Rational Curves On K3 Surfaces
|
|
- Easter Parrish
- 6 years ago
- Views:
Transcription
1 Rational Curves On K3 Surfaces Jun Li Department of Mathematics Stanford University Conference in honor of Peter Li
2 Overview of the talk The problem: existence of rational curves on a K3 surface The conjecture: infinitely many rational curves on an algebraic K3 The technique: lifting from a special K3 to a general K3 The new approach: of Bogomolov-Hassett-Tschinkel, using the jumping of Picard ranks Theorem (L-Liedtke): When ρ(x ) is odd, then X contains infinitely many rational curves.
3 The Problem Problem: Find rational curves in a complex K3 surfaces. K3 surfaces: smooth, compact complex surface S satisfying π 1 (S) = {1}; K S = OS ; Examples: this is 2 T C X = O S ; By Yau s celebrated theorem on Calabi conjecture, Complex K3 surfaces are Ricci flat Kahler metrics, and vice versa. degree four hypersurfaces: S = (p 4 = 0) CP 3 ; resolutions of a torus quotient by involution: S = (T 2 C / τ )desing.
4 Rational curves: image of a u : CP 1 S non-constant; generic one-one onto image; holomorphic; distinct rational curves if they have different images. Distinct rational curves: if two have distinct images.
5 Main questions: given a compact, complex K3 surface S, Q1 : does S contain a rational curve? Q2 : if does, does it contain more than one rational curves? Q1 : if does, does it contain infinitely many rational curves? Q1 : if does, does it contain a continuous family of rational curves?
6 The existence of (at least) one rational curve Theorem (Bogomolov-Mumford) The existence of L = OS produces rational curves. Outline L = OS produces s 0 H 0 (L n ), C = s 1 (0) a curve in S; c 1 (L) 2 = 0 or = 2 are ok; (elliptic fibration; ( 2)-curves); c 1 (L) 2 > 0 implies S is algebraic (projective).
7 For S projective, find a K3 X 0 so that it has two smooth rationals C 1 and C 2 so that they intersects transversally; let L 0 = O X0 (C 1 + C 2 ); find a deformation (X t, L t ) of (X 0, L 0 ), with (X 1, L 1 ) = (S, L); show that C 1 C 2 X 0 deforms to rationals C t X t, for general t; (this argument will be revisited) the limit of rational curves are union of rational curves, so X 1 = S contains rational curves.
8 Theorem (Lefschetz) A simply connected compact complex manifold S has a line bundle L = OS if (the Hodge classes) H 1,1 (S, C) H 2 (S, Z) {0}. Answer to Q1: S has a rational curve if H 1,1 (S, C) H 2 (S, Z) {0}.
9 Remarks We have three different proofs of this existence theorem Bogomolov-Mumford: find the existence of a special K3, using deformation to prove the existence for general K3, plus limit of rational curves is a union rational curves; Mori: using reduction to char= p technique, not relying on deformation of K3 surfaces; Yau-Zaslow: using moduli of sheaves on K3, plus beautiful geometry of Hyperekahler manifolds; not relying on deformation of K3 surfaces; the theorem holds for any field k = k.
10 The existence of more than one rational curves Corollary: The space H 1,1 (S, C) H 2 (S, Q) {0} is spanned by classes generated by rational curves. Answer to Q2: The number of rational curves is # dim H 1,1 (S, C) H 2 (S, Q).
11 No family of rational curves Answer to Q4: There can be no continuous family of rational curves. Otherwise, a. we can find a birational morphism ϕ : X S, X is a ruled surface (generic P 1 -bundle); b. S K3 implies there is a non-zero (2, 0)-form on S; c. ϕ (Ω) = 0 from (b), 0 from (a), a contradiction.
12 The Conjecture (on Q3) Conjecture: Any smooth complex K3 surface S contains infinitely many rational curves. This is motivated by Lang s conjecture: Lang Conjecture: Let X be a general type complex manifold. Then the union of the images of holomorphic u : C X lies in a finite union of proper subvarieties of X.
13 Key to the existence of rational curves: A class α 0 H 2 (S, Z) is Hodge (i.e. H 1,1 (S, C) H 2 (S, Q)) is necessary and sufficient for the existence of a union of rational curves C i so that [C i ] = α. Example: Say we can have a family S t, t disk, α H 1,1 (S 0, C) H 2 (S 0, Q) so that S 0 has C 0 = CP 1 S 0 with [C 0 ] = α; in case α H 1,1 (S t, C) for general t, then CP 1 = C 0 S 0 can not be extended to holomorphic u t : CP 1 S t.
14 Preparations We will consider polarized K3 surfaces (S, H), c 1 (H) > 0; we can group them according to H 2 = 2d: M 2d = {(S, H) H 2 = 2d}. each M 2d is smooth, of dimension 19; each M 2d is defined over Z. (defined by equation with coefficients in Z.) to show that (S, H) contains infinitely many rational curves, it suffices to show that for any N, there is a rational curve R S so that [R] H N. we define ρ(s) = dim H 1,1 (S, C) H 2 (S, Q), called the rank of the Picard group of S.
15 The existence of rational curves on complex algebraic K3 surfaces Theorem (Chen, 99): Very general polarized complex K3 surfaces (i.e. (S, H)) contains infinitely many rational curves. Theorem: A complex elliptic K3 surface contains infinitely many rational curves. Theorem: A complex elliptic K3 surface with infinite automorphism group contains infinitely many rational curves.
16 Theorem (Bogomolov-Hassett-Tschinkel, 10) A degree 2 polarized complex K3 surface (S, H) (i.e. H 2 = 2) having ρ(s) = 1 contains infinitely many rational curves. Extending their method, we prove Theorem (L Liedtke, 11) A polarized complex K3 surface (S, H) having ρ(s) = 1 contains infinitely many rational curves. ρ can run between 1 and 20; A K3 with ρ 5 is an elliptic K3, thus covered by an earlier theorem.
17 Technical approach Extension Problem: a family of K3 surface S t, t T (a parameter space); C 0 S 0 a union of rational curves; let α = [C 0 ] H 2 (S, Z); (it is a Hodge class); suppose α H 1,1 (S t, C) for all t T ; We like to show that exists a family of curves C t S t, such that Ct are union of rational curves; C0 = C 0.
18 Use moduli of genus 0 stable maps A genus zero stable map is a holomorphic map u : C X such that C is a nodal, genus 0 complex curve (i.e. a tree of CP 1 s); for any Σ = CP 1 C with u Σ = const., Σ contains 3 nodes of C.
19 Moduli of genus 0 stable maps A genus zero stable map is a holomorphic map u : C X such that C is a nodal, genus 0 complex curve (i.e. a tree of CP 1 s); for any Σ = CP 1 C with u Σ = const., Σ contains 3 nodes of C. Let S be a K3 surface, α H 2 (S, Z), M 0 (S, α) = {[u : C S] genus 0 stable maps with u ([C]) = α} is an algebraic space; has exp.dimension 1. (expected, as when α is not Hodge, then M0 =.)
20 Extension Lemma S t, t T, be a family of K3 surfaces; α H 2 (S 0, Z) is Hodge for all t; [u 0 ] M 0 (S 0, α). Extension Lemma (Ran, Bogomolov-Tschinkel, ): Suppose [u 0 ] M 0 (S 0, α) is isolated, then u 0 extends to u t M 0 (S t, α) for general t T.
21 Extension Lemma (Ran, Bogomolov-Tschinkel, ): Suppose [u 0 ] M 0 (S 0, α) is isolated, then u 0 extends to u t M 0 (S t, α) for general t T. Outline of the proof let U be an open neighborhood of all deformations of S 0 ; U is a smooth complex manifold of C dimension 20; let X U be the tautological family of K3 surfaces; form M 0 (X, α), the moduli of genus zero stable morphisms u : C X z, for some z U. exp. dim M 0 (S 0, α) = 1 plus dim U = 20 implies exp. dim M 0 (X, α) = 19. Let [u 0 ] W M 0 (X, α) be an irreducible component; dim W 19.
22
23 Look at π : W U, π(w ) U has dimension at least 19; U(α) U is the locus where α H 2 (S z, Z) remain Hodge; dim U(α) = 19; π(w ) U(α); by assumption, T U(α); This implies that T π(w ); thus u 0 lifts to a family u t : C t S t, t T.
24 Extension criterion Definition: We say a map [u] M 0 (S, α) rigid if [u] is an isolated point in M 0 (S, α). Extension principle: In case (a genus zero stable map) u : C S is rigid, then u extends to nearby K3 surfaces as long as the class u [C] H 2 (S, Z) remains Hodge.
25 Lifting to a general K3 surface Given an (S, H) and an integer N, to construct a rational R S of [R] H N, we will find a family of K3 surfaces (X t, H t ), t T a parameter space, such that (S, H) is a general member of (Xt, H t); for a 0 T, we have C0 X 0 a union of rational curves, and an irreducible R 0 C 0 so that [R 0] H 0 N; [C0] = nh 0; we can represent C0 as the image of a rigid genus zero stable map [u 0]. By Extension criterion: we can extend u 0 to u t for general t; since t is general, there is an irreducible component R t of image(u t ) so that [R t ] H t [R 0 ] H 0 N; since (S, H) is a general member, we can assume (S, H) = (X t, H t ).
26
27 Theorem (Chen) Very general K3 (S, H) M 2d contains infinitely many rational curves. Proof Form a family, so that (X 0, H 0 ) is a singular K3 surface, thus C 0 X 0 can be constructed explicitly. Applying the lifting property. So for any N, all K3 s in a dense open subset in M 2d contains rational curves of degree N.
28 Theorem (Bogomolov-Hassett-Tschikel). Fix an r. Assume that every K3 (X, H) M 2d (of ρ(x ) = r) defined over a number field K contains infinitely many rational curves in X Q, then every (X, H) M 2d contains infinitely many rational curves. Let X be any K3, not defined over a number field, then X is the generic fiber of X S, S a variety over K; Example X = (πx x x x 4 4 = 0) CP3. Switch π by t, we define X = (tx x x x 4 4 = 0) P3 A 1, S = A 1.
29 Theorem (Bogomolov-Hassett-Tschikel). Fix an r. Assume that every K3 (X, H) M 2d (of ρ(x ) = r) defined over a number field K contains infinitely many rational curves in X Q, then every complex (X, H) M 2d contains infinitely many rational curves. Let X be any K3. Then X is the generic fiber of X S, S a variety over K; By (Deligne), we can find ξ S, over L/K, s.t. PicX = PicX ξ ; By assumption, C 0 X ξ, high degree; [C 0 X ξ ] M 0 (X ξ ) rigid; applying lifting criterion, [C 0 X ξ ] lifts to generic fiber of X B; thus lifts to C X, of high degree.
30 When (S, H) is defined over a number field, say Q, after deleting a finite prime numbers, (S, H) is the generic member of a family of K3 over T = specz finite places:
31 Theorem (Bogomolov-Zarhin, Nygaard-Ogus). Let X be a K3 over a number field K. Then 1. for all but finitely many places p of K, the reductions X p are smooth, not supersingular; 2. for all such places with char p 5, ρ((x p ) F p ) are even. Proof Use Tate Conjecture for K3; Use Weil Conjecture for K3.
32 Theorem (Bogomolov-Hassett-Tschinkel). Let (X, H) be a polarized K3 in M 2. Suppose ρ(x ) = 1, then X contains infinitely many rational curves. Sketch we only need to prove the case where X is defined over K, say K = Q; for any places p, ρ((x p ) F p ) 2 imply D p (X p ) F p rational curves, not in ZH; if D p H N uniformly, implies ρ(x Q ) 2, a contradiction;
33 Sketch places p, D p (X p ) F p rational curves, not in ZH, D p H arbitrary large; using H 2 = 2, find D p (X p ) F p rational curves, D p + D p n p H ; D p + D p is the image of a rigid genus zero stable map. Applying extension criterion, we can find a rational curve of degree N in the generic fiber, which is X.
34 Theorem ( Liedtke) Let (X, H) be a polarized complex K3 surface such that ρ(x ) is odd. Then X contains infinitely many rational curves. Outline of proof We only need to prove the Theorem for (X, H) defined over a number field K; say K = Q, we get a family X p for every prime p Z, X is the generic member of this family; p, exists D p X p, D p ZH, we have sup Dp H ; pick C p X p union of rationals, D p + C p n p H Difficulty: D p + C p may not be representable as the image of a rigid genus zero stable map.
35 Solution: Suppose we can find a nodal rational curves R X, of class kh for some k, then for some large m we can represent C p + D p + mr, which is a class in (n + mk)h, by a rigid genus zero stable map.
36 End of the proof: In general, X may not contain any nodal rational curve in kh. However, we know a small deformation of X in M 2d contains nodal rational curves in kh (a Theorem of Chen). Using this, plus some further algebraic geometry argument, we can complete the proof.
37 Happy Birthday, Peter.
Gromov-Witten invariants and Algebraic Geometry (II) Jun Li
Gromov-Witten invariants and Algebraic Geometry (II) Shanghai Center for Mathematical Sciences and Stanford University GW invariants of quintic Calabi-Yau threefolds Quintic Calabi-Yau threefolds: X =
More informationA NEW FAMILY OF SYMPLECTIC FOURFOLDS
A NEW FAMILY OF SYMPLECTIC FOURFOLDS OLIVIER DEBARRE This is joint work with Claire Voisin. 1. Irreducible symplectic varieties It follows from work of Beauville and Bogomolov that any smooth complex compact
More informationCorollary. Let X Y be a dominant map of varieties, with general fiber F. If Y and F are rationally connected, then X is.
1 Theorem. Let π : X B be a proper morphism of varieties, with B a smooth curve. If the general fiber F of f is rationally connected, then f has a section. Corollary. Let X Y be a dominant map of varieties,
More informationFAKE PROJECTIVE SPACES AND FAKE TORI
FAKE PROJECTIVE SPACES AND FAKE TORI OLIVIER DEBARRE Abstract. Hirzebruch and Kodaira proved in 1957 that when n is odd, any compact Kähler manifold X which is homeomorphic to P n is isomorphic to P n.
More informationSpace of surjective morphisms between projective varieties
Space of surjective morphisms between projective varieties -Talk at AMC2005, Singapore- Jun-Muk Hwang Korea Institute for Advanced Study 207-43 Cheongryangri-dong Seoul, 130-722, Korea jmhwang@kias.re.kr
More informationThe geometry of Landau-Ginzburg models
Motivation Toric degeneration Hodge theory CY3s The Geometry of Landau-Ginzburg Models January 19, 2016 Motivation Toric degeneration Hodge theory CY3s Plan of talk 1. Landau-Ginzburg models and mirror
More informationHYPER-KÄHLER FOURFOLDS FIBERED BY ELLIPTIC PRODUCTS
HYPER-KÄHLER FOURFOLDS FIBERED BY ELLIPTIC PRODUCTS LJUDMILA KAMENOVA Abstract. Every fibration of a projective hyper-kähler fourfold has fibers which are Abelian surfaces. In case the Abelian surface
More informationThe Cone Theorem. Stefano Filipazzi. February 10, 2016
The Cone Theorem Stefano Filipazzi February 10, 2016 These notes are supposed to be a handout for the student seminar in algebraic geometry at the University of Utah. In this seminar, we will give an overview
More informationVanishing theorems and holomorphic forms
Vanishing theorems and holomorphic forms Mihnea Popa Northwestern AMS Meeting, Lansing March 14, 2015 Holomorphic one-forms and geometry X compact complex manifold, dim C X = n. Holomorphic one-forms and
More informationDensity of rational points on Enriques surfaces
Density of rational points on Enriques surfaces F. A. Bogomolov Courant Institute of Mathematical Sciences, N.Y.U. 251 Mercer str. New York, (NY) 10012, U.S.A. e-mail: bogomolo@cims.nyu.edu and Yu. Tschinkel
More informationOn Mordell-Lang in Algebraic Groups of Unipotent Rank 1
On Mordell-Lang in Algebraic Groups of Unipotent Rank 1 Paul Vojta University of California, Berkeley and ICERM (work in progress) Abstract. In the previous ICERM workshop, Tom Scanlon raised the question
More informationSPACES OF RATIONAL CURVES IN COMPLETE INTERSECTIONS
SPACES OF RATIONAL CURVES IN COMPLETE INTERSECTIONS ROYA BEHESHTI AND N. MOHAN KUMAR Abstract. We prove that the space of smooth rational curves of degree e in a general complete intersection of multidegree
More informationDynamics and Canonical Heights on K3 Surfaces with Noncommuting Involutions Joseph H. Silverman
Dynamics and Canonical Heights on K3 Surfaces with Noncommuting Involutions Joseph H. Silverman Brown University Conference on the Arithmetic of K3 Surfaces Banff International Research Station Wednesday,
More informationON A THEOREM OF CAMPANA AND PĂUN
ON A THEOREM OF CAMPANA AND PĂUN CHRISTIAN SCHNELL Abstract. Let X be a smooth projective variety over the complex numbers, and X a reduced divisor with normal crossings. We present a slightly simplified
More informationGeneralized Tian-Todorov theorems
Generalized Tian-Todorov theorems M.Kontsevich 1 The classical Tian-Todorov theorem Recall the classical Tian-Todorov theorem (see [4],[5]) about the smoothness of the moduli spaces of Calabi-Yau manifolds:
More informationLAGRANGIAN HOMOLOGY CLASSES WITHOUT REGULAR MINIMIZERS
LAGRANGIAN HOMOLOGY CLASSES WITHOUT REGULAR MINIMIZERS JON WOLFSON Abstract. We show that there is an integral homology class in a Kähler-Einstein surface that can be represented by a lagrangian twosphere
More informationON THE NUMBER AND BOUNDEDNESS OF LOG MINIMAL MODELS OF GENERAL TYPE
ON THE NUMBER AND BOUNDEDNESS OF LOG MINIMAL MODELS OF GENERAL TYPE DILETTA MARTINELLI, STEFAN SCHREIEDER, AND LUCA TASIN Abstract. We show that the number of marked minimal models of an n-dimensional
More informationSPACES OF RATIONAL CURVES ON COMPLETE INTERSECTIONS
SPACES OF RATIONAL CURVES ON COMPLETE INTERSECTIONS ROYA BEHESHTI AND N. MOHAN KUMAR Abstract. We prove that the space of smooth rational curves of degree e on a general complete intersection of multidegree
More informationlocus such that Γ + F = 0. Then (Y, Γ = Γ + F ) is kawamata log terminal, and K Y + Γ = π (K X + ) + E + F. Pick H ample such that K Y + Γ + H is nef
17. The MMP Using the cone (16.6) and contraction theorem (16.3), one can define the MMP in all dimensions: (1) Start with a kawamata log terminal pair (X, ). (2) Is K X + nef? Is yes, then stop. (3) Otherwise
More informationSupersingular K3 Surfaces are Unirational
Supersingular K3 Surfaces are Unirational Christian Liedtke (Technical University Munich) August 7, 2014 Seoul ICM 2014 Satellite Conference, Daejeon Lüroth s theorem Given a field k a field, Lüroth s
More informationON THE SLOPE_AND KODAIRA DIMENSION OF M g. FOR SMALL g
J. DIFFERENTIAL GEOMETRY 34(11) 26-24 ON THE SLOPE_AND KODAIRA DIMENSION OF M g FOR SMALL g MEI-CHU CHANG & ZIV RAN Let M denote the moduli space of smooth curves of genus g and let σ be its stable compactification;
More informationPullbacks of hyperplane sections for Lagrangian fibrations are primitive
Pullbacks of hyperplane sections for Lagrangian fibrations are primitive Ljudmila Kamenova, Misha Verbitsky 1 Dedicated to Professor Claire Voisin Abstract. Let p : M B be a Lagrangian fibration on a hyperkähler
More informationStable bundles on CP 3 and special holonomies
Stable bundles on CP 3 and special holonomies Misha Verbitsky Géométrie des variétés complexes IV CIRM, Luminy, Oct 26, 2010 1 Hyperkähler manifolds DEFINITION: A hyperkähler structure on a manifold M
More informationON THE NUMBER AND BOUNDEDNESS OF LOG MINIMAL MODELS OF GENERAL TYPE
ON THE NUMBER AND BOUNDEDNESS OF LOG MINIMAL MODELS OF GENERAL TYPE DILETTA MARTINELLI, STEFAN SCHREIEDER, AND LUCA TASIN Abstract. We show that the number of marked minimal models of an n-dimensional
More informationLIFTABILITY OF FROBENIUS
LIFTABILITY OF FROBENIUS (JOINT WITH J. WITASZEK AND M. ZDANOWICZ) Throughout the tal we fix an algebraically closed field of characteristic p > 0. 1. Liftability to(wards) characteristic zero It often
More informationStructure theorems for compact Kähler manifolds
Structure theorems for compact Kähler manifolds Jean-Pierre Demailly joint work with Frédéric Campana & Thomas Peternell Institut Fourier, Université de Grenoble I, France & Académie des Sciences de Paris
More informationAlgebraic geometry over quaternions
Algebraic geometry over quaternions Misha Verbitsky November 26, 2007 Durham University 1 History of algebraic geometry. 1. XIX centrury: Riemann, Klein, Poincaré. Study of elliptic integrals and elliptic
More informationEXAMPLES OF CALABI-YAU 3-MANIFOLDS WITH COMPLEX MULTIPLICATION
EXAMPLES OF CALABI-YAU 3-MANIFOLDS WITH COMPLEX MULTIPLICATION JAN CHRISTIAN ROHDE Introduction By string theoretical considerations one is interested in Calabi-Yau manifolds since Calabi-Yau 3-manifolds
More informationKähler manifolds and variations of Hodge structures
Kähler manifolds and variations of Hodge structures October 21, 2013 1 Some amazing facts about Kähler manifolds The best source for this is Claire Voisin s wonderful book Hodge Theory and Complex Algebraic
More informationTopics in Geometry: Mirror Symmetry
MIT OpenCourseWare http://ocw.mit.edu 18.969 Topics in Geometry: Mirror Symmetry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MIRROR SYMMETRY:
More informationCohomology jump loci of quasi-projective varieties
Cohomology jump loci of quasi-projective varieties Botong Wang joint work with Nero Budur University of Notre Dame June 27 2013 Motivation What topological spaces are homeomorphic (or homotopy equivalent)
More informationLogarithmic geometry and rational curves
Logarithmic geometry and rational curves Summer School 2015 of the IRTG Moduli and Automorphic Forms Siena, Italy Dan Abramovich Brown University August 24-28, 2015 Abramovich (Brown) Logarithmic geometry
More informationRATIONALLY INEQUIVALENT POINTS ON HYPERSURFACES IN P n
RATIONALLY INEQUIVALENT POINTS ON HYPERSURFACES IN P n XI CHEN, JAMES D. LEWIS, AND MAO SHENG Abstract. We prove a conjecture of Voisin that no two distinct points on a very general hypersurface of degree
More informationALGEBRAICALLY TRIVIAL, BUT TOPOLOGICALLY NON-TRIVIAL MAP. Contents 1. Introduction 1
ALGEBRAICALLY TRIVIAL, BUT TOPOLOGICALLY NON-TRIVIAL MAP HONG GYUN KIM Abstract. I studied the construction of an algebraically trivial, but topologically non-trivial map by Hopf map p : S 3 S 2 and a
More informationCANONICAL BUNDLE FORMULA AND VANISHING THEOREM
CANONICAL BUNDLE FORMULA AND VANISHING THEOREM OSAMU FUJINO Abstract. In this paper, we treat two different topics. We give sample computations of our canonical bundle formula. They help us understand
More informationCOUNTING ELLIPTIC PLANE CURVES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 12, December 1997, Pages 3471 3479 S 0002-9939(97)04136-1 COUNTING ELLIPTIC PLANE CURVES WITH FIXED j-invariant RAHUL PANDHARIPANDE (Communicated
More informationPRIME EXCEPTIONAL DIVISORS ON HOLOMORPHIC SYMPLECTIC VARIETIES AND MONODROMY-REFLECTIONS
PRIME EXCEPTIONAL DIVISORS ON HOLOMORPHIC SYMPLECTIC VARIETIES AND MONODROMY-REFLECTIONS EYAL MARKMAN Abstract. Let X be a projective irreducible holomorphic symplectic variety. The second integral cohomology
More informationThis theorem gives us a corollary about the geometric height inequality which is originally due to Vojta [V].
694 KEFENG LIU This theorem gives us a corollary about the geometric height inequality which is originally due to Vojta [V]. Corollary 0.3. Given any ε>0, there exists a constant O ε (1) depending on ε,
More informationThe Strominger Yau Zaslow conjecture
The Strominger Yau Zaslow conjecture Paul Hacking 10/16/09 1 Background 1.1 Kähler metrics Let X be a complex manifold of dimension n, and M the underlying smooth manifold with (integrable) almost complex
More informationHYPERKÄHLER MANIFOLDS
HYPERKÄHLER MANIFOLDS PAVEL SAFRONOV, TALK AT 2011 TALBOT WORKSHOP 1.1. Basic definitions. 1. Hyperkähler manifolds Definition. A hyperkähler manifold is a C Riemannian manifold together with three covariantly
More informationThe geography of irregular surfaces
Università di Pisa Classical Algebraic Geometry today M.S.R.I., 1/26 1/30 2008 Summary Surfaces of general type 1 Surfaces of general type 2 3 and irrational pencils Surface = smooth projective complex
More informationHolomorphic line bundles
Chapter 2 Holomorphic line bundles In the absence of non-constant holomorphic functions X! C on a compact complex manifold, we turn to the next best thing, holomorphic sections of line bundles (i.e., rank
More information7. Classification of Surfaces The key to the classification of surfaces is the behaviour of the canonical
7. Classification of Surfaces The key to the classification of surfaces is the behaviour of the canonical divisor. Definition 7.1. We say that a smooth projective surface is minimal if K S is nef. Warning:
More informationConstructing compact 8-manifolds with holonomy Spin(7)
Constructing compact 8-manifolds with holonomy Spin(7) Dominic Joyce, Oxford University Simons Collaboration meeting, Imperial College, June 2017. Based on Invent. math. 123 (1996), 507 552; J. Diff. Geom.
More informationSIMONS SYMPOSIUM 2015: OPEN PROBLEM SESSIONS
SIMONS SYMPOSIUM 2015: OPEN PROBLEM SESSIONS The 2015 Simons Symposium on Tropical and Nonarchimedean Geometry included two open problem sessions, in which leading experts shared questions and conjectures
More informationPROJECTIVE BUNDLES OF SINGULAR PLANE CUBICS STEFAN KEBEKUS
PROJECTIVE BUNDLES OF SINGULAR PLANE CUBICS STEFAN KEBEKUS ABSTRACT. Classification theory and the study of projective varieties which are covered by rational curves of minimal degrees naturally leads
More informationALGEBRAIC HYPERBOLICITY OF THE VERY GENERAL QUINTIC SURFACE IN P 3
ALGEBRAIC HYPERBOLICITY OF THE VERY GENERAL QUINTIC SURFACE IN P 3 IZZET COSKUN AND ERIC RIEDL Abstract. We prove that a curve of degree dk on a very general surface of degree d 5 in P 3 has geometric
More informationASIAN J. MATH. cfl 2002 International Press Vol. 6, No. 1, pp , March THE WEAK LEFSCHETZ PRINCIPLE IS FALSE FOR AMPLE CONES Λ BRENDAN
ASIAN J. MATH. cfl 2002 International Press Vol. 6, No. 1, pp. 095 100, March 2002 005 THE WEAK LEFSCHETZ PRINCIPLE IS FALSE FOR AMPLE CONES Λ BRENDAN HASSETT y, HUI-WEN LIN z, AND CHIN-LUNG WANG x 1.
More informationThe hyperbolic Ax-Lindemann-Weierstraß conjecture
The hyperbolic Ax-Lindemann-Weierstraß conjecture B. Klingler (joint work with E.Ullmo and A.Yafaev) Université Paris 7 ICM Satellite Conference, Daejeon Plan of the talk: (1) Motivation: Hodge locus and
More informationTHE QUANTUM CONNECTION
THE QUANTUM CONNECTION MICHAEL VISCARDI Review of quantum cohomology Genus 0 Gromov-Witten invariants Let X be a smooth projective variety over C, and H 2 (X, Z) an effective curve class Let M 0,n (X,
More informationMIXED HODGE MODULES PAVEL SAFRONOV
MIED HODGE MODULES PAVEL SAFRONOV 1. Mixed Hodge theory 1.1. Pure Hodge structures. Let be a smooth projective complex variety and Ω the complex of sheaves of holomorphic differential forms with the de
More informationLinear systems and Fano varieties: introduction
Linear systems and Fano varieties: introduction Caucher Birkar New advances in Fano manifolds, Cambridge, December 2017 References: [B-1] Anti-pluricanonical systems on Fano varieties. [B-2] Singularities
More informationBjorn Poonen. MSRI Introductory Workshop on Rational and Integral Points on Higher-dimensional Varieties. January 17, 2006
University of California at Berkeley MSRI Introductory Workshop on Rational and Integral Points on Higher-dimensional (organized by Jean-Louis Colliot-Thélène, Roger Heath-Brown, János Kollár,, Alice Silverberg,
More informationON THE INTEGRAL HODGE CONJECTURE FOR REAL THREEFOLDS. 1. Motivation
ON THE INTEGRAL HODGE CONJECTURE FOR REAL THREEFOLDS OLIVIER WITTENBERG This is joint work with Olivier Benoist. 1.1. Work of Kollár. 1. Motivation Theorem 1.1 (Kollár). If X is a smooth projective (geometrically)
More informationHyperkähler manifolds
Hyperkähler manifolds Kieran G. O Grady May 20 2 Contents I K3 surfaces 5 1 Examples 7 1.1 The definition..................................... 7 1.2 A list of examples...................................
More informationLAGRANGIAN SUBVARIETIES OF ABELIAN FOURFOLDS. Fedor Bogomolov and Yuri Tschinkel
LAGRANGIAN SUBVARIETIES OF ABELIAN FOURFOLDS by Fedor Bogomolov and Yuri Tschinkel 2 FEDOR BOGOMOLOV and YURI TSCHINKEL Dedicated to Professor K. Kodaira 1. Introduction Let (W, ω) be a smooth projective
More informationBRILL-NOETHER THEORY. This article follows the paper of Griffiths and Harris, "On the variety of special linear systems on a general algebraic curve.
BRILL-NOETHER THEORY TONY FENG This article follows the paper of Griffiths and Harris, "On the variety of special linear systems on a general algebraic curve." 1. INTRODUCTION Brill-Noether theory is concerned
More informationFinite generation of pluricanonical rings
University of Utah January, 2007 Outline of the talk 1 Introduction Outline of the talk 1 Introduction 2 Outline of the talk 1 Introduction 2 3 The main Theorem Curves Surfaces Outline of the talk 1 Introduction
More informationGeometry of the Calabi-Yau Moduli
Geometry of the Calabi-Yau Moduli Zhiqin Lu 2012 AMS Hawaii Meeting Department of Mathematics, UC Irvine, Irvine CA 92697 March 4, 2012 Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 1/51
More informationHOW TO CLASSIFY FANO VARIETIES?
HOW TO CLASSIFY FANO VARIETIES? OLIVIER DEBARRE Abstract. We review some of the methods used in the classification of Fano varieties and the description of their birational geometry. Mori theory brought
More informationMathematical Research Letters 2, (1995) A VANISHING THEOREM FOR SEIBERG-WITTEN INVARIANTS. Shuguang Wang
Mathematical Research Letters 2, 305 310 (1995) A VANISHING THEOREM FOR SEIBERG-WITTEN INVARIANTS Shuguang Wang Abstract. It is shown that the quotients of Kähler surfaces under free anti-holomorphic involutions
More informationGeometry of moduli spaces
Geometry of moduli spaces 20. November 2009 1 / 45 (1) Examples: C: compact Riemann surface C = P 1 (C) = C { } (Riemann sphere) E = C / Z + Zτ (torus, elliptic curve) 2 / 45 (2) Theorem (Riemann existence
More informationLECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS
LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS WEIMIN CHEN, UMASS, SPRING 07 1. Basic elements of J-holomorphic curve theory Let (M, ω) be a symplectic manifold of dimension 2n, and let J J (M, ω) be
More informationCHAPTER 1. TOPOLOGY OF ALGEBRAIC VARIETIES, HODGE DECOMPOSITION, AND APPLICATIONS. Contents
CHAPTER 1. TOPOLOGY OF ALGEBRAIC VARIETIES, HODGE DECOMPOSITION, AND APPLICATIONS Contents 1. The Lefschetz hyperplane theorem 1 2. The Hodge decomposition 4 3. Hodge numbers in smooth families 6 4. Birationally
More informationHodge Theory of Maps
Hodge Theory of Maps Migliorini and de Cataldo June 24, 2010 1 Migliorini 1 - Hodge Theory of Maps The existence of a Kähler form give strong topological constraints via Hodge theory. Can we get similar
More informationThe structure of algebraic varieties
The structure of algebraic varieties János Kollár Princeton University ICM, August, 2014, Seoul with the assistance of Jennifer M. Johnson and Sándor J. Kovács (Written comments added for clarity that
More informationTitleOn manifolds with trivial logarithm. Citation Osaka Journal of Mathematics. 41(2)
TitleOn manifolds with trivial logarithm Author(s) Winkelmann, Jorg Citation Osaka Journal of Mathematics. 41(2) Issue 2004-06 Date Text Version publisher URL http://hdl.handle.net/11094/7844 DOI Rights
More informationContributors. Preface
Contents Contributors Preface v xv 1 Kähler Manifolds by E. Cattani 1 1.1 Complex Manifolds........................... 2 1.1.1 Definition and Examples.................... 2 1.1.2 Holomorphic Vector Bundles..................
More informationRational curves on general type hypersurfaces
Rational curves on general type hypersurfaces Eric Riedl and David Yang May 5, 01 Abstract We develop a technique that allows us to prove results about subvarieties of general type hypersurfaces. As an
More informationarxiv:math/ v1 [math.ag] 2 Dec 1998
THE CHARACTERISTIC NUMBERS OF QUARTIC PLANE CURVES RAVI VAKIL arxiv:math/9812018v1 [math.ag] 2 Dec 1998 Abstract. The characteristic numbers of smooth plane quartics are computed using intersection theory
More informationBOUNDEDNESS OF MODULI OF VARIETIES OF GENERAL TYPE
BOUNDEDNESS OF MODULI OF VARIETIES OF GENERAL TYPE CHRISTOPHER D. HACON, JAMES M C KERNAN, AND CHENYANG XU Abstract. We show that the family of semi log canonical pairs with ample log canonical class and
More informationVojta s conjecture and level structures on abelian varieties
Vojta s conjecture and level structures on abelian varieties Dan Abramovich, Brown University Joint work with Anthony Várilly-Alvarado and Keerthi Padapusi Pera ICERM workshop on Birational Geometry and
More informationPERIOD INTEGRALS OF LOCAL COMPLETE INTERSECTIONS AND TAUTOLOGICAL SYSTEMS
PERIOD INTEGRALS OF LOCAL COMPLETE INTERSECTIONS AND TAUTOLOGICAL SYSTEMS AN HUANG, BONG LIAN, SHING-TUNG YAU AND CHENGLONG YU Abstract. Tautological systems developed in [6],[7] are Picard-Fuchs type
More informationUNEXPECTED ISOMORPHISMS BETWEEN HYPERKÄHLER FOURFOLDS
UNEXPECTED ISOMORPHISMS BETWEEN HYPERKÄHLER FOURFOLDS OLIVIER DEBARRE Abstract. Using Verbitsky s Torelli theorem, we show the existence of various isomorphisms between certain hyperkähler fourfolds. This
More informationLecture VI: Projective varieties
Lecture VI: Projective varieties Jonathan Evans 28th October 2010 Jonathan Evans () Lecture VI: Projective varieties 28th October 2010 1 / 24 I will begin by proving the adjunction formula which we still
More informationSEPARABLE RATIONAL CONNECTEDNESS AND STABILITY
SEPARABLE RATIONAL CONNECTEDNESS AND STABILIT ZHIU TIAN Abstract. In this short note we prove that in many cases the failure of a variety to be separably rationally connected is caused by the instability
More informationwhere m is the maximal ideal of O X,p. Note that m/m 2 is a vector space. Suppose that we are given a morphism
8. Smoothness and the Zariski tangent space We want to give an algebraic notion of the tangent space. In differential geometry, tangent vectors are equivalence classes of maps of intervals in R into the
More informationToric Varieties and the Secondary Fan
Toric Varieties and the Secondary Fan Emily Clader Fall 2011 1 Motivation The Batyrev mirror symmetry construction for Calabi-Yau hypersurfaces goes roughly as follows: Start with an n-dimensional reflexive
More informationMODULI SPACES OF CURVES
MODULI SPACES OF CURVES SCOTT NOLLET Abstract. My goal is to introduce vocabulary and present examples that will help graduate students to better follow lectures at TAGS 2018. Assuming some background
More informationMinimal Model Theory for Log Surfaces
Publ. RIMS Kyoto Univ. 48 (2012), 339 371 DOI 10.2977/PRIMS/71 Minimal Model Theory for Log Surfaces by Osamu Fujino Abstract We discuss the log minimal model theory for log surfaces. We show that the
More informationMonodromy of the Dwork family, following Shepherd-Barron X n+1. P 1 λ. ζ i = 1}/ (µ n+1 ) H.
Monodromy of the Dwork family, following Shepherd-Barron 1. The Dwork family. Consider the equation (f λ ) f λ (X 0, X 1,..., X n ) = λ(x n+1 0 + + X n+1 n ) (n + 1)X 0... X n = 0, where λ is a free parameter.
More informationDiagonal Subschemes and Vector Bundles
Pure and Applied Mathematics Quarterly Volume 4, Number 4 (Special Issue: In honor of Jean-Pierre Serre, Part 1 of 2 ) 1233 1278, 2008 Diagonal Subschemes and Vector Bundles Piotr Pragacz, Vasudevan Srinivas
More informationMinimal model program and moduli spaces
Minimal model program and moduli spaces Caucher Birkar Algebraic and arithmetic structures of moduli spaces, Hokkaido University, Japan 3-7 Sep 2007 Minimal model program In this talk, all the varieties
More informationPacific Journal of Mathematics
Pacific Journal of Mathematics STABLE REFLEXIVE SHEAVES ON SMOOTH PROJECTIVE 3-FOLDS PETER VERMEIRE Volume 219 No. 2 April 2005 PACIFIC JOURNAL OF MATHEMATICS Vol. 219, No. 2, 2005 STABLE REFLEXIVE SHEAVES
More informationHolomorphic Symplectic Manifolds
Holomorphic Symplectic Manifolds Marco Andreatta joint project with J. Wiśniewski Dipartimento di Matematica Universitá di Trento Holomorphic Symplectic Manifolds p.1/35 Definitions An holomorphic symplectic
More informationA TOOL FOR STABLE REDUCTION OF CURVES ON SURFACES
A TOOL FOR STABLE REDUCTION OF CURVES ON SURFACES RAVI VAKIL Abstract. In the study of the geometry of curves on surfaces, the following question often arises: given a one-parameter family of divisors
More informationGAUGED LINEAR SIGMA MODEL SPACES
GAUGED LINEAR SIGMA MODEL SPACES FELIPE CASTELLANO-MACIAS ADVISOR: FELIX JANDA Abstract. The gauged linear sigma model (GLSM) originated in physics but it has recently made it into mathematics as an enumerative
More informationUnirational threefolds with no universal codimension 2 cycle
Unirational threefolds with no universal codimension 2 cycle Claire Voisin CNRS and École Polytechnique Abstract We prove that the general quartic double solid with k 7 nodes does not admit a Chow theoretic
More informationGeometry of curves with exceptional secant planes
Geometry of curves with exceptional secant planes 15 October 2007 Reference for this talk: arxiv.org/0706.2049 I. Linear series and secant planes In my thesis, I studied linear series on curves, and tried
More informationON NODAL PRIME FANO THREEFOLDS OF DEGREE 10
ON NODAL PRIME FANO THREEFOLDS OF DEGREE 10 OLIVIER DEBARRE, ATANAS ILIEV, AND LAURENT MANIVEL Abstract. We study the geometry and the period map of nodal complex prime Fano threefolds with index 1 and
More informationTwo simple ideas from calculus applied to Riemannian geometry
Calibrated Geometries and Special Holonomy p. 1/29 Two simple ideas from calculus applied to Riemannian geometry Spiro Karigiannis karigiannis@math.uwaterloo.ca Department of Pure Mathematics, University
More informationTHE MODULI SPACE OF TWISTED CANONICAL DIVISORS
THE MODULI SPACE OF TWISTED CANONICAL DIVISORS GAVRIL FARKAS AND RAHUL PANDHARIPANDE ABSTRACT. The moduli space of canonical divisors (with prescribed zeros and poles) on nonsingular curves is not compact
More informationHomological mirror symmetry via families of Lagrangians
Homological mirror symmetry via families of Lagrangians String-Math 2018 Mohammed Abouzaid Columbia University June 17, 2018 Mirror symmetry Three facets of mirror symmetry: 1 Enumerative: GW invariants
More informationLECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY
LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY WEIMIN CHEN, UMASS, SPRING 07 1. Blowing up and symplectic cutting In complex geometry the blowing-up operation amounts to replace a point in
More informationGeometry dictates arithmetic
Geometry dictates arithmetic Ronald van Luijk February 21, 2013 Utrecht Curves Example. Circle given by x 2 + y 2 = 1 (or projective closure in P 2 ). Curves Example. Circle given by x 2 + y 2 = 1 (or
More informationAlgebraic geometry versus Kähler geometry
Algebraic geometry versus Kähler geometry Claire Voisin CNRS, Institut de mathématiques de Jussieu Contents 0 Introduction 1 1 Hodge theory 2 1.1 The Hodge decomposition............................. 2
More informationCOMPLEX ALGEBRAIC SURFACES CLASS 15
COMPLEX ALGEBRAIC SURFACES CLASS 15 RAVI VAKIL CONTENTS 1. Every smooth cubic has 27 lines, and is the blow-up of P 2 at 6 points 1 1.1. An alternate approach 4 2. Castelnuovo s Theorem 5 We now know that
More informationGeometric motivic integration
Université Lille 1 Modnet Workshop 2008 Introduction Motivation: p-adic integration Kontsevich invented motivic integration to strengthen the following result by Batyrev. Theorem (Batyrev) If two complex
More informationIN POSITIVE CHARACTERISTICS: 3. Modular varieties with Hecke symmetries. 7. Foliation and a conjecture of Oort
FINE STRUCTURES OF MODULI SPACES IN POSITIVE CHARACTERISTICS: HECKE SYMMETRIES AND OORT FOLIATION 1. Elliptic curves and their moduli 2. Moduli of abelian varieties 3. Modular varieties with Hecke symmetries
More informationMini-Course on Moduli Spaces
Mini-Course on Moduli Spaces Emily Clader June 2011 1 What is a Moduli Space? 1.1 What should a moduli space do? Suppose that we want to classify some kind of object, for example: Curves of genus g, One-dimensional
More information