A Numerical Criterion for Lower bounds on K-energy maps of Algebraic manifolds

Size: px
Start display at page:

Download "A Numerical Criterion for Lower bounds on K-energy maps of Algebraic manifolds"

Transcription

1 A Numerical Criterion for Lower bounds on K-energy maps of Algebraic manifolds Sean Timothy Paul University of Wisconsin, Madison

2 Outline Formulation of the problem: To bound the Mabuchi energy from below on the space of Kähler metrics in a given Kähler class [ω]. Tian s program 88-97: In algebraic case should restrict K-energy to Bergman metrics". Representation theory : Toric Morphisms and Equivariant embeddings. Discriminants and resultants of projective varieties: Hyperdiscriminants and Cayley- Chow forms. Output: A complete description of the extremal properties of the Mabuchi energy restricted to the space of Bergman metrics.

3 Formulating the problem Set up and notation: (X n, ω) closed Kähler manifold H ω := {ϕ C (X) ω ϕ > 0} (the space of Kähler metrics in the class [ω] ) ω ϕ := ω + 1 2π ϕ Scal(ω): = scalar curvature of ω µ = 1 V X Scal(ω)ωn (average of the scalar curvature) V =volume

4 Definition. (Mabuchi 1986 ) The K-energy map ν ω : H ω R is given by ν ω (ϕ) := 1 V 1 0 X ϕ t(scal(ω ϕt ) µ)ω n t dt ϕ t is a C 1 path in H ω satisfying ϕ 0 = 0, ϕ 1 = ϕ Observe : ϕ is a critical point for ν ω iff Scal(ω ϕ ) µ (a constant) Basic Theorem (Bando-Mabuchi, Donaldson,..., Chen-Tian) If there is a ψ H ω with constant scalar curvature then there exits C 0 such that ν ω (ϕ) C for all ϕ H ω.

5 Question ( ) : Given [ω] how to detect when ν ω is bounded below on H ω? N.B. : In general we do not know (!) if there is a constant scalar curvature metric in the class [ω]. Special Case: Assume that [ω] is an integral class, i.e. there is an ample divisor L on X such that [ω] = c 1 (L) We may assume that X P N (embedded) and ω = ω F S X

6 Observe that for σ G := SL(N + 1, C) there is a ϕ σ C (P N ) such that σ ω F S = ω F S + This gives a map 1 2π ϕ σ > 0 G σ ϕ σ H ω The space of Bergman Metrics is the image of this map B := {ϕ σ σ G} H ω. Tian s idea: RESTRICT THE K-ENERGY TO B

7 Question ( ) : Given X P N how to detect when ν ω is bounded below on B?

8 Definition. Let (G) be the space of algebraic one parameter subgroups λ of G. These are algebraic homomorphisms λ : C G λ ij C[ t, t 1 ]. Definition. (The space of degenerations in B) (B) := {C ϕ λ B ; λ (G)}.

9 Theorem. ( Paul 2012 ) Assume that for every degeneration λ in B there is a (finite) constant C(λ) such that lim α 0 ν ω(ϕ λ(α) ) C(λ). Then there is a uniform constant C such that for all ϕ σ B we have the lower bound ν ω (ϕ σ ) C.

10 Equivariant Embeddings of Algebraic Homogeneous Spaces G reductive complex linear algebraic group: G = GL(N + 1, C), SL(N + 1, C), (C ) N, SO(N, C), Sp 2n (C). H := Zariski closed subgroup. O := G/H associated homogeneous space.

11 Definition. An embedding of O is an irreducible G variety X together with a G-equivariant embedding i : O X such that i(o) is an open dense orbit of X.

12 Let (X 1, i 1 ) and (X 2, i 2 ) be two embeddings of O. Definition. A morphism ϕ from (X 1, i 1 ) to (X 2, i 2 ) is a G equivariant regular map ϕ : X 1 X 2 such that the diagram commutes. O i 1 i 2 X 1 ϕ X 2 One says that (X 1, i 1 ) dominates (X 2, i 2 ).

13 Assume these embeddings are both projective (hence complete) with very ample linearizations satisfying L 1 Pic(X 1 ) G, L 2 Pic(X 2 ) G ϕ (L 2 ) = L 1. Get injective map of G modules ϕ : H 0 (X 2, L 2 ) H 0 (X 1, L 1 )

14 The adjoint (ϕ ) t : H 0 (X 1, L 1 ) H 0 (X 2, L 2 ) is surjective and gives a rational map : O i 1 X 1 ϕ P(H 0 (X 1, L 1 ) ) (ϕ ) t i 2 X 2 P(H 0 (X 2, L 2 ) )

15 We abstract this situation : 1. V, W finite dimensional rational G-modules 2. v, w nonzero vectors in V, W respectively 3. Linear span of G v coincides with V (same for w) 4. [v] corresponding line through v = point in P(V) 5. O v := G [v] P(V) ( projective orbit ) 6. O v = Zariski closure in P(V).

16 Definition. (V; v) dominates (W; w) if and only if there exists π Hom(V, W) G such that π(v) = w and the rational map π : P(V) P(W) induces a regular finite morphism π : G [v] G [w] O i v O v π P(V) π i w O w P(W)

17 Observe that the map π extends to the boundary if and only if ( ) G [v] P(ker π) =. π(v) = W V = ker(π) W (G-module splitting) Identify π with projection onto W v = (v π, w) v π 0 ( ) is equivalent to ( ) G [(v π, w)] G [(v π, 0)] = (Zariski closure inside P(ker(π) W ) )

18 Given (v, w) V W set O vw := G [(v, w)] P(V W) O v := G [(v, 0)] P(V {0}) This motivates: Definition. (Paul 2010) The pair (v, w) is semistable if and only if O vw O v =

19 Example. Let V e and V d be irreducible SL(2, C) modules with highest weights e, d N = homogeneous polynomials in two variables. Let f and g in V e \ {0} and W d \ {0} respectively. Claim. (f, g) is semistable if and only if e d and for all p P 1 ord p (g) ord p (f) d e 2. When e = 0 and f = 1 conclude that (1, g) is semistable if and only if ord p (g) d 2 for all p P1.

20 Toric Morphisms If the pair (v, w) is semistable then we certainly have that T [(v, w)] T [(v, 0)] = for all maximal algebraic tori T G. Therefore there exists a morphism of projective toric varieties. T T [(v, w)] π P(V W) π T [(0, w)] P(W) We expect that the existence of such a morphism is completely dictated by the weight polyhedra : N (v) and N (w).

21 Theorem. (Paul 2012) The following statements are equivalent. 1. (v, w) is semistable. Recall that this means O vw O v = 2. N (v) N (w) for all maximal tori H G. We say that (v, w) is numerically semistable. 3. For every maximal algebraic torus H G and χ A H (v) there exists an integer d > 0 and a relative invariant f C d [ V W ] H dχ such that f(v, w) 0 and f V 0.

22 Corollary A. If O vw O v then there exists an alg. 1psg λ (G) such that lim α 0 λ(α) [(v, w)] O v.

23 Equip V and W with Hermitian norms. The energy of the pair (v, w) is the function on G defined by G σ p vw (σ) := log σ w 2 log σ v 2. Corollary B. inf p σ G vw (σ) = if and only if there is a degeneration λ (G) such that lim α 0 p vw (λ(α)) =.

24 Hilbert-Mumford Semistability Semistable Pairs For all H G d Z >0 and For all H G and χ A H (v) f C d [ W ] H such that d Z >0 and f C d [ V W ] H dχ f(w) 0 and f(0) = 0 such that f(v, w) 0 and f V 0 0 / G w O vw O v = w λ (w) 0 w λ (w) w λ (v) 0 for all 1psg s λ of G for all 1psg s λ of G 0 N (w) all H G N (v) N (w) all H G C 0 such that log σ w 2 C all σ G C 0 such that log σ w 2 log σ v 2 C all σ G

25 To summarize, the context for the study of SEMISTABLE PAIRS is 1. A reductive linear algebraic group G. 2. A pair V, W of finite dimensional rational G- modules. 3. A pair of (non-zero) vectors (v, w) V W.

26 Resultants and Discriminants Let X be a smooth linearly normal variety X P N Consider two polynomials: R X := X-resultant X P n 1 := X-hyperdiscriminant Let s normalize the degrees of these polynomials X R = R(X) := R deg( X P n 1 ) X X = (X) := deg(r X) X P n 1

27 It is known that R(X) E λ \ {0}, (n + 1)λ = ( n+1 N n {}}{{}}{ r, r,..., r, 0,..., 0 ). (X) E µ \ {0}, nµ = ( {}} n N+1 n {{}}{ r, r,..., r, 0,..., 0 ). r = deg(r(x)) = deg( (X)). E λ and E µ are irreducible G modules. The associations X R(X), X (X) are G equivariant: R(σ X) = σ R(X) (σ X) = σ (X).

28 K-Energy maps and Semistable Pairs Let P be a numerical polynomial P (T ) = c ( T ) ( T ) n + cn 1 + O(T n 2 ) c n Z >0. n n 1 Consider the Hilbert scheme H P P N := { all (smooth) X P N with Hilbert polynomial P }. Recall the G-equivariant morphisms R, : H P P N P(E λ ), P(E µ ).

29 Theorem (Paul 2012 ) There is a constant M depending only on c n, c n 1 and the Fubini Study metric such that for all [X] H P P N and all σ G we have ν ωf S X (ϕ σ ) p R(X) (X) (σ) M.

Uniform K-stability of pairs

Uniform K-stability of pairs Uniform K-stability of pairs Gang Tian Peking University Let G = SL(N + 1, C) with two representations V, W over Q. For any v V\{0} and any one-parameter subgroup λ of G, we can associate a weight w λ

More information

K-stability and Kähler metrics, I

K-stability and Kähler metrics, I K-stability and Kähler metrics, I Gang Tian Beijing University and Princeton University Let M be a Kähler manifold. This means that M be a complex manifold together with a Kähler metric ω. In local coordinates

More information

The Yau-Tian-Donaldson Conjectuture for general polarizations

The Yau-Tian-Donaldson Conjectuture for general polarizations The Yau-Tian-Donaldson Conjectuture for general polarizations Toshiki Mabuchi, Osaka University 2015 Taipei Conference on Complex Geometry December 22, 2015 1. Introduction 2. Background materials Table

More information

Remarks on hypersurface K-stability. Complex Geometry: A Conference Honoring Simon Donaldson

Remarks on hypersurface K-stability. Complex Geometry: A Conference Honoring Simon Donaldson Remarks on hypersurface K-stability Zhiqin Lu, UC Irvine Complex Geometry: A Conference Honoring Simon Donaldson October 26, 2009 Zhiqin Lu, UC. Irvine Hypersurface K-stability 1/42 The Result Theorem

More information

Representations and Linear Actions

Representations and Linear Actions Representations and Linear Actions Definition 0.1. Let G be an S-group. A representation of G is a morphism of S-groups φ G GL(n, S) for some n. We say φ is faithful if it is a monomorphism (in the category

More information

The Hilbert-Mumford Criterion

The Hilbert-Mumford Criterion The Hilbert-Mumford Criterion Klaus Pommerening Johannes-Gutenberg-Universität Mainz, Germany January 1987 Last change: April 4, 2017 The notions of stability and related notions apply for actions of algebraic

More information

Kähler-Einstein metrics and K-stability

Kähler-Einstein metrics and K-stability May 3, 2012 Table Of Contents 1 Preliminaries 2 Continuity method 3 Review of Tian s program 4 Special degeneration and K-stability 5 Thanks Basic Kähler geometry (X, J, g) (X, J, ω g ) g(, ) = ω g (,

More information

A Bird Eye s view: recent update to Extremal metrics

A Bird Eye s view: recent update to Extremal metrics A Bird Eye s view: recent update to Extremal metrics Xiuxiong Chen Department of Mathematics University of Wisconsin at Madison January 21, 2009 A Bird Eye s view: recent update to Extremal metrics Xiuxiong

More information

Reductive group actions and some problems concerning their quotients

Reductive group actions and some problems concerning their quotients Reductive group actions and some problems concerning their quotients Brandeis University January 2014 Linear Algebraic Groups A complex linear algebraic group G is an affine variety such that the mappings

More information

(1) is an invertible sheaf on X, which is generated by the global sections

(1) is an invertible sheaf on X, which is generated by the global sections 7. Linear systems First a word about the base scheme. We would lie to wor in enough generality to cover the general case. On the other hand, it taes some wor to state properly the general results if one

More information

LECTURE 11: SYMPLECTIC TORIC MANIFOLDS. Contents 1. Symplectic toric manifolds 1 2. Delzant s theorem 4 3. Symplectic cut 8

LECTURE 11: SYMPLECTIC TORIC MANIFOLDS. Contents 1. Symplectic toric manifolds 1 2. Delzant s theorem 4 3. Symplectic cut 8 LECTURE 11: SYMPLECTIC TORIC MANIFOLDS Contents 1. Symplectic toric manifolds 1 2. Delzant s theorem 4 3. Symplectic cut 8 1. Symplectic toric manifolds Orbit of torus actions. Recall that in lecture 9

More information

Moduli spaces of log del Pezzo pairs and K-stability

Moduli spaces of log del Pezzo pairs and K-stability Report on Research in Groups Moduli spaces of log del Pezzo pairs and K-stability June 20 - July 20, 2016 Organizers: Patricio Gallardo, Jesus Martinez-Garcia, Cristiano Spotti. In this report we start

More information

CANONICAL METRICS AND STABILITY OF PROJECTIVE VARIETIES

CANONICAL METRICS AND STABILITY OF PROJECTIVE VARIETIES CANONICAL METRICS AND STABILITY OF PROJECTIVE VARIETIES JULIUS ROSS This short survey aims to introduce some of the ideas and conjectures relating stability of projective varieties to the existence of

More information

GEOMETRIC STRUCTURES OF SEMISIMPLE LIE ALGEBRAS

GEOMETRIC STRUCTURES OF SEMISIMPLE LIE ALGEBRAS GEOMETRIC STRUCTURES OF SEMISIMPLE LIE ALGEBRAS ANA BALIBANU DISCUSSED WITH PROFESSOR VICTOR GINZBURG 1. Introduction The aim of this paper is to explore the geometry of a Lie algebra g through the action

More information

EKT of Some Wonderful Compactifications

EKT of Some Wonderful Compactifications EKT of Some Wonderful Compactifications and recent results on Complete Quadrics. (Based on joint works with Soumya Banerjee and Michael Joyce) Mahir Bilen Can April 16, 2016 Mahir Bilen Can EKT of Some

More information

Math 210C. The representation ring

Math 210C. The representation ring Math 210C. The representation ring 1. Introduction Let G be a nontrivial connected compact Lie group that is semisimple and simply connected (e.g., SU(n) for n 2, Sp(n) for n 1, or Spin(n) for n 3). Let

More information

Hilbert series and obstructions to asymptotic Chow semistability

Hilbert series and obstructions to asymptotic Chow semistability Hilbert series and obstructions to asymptotic Chow semistability Akito Futaki Tokyo Institute of Technology Kähler and Sasakian Geometry in Rome Rome, June 16th-19th, 2009 In memory of Krzysztof Galicki

More information

LECTURE 7: STABLE RATIONALITY AND DECOMPOSITION OF THE DIAGONAL

LECTURE 7: STABLE RATIONALITY AND DECOMPOSITION OF THE DIAGONAL LECTURE 7: STABLE RATIONALITY AND DECOMPOSITION OF THE DIAGONAL In this lecture we discuss a criterion for non-stable-rationality based on the decomposition of the diagonal in the Chow group. This criterion

More information

Quadratic reciprocity (after Weil) 1. Standard set-up and Poisson summation

Quadratic reciprocity (after Weil) 1. Standard set-up and Poisson summation (September 17, 010) Quadratic reciprocity (after Weil) Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ I show that over global fields (characteristic not ) the quadratic norm residue

More information

SYMPLECTIC GEOMETRY: LECTURE 5

SYMPLECTIC GEOMETRY: LECTURE 5 SYMPLECTIC GEOMETRY: LECTURE 5 LIAT KESSLER Let (M, ω) be a connected compact symplectic manifold, T a torus, T M M a Hamiltonian action of T on M, and Φ: M t the assoaciated moment map. Theorem 0.1 (The

More information

THE HITCHIN FIBRATION

THE HITCHIN FIBRATION THE HITCHIN FIBRATION Seminar talk based on part of Ngô Bao Châu s preprint: Le lemme fondamental pour les algèbres de Lie [2]. Here X is a smooth connected projective curve over a field k whose genus

More information

Spherical varieties and arc spaces

Spherical varieties and arc spaces Spherical varieties and arc spaces Victor Batyrev, ESI, Vienna 19, 20 January 2017 1 Lecture 1 This is a joint work with Anne Moreau. Let us begin with a few notations. We consider G a reductive connected

More information

LECTURE 3: REPRESENTATION THEORY OF SL 2 (C) AND sl 2 (C)

LECTURE 3: REPRESENTATION THEORY OF SL 2 (C) AND sl 2 (C) LECTURE 3: REPRESENTATION THEORY OF SL 2 (C) AND sl 2 (C) IVAN LOSEV Introduction We proceed to studying the representation theory of algebraic groups and Lie algebras. Algebraic groups are the groups

More information

Rings and groups. Ya. Sysak

Rings and groups. Ya. Sysak Rings and groups. Ya. Sysak 1 Noetherian rings Let R be a ring. A (right) R -module M is called noetherian if it satisfies the maximum condition for its submodules. In other words, if M 1... M i M i+1...

More information

GEOMETRIC INVARIANT THEORY AND SYMPLECTIC QUOTIENTS

GEOMETRIC INVARIANT THEORY AND SYMPLECTIC QUOTIENTS GEOMETRIC INVARIANT THEORY AND SYMPLECTIC QUOTIENTS VICTORIA HOSKINS 1. Introduction In this course we study methods for constructing quotients of group actions in algebraic and symplectic geometry and

More information

June 2014 Written Certification Exam. Algebra

June 2014 Written Certification Exam. Algebra June 2014 Written Certification Exam Algebra 1. Let R be a commutative ring. An R-module P is projective if for all R-module homomorphisms v : M N and f : P N with v surjective, there exists an R-module

More information

Delzant s Garden. A one-hour tour to symplectic toric geometry

Delzant s Garden. A one-hour tour to symplectic toric geometry Delzant s Garden A one-hour tour to symplectic toric geometry Tour Guide: Zuoqin Wang Travel Plan: The earth America MIT Main building Math. dept. The moon Toric world Symplectic toric Delzant s theorem

More information

COURSE SUMMARY FOR MATH 504, FALL QUARTER : MODERN ALGEBRA

COURSE SUMMARY FOR MATH 504, FALL QUARTER : MODERN ALGEBRA COURSE SUMMARY FOR MATH 504, FALL QUARTER 2017-8: MODERN ALGEBRA JAROD ALPER Week 1, Sept 27, 29: Introduction to Groups Lecture 1: Introduction to groups. Defined a group and discussed basic properties

More information

Mathematics 7800 Quantum Kitchen Sink Spring 2002

Mathematics 7800 Quantum Kitchen Sink Spring 2002 Mathematics 7800 Quantum Kitchen Sink Spring 2002 4. Quotients via GIT. Most interesting moduli spaces arise as quotients of schemes by group actions. We will first analyze such quotients with geometric

More information

12. Linear systems Theorem Let X be a scheme over a ring A. (1) If φ: X P n A is an A-morphism then L = φ O P n

12. Linear systems Theorem Let X be a scheme over a ring A. (1) If φ: X P n A is an A-morphism then L = φ O P n 12. Linear systems Theorem 12.1. Let X be a scheme over a ring A. (1) If φ: X P n A is an A-morphism then L = φ O P n A (1) is an invertible sheaf on X, which is generated by the global sections s 0, s

More information

Algebra Homework, Edition 2 9 September 2010

Algebra Homework, Edition 2 9 September 2010 Algebra Homework, Edition 2 9 September 2010 Problem 6. (1) Let I and J be ideals of a commutative ring R with I + J = R. Prove that IJ = I J. (2) Let I, J, and K be ideals of a principal ideal domain.

More information

On exceptional completions of symmetric varieties

On exceptional completions of symmetric varieties Journal of Lie Theory Volume 16 (2006) 39 46 c 2006 Heldermann Verlag On exceptional completions of symmetric varieties Rocco Chirivì and Andrea Maffei Communicated by E. B. Vinberg Abstract. Let G be

More information

ASYMPTOTIC CHOW SEMI-STABILITY AND INTEGRAL INVARIANTS

ASYMPTOTIC CHOW SEMI-STABILITY AND INTEGRAL INVARIANTS ASYPTOTIC CHOW SEI-STABILITY AND INTEGRAL INVARIANTS AKITO FUTAKI Abstract. We define a family of integral invariants containing those which are closely related to asymptotic Chow semi-stability of polarized

More information

arxiv: v1 [math.ag] 14 Mar 2019

arxiv: v1 [math.ag] 14 Mar 2019 ASYMPTOTIC CONSTRUCTIONS AND INVARIANTS OF GRADED LINEAR SERIES ariv:1903.05967v1 [math.ag] 14 Mar 2019 CHIH-WEI CHANG AND SHIN-YAO JOW Abstract. Let be a complete variety of dimension n over an algebraically

More information

Normality of secant varieties

Normality of secant varieties Normality of secant varieties Brooke Ullery Joint Mathematics Meetings January 6, 2016 Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, 2016 1 / 11 Introduction Let X

More information

We can choose generators of this k-algebra: s i H 0 (X, L r i. H 0 (X, L mr )

We can choose generators of this k-algebra: s i H 0 (X, L r i. H 0 (X, L mr ) MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 43 5.3. Linearisations. An abstract projective scheme X does not come with a pre-specified embedding in a projective space. However, an ample line bundle

More information

QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday 10 February 2004 (Day 1)

QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday 10 February 2004 (Day 1) Tuesday 10 February 2004 (Day 1) 1a. Prove the following theorem of Banach and Saks: Theorem. Given in L 2 a sequence {f n } which weakly converges to 0, we can select a subsequence {f nk } such that the

More information

Lecture 1. Toric Varieties: Basics

Lecture 1. Toric Varieties: Basics Lecture 1. Toric Varieties: Basics Taras Panov Lomonosov Moscow State University Summer School Current Developments in Geometry Novosibirsk, 27 August1 September 2018 Taras Panov (Moscow University) Lecture

More information

Math 429/581 (Advanced) Group Theory. Summary of Definitions, Examples, and Theorems by Stefan Gille

Math 429/581 (Advanced) Group Theory. Summary of Definitions, Examples, and Theorems by Stefan Gille Math 429/581 (Advanced) Group Theory Summary of Definitions, Examples, and Theorems by Stefan Gille 1 2 0. Group Operations 0.1. Definition. Let G be a group and X a set. A (left) operation of G on X is

More information

Geometric Structure and the Local Langlands Conjecture

Geometric Structure and the Local Langlands Conjecture Geometric Structure and the Local Langlands Conjecture Paul Baum Penn State Representations of Reductive Groups University of Utah, Salt Lake City July 9, 2013 Paul Baum (Penn State) Geometric Structure

More information

PART I: GEOMETRY OF SEMISIMPLE LIE ALGEBRAS

PART I: GEOMETRY OF SEMISIMPLE LIE ALGEBRAS PART I: GEOMETRY OF SEMISIMPLE LIE ALGEBRAS Contents 1. Regular elements in semisimple Lie algebras 1 2. The flag variety and the Bruhat decomposition 3 3. The Grothendieck-Springer resolution 6 4. The

More information

LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS

LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS IVAN LOSEV 16. Symplectic resolutions of µ 1 (0)//G and their deformations 16.1. GIT quotients. We will need to produce a resolution of singularities for C 2n

More information

REPRESENTATION THEORY. WEEKS 10 11

REPRESENTATION THEORY. WEEKS 10 11 REPRESENTATION THEORY. WEEKS 10 11 1. Representations of quivers I follow here Crawley-Boevey lectures trying to give more details concerning extensions and exact sequences. A quiver is an oriented graph.

More information

INTRODUCTION TO GEOMETRIC INVARIANT THEORY

INTRODUCTION TO GEOMETRIC INVARIANT THEORY INTRODUCTION TO GEOMETRIC INVARIANT THEORY JOSÉ SIMENTAL Abstract. These are the expanded notes for a talk at the MIT/NEU Graduate Student Seminar on Moduli of sheaves on K3 surfaces. We give a brief introduction

More information

APPENDIX 3: AN OVERVIEW OF CHOW GROUPS

APPENDIX 3: AN OVERVIEW OF CHOW GROUPS APPENDIX 3: AN OVERVIEW OF CHOW GROUPS We review in this appendix some basic definitions and results that we need about Chow groups. For details and proofs we refer to [Ful98]. In particular, we discuss

More information

Stability of algebraic varieties and algebraic geometry. AMS Summer Research Institute in Algebraic Geometry

Stability of algebraic varieties and algebraic geometry. AMS Summer Research Institute in Algebraic Geometry Stability of algebraic varieties and algebraic geometry AMS Summer Research Institute in Algebraic Geometry Table of Contents I Background Kähler metrics Geometric Invariant theory, Kempf-Ness etc. Back

More information

Pacific Journal of Mathematics

Pacific Journal of Mathematics Pacific Journal of Mathematics GROUP ACTIONS ON POLYNOMIAL AND POWER SERIES RINGS Peter Symonds Volume 195 No. 1 September 2000 PACIFIC JOURNAL OF MATHEMATICS Vol. 195, No. 1, 2000 GROUP ACTIONS ON POLYNOMIAL

More information

MODULI SPACES OF CURVES

MODULI 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 information

Math 249B. Nilpotence of connected solvable groups

Math 249B. Nilpotence of connected solvable groups Math 249B. Nilpotence of connected solvable groups 1. Motivation and examples In abstract group theory, the descending central series {C i (G)} of a group G is defined recursively by C 0 (G) = G and C

More information

REPRESENTATION THEORY WEEK 7

REPRESENTATION THEORY WEEK 7 REPRESENTATION THEORY WEEK 7 1. Characters of L k and S n A character of an irreducible representation of L k is a polynomial function constant on every conjugacy class. Since the set of diagonalizable

More information

A Gauss-Bonnet theorem for constructible sheaves on reductive groups

A Gauss-Bonnet theorem for constructible sheaves on reductive groups A Gauss-Bonnet theorem for constructible sheaves on reductive groups V. Kiritchenko 1 Introduction In this paper, we prove an analog of the Gauss-Bonnet formula for constructible sheaves on reductive groups.

More information

Constant Scalar Curvature Kähler Metric Obtains the Minimum of K-energy

Constant Scalar Curvature Kähler Metric Obtains the Minimum of K-energy Li, C. (20) Constant Scalar Curvature Kähler Metric Obtains the Minimum of K-energy, International Mathematics Research Notices, Vol. 20, No. 9, pp. 26 275 Advance Access publication September, 200 doi:0.093/imrn/rnq52

More information

The Gelfand-Tsetlin Basis (Too Many Direct Sums, and Also a Graph)

The Gelfand-Tsetlin Basis (Too Many Direct Sums, and Also a Graph) The Gelfand-Tsetlin Basis (Too Many Direct Sums, and Also a Graph) David Grabovsky June 13, 2018 Abstract The symmetric groups S n, consisting of all permutations on a set of n elements, naturally contain

More information

Hyperkähler geometry lecture 3

Hyperkähler geometry lecture 3 Hyperkähler geometry lecture 3 Misha Verbitsky Cohomology in Mathematics and Physics Euler Institute, September 25, 2013, St. Petersburg 1 Broom Bridge Here as he walked by on the 16th of October 1843

More information

Stable bundles on CP 3 and special holonomies

Stable 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 information

Problems on Minkowski sums of convex lattice polytopes

Problems on Minkowski sums of convex lattice polytopes arxiv:08121418v1 [mathag] 8 Dec 2008 Problems on Minkowski sums of convex lattice polytopes Tadao Oda odatadao@mathtohokuacjp Abstract submitted at the Oberwolfach Conference Combinatorial Convexity and

More information

IN POSITIVE CHARACTERISTICS: 3. Modular varieties with Hecke symmetries. 7. Foliation and a conjecture of Oort

IN 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 information

Combinatorics and geometry of E 7

Combinatorics and geometry of E 7 Combinatorics and geometry of E 7 Steven Sam University of California, Berkeley September 19, 2012 1/24 Outline Macdonald representations Vinberg representations Root system Weyl group 7 points in P 2

More information

Math 797W Homework 4

Math 797W Homework 4 Math 797W Homework 4 Paul Hacking December 5, 2016 We work over an algebraically closed field k. (1) Let F be a sheaf of abelian groups on a topological space X, and p X a point. Recall the definition

More information

ASSOCIATED FORM MORPHISM

ASSOCIATED FORM MORPHISM ASSOCIATED FORM MORPHISM MAKSYM FEDORCHUK AND ALEXANDER ISAEV Abstract. We study the geometry of the morphism that sends a smooth hypersurface of degree d + 1 in P n 1 to its associated hypersurface of

More information

ALGEBRAIC GROUPS. Disclaimer: There are millions of errors in these notes!

ALGEBRAIC GROUPS. Disclaimer: There are millions of errors in these notes! ALGEBRAIC GROUPS Disclaimer: There are millions of errors in these notes! 1. Some algebraic geometry The subject of algebraic groups depends on the interaction between algebraic geometry and group theory.

More information

4.5 Hilbert s Nullstellensatz (Zeros Theorem)

4.5 Hilbert s Nullstellensatz (Zeros Theorem) 4.5 Hilbert s Nullstellensatz (Zeros Theorem) We develop a deep result of Hilbert s, relating solutions of polynomial equations to ideals of polynomial rings in many variables. Notation: Put A = F[x 1,...,x

More information

Introduction to toric geometry

Introduction to toric geometry Introduction to toric geometry Ugo Bruzzo Scuola Internazionale Superiore di Studi Avanzati and Istituto Nazionale di Fisica Nucleare Trieste ii Instructions for the reader These are work-in-progress notes

More information

Holomorphic line bundles

Holomorphic 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 information

COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY

COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY BRIAN OSSERMAN Classical algebraic geometers studied algebraic varieties over the complex numbers. In this setting, they didn t have to worry about the Zariski

More information

ne varieties (continued)

ne varieties (continued) Chapter 2 A ne varieties (continued) 2.1 Products For some problems its not very natural to restrict to irreducible varieties. So we broaden the previous story. Given an a ne algebraic set X A n k, we

More information

Lemma 1.3. The element [X, X] is nonzero.

Lemma 1.3. The element [X, X] is nonzero. Math 210C. The remarkable SU(2) Let G be a non-commutative connected compact Lie group, and assume that its rank (i.e., dimension of maximal tori) is 1; equivalently, G is a compact connected Lie group

More information

Zero cycles on twisted Cayley plane

Zero cycles on twisted Cayley plane Zero cycles on twisted Cayley plane V. Petrov, N. Semenov, K. Zainoulline August 8, 2005 Abstract In the present paper we compute the group of zero-cycles modulo rational equivalence of a twisted form

More information

Eigenvalue problem for Hermitian matrices and its generalization to arbitrary reductive groups

Eigenvalue problem for Hermitian matrices and its generalization to arbitrary reductive groups Eigenvalue problem for Hermitian matrices and its generalization to arbitrary reductive groups Shrawan Kumar Talk given at AMS Sectional meeting held at Davidson College, March 2007 1 Hermitian eigenvalue

More information

On Mordell-Lang in Algebraic Groups of Unipotent Rank 1

On 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 information

Real Analysis Prelim Questions Day 1 August 27, 2013

Real Analysis Prelim Questions Day 1 August 27, 2013 Real Analysis Prelim Questions Day 1 August 27, 2013 are 5 questions. TIME LIMIT: 3 hours Instructions: Measure and measurable refer to Lebesgue measure µ n on R n, and M(R n ) is the collection of measurable

More information

Varieties of Characters

Varieties of Characters Sean Lawton George Mason University Fall Eastern Sectional Meeting September 25, 2016 Lawton (GMU) (AMS, September 2016) Step 1: Groups 1 Let Γ be a finitely generated group. Lawton (GMU) (AMS, September

More information

Algebraic Hecke Characters

Algebraic Hecke Characters Algebraic Hecke Characters Motivation This motivation was inspired by the excellent article [Serre-Tate, 7]. Our goal is to prove the main theorem of complex multiplication. The Galois theoretic formulation

More information

Quaternionic Complexes

Quaternionic Complexes Quaternionic Complexes Andreas Čap University of Vienna Berlin, March 2007 Andreas Čap (University of Vienna) Quaternionic Complexes Berlin, March 2007 1 / 19 based on the joint article math.dg/0508534

More information

Kähler configurations of points

Kähler configurations of points Kähler configurations of points Simon Salamon Oxford, 22 May 2017 The Hesse configuration 1/24 Let ω = e 2πi/3. Consider the nine points [0, 1, 1] [0, 1, ω] [0, 1, ω 2 ] [1, 0, 1] [1, 0, ω] [1, 0, ω 2

More information

10. Smooth Varieties. 82 Andreas Gathmann

10. Smooth Varieties. 82 Andreas Gathmann 82 Andreas Gathmann 10. Smooth Varieties Let a be a point on a variety X. In the last chapter we have introduced the tangent cone C a X as a way to study X locally around a (see Construction 9.20). It

More information

Part II Galois Theory

Part II Galois Theory Part II Galois Theory Theorems Based on lectures by C. Birkar Notes taken by Dexter Chua Michaelmas 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after

More information

Proof of the Shafarevich conjecture

Proof of the Shafarevich conjecture Proof of the Shafarevich conjecture Rebecca Bellovin We have an isogeny of degree l h φ : B 1 B 2 of abelian varieties over K isogenous to A. We wish to show that h(b 1 ) = h(b 2 ). By filtering the kernel

More information

Construction of M B, M Dol, M DR

Construction of M B, M Dol, M DR Construction of M B, M Dol, M DR Hendrik Orem Talbot Workshop, Spring 2011 Contents 1 Some Moduli Space Theory 1 1.1 Moduli of Sheaves: Semistability and Boundedness.............. 1 1.2 Geometric Invariant

More information

where 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

where 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 information

MA 206 notes: introduction to resolution of singularities

MA 206 notes: introduction to resolution of singularities MA 206 notes: introduction to resolution of singularities Dan Abramovich Brown University March 4, 2018 Abramovich Introduction to resolution of singularities 1 / 31 Resolution of singularities Let k be

More information

Algebraic v.s. Analytic Point of View

Algebraic v.s. Analytic Point of View Algebraic v.s. Analytic Point of View Ziwen Zhu September 19, 2015 In this talk, we will compare 3 different yet similar objects of interest in algebraic and complex geometry, namely algebraic variety,

More information

A short proof of Klyachko s theorem about rational algebraic tori

A short proof of Klyachko s theorem about rational algebraic tori A short proof of Klyachko s theorem about rational algebraic tori Mathieu Florence Abstract In this paper, we give another proof of a theorem by Klyachko ([?]), which asserts that Zariski s conjecture

More information

LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI. 1. Maximal Tori

LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI. 1. Maximal Tori LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI 1. Maximal Tori By a torus we mean a compact connected abelian Lie group, so a torus is a Lie group that is isomorphic to T n = R n /Z n. Definition 1.1.

More information

On some smooth projective two-orbit varieties with Picard number 1

On some smooth projective two-orbit varieties with Picard number 1 On some smooth projective two-orbit varieties with Picard number 1 Boris Pasquier March 3, 2009 Abstract We classify all smooth projective horospherical varieties with Picard number 1. We prove that the

More information

Demushkin s Theorem in Codimension One

Demushkin s Theorem in Codimension One Universität Konstanz Demushkin s Theorem in Codimension One Florian Berchtold Jürgen Hausen Konstanzer Schriften in Mathematik und Informatik Nr. 176, Juni 22 ISSN 143 3558 c Fachbereich Mathematik und

More information

(1) A frac = b : a, b A, b 0. We can define addition and multiplication of fractions as we normally would. a b + c d

(1) A frac = b : a, b A, b 0. We can define addition and multiplication of fractions as we normally would. a b + c d The Algebraic Method 0.1. Integral Domains. Emmy Noether and others quickly realized that the classical algebraic number theory of Dedekind could be abstracted completely. In particular, rings of integers

More information

The Grothendieck-Katz Conjecture for certain locally symmetric varieties

The Grothendieck-Katz Conjecture for certain locally symmetric varieties The Grothendieck-Katz Conjecture for certain locally symmetric varieties Benson Farb and Mark Kisin August 20, 2008 Abstract Using Margulis results on lattices in semi-simple Lie groups, we prove the Grothendieck-

More information

Weyl Group Representations and Unitarity of Spherical Representations.

Weyl Group Representations and Unitarity of Spherical Representations. Weyl Group Representations and Unitarity of Spherical Representations. Alessandra Pantano, University of California, Irvine Windsor, October 23, 2008 β ν 1 = ν 2 S α S β ν S β ν S α ν S α S β S α S β ν

More information

ALGEBRAIC DENSITY PROPERTY OF HOMOGENEOUS SPACES

ALGEBRAIC DENSITY PROPERTY OF HOMOGENEOUS SPACES Transformation Groups c Birkhäuser Boston (2010) ALGEBRAIC DENSITY PROPERTY OF HOMOGENEOUS SPACES F. DONZELLI Institute of Mathematical Sciences Stony Brook University Stony Brook, NY 11794, USA fabrizio@math.sunysb.edu

More information

CHEVALLEY S THEOREM AND COMPLETE VARIETIES

CHEVALLEY S THEOREM AND COMPLETE VARIETIES CHEVALLEY S THEOREM AND COMPLETE VARIETIES BRIAN OSSERMAN In this note, we introduce the concept which plays the role of compactness for varieties completeness. We prove that completeness can be characterized

More information

Abelian varieties. Chapter Elliptic curves

Abelian varieties. Chapter Elliptic curves Chapter 3 Abelian varieties 3.1 Elliptic curves An elliptic curve is a curve of genus one with a distinguished point 0. Topologically it is looks like a torus. A basic example is given as follows. A subgroup

More information

Characteristic classes in the Chow ring

Characteristic classes in the Chow ring arxiv:alg-geom/9412008v1 10 Dec 1994 Characteristic classes in the Chow ring Dan Edidin and William Graham Department of Mathematics University of Chicago Chicago IL 60637 Let G be a reductive algebraic

More information

Algebraic Cobordism. 2nd German-Chinese Conference on Complex Geometry East China Normal University Shanghai-September 11-16, 2006.

Algebraic Cobordism. 2nd German-Chinese Conference on Complex Geometry East China Normal University Shanghai-September 11-16, 2006. Algebraic Cobordism 2nd German-Chinese Conference on Complex Geometry East China Normal University Shanghai-September 11-16, 2006 Marc Levine Outline: Describe the setting of oriented cohomology over a

More information

AN EXPOSITION OF THE RIEMANN ROCH THEOREM FOR CURVES

AN EXPOSITION OF THE RIEMANN ROCH THEOREM FOR CURVES AN EXPOSITION OF THE RIEMANN ROCH THEOREM FOR CURVES DOMINIC L. WYNTER Abstract. We introduce the concepts of divisors on nonsingular irreducible projective algebraic curves, the genus of such a curve,

More information

Toroidal Embeddings and Desingularization

Toroidal Embeddings and Desingularization California State University, San Bernardino CSUSB ScholarWorks Electronic Theses, Projects, and Dissertations Office of Graduate Studies 6-2018 Toroidal Embeddings and Desingularization LEON NGUYEN 003663425@coyote.csusb.edu

More information

Modern Computer Algebra

Modern Computer Algebra Modern Computer Algebra Exercises to Chapter 25: Fundamental concepts 11 May 1999 JOACHIM VON ZUR GATHEN and JÜRGEN GERHARD Universität Paderborn 25.1 Show that any subgroup of a group G contains the neutral

More information

Math 213br HW 12 solutions

Math 213br HW 12 solutions Math 213br HW 12 solutions May 5 2014 Throughout X is a compact Riemann surface. Problem 1 Consider the Fermat quartic defined by X 4 + Y 4 + Z 4 = 0. It can be built from 12 regular Euclidean octagons

More information

FILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS.

FILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS. FILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS. Let A be a ring, for simplicity assumed commutative. A filtering, or filtration, of an A module M means a descending sequence of submodules M = M 0

More information

Representation Theory

Representation Theory Representation Theory Representations Let G be a group and V a vector space over a field k. A representation of G on V is a group homomorphism ρ : G Aut(V ). The degree (or dimension) of ρ is just dim

More information