DOCSLIB.ORG

D-Modules in Birational Geometry

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
D-Modules in Birational Geometry D-modules in birational geometry Contents Introduction vii Chapter 1. Background on filtered D-modules1 1.1. Generalities on D-modules1 1.2. Filtered D-modules4 1.3. Push-forward of D-modules7 1.4. Induced D-modules and filtered differential morphisms 11 1.5. Push-forward of filtered D-modules 18 1.6. Strictness 22 Chapter 2. The Bernstein-Sato polynomial 25 2.1. Push-forward via a graph embedding 25 2.2. Definition, existence, and examples 28 2.3. Quasi-homogeneous singularities 33 2.4. Analytic continuation of the archimedean zeta function 38 2.5. Kashiwara's rationality theorem 42 2.6. Log resolutions, log canonical thresholds, multiplier ideals 47 2.7. Log canonical threshold and jumping numbers as roots 55 Chapter 3. The V -filtration, and more on Bernstein-Sato polynomials 59 3.1. V -filtration: the smooth case 59 3.2. V -filtration: the general case 63 3.3. Nearby and vanishing cycles 69 3.4. Sabbah's description of the V -filtration 73 3.5. Lichtin's theorem and generalizations 78 3.6. Multiplier ideals vs. V -filtration on OX 85 3.7. Minimal exponent 88 Chapter 4. Filtered D-modules and Hodge D-modules 93 4.1. F -filtration and V -filtration 93 4.2. D-modules with Q-structure 97 4.3. Pure Hodge modules 101 4.4. Mixed Hodge modules 105 4.5. Saito's vanishing theorem 109 Chapter 5. Applications to holomorphic forms and families of varieties 113 5.1. Singular loci of morphisms 113 5.2. Zeros of holomorphic 1-forms 115 5.3. Families of varieties of general type 121 v vi CONTENTS Chapter 6. Localization and Hodge ideals 123 6.1. Definition and basic properties 123 6.2. Local properties 135 6.3. Topological vanishing 136 6.4. Vanishing theorem for Hodge ideals 137 6.5. Applications to hypersurfaces in Pn 143 6.6. Applications to singularities of theta divisors 145 Bibliography 149 INTRODUCTION vii Introduction These are notes written for a course I taught at Northwestern during the Winter and Spring quarters of 2020, as well as for a course I am currently (Spring semester 2021) teaching at Harvard. They cover various topics on D-modules, Hodge modules, and especially their connections to birational geometry. At the moment they are limited to the material I was able to cover during those two quarters. My intention is however to gradually add more material, especially on various properties and applications of Hodge modules. Among the topics to be added: vanishing and positivity package for Hodge mod- ules; applications of Hodge modules to generic vanishing, holomorphic forms, families of varieties; Hodge ideals and applications; Hodge filtration on local cohomology. What these notes are not meant to be is a careful introduction to D-modules, or to mixed Hodge modules (or to birational geometry for that matter). For general D-module theory there are several excellent references (as well as lots of online lecture notes). Among these, the main reference that covers essentially all the basic facts needed here is [HTT]. Nevertheless as these notes develop I will include more precise references, and also an introductory chapter with some background on D-modules, as well as on constructions in birational geometry. As for the theory of mixed Hodge modules, the fundamental references are Saito's papers [Sa1] and [Sa2]. There are also his more recent improvement [Sa7] and survey [Sa8]. Schnell's overview of the theory [Sch] is an excellent quick introduction to the topic, while the developing MHM project [MHM] of Sabbah and Schnell is meant to gradually become a comprehensive reference. CHAPTER 1 Background on filtered D-modules This chapter contains a very brief review of basic definitions and facts from the theory of D-modules. The main source for this material, including proofs and further details, is the book [HTT]. For more specialized facts regarding the theory of filtered D-modules, the main source is [Sa1]. 1.1. Generalities on D-modules We consider a smooth complex variety X of dimension n. We denote by DX the sheaf of differential operators on X; this is the sheaf of C-subalgebras of EndC(OX ) generated by OX and TX , where OX acts by multiplication by functions, and we think of the tangent sheaf TX as being the sheaf of derivations DerC(OX ) ⊂ EndC(OX ). Locally in algebraic coordinates x1; : : : ; xn on an affine neighborhood U around a point x 2 X, it can be described as follows. Consider the corresponding local basis @1;:::;@n of vector fields, dual to dx1; : : : ; dxn. They satisfy [@i;@j] = 0 and [@i; xj] = δi;j n α α1 αn for all i and j. If we denote α = (α1; : : : ; αn) 2 N and @ = @1 ··· @n , then we have M α DX (U) = OX (U)@ : α2Nn P α Therefore locally we will write sections of DX as finite sums P = α aα@ , with aα 2 OX (U). The order of P is the maximal jαj = α1 + ··· + αn appearing in this sum. n Remark 1.1.1 (Affine space). When X = A , we can take x1; : : : ; xn to be global coordinates, and we have DX (X) = An, the n-th Weyl algebra. In this case the theory is essentially equivalent to the theory of modules over An; an excellent introduction, including some special features in this case, can be found in [Co]. Definition 1.1.2. The order filtration on DX is defined (locally) by X α FkDX := f aα@ j jαj ≤ k g: α It can be easily checked that this induces a global filtration on X, and FkDX is called the sheaf of differential operators on X of order at most k. It satisfies the following properties: (1) F•DX is an increasing, exhaustive filtration by locally free sheaves of finite rank. (2) F0DX = OX and F1DX ' OX ⊕ TX . (3) FkDX · F`DX ⊆ Fk+`DX for all k and `. (4) If P 2 FkDX and Q 2 F`DX , then [P; Q] 2 Fk+`−1DX . 1 2 1. BACKGROUND ON FILTERED D-MODULES We use the notation F F M F grk DX := FkDX =Fk−1DX and gr• DX := grk DX k2N for the individual, respectively total, associated graded objects. The natural extension to differential operators of the assignment TX (U) −! OX (U)[y1; : : : ; yn];@i 7! yi induces isomorphisms F k F • grk DX ' Sym TX and gr• DX ' Sym TX ; • where Sym TX is the symmetric algebra of the tangent sheaf of X; see [HTT, x1.1]. In particular, if π : T ∗X −! X is the cotangent bundle of X, since by the standard correspondence between vector bundles locally free sheaves (see e.g. [Ha, Ch.II, Exer. ∗ • 5.17, 5.18]) we have T X ' Spec(Sym TX ), it follows that F gr• DX ' π∗OT ∗X : Definition 1.1.3. A sheaf of OX -modules M is a left DX -module if for every open set U ⊆ X, M(U) has a left DX (U)-module structure, compatible with restrictions. Informally, M admits an action by differentiation. An analogous definition can be made for right DX -modules. It is well known that the data of a DX -module structure on M is equivalent to the data of a C-linear map 1 r: M ! M ⊗ ΩX satisfying the properties of an integrable connection. Concretely, for every open set U ⊆ X, and f 2 OX (U) and s 2 M(U) we have (1.1.1) r(fs) = fr(s) + s ⊗ df; and in addition r ◦ r = 0. See for instance [HTT, Lemma 1.2.1]. This interpretation leads to one of the key objects associated to a DX -module: Definition 1.1.4. The de Rham complex of M is the complex h 1 n i DR(M) = M! ΩX ⊗ M ! · · · ! ΩX ⊗ M : with C-linear differentials induced by iterating r. We consider it to be placed in degrees −n; : : : ; 0.1 We next give a few first examples of DX -modules of a geometric nature. They will play an important role throughout. 1We will work with this convention, even though strictly speaking as such it is usually considered to be the de Rham complex associated to the corresponding right D-module. 1.1. GENERALITIES ON D-MODULES 3 Example 1.1.5. (1) The structure sheaf OX is a left DX -module, via the usual differentiation of functions. It is sometimes called the trivial DX -module. (2) More generally, a vector bundle E on X endowed with an integrable (or flat) connection 1 r: E ! E ⊗ ΩX is an example of a DX -module according to the discussion above. Example (1) is the 1 special case corresponding to the standard differential d: OX ! ΩX . It turns out that this is the only way in which a DX -module can be coherent as an OX -module; see [HTT, Theorem 1.4.10]: Theorem 1.1.6. If M is a DX -module which is coherent as an OX -module, then M is locally free, hence an integrable connection. We now turn to examples that are not coherent as OX -modules. (3) Let D be an effective divisor on X, and consider the quasi-coherent OX -module of rational functions which are regular away from D and have with poles of arbitrary order along D, i.e [ OX (∗D) = OX (kD): k≥0 This is sometimes called the localization of OX along D; indeed if U = Spec(R) is an affine open set in X in which D = Z(f) with f 2 R, then OX (∗D) = Rf , the localization of R at f.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Interaction Between Singularity Theory and the Minimal Model
    INTERACTION BETWEEN SINGULARITY THEORY AND THE MINIMAL MODEL PROGRAM CHENYANG XU Abstract. We survey some recent topics on singularities, with a focus on their connection to the minimal model program. This includes the construction and properties of dual complexes, the proof of the ACC conjecture for log canonical thresholds and the recent progress on the ‘local stability theory’ of an arbitrary Kawamata log terminal singularity. 1. Introduction Through out this paper, we will consider algebraic singularities in characteristic 0. It is well known that even if we are mostly interested in smooth varieties, for many different reasons, we have to deal with singular varieties. For the minimal model program (MMP) (also known as Mori’s program), the reason is straightforward, mildly singular varieties are built into the MMP process, and there is no good way to avoid them (see e.g. [KM98]). In fact, the development of the MMP has been intertwined with the progress of our understanding of the corresponding singularity theory, in particular for the classes of singularities preserved by an MMP sequence. This is one of the main reasons why when the theory was started around four decades ago, people spent a lot of time to classify these singularities. However, once we move beyond dimension three, an explicit characterisation of these singularities is often too complicated, and we have to search for a more intrinsic and qualitative method. It turns out that MMP theory itself provides many new tools for the study of singularities. In this note, we will survey some recent progress along these lines. More precisely, we will discuss the construction and properties of dual complexes, the proof of the ACC conjecture for log canonical thresholds, and the recently developed concept of ‘local stability theory’ of an arbitrary Kawamata log terminal singularity.
    [Show full text]
  • Projectivity and Birational Geometry of Bridgeland Moduli Spaces 3
    PROJECTIVITY AND BIRATIONAL GEOMETRY OF BRIDGELAND MODULI SPACES AREND BAYER AND EMANUELE MACR`I Abstract. We construct a family of nef divisor classes on every moduli space of stable complexes in the sense of Bridgeland. This divisor class varies naturally with the Bridge- land stability condition. For a generic stability condition on a K3 surface, we prove that this class is ample, thereby generalizing a result of Minamide, Yanagida, and Yoshioka. Our result also gives a systematic explanation of the relation between wall-crossing for Bridgeland-stability and the minimal model program for the moduli space. We give three applications of our method for classical moduli spaces of sheaves on a K3 surface: 1. We obtain a region in the ample cone in the moduli space of Gieseker-stable sheaves only depending on the lattice of the K3. 2. We determine the nef cone of the Hilbert scheme of n points on a K3 surface of Picard rank one when n is large compared to the genus. 3. We verify the “Hassett-Tschinkel/Huybrechts/Sawon” conjecture on the existence of a birational Lagrangian fibration for the Hilbert scheme in a new family of cases. Contents 1. Introduction 2 2. Review: Bridgeland stability conditions 8 3. Positivity 11 4. A natural nef divisor class on the moduli space, and comparison 15 5. Review: Moduli spaces for stable sheaves on K3 surfaces 17 6. Review: Stability conditions and moduli spaces for objects on K3 surfaces 20 7. K3 surfaces: Projectivity of moduli spaces 24 arXiv:1203.4613v2 [math.AG] 10 Oct 2013 8.
    [Show full text]
  • A Geometric Approach to Problems in Birational Geometry
    A geometric approach to problems in birational geometry Chen-Yu Chi1 and Shing-Tung Yau Department of Mathematics, Harvard University, Cambridge, MA 02138; Edited by Richard V. Kadison, University of Pennsylvania, Philadelphia, PA, and approved September 15, 2008 (received for review July 18, 2008) A classical set of birational invariants of a variety are its spaces We call this kind of theorem a Torelli-type theorem because of pluricanonical forms and some of their canonically defined sub- the classical Torelli theorem says that the periods of inte- spaces. Each of these vector spaces admits a typical metric struc- grals determine an algebraic curve. This remarkable theorem ture which is also birationally invariant. These vector spaces so was generalized to higher-dimensional algebraic varieties. The metrized will be referred to as the pseudonormed spaces of the most notable one was the work of Piatetsky-Shapiro and Sha- original varieties. A fundamental question is the following: Given farevich (3) for algebraic K3 surfaces, which was generalized two mildly singular projective varieties with some of the first vari- to Kähler K3s by Burns and Rapoport (4), where they proved ety’s pseudonormed spaces being isometric to the corresponding the injectivity of period maps. The surjectivity of period maps ones of the second variety’s, can one construct a birational map for K3 surfaces was done using Ricci flat metrics by Siu (5) between them that induces these isometries? In this work, a posi- and Todorov (6) following the work of Kulikov (7) and of tive answer to this question is given for varieties of general type.
    [Show full text]
  • THE BIRATIONAL GEOMETRY of TROPICAL COMPACTIFICATIONS Colin Diemer a Dissertation in Mathematics Presented to the Faculties of T
    THE BIRATIONAL GEOMETRY OF TROPICAL COMPACTIFICATIONS Colin Diemer A Dissertation in Mathematics Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy 2010 Antonella Grassi, Professor of Mathematics Supervisor of Dissertation Tony Pantev, Professor of Mathematics Graduate Group Chairperson Dissertation Committee Antonella Grassi, Professor of Mathematics Tony Pantev, Professor of Mathematics Ted Chinburg, Professor of Mathematics Acknowledgments I would like thank my advisor Antonella Grassi for suggesting tropical geometry as a possible dissertation topic (and for her truly commendable patience and willingness to help). Ron Donagi, Tony Pantev, Jonathan Block, and Erik van Erp also helped me find my way mathematically here at Penn. Conversations with Paul Hacking, Jenia Tevelev, Eric Katz, Ilia Zharkov, Eduardo Cattani, and Alicia Dickenstein were beneficial during the preparation of this thesis. This thesis is dedicated to my father. ii ABSTRACT THE BIRATIONAL GEOMETRY OF TROPICAL COMPACTIFICATIONS Colin Diemer Antonella Grassi, Advisor We study compactifications of subvarieties of algebraic tori using methods from the still developing subject of tropical geometry. Associated to each \tropical" compactification is a polyhedral object called a tropical fan. Techniques developed by Hacking, Keel, and Tevelev [19, 45] relate the polyhedral geometry of the tropical variety to the algebraic geometry of the compactification. We compare these constructions to similar classical constructions. The main results of this thesis involve the application of methods from logarithmic geometry in the sense of Iitaka [22] to these compactifications. We derive a precise formula for the log Kodaira dimension and log irregularity in terms of polyhedral geometry.
    [Show full text]