and the smallest integer so that the pluricanonical system of F is non-empty, such that the M-th pluricanonical map of X is birational to the Iitaka fibration.
    [Show full text]
  • MINIMAL MODEL THEORY of NUMERICAL KODAIRA DIMENSION ZERO Contents 1. Introduction 1 2. Preliminaries 3 3. Log Minimal Model Prog
    MINIMAL MODEL THEORY OF NUMERICAL KODAIRA DIMENSION ZERO YOSHINORI GONGYO Abstract. We prove the existence of good minimal models of numerical Kodaira dimension 0. Contents 1. Introduction 1 2. Preliminaries 3 3. Log minimal model program with scaling 7 4. Zariski decomposition in the sense of Nakayama 9 5. Existence of minimal model in the case where º = 0 10 6. Abundance theorem in the case where º = 0 12 References 14 1. Introduction Throughout this paper, we work over C, the complex number ¯eld. We will make use of the standard notation and de¯nitions as in [KM] and [KMM]. The minimal model conjecture for smooth varieties is the following: Conjecture 1.1 (Minimal model conjecture). Let X be a smooth pro- jective variety. Then there exists a minimal model or a Mori ¯ber space of X. This conjecture is true in dimension 3 and 4 by Kawamata, Koll¶ar, Mori, Shokurov and Reid (cf. [KMM], [KM] and [Sho2]). In the case where KX is numerically equivalent to some e®ective divisor in dimen- sion 5, this conjecture is proved by Birkar (cf. [Bi1]). When X is of general type or KX is not pseudo-e®ective, Birkar, Cascini, Hacon Date: 2010/9/21, version 4.03. 2000 Mathematics Subject Classi¯cation. 14E30. Key words and phrases. minimal model, the abundance conjecture, numerical Kodaira dimension. 1 2 YOSHINORI GONGYO and McKernan prove Conjecture 1.1 for arbitrary dimension ([BCHM]). Moreover if X has maximal Albanese dimension, Conjecture 1.1 is true by [F2]. In this paper, among other things, we show Conjecture 1.1 in the case where º(KX ) = 0 (for the de¯nition of º, see De¯nition 2.6): Theorem 1.2.
    [Show full text]
  • Arxiv:2008.08852V2 [Math.AG] 15 Oct 2020 Assuming Theorem 2, We Conclude That M16 Cannot Be of General Type, Thus Es- Tablishing Theorem 1
    ON THE KODAIRA DIMENSION OF M16 GAVRIL FARKAS AND ALESSANDRO VERRA ABSTRACT. We prove that the moduli space of curves of genus 16 is not of general type. The problem of determining the nature of the moduli space Mg of stable curves of genus g has long been one of the key questions in the field, motivating important developments in moduli theory. Severi [Sev] observed that Mg is unirational for g ≤ 10, see [AC] for a modern presentation. Much later, in the celebrated series of papers [HM], [H], [EH], Harris, Mumford and Eisenbud showed that Mg is of general type for g ≥ 24. Very recently, it has been showed in [FJP] that both M22 and M23 are of general type. On the other hand, due to work of Sernesi [Ser], Chang-Ran [CR1], [CR2] and Verra [Ve] it is known that Mg is unirational also for 11 ≤ g ≤ 14. Finally, Bruno and Verra [BV] proved that M15 is rationally connected. Our result is the following: Theorem 1. The moduli space M16 of stable curves of genus 16 is not of general type. A few comments are in order. The main result of [CR3] claims that M16 is unir- uled. It has been however recently pointed out by Tseng [Ts] that the key calculation in [CR3] contains a fatal error, which genuinely reopens this problem (after 28 years)! Before explaining our strategy of proving Theorem 1, recall the standard notation ∆0;:::; ∆ g for the irreducible boundary divisors on Mg, see [HM]. Here ∆0 denotes b 2 c the closure in Mg of the locus of irreducible 1-nodal curves of arithmetic genus g.
    [Show full text]
  • The Kodaira Dimension of Siegel Modular Varieties of Genus 3 Or Higher
    The Kodaira dimension of Siegel modular varieties of genus 3 or higher Vom Fachbereich Mathematik der Universitat¨ Hannover zur Erlangung des Grades Doktor der Naturwissenschaften Dr. rer. nat. genehmigte Dissertation von Dipl.-Math. Eric Schellhammer geboren am 18. November 1973 in Stuttgart 2004 [email protected] Universitat¨ Hannover – Fachbereich Mathematik February 25, 2004 Referent: Prof. Dr. Klaus Hulek, Universitat¨ Hannover Koreferent: Prof. Dr. Herbert Lange, Universitat¨ Erlangen Tag der Promotion: 6. Februar 2004 2 Abstract An Abelian variety is a g-dimensional complex torus which is a projective variety. In order to obtain an embedding into projective space one has to chose an ample line bundle. To each such line bundle one can associate a polarisation which only depends on the class of the line bundle in the Neron-Se´ veri group. The type of the polarisation is given by a g-tuple of integers e1;::: ;eg with the property that ei ei+1 for i = 1;:::;g 1. j ei+1 − If e1 = = eg = 1, the polarisation is said to be principal. If the values are ··· ei pairwise coprime, the polarisation is said to be coprime. Furthermore, we can give a symplectic basis for the group of n-torsion points, which is then called a level-n structure. Not only Abelian varieties but also their moduli spaces have been of interest for many years. The moduli space of Abelian varieties with a fixed polarisation (and op- tionally a level-n structure for a fixed n) can be constructed from the Siegel upper half space by dividing out the action of the appropriate arithmetic symplectic group.
    [Show full text]
  • Kodaira Dimension in Low Dimensional Topology 3
    KODAIRA DIMENSION IN LOW DIMENSIONAL TOPOLOGY TIAN-JUN LI Abstract. This is a survey on the various notions of Kodaira dimen- sion in low dimensional topology. The focus is on progress after the 2006 survey [78]. Contents 1. Introduction 1 2. κt and κs 2 2.1. The topological Kod dim κt for manifolds up to dimension 3 2 2.2. The symplectic Kod dim κs for 4−manifolds 4 2.3. Symplectic manifolds of dimension 6 and higher 6 3. Calculating κs 6 3.1. Additivity and subadditivity 6 3.2. Behavior under Surgeries 7 4. Main problems and progress in each class 8 4.1. Surfaces and symmetry of κ = −∞ manifolds 8 4.2. Towards the classification of κ = 0 manifolds 11 4.3. Euler number and decomposition of κs = 1 manifolds 15 4.4. Geography and exotic geography of κs = 2 manifolds 16 5. Extensions 17 5.1. Relative Kodaira dim for a symplectic pair 17 5.2. Symplectic manifolds with concave boundary 19 References 19 arXiv:1511.04831v1 [math.SG] 16 Nov 2015 1. Introduction Roughly speaking, a Kodaira dimension type invariant on a class of n−dimensional manifolds is a numerical invariant taking values in the fi- nite set n {−∞, 0, 1, · · · , ⌊ ⌋}, 2 where ⌊x⌋ is the largest integer bounded by x. Date: November 17, 2015. 1 2 TIAN-JUN LI The first invariant of this type is due to Kodaira ([64]) for smooth alge- braic varieties (naturally extended to complex manifolds): Suppose (M, J) is a complex manifold of real dimension 2m. The holomorphic Kodaira dimension κh(M, J) is defined as follows: −∞ if Pl(M, J) = 0 for all l ≥ 1, h κ (M, J)= 0 if Pl(M, J) ∈ {0, 1}, but 6≡ 0 for all l ≥ 1, k k if Pl(M, J) ∼ cl ; c> 0.
    [Show full text]
  • [Math.DG] 27 Jun 2021 Kodaira Dimension & the Yamabe Problem, II
    Kodaira Dimension & the Yamabe Problem, II Michael Albanese and Claude LeBrun∗ June 27, 2021 Abstract For compact complex surfaces (M 4, J) of K¨ahler type, it was pre- viously shown [31] that the sign of the Yamabe invariant Y (M) only depends on the Kodaira dimension Kod(M, J). In this paper, we prove that this pattern in fact extends to all compact complex surfaces ex- cept those of class VII. In the process, we also reprove a result from [2] that explains why the exclusion of class VII is essential here. The Yamabe invariant Y (M) of a smooth compact n-manifold M is a dif- feomorphism invariant that largely captures the global behavior of the scalar curvature s = sg for all possible Riemannian metrics g on M. A cartoon ver- sion of the idea would be to just take the supremum of the scalar curvatures of all unit-volume constant-scalar-curvature metrics on M. However, in the precise definition of the invariant, one only considers those constant-scalar- curvature metrics that are Yamabe minimizers; this does not change the sign of the supremum, but it does in particular guarantee that it is always finite. Thus the Yamabe invariant of M is defined to be arXiv:2106.14333v1 [math.DG] 27 Jun 2021 s dµ Y M g g (M) := sup inf n−2 (1) γ g∈γ ´ n [ M dµg] ´ where dµg denotes the metric volume measure and γ varies over all conformal classes of Riemannian metrics g on M. With this convention, Y (M) > 0 iff M admits a metric of scalar curvature s > 0, whereas Y (M) 0 iff M admits a unit-volume metric of scalar curvature s> ǫ for every ǫ>≥ 0.
    [Show full text]
  • SINTESI DELLA TESI Enriques-Kodaira Classification Of
    Universit`adegli Studi Roma Tre Facolta` di Scienze Matematiche, Fisiche e Naturali Corso di Laurea Magistrale in Matematica SINTESI DELLA TESI Enriques-Kodaira classification of Complex Algebraic Surfaces Candidato Relatore Cristian Minoccheri Prof.ssa Lucia Caporaso Anno Accademico 2011-12 Sessione di Luglio 2012 Our principal aim is to describe and prove the classification of complex algebraic surfaces, a result due to the Italian school of Algebraic Geometry at the beginning of the XX century, and in particular to Enriques. This classification is done up to birational maps, but it is different from the classi- fication of curves. In fact, in the case of curves there is a unique nonsingular projective model for each equivalence class, whereas in the case of surfaces this model is not unique. Therefore we will need to deal with something dif- ferent: the minimal models. In fact, apart from the case of ruled surfaces (i.e. those birational to the product of a curve and the projective line), these are unique and allow us to obtain birationally equivalent surfaces by successive blow-ups. Thus the classification problem will split in two parts: ruled surfaces, which require special considerations, and non-ruled ones, for which it will suffice to classify minimal models. For this, we will need some birational invariants, which capture the geometric peculiarities of each class, as we will see. A first rough classification is achieved by means of Kodaira dimension, κ, which is defined as the largest dimension of the image of the surface in a projective space by the rational map determined by the linear system jnKj, or as −1 if jnKj = ? for every n.
    [Show full text]
  • The Subadditivity of the Kodaira Dimension for Fibrations of Relative
    Math. Res. Lett. Volume 22, Number 3, 675–696, 2015 The subadditivity of the Kodaira dimension for fibrations of relative dimension one in positive characteristics Yifei Chen and Lei Zhang Let f : X → Z be a separable fibration of relative dimension 1 between smooth projective varieties over an algebraically closed field k of positive characteristic. We prove the subadditivity of Kodaira dimension κ(X) ≥ κ(Z)+κ(F ), where F is the generic geometric fiber of f,andκ(F ) is the Kodaira dimension of the normalization of F . Moreover, if dim X =2 anddimZ =1,we have a stronger inequality κ(X) ≥ κ(Z)+κ1(F )whereκ1(F )= o o κ(F, ωF ) is the Kodaira dimension of the dualizing sheaf ωF . 1. Introduction The following conjecture due to Iitaka is of fundamental importance in the classification theory of algebraic varieties over C, the field of complex num- bers: Conjecture 1.1 (Cn,m). Let f : X → Z be a surjective morphism of proper, smooth varieties over C,wheren =dimX and m =dimZ. Assuming the generic geometric fibre F of f is connected, then κ(X) ≥ κ(Z)+κ(F ). This conjecture is proved in the following cases: (1) dim F =1, 2 by Viehweg ([30], [33]); (2) Z is of general type by Kawamata and Viehweg ([19] Theorem 3, [32]); (3) dim Z = 1 by Kawamata ([20]); (4) F has a good minimal model by Kawamata ([21]); (5) F is of general type by Koll´ar ([23]); 675 676 Y. Chen and L. Zhang (6) Z is of maximal Albanese dimension by J.
    [Show full text]
  • Harmonic Theta Series and the Kodaira Dimension of A6
    Harmonic theta series and the Kodaira dimension of A6 Moritz Dittmann1, Riccardo Salvati Manni2, Nils R. Scheithauer1 We construct a basis of the space S14(Sp12(Z)) of Siegel cusp forms of degree 6 and weight 14 consisting of harmonic theta series. One of these functions has vanishing order 2 at the boundary which implies that the Kodaira dimension of A6 is non-negative. 1 Introduction Siegel modular forms are natural generalizations of elliptic modular forms on SL2(Z). They play an important role in number theory and algebraic geometry. One of the main problems in the theory is that it is very difficult to determine the Fourier coefficients of these forms even in simple examples. In this paper we construct a basis of the space S14(Sp12(Z)) consisting of harmonic theta series for Niemeier lattices and calculate some of their Fourier coefficients (see Theorems 4.2 and 4.3). As a consistency check we also compute the Fourier coefficients of the Ikeda lift of the unique normalized cusp form in S22(SL2(Z)). The quotient An = Sp2n(Z)nHn is the (coarse) moduli space of principally polarized complex abelian varieties of dimension n. It was believed for a long time that the spaces An are unirational. This was disproved by Freitag for n = 1 mod 8, n ≥ 17 [F1] and for n = 0 mod 24 [F2]. In the latter work he also showed that the Kodaira dimension of An tends to infinity for n = 0 mod 24. A few years later Tai proved that An is of general type for all n ≥ 9 [Tai].
    [Show full text]
  • [Math.AG] 26 Jan 2003
    NEF DIMENSION OF MINIMAL MODELS FLORIN AMBRO Abstract. We reduce the Abundance Conjecture in dimension 4 to the following numerical statement: if the canonical divisor K is nef and has maximal nef dimension, then K is big. From this point of view, we “classify” in dimension 2 nef divisors which have maximal nef dimension, but which are not big. 0. Introduction A minimal model is a complex projective variety X with at most ter- minal singularities, whose canonical divisor K is numerically effective (nef): K · C ≥ 0 for every curve C ⊂ X. Up to dimension three, min- imal models have a geometrical characterization (Kawamata [9, 12], Miyaoka [14, 15, 16]) : Abundance Conjecture . [11] Let X be a minimal model. Then the linear system |kK| is base point free, for some positive integer k. In dimension four, it is enough to show that X has positive Kodaira dimension if K is not numerically trivial (Kawamata [9], Mori [17]). A direct approach is to first construct the morphism associated to the expected base point free pluricanonical linear systems: 0 f : X → Proj(⊕k≥0H (X,kK)). Since K is nef, f is the unique morphism with connected fibers which contracts exactly the curves C ⊂ X with K · C = 0. Tsuji [22] and Bauer et al [2] have recently solved this existence problem birationally: arXiv:math/0301305v1 [math.AG] 26 Jan 2003 for any nef divisor D on X, there exists a rational dominant map f : X− → Y such that f is regular over the generic point of Y and a very general curve C is contracted by f if and only if D · C = 0.
    [Show full text]
  • Kodaira Dimension and Zeros of Holomorphic One-Forms
    KODAIRA DIMENSION AND ZEROS OF HOLOMORPHIC ONE-FORMS MIHNEA POPA AND CHRISTIAN SCHNELL Abstract. We show that every holomorphic one-form on a smooth complex projective variety of general type must vanish at some point. The proof uses generic vanishing theory for Hodge modules on abelian varieties. Introduction 1. In this article, we use results about Hodge modules on abelian varieties [PS13] to prove two conjectures about zeros of holomorphic one-forms on smooth complex projective varieties. One conjecture was formulated by Hacon and Kov´acs[HK05] and by Luo and Zhang [LZ05]; the former partly attribute the question to Carrell [Car74], and also explain why it is natural to consider varieties of general type. Conjecture 1.1 (Hacon-Kov´acs,Luo-Zhang). If X is a smooth complex projective variety of general type, then the zero locus Z(!) = x 2 X !(TxX) = 0 0 1 of every global holomorphic one-form ! 2 H (X; ΩX ) is nonempty. The statement is a tautology in the case of curves; for surfaces, it was proved by Carrell [Car74], and for threefolds by Luo and Zhang [LZ05]. It was also known to be true if the canonical bundle of X is ample [Zha97], and more generally when X is minimal [HK05]. For varieties that are not necessarily of general type, Luo and Zhang proposed the following more general version of the conjecture, in terms of the Kodaira dimension κ(X), and proved it in the case of threefolds [LZ05]. Conjecture 1.2 (Luo-Zhang). Let X be a smooth complex projective variety, and 0 1 let W ⊆ H (X; ΩX ) be a linear subspace such that Z(!) is empty for every nonzero one-form ! 2 W .
    [Show full text]
  • A SNAPSHOT of the MINIMAL MODEL PROGRAM 1. Introduction
    A SNAPSHOT OF THE MINIMAL MODEL PROGRAM BRIAN LEHMANN Abstract. We briefly review the main goals of the minimal model pro- gram. We then discuss which results are known unconditionally, which are known conditionally, and which are still open. 1. Introduction This survey paper reviews the main goals and results of the Minimal Model Program (henceforth MMP). The paper has three parts. In Section 2, we give a very gentle introduction to the main ideas and conjectures of the MMP. We emphasize why the results are useful for many different areas of algebraic geometry. In Sections 3-6, we take a \snapshot" of the MMP: we describe which results are currently known unconditionally, which are known conditionally, and which are wide open. We aim to state these results precisely, but in a manner which is as useful as possible to as wide a range of mathematicians as possible. Accordingly, this paper becomes more and more technical as we go. The hope is this paper will be helpful for mathematicians looking to apply the results of the MMP in their research. In Section 7, we briefly overview current directions of research which use the MMP. Since the foundations of the MMP are discussed previously, this section focuses on applications. The choice of topics is not comprehensive and is idiosyncratically based on my own knowledge. These notes are not intended to be a technical introduction to the MMP. There are many good introductions to the techniques of the MMP already: [KM98], [Mat02], [HK10], [Kol13b], and many others. These notes are also not intended to be a historical introduction.
    [Show full text]