Lecture 1. Toric Varieties: Basics

Size: px
Start display at page:

Download "Lecture 1. Toric Varieties: Basics"

Transcription

1 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 1. Toric Varieties 2730 August / 27

2 An algebraic torus is a commutative complex algebraic group isomorphic to a product (C ) n of copies of the multiplicative group C = C \ {0}. It contains a compact torus T n as a Lie (but not algebraic) subgroup. A toric variety is a normal complex algebraic variety V containing an algebraic torus (C ) n as a Zariski open subset in such a way that the natural action of (C ) n on itself extends to an action on V. It follows that (C ) n acts on V with a dense orbit. Toric varieties originally appeared as equivariant compactications of an algebraic torus, although non-compact (e.g., ane) examples are now of equal importance. Example The algebraic torus (C ) n and the ane space C n are the simplest examples of toric varieties. A compact example is given by the projective space CP n on which the torus acts in homogeneous coordinates as (t 1,..., t n ) (z 0 : z 1 :... : z n ) = (z 0 : t 1 z 1 :... : t n z n ). Taras Panov (Moscow University) Lecture 1. Toric Varieties 2730 August / 27

3 Polyhedral cones A convex polyhedral cone generated by a1,..., ak R n : σ = R a1,..., ak = {µ 1 a1 + + µ k ak : µ i R }. Given a cone σ, can choose a minimal generating set a1,... ak. It is dened up to multiplication of vectors by positive constants. A cone is rational if its generators ai can be chosen from the integer lattice Z n R n. Then assume that each ai Z n is primitive, i. e. it is the smallest lattice vector in the ray dened by it. A cone is strongly convex if it does not contain a line. A cone is simplicial if it is generated by part of a basis of R n. A cone is regular if it is generated by part of a basis of Z n. Taras Panov (Moscow University) Lecture 1. Toric Varieties 2730 August / 27

4 Fans A fan is a nite collection Σ = {σ 1,..., σ s } of strongly convex cones in some R n such that every face of a cone in Σ belongs to Σ, and the intersection of any two cones in Σ is a face of each. A fan Σ is rational (respectively, simplicial, regular) if every cone in Σ is rational (respectively, simplicial, regular). A fan Σ = {σ 1,..., σ s } is called complete if σ 1 σ s = R n. Taras Panov (Moscow University) Lecture 1. Toric Varieties 2730 August / 27

5 Algebraic geometry of toric varieties is translated completely into the language of combinatorial and convex geometry. Namely, there is a bijective correspondence between rational fans in n-dimensional space and complex n-dimensional toric varieties. Under this correspondence, cones ane varieties complete fans complete (compact) varieties normal fans of polytopes projective varieties regular fans nonsingular varieties simplicial fans orbifolds N a lattice of rank n (isomorphic to Z n ), N R = N Z R = R n its ambient n-dimensional real vector space. C N = N Z C = (C ) n the algebraic torus. All cones and fans are assumed to be rational. Taras Panov (Moscow University) Lecture 1. Toric Varieties 2730 August / 27

6 Ane toric varieties Given a cone σ N R, its dual cone is σ v = { x NR : u, x 0 for all u σ}. We have (σ v ) v = σ, and σ v is strongly convex i dim σ = n. S σ := σ v N a nitely generated semigroup (with respect to addition). A σ = C[S σ ] the semigroup ring of S σ. It is a commutative nitely generated C-algebra, with a C-vector space basis {χ u : u S σ }, and multiplication χ u χ u = χ u+u, so χ 0 is the unit. V σ := Spec(A σ ) the ane toric variety corresponding to σ, A σ = C[V σ ] is the algebra of regular functions on V σ. Taras Panov (Moscow University) Lecture 1. Toric Varieties 2730 August / 27

7 Example Let N = Z n and σ = R e1,..., ek, where 0 k n. Then S σ = σ v N is generated by e 1,..., e k and ±e k+1,..., ±e n, and A σ = C[x1,..., x k, x k+1, x 1 k+1,..., x n, x 1 n ], where x i := χ e i. The corresponding ane variety is V σ = C C C C = C k (C ) n k. In particular, for k = n we obtain V σ = C n, and for k = 0 (i. e. σ = {0}) we obtain V σ = (C ) n, the torus. Taras Panov (Moscow University) Lecture 1. Toric Varieties 2730 August / 27

8 By choosing a multiplicative generator set in A σ we represent it as A σ = C[x 1,..., x r ]/I; then the variety V σ is the common zero set of polynomials from the ideal I. Example Let σ = R e2, 2e1 e2 in R 2. The two vectors do not span Z 2, so the cone is not regular. Then σ v = R e 1, e 1 + 2e 2. The semigroup S σ is generated by e 1, e 1 + e 2 and e 1 + 2e 2, with one relation among them.therefore, A σ = C[x, xy, xy 2 ] = C[u, v, w]/(v 2 uw) and V σ is a quadratic cone (a singular variety). Taras Panov (Moscow University) Lecture 1. Toric Varieties 2730 August / 27

9 Toric varieties from fans If τ is a face of σ, then σ v τ v, and the inclusion of algebras C[S σ ] C[S τ ] induces an inclusion V τ V σ of a Zariski open subset. This allows us to glue the ane varieties V σ corresponding to all cones σ in a fan Σ into a toric variety V Σ = colim σ Σ V σ. Here is the crucial point: the fact that the cones σ patch into a fan Σ guarantees that the variety V Σ obtained by gluing the pieces V σ is Hausdor in the usual topology. In algebraic geometry, the Hausdorness is replaced by the related notion of separatedness: a variety V is separated if the image of the diagonal map : V V V is Zariski closed. Lemma If Σ = {σ} is a fan, then the variety V Σ = colim σ Σ V σ is separated. Taras Panov (Moscow University) Lecture 1. Toric Varieties 2730 August / 27

10 The torus action The variety V σ carries an algebraic action of the torus C N = N Z C which is dened as follows. C N V σ V σ, (t, x) t x A point t C N gives a homomorphism N C, u = (u 1,..., u n ) t(u) = t u 1 1 tun n. Points of V σ correspond to algebra morphisms A σ C, or to semigroup homomorphisms S σ Cm, where Cm = C {0} is the multiplicative semigroup of complex numbers. Then dene t x as the point in V σ corresponding to the semigroup homomorphism S σ Cm given by u t(u)x(u). Taras Panov (Moscow University) Lecture 1. Toric Varieties 2730 August / 27

11 The morphism of algebras dual to the action C N V σ V σ is given by A σ C[N ] A σ, χ u χ u χ u for u S σ. If σ = {0}, then we obtain the multiplication in the algebraic group C N. The actions on the varieties V σ are compatible with the inclusions of open sets V τ V σ corresponding to the inclusions of faces τ σ. Therefore, for each fan Σ we obtain a C N -action on the variety V Σ, which extends the C N-action on itself. Taras Panov (Moscow University) Lecture 1. Toric Varieties 2730 August / 27

12 e2 e1 e1 e2 (a) e1 + k e2 e2 e2 (b) e1 Example (complex projective plane CP 2 ) Σ has three maximal cones: σ 0 = R e1, e2, σ 1 = R e2, e1 e2, σ 2 = R e1 e2, e1, see Figure (a). Then each V σi is C 2, with coordinates (x, y) for σ 0, (x 1, x 1 y) for σ 1, (y 1, xy 1 ) for σ 2. These three ane charts glue together into V Σ = CP 2 with homogeneous coordinates [z 0 : z 1 : z 2 ], x = z 1 /z 0, y = z 2 /z 0. Taras Panov (Moscow University) Lecture 1. Toric Varieties 2730 August / 27

13 e2 e1 e1 e2 (a) e1 + k e2 e2 e2 (b) e1 Example (Hirzebruch surfaces) Fix k Z and consider the complete fan in R 2 with the four two-dimensional cones generated by (e1, e2), (e1, e2), ( e1 + k e2, e2), ( e1 + k e2, e2), see Figure (b). Tthe corresponding toric variety F k is the projectivisation CP(C O(k)) of the sum of a trivial line bundle C and the kth power O(k) = η k of the conjugate tautological line bundle η over CP 1. These 2-dimensional complex varieties F k are known as Hirzebruch surfaces. Taras Panov (Moscow University) Lecture 1. Toric Varieties 2730 August / 27

14 Orbits and invariant subvarieties A toric variety V Σ is a disjoint union of its orbits by the action of the algebraic torus C N. There is one such orbit O σ for each cone σ Σ, and we have O σ = (C ) n k if dim σ = k. In particular, n-dimensional cones correspond to xed points, and the apex (the zero cone) corresponds to the dense orbit C N. The orbit closure O σ is a closed irreducible C N -invariant subvariety of V σ, and it is itself a toric variety. In fact, O σ consists of those orbits O τ for which τ contains σ as a face. Any irreducible invariant subvariety of V Σ can be obtained in this way. In particular, irreducible invariant divisors D 1,..., D m of V Σ correspond to edges of Σ. Taras Panov (Moscow University) Lecture 1. Toric Varieties 2730 August / 27

15 Polytopes and normal fans P a convex polytope with vertices in the lattice N (a lattice polytope): P = {x NR : a i, x + b i 0 for i = 1,..., m} where b i Z and ai N is the primitive normal to the facet F i = {x P : ai, x + b i = 0}, i = 1,..., m. Given a face Q P, dene the cone σ Q = {u N R : u, x u, x for all x Q and x P}. The dual cone σq v is the `polyhedral angle' at the face Q; it is generated by all vectors x x pointing from x Q to x P. The cone σ Q is generated by those ai which are normal to Q. Then Σ P = {σ Q : Q is a face of P} is a complete fan Σ P in N R, called the normal fan of P. If 0 int P, then Σ P consists of cones over the faces of the polar P. Taras Panov (Moscow University) Lecture 1. Toric Varieties 2730 August / 27

16 The normal fan Σ P has a maximal cone σ v for each vertex v P. The dual cone σv is the `vertex cone' at v, generated by all vectors pointing from v to other points of P. The normal fan Σ P is simplicial if and only if P is simple, i. e. there are precisely n = dim P facets meeting at each vertex of P. In this case, the cones of Σ P are generated by those sets of normals {ai 1,..., ai k } for which the intersection of facets F i1 F ik is nonempty. The normal fan Σ P of a polytope P contains the information about the normals to the facets (the generators ai of the edges of Σ P ) and the combinatorial structure of P (which sets of vectors ai span a cone of Σ P is determined by which facets intersect at a face). However the scalars b i in the presentation of P by inequalities are lost. Not every complete fan can be obtained by `forgetting the numbers b i ' from a presentation of a polytope by inequalities, i. e. not every complete fan is a normal fan. This is fails even for regular fans. Taras Panov (Moscow University) Lecture 1. Toric Varieties 2730 August / 27

17 Projective toric varieties Given a lattice polytope P N R, dene the toric variety V P = V ΣP. Since the normal fan Σ P does not depend on the linear size of the polytope, we may assume that for each vertex v the semigroup S σv is generated by the lattice points of the polytope (this can always be achieved by replacing P by kp with suciently large k). Since N = Hom Z (C N, C ), the lattice points of the polytope P N dene an embedding i P : C N (C ) N P, where N P is the number of lattice points in P. Proposition The toric variety V P is identied with the projective closure i P (C N ) CP N P. Taras Panov (Moscow University) Lecture 1. Toric Varieties 2730 August / 27

18 It follows that toric varieties V P arising from lattice polytopes P are projective, i. e. can be dened by a set of homogeneous equations in a projective space. The converse is also true: the fan corresponding to a projective toric variety is the normal fan of a lattice polytope. A polytope carries more geometric information than its normal fan: dierent lattice polytopes with the same normal fan Σ correspond to dierent projective embeddings of the toric variety V Σ. If D 1,..., D m are the invariant divisors corresponding to the facets of P, then D P = b 1 D b m D m is an ample divisor on V P. This means that, when k is suciently large, kd P is a hyperplane section divisor for a projective embedding V P CP r. The space of sections H 0 (V P, kd P ) of (the line bundle corresponding to) kd P has basis corresponding to the lattice points in kp. The embedding of V P into the projectivisation of H 0 (V P, kd P ) is exactly the projective embedding described above. Taras Panov (Moscow University) Lecture 1. Toric Varieties 2730 August / 27

19 Cohomology of toric manifolds A toric manifold is a smooth complete (compact) toric variety. Toric manifolds V Σ correspond to complete regular fans Σ. Projective toric manifolds V P correspond to lattice polytopes P whose normal fans are regular. The cohomology of a toric manifold V Σ can be calculated eectively from the fan Σ. The Betti numbers are determined by the combinatorics of Σ only, while the ring structure of H (V Σ ) depends on the geometric data. The latter consist of the primitive generators a1,..., am of one-dimensional cones (edges) of Σ. Taras Panov (Moscow University) Lecture 1. Toric Varieties 2730 August / 27

20 f i = f i (Σ) the number of (i + 1)-dimensional cones of Σ. If Σ = Σ P is the normal fan of an n-dimensional polytope P, then f i is the number of (n i 1)-dimensional faces of P. f 1 = 1, f 0 is the number of edges of Σ (facets of P), f n 1 is the number of maximal cones of Σ (vertices of P). f (Σ) = (f 0, f 1,..., f n 1 ) the f -vector of Σ. The h-vector h(σ) = (h 0, h 1..., h n ) is dened from the identity h 0 t n + h 1 t n h n = (t 1) n + f 0 (t 1) n f n 1. Taras Panov (Moscow University) Lecture 1. Toric Varieties 2730 August / 27

21 Theorem (DanilovJurkiewicz) Let V Σ be the toric manifold corresponding to a complete regular fan Σ in N R. The cohomology ring of V Σ is given by H (V Σ ) = Z[v 1,..., v m ]/I, where v 1,..., v m H 2 (V Σ ) are the cohomology classes dual to the invariant divisors corresponding to the one-dimensional cones of Σ, and I is the ideal generated by elements of the following two types: (a) v i1 v ik, where ai 1,..., ai k do not span a cone of Σ; m (b) aj, u v j, for any u N. j=1 The homology groups of V Σ vanish in odd dimensions, and are free abelian in even dimensions, with ranks given by b 2i (V Σ ) = h i, where h i, i = 0, 1,..., n, are the components of the h-vector of Σ. Taras Panov (Moscow University) Lecture 1. Toric Varieties 2730 August / 27

22 To obtain an explicit presentation of the ring H (V Σ ) we choose a basis of N write the coordinates of ai in the columns of the integer n m-matrix a 11 a 1m Λ =..... a n1 a nm Then the n linear forms a j1 v a jm v m corresponding to the rows of Λ vanish in H (V ; Z). Example (complex projective space CP n ) The corresponding polytope is an n-simplex P = n, and Λ = The cohomology ring H (CP n ) is given by Z[v 1,..., v n+1 ]/(v 1 v n+1, v 1 v n+1,..., v n v n+1 ) = Z[v]/(v n+1 ) Taras Panov (Moscow University) Lecture 1. Toric Varieties 2730 August / 27

23 The DanilovJurkiewicz Theorem remains valid for complete simplicial fans and corresponding toric orbifolds if the integer coecients are replaced by the rationals. The integral cohomology of toric orbifolds often has torsion, and the integer ring structure is subtle even in the simplest case of weighted projective spaces. The cohomology ring of a toric manifold V Σ is generated by two-dimensional classes. This is the rst property to check if one wishes to determine whether a given algebraic variety or smooth manifold has a structure of a toric manifold. For instance, this rules out ag varieties and Grassmannians dierent from projective spaces. Taras Panov (Moscow University) Lecture 1. Toric Varieties 2730 August / 27

24 Theorem (Hard Lefschetz Theorem for toric orbifolds) Let P be a lattice simple polytope, V P the corresponding projective toric variety with ample divisor b 1 D b m D m, and ω = b 1 v b m v m H 2 (V P ; C) the corresponding cohomology class. Then the maps are isomorphisms for all i = 1,..., n. H n i (V P ; C) ωi H n+i (V P ; C) If V P is smooth, then it is K ahler, and ω is the class of the K ahler 2-form. Taras Panov (Moscow University) Lecture 1. Toric Varieties 2730 August / 27

25 Question How to characterise the f -vectors (or h-vectors) for simplicial polytopes, simplicial fans or triangulated spheres? Theorem (BilleraLee, Stanley) The following conditions are necessary and sucient for a collection (f 0, f 1,..., f n 1 ) to be the f -vector of a simplicial polytope: (a) h i = h n i for i = 0,..., n; (b) h 0 h 1 h 2 h [n/2] ; (c)... (a restriction on the growth of h i ). Stanley's argument: realise the dual simple polytope as a lattice polytope P and consider the projective toric variety V P. We have dim H 2i (V P, Q) = h i. Then (a) is Poincar e duality, while (b) and (c) follow from the Hard Lefschetz Theorem. Taras Panov (Moscow University) Lecture 1. Toric Varieties 2730 August / 27

26 Conjecture (McMullen) Is it true that the same conditions (a)(c) characterise the f -vectors of triangulated spheres? This is open even for complete simplicial fans (star-shaped spheres). Taras Panov (Moscow University) Lecture 1. Toric Varieties 2730 August / 27

27 Literature [1] Victor Buchstaber and Taras Panov. Toric Topology. Mathematical Surveys and Monographs, vol. 204, Amer. Math. Soc., Providence, RI, [2] David A. Cox, John B. Little and Henry K. Schenck. Toric varieties. Graduate Studies in Mathematics, 124. Amer. Math. Soc., Providence, RI, [3] William Fulton. Introduction to Toric Varieties. Annals of Mathematics Studies, 131. Princeton Univ. Press, Princeton, NJ, Taras Panov (Moscow University) Lecture 1. Toric Varieties 2730 August / 27

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

New constructions of Hamiltonian-minimal Lagrangian submanifolds

New constructions of Hamiltonian-minimal Lagrangian submanifolds New constructions of Hamiltonian-minimal Lagrangian submanifolds based on joint works with Andrey E. Mironov Taras Panov Moscow State University Integrable Systems CSF Ascona, 19-24 June 2016 Taras Panov

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

Model categories and homotopy colimits in toric topology

Model categories and homotopy colimits in toric topology Model categories and homotopy colimits in toric topology Taras Panov Moscow State University joint work with Nigel Ray 1 1. Motivations. Object of study of toric topology : torus actions on manifolds or

More information

Homogeneous Coordinate Ring

Homogeneous Coordinate Ring Students: Kaiserslautern University Algebraic Group June 14, 2013 Outline Quotients in Algebraic Geometry 1 Quotients in Algebraic Geometry 2 3 4 Outline Quotients in Algebraic Geometry 1 Quotients in

More information

Monomial equivariant embeddings of quasitoric manifolds and the problem of existence of invariant almost complex structures.

Monomial equivariant embeddings of quasitoric manifolds and the problem of existence of invariant almost complex structures. Monomial equivariant embeddings of quasitoric manifolds and the problem of existence of invariant almost complex structures. Andrey Kustarev joint work with V. M. Buchstaber, Steklov Mathematical Institute

More information

Toric Geometry. An introduction to toric varieties with an outlook towards toric singularity theory

Toric Geometry. An introduction to toric varieties with an outlook towards toric singularity theory Toric Geometry An introduction to toric varieties with an outlook towards toric singularity theory Thesis project for the research track of the Master Mathematics Academic year 2014/2015 Author: Geert

More information

7.3 Singular Homology Groups

7.3 Singular Homology Groups 184 CHAPTER 7. HOMOLOGY THEORY 7.3 Singular Homology Groups 7.3.1 Cycles, Boundaries and Homology Groups We can define the singular p-chains with coefficients in a field K. Furthermore, we can define the

More information

TORIC REDUCTION AND TROPICAL GEOMETRY A.

TORIC REDUCTION AND TROPICAL GEOMETRY A. Mathematisches Institut, Seminars, (Y. Tschinkel, ed.), p. 109 115 Universität Göttingen, 2004-05 TORIC REDUCTION AND TROPICAL GEOMETRY A. Szenes ME Institute of Mathematics, Geometry Department, Egry

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

Cohomological rigidity of 6-dimensional quasitoric manifolds

Cohomological rigidity of 6-dimensional quasitoric manifolds Cohomological rigidity of 6-dimensional quasitoric manifolds Seonjeong Park 1 Joint work with Buchstaber 2, Erokhovets 2, Masuda 1, and Panov 2 1 Osaka City University, Japan 2 Moscow State University,

More information

INTRODUCTION TO TORIC VARIETIES

INTRODUCTION TO TORIC VARIETIES INTRODUCTION TO TORIC VARIETIES Jean-Paul Brasselet IML Luminy Case 97 F-13288 Marseille Cedex 9 France jpb@iml.univ-mrs.fr The course given during the School and Workshop The Geometry and Topology of

More information

Analogs of Hodge Riemann relations

Analogs of Hodge Riemann relations Analogs of Hodge Riemann relations in algebraic geometry, convex geometry and linear algebra V. Timorin State University of New York at Stony Brook March 28, 2006 Hodge Riemann form Let X be a compact

More information

arxiv:math.ag/ v1 7 Jan 2005

arxiv:math.ag/ v1 7 Jan 2005 arxiv:math.ag/0501104 v1 7 Jan 2005 Asymptotic cohomological functions of toric divisors Milena Hering, Alex Küronya, Sam Payne January 7, 2005 Abstract We study functions on the class group of a toric

More information

HILBERT BASIS OF THE LIPMAN SEMIGROUP

HILBERT BASIS OF THE LIPMAN SEMIGROUP Available at: http://publications.ictp.it IC/2010/061 United Nations Educational, Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL

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

Toric Varieties and the Secondary Fan

Toric 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 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

arxiv: v1 [quant-ph] 19 Jan 2010

arxiv: v1 [quant-ph] 19 Jan 2010 A toric varieties approach to geometrical structure of multipartite states Hoshang Heydari Physics Department, Stockholm university 10691 Stockholm Sweden arxiv:1001.3245v1 [quant-ph] 19 Jan 2010 Email:

More information

Topology of Toric Varieties, Part II

Topology of Toric Varieties, Part II Topology of Toric Varieties, Part II Daniel Chupin April 2, 2018 Abstract Notes for a talk leading up to a discussion of the Hirzebruch-Riemann-Roch (HRR) theorem for toric varieties, and some consequences

More information

TORIC WEAK FANO VARIETIES ASSOCIATED TO BUILDING SETS

TORIC WEAK FANO VARIETIES ASSOCIATED TO BUILDING SETS TORIC WEAK FANO VARIETIES ASSOCIATED TO BUILDING SETS YUSUKE SUYAMA Abstract. We give a necessary and sufficient condition for the nonsingular projective toric variety associated to a building set to be

More information

Hodge theory for combinatorial geometries

Hodge theory for combinatorial geometries Hodge theory for combinatorial geometries June Huh with Karim Adiprasito and Eric Katz June Huh 1 / 48 Three fundamental ideas: June Huh 2 / 48 Three fundamental ideas: The idea of Bernd Sturmfels that

More information

Hamiltonian Toric Manifolds

Hamiltonian Toric Manifolds Hamiltonian Toric Manifolds JWR (following Guillemin) August 26, 2001 1 Notation Throughout T is a torus, T C is its complexification, V = L(T ) is its Lie algebra, and Λ V is the kernel of the exponential

More information

1 Moduli spaces of polarized Hodge structures.

1 Moduli spaces of polarized Hodge structures. 1 Moduli spaces of polarized Hodge structures. First of all, we briefly summarize the classical theory of the moduli spaces of polarized Hodge structures. 1.1 The moduli space M h = Γ\D h. Let n be an

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

Gorenstein rings through face rings of manifolds.

Gorenstein rings through face rings of manifolds. Gorenstein rings through face rings of manifolds. Isabella Novik Department of Mathematics, Box 354350 University of Washington, Seattle, WA 98195-4350, USA, novik@math.washington.edu Ed Swartz Department

More information

On the Parameters of r-dimensional Toric Codes

On the Parameters of r-dimensional Toric Codes On the Parameters of r-dimensional Toric Codes Diego Ruano Abstract From a rational convex polytope of dimension r 2 J.P. Hansen constructed an error correcting code of length n = (q 1) r over the finite

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

Cohomology groups of toric varieties

Cohomology groups of toric varieties Cohomology groups of toric varieties Masanori Ishida Mathematical Institute, Tohoku University 1 Fans and Complexes Although we treat real fans later, we begin with fans consisting of rational cones which

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

Math 530 Lecture Notes. Xi Chen

Math 530 Lecture Notes. Xi Chen Math 530 Lecture Notes Xi Chen 632 Central Academic Building, University of Alberta, Edmonton, Alberta T6G 2G1, CANADA E-mail address: xichen@math.ualberta.ca 1991 Mathematics Subject Classification. Primary

More information

arxiv: v2 [math.at] 18 Mar 2012

arxiv: v2 [math.at] 18 Mar 2012 TODD GENERA OF COMPLEX TORUS MANIFOLDS arxiv:1203.3129v2 [math.at] 18 Mar 2012 HIROAKI ISHIDA AND MIKIYA MASUDA Abstract. In this paper, we prove that the Todd genus of a compact complex manifold X of

More information

CLASSIFICATION OF TORIC MANIFOLDS OVER AN n-cube WITH ONE VERTEX CUT

CLASSIFICATION OF TORIC MANIFOLDS OVER AN n-cube WITH ONE VERTEX CUT CLASSIFICATION OF TORIC MANIFOLDS OVER AN n-cube WITH ONE VERTEX CUT SHO HASUI, HIDEYA KUWATA, MIKIYA MASUDA, AND SEONJEONG PARK Abstract We say that a complete nonsingular toric variety (called a toric

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

An Introduction to Toric Varieties

An Introduction to Toric Varieties An Introduction to Toric Varieties submitted by Christopher Eur in partial fulfillment of the requirements for the degree of Bachelor of Arts with Honors Advisor: Melody Chan March 23, 2015 Harvard University

More information

Toric Varieties. Madeline Brandt. April 26, 2017

Toric Varieties. Madeline Brandt. April 26, 2017 Toric Varieties Madeline Brandt April 26, 2017 Last week we saw that we can define normal toric varieties from the data of a fan in a lattice. Today I will review this idea, and also explain how they can

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

Combinatorics for algebraic geometers

Combinatorics for algebraic geometers Combinatorics for algebraic geometers Calculations in enumerative geometry Maria Monks March 17, 214 Motivation Enumerative geometry In the late 18 s, Hermann Schubert investigated problems in what is

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

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

Oral exam practice problems: Algebraic Geometry

Oral exam practice problems: Algebraic Geometry Oral exam practice problems: Algebraic Geometry Alberto García Raboso TP1. Let Q 1 and Q 2 be the quadric hypersurfaces in P n given by the equations f 1 x 2 0 + + x 2 n = 0 f 2 a 0 x 2 0 + + a n x 2 n

More information

Geometry and combinatorics of spherical varieties.

Geometry and combinatorics of spherical varieties. Geometry and combinatorics of spherical varieties. Notes of a course taught by Guido Pezzini. Abstract This is the lecture notes from a mini course at the Winter School Geometry and Representation Theory

More information

EQUIVARIANT COHOMOLOGY IN ALGEBRAIC GEOMETRY APPENDIX A: ALGEBRAIC TOPOLOGY

EQUIVARIANT COHOMOLOGY IN ALGEBRAIC GEOMETRY APPENDIX A: ALGEBRAIC TOPOLOGY EQUIVARIANT COHOMOLOGY IN ALGEBRAIC GEOMETRY APPENDIX A: ALGEBRAIC TOPOLOGY WILLIAM FULTON NOTES BY DAVE ANDERSON In this appendix, we collect some basic facts from algebraic topology pertaining to the

More information

arxiv: v2 [math.ac] 19 Oct 2014

arxiv: v2 [math.ac] 19 Oct 2014 A duality theorem for syzygies of Veronese ideals arxiv:1311.5653v2 [math.ac] 19 Oct 2014 Stepan Paul California Polytechnic State University San Luis Obispo, CA 93407 0403 (805) 756 5552 stpaul@calpoly.edu

More information

Algebraic Varieties. Notes by Mateusz Micha lek for the lecture on April 17, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra

Algebraic Varieties. Notes by Mateusz Micha lek for the lecture on April 17, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra Algebraic Varieties Notes by Mateusz Micha lek for the lecture on April 17, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra Algebraic varieties represent solutions of a system of polynomial

More information

MANIN-MUMFORD AND LATTÉS MAPS

MANIN-MUMFORD AND LATTÉS MAPS MANIN-MUMFORD AND LATTÉS MAPS JORGE PINEIRO Abstract. The present paper is an introduction to the dynamical Manin-Mumford conjecture and an application of a theorem of Ghioca and Tucker to obtain counterexamples

More information

C n.,..., z i 1., z i+1., w i+1,..., wn. =,..., w i 1. : : w i+1. :... : w j 1 1.,..., w j 1. z 0 0} = {[1 : w] w C} S 1 { },

C n.,..., z i 1., z i+1., w i+1,..., wn. =,..., w i 1. : : w i+1. :... : w j 1 1.,..., w j 1. z 0 0} = {[1 : w] w C} S 1 { }, Complex projective space The complex projective space CP n is the most important compact complex manifold. By definition, CP n is the set of lines in C n+1 or, equivalently, CP n := (C n+1 \{0})/C, where

More information

The rational cohomology of real quasi-toric manifolds

The rational cohomology of real quasi-toric manifolds The rational cohomology of real quasi-toric manifolds Alex Suciu Northeastern University Joint work with Alvise Trevisan (VU Amsterdam) Toric Methods in Homotopy Theory Queen s University Belfast July

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

Hard Lefschetz theorem for simple polytopes

Hard Lefschetz theorem for simple polytopes J Algebr Comb (2010) 32: 227 239 DOI 10.1007/s10801-009-0212-1 Hard Lefschetz theorem for simple polytopes Balin Fleming Kalle Karu Received: 16 March 2009 / Accepted: 24 November 2009 / Published online:

More information

Logarithmic functional and reciprocity laws

Logarithmic functional and reciprocity laws Contemporary Mathematics Volume 00, 1997 Logarithmic functional and reciprocity laws Askold Khovanskii Abstract. In this paper, we give a short survey of results related to the reciprocity laws over the

More information

Betti numbers of abelian covers

Betti numbers of abelian covers Betti numbers of abelian covers Alex Suciu Northeastern University Geometry and Topology Seminar University of Wisconsin May 6, 2011 Alex Suciu (Northeastern University) Betti numbers of abelian covers

More information

H(G(Q p )//G(Z p )) = C c (SL n (Z p )\ SL n (Q p )/ SL n (Z p )).

H(G(Q p )//G(Z p )) = C c (SL n (Z p )\ SL n (Q p )/ SL n (Z p )). 92 19. Perverse sheaves on the affine Grassmannian 19.1. Spherical Hecke algebra. The Hecke algebra H(G(Q p )//G(Z p )) resp. H(G(F q ((T ))//G(F q [[T ]])) etc. of locally constant compactly supported

More information

EXCLUDED HOMEOMORPHISM TYPES FOR DUAL COMPLEXES OF SURFACES

EXCLUDED HOMEOMORPHISM TYPES FOR DUAL COMPLEXES OF SURFACES EXCLUDED HOMEOMORPHISM TYPES FOR DUAL COMPLEXES OF SURFACES DUSTIN CARTWRIGHT Abstract. We study an obstruction to prescribing the dual complex of a strict semistable degeneration of an algebraic surface.

More information

LECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM

LECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM LECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM Contents 1. The Atiyah-Guillemin-Sternberg Convexity Theorem 1 2. Proof of the Atiyah-Guillemin-Sternberg Convexity theorem 3 3. Morse theory

More information

Vector bundles in Algebraic Geometry Enrique Arrondo. 1. The notion of vector bundle

Vector bundles in Algebraic Geometry Enrique Arrondo. 1. The notion of vector bundle Vector bundles in Algebraic Geometry Enrique Arrondo Notes(* prepared for the First Summer School on Complex Geometry (Villarrica, Chile 7-9 December 2010 1 The notion of vector bundle In affine geometry,

More information

Invariants of Normal Surface Singularities

Invariants of Normal Surface Singularities Rényi Institute of Mathematics, Budapest June 19, 2009 Notations Motivation. Questions. Normal surface singularities (X, o) = a normal surface singularity M = its link (oriented 3manifold) assume: M is

More information

Resolution of Singularities in Algebraic Varieties

Resolution of Singularities in Algebraic Varieties Resolution of Singularities in Algebraic Varieties Emma Whitten Summer 28 Introduction Recall that algebraic geometry is the study of objects which are or locally resemble solution sets of polynomial equations.

More information

Surprising Examples of Manifolds in Toric Topology!

Surprising Examples of Manifolds in Toric Topology! Surprising Examples of Manifolds in Toric Topology! Djordje Baralić Mathematical Institute SASA Belgrade Serbia Lazar Milenković Union University Faculty of Computer Science Belgrade Serbia arxiv:17005932v1

More information

(1.) For any subset P S we denote by L(P ) the abelian group of integral relations between elements of P, i.e. L(P ) := ker Z P! span Z P S S : For ea

(1.) For any subset P S we denote by L(P ) the abelian group of integral relations between elements of P, i.e. L(P ) := ker Z P! span Z P S S : For ea Torsion of dierentials on toric varieties Klaus Altmann Institut fur reine Mathematik, Humboldt-Universitat zu Berlin Ziegelstr. 13a, D-10099 Berlin, Germany. E-mail: altmann@mathematik.hu-berlin.de Abstract

More information

Combinatorial Commutative Algebra and D-Branes

Combinatorial Commutative Algebra and D-Branes Combinatorial Commutative Algebra and D-Branes Chirag Lakhani May 13, 2009 Abstract This is a survey paper written for Professor Ezra Miller s Combinatorial Commutative Algebra course in the Spring of

More information

INTRODUCTION TO TROPICAL ALGEBRAIC GEOMETRY

INTRODUCTION TO TROPICAL ALGEBRAIC GEOMETRY INTRODUCTION TO TROPICAL ALGEBRAIC GEOMETRY DIANE MACLAGAN These notes are the lecture notes from my lectures on tropical geometry at the ELGA 2011 school on Algebraic Geometry and Applications in Buenos

More information

Reciprocal domains and Cohen Macaulay d-complexes in R d

Reciprocal domains and Cohen Macaulay d-complexes in R d Reciprocal domains and Cohen Macaulay d-complexes in R d Ezra Miller and Victor Reiner School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA ezra@math.umn.edu, reiner@math.umn.edu

More information

Affine Geometry and Discrete Legendre Transform

Affine Geometry and Discrete Legendre Transform Affine Geometry and Discrete Legendre Transform Andrey Novoseltsev April 24, 2008 Abstract The combinatorial duality used in Gross-Siebert approach to mirror symmetry is the discrete Legendre transform

More information

Boundary of Cohen-Macaulay cone and asymptotic behavior of system of ideals

Boundary of Cohen-Macaulay cone and asymptotic behavior of system of ideals Boundary of Cohen-Macaulay cone and asymptotic behavior of system of ideals Kazuhiko Kurano Meiji University 1 Introduction On a smooth projective variety, we can define the intersection number for a given

More information

Three Descriptions of the Cohomology of Bun G (X) (Lecture 4)

Three Descriptions of the Cohomology of Bun G (X) (Lecture 4) Three Descriptions of the Cohomology of Bun G (X) (Lecture 4) February 5, 2014 Let k be an algebraically closed field, let X be a algebraic curve over k (always assumed to be smooth and complete), and

More information

Secant varieties of toric varieties

Secant varieties of toric varieties Journal of Pure and Applied Algebra 209 (2007) 651 669 www.elsevier.com/locate/jpaa Secant varieties of toric varieties David Cox a, Jessica Sidman b, a Department of Mathematics and Computer Science,

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

ADE SURFACE SINGULARITIES, CHAMBERS AND TORIC VARIETIES. Meral Tosun

ADE SURFACE SINGULARITIES, CHAMBERS AND TORIC VARIETIES. Meral Tosun Séminaires & Congrès 10, 2005, p. 341 350 ADE SURFACE SINGULARITIES, CHAMBERS AND TORIC VARIETIES by Meral Tosun Abstract. We study the link between the positive divisors supported on the exceptional divisor

More information

PROBLEMS, MATH 214A. Affine and quasi-affine varieties

PROBLEMS, MATH 214A. Affine and quasi-affine varieties PROBLEMS, MATH 214A k is an algebraically closed field Basic notions Affine and quasi-affine varieties 1. Let X A 2 be defined by x 2 + y 2 = 1 and x = 1. Find the ideal I(X). 2. Prove that the subset

More information

arxiv: v1 [math.at] 9 Sep 2016

arxiv: v1 [math.at] 9 Sep 2016 September 13, 2016 Lecture Note Series, IMS, NUS Review Vol. 9in x 6in Lectures-F page 1 FULLERENES, POLYTOPES AND TORIC TOPOLOGY arxiv:1609.02949v1 [math.at] 9 Sep 2016 Victor M. Buchstaber Steklov Mathematical

More information

The tangent space to an enumerative problem

The tangent space to an enumerative problem The tangent space to an enumerative problem Prakash Belkale Department of Mathematics University of North Carolina at Chapel Hill North Carolina, USA belkale@email.unc.edu ICM, Hyderabad 2010. Enumerative

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

Multiplicity of singularities is not a bi-lipschitz invariant

Multiplicity of singularities is not a bi-lipschitz invariant Multiplicity of singularities is not a bi-lipschitz invariant Misha Verbitsky Joint work with L. Birbrair, A. Fernandes, J. E. Sampaio Geometry and Dynamics Seminar Tel-Aviv University, 12.12.2018 1 Zariski

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

THE QUANTUM CONNECTION

THE 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 information

arxiv:math/ v1 [math.ag] 18 Oct 2003

arxiv:math/ v1 [math.ag] 18 Oct 2003 Proc. Indian Acad. Sci. (Math. Sci.) Vol. 113, No. 2, May 2003, pp. 139 152. Printed in India The Jacobian of a nonorientable Klein surface arxiv:math/0310288v1 [math.ag] 18 Oct 2003 PABLO ARÉS-GASTESI

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

AG NOTES MARK BLUNK. 3. Varieties

AG NOTES MARK BLUNK. 3. Varieties AG NOTES MARK BLUNK Abstract. Some rough notes to get you all started. 1. Introduction Here is a list of the main topics that are discussed and used in my research talk. The information is rough and brief,

More information

arxiv: v2 [math.ag] 29 Aug 2009

arxiv: v2 [math.ag] 29 Aug 2009 LOGARITHMIC GEOMETRY, MINIMAL FREE RESOLUTIONS AND TORIC ALGEBRAIC STACKS arxiv:0707.2568v2 [math.ag] 29 Aug 2009 ISAMU IWANARI Abstract. In this paper we will introduce a certain type of morphisms of

More information

Part II. Algebraic Topology. Year

Part II. Algebraic Topology. Year Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 Paper 3, Section II 18I The n-torus is the product of n circles: 5 T n = } S 1. {{.. S } 1. n times For all n 1 and 0

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

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

Finite generation of pluricanonical rings

Finite 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 information

SPACES OF RATIONAL CURVES IN COMPLETE INTERSECTIONS

SPACES 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 information

Igusa fibre integrals over local fields

Igusa fibre integrals over local fields March 25, 2017 1 2 Standard definitions The map ord S The map ac S 3 Toroidal constructible functions Main Theorem Some conjectures 4 Toroidalization Key Theorem Proof of Key Theorem Igusa fibre integrals

More information

CW-complexes. Stephen A. Mitchell. November 1997

CW-complexes. Stephen A. Mitchell. November 1997 CW-complexes Stephen A. Mitchell November 1997 A CW-complex is first of all a Hausdorff space X equipped with a collection of characteristic maps φ n α : D n X. Here n ranges over the nonnegative integers,

More information

Intersection Theory course notes

Intersection Theory course notes Intersection Theory course notes Valentina Kiritchenko Fall 2013, Faculty of Mathematics, NRU HSE 1. Lectures 1-2: examples and tools 1.1. Motivation. Intersection theory had been developed in order to

More information

1 Structures 2. 2 Framework of Riemann surfaces Basic configuration Holomorphic functions... 3

1 Structures 2. 2 Framework of Riemann surfaces Basic configuration Holomorphic functions... 3 Compact course notes Riemann surfaces Fall 2011 Professor: S. Lvovski transcribed by: J. Lazovskis Independent University of Moscow December 23, 2011 Contents 1 Structures 2 2 Framework of Riemann surfaces

More information

Affine Geometry and the Discrete Legendre Transfrom

Affine Geometry and the Discrete Legendre Transfrom Affine Geometry and the Discrete Legendre Transfrom Andrey Novoseltsev Department of Mathematics University of Washington April 25, 2008 Andrey Novoseltsev (UW) Affine Geometry April 25, 2008 1 / 24 Outline

More information

A Z q -Fan theorem. 1 Introduction. Frédéric Meunier December 11, 2006

A Z q -Fan theorem. 1 Introduction. Frédéric Meunier December 11, 2006 A Z q -Fan theorem Frédéric Meunier December 11, 2006 Abstract In 1952, Ky Fan proved a combinatorial theorem generalizing the Borsuk-Ulam theorem stating that there is no Z 2-equivariant map from the

More information

arxiv: v1 [math.ag] 13 Mar 2019

arxiv: v1 [math.ag] 13 Mar 2019 THE CONSTRUCTION PROBLEM FOR HODGE NUMBERS MODULO AN INTEGER MATTHIAS PAULSEN AND STEFAN SCHREIEDER arxiv:1903.05430v1 [math.ag] 13 Mar 2019 Abstract. For any integer m 2 and any dimension n 1, we show

More information

On the singular elements of a semisimple Lie algebra and the generalized Amitsur-Levitski Theorem

On the singular elements of a semisimple Lie algebra and the generalized Amitsur-Levitski Theorem On the singular elements of a semisimple Lie algebra and the generalized Amitsur-Levitski Theorem Bertram Kostant, MIT Conference on Representations of Reductive Groups Salt Lake City, Utah July 10, 2013

More information

Complex Bordism and Cobordism Applications

Complex Bordism and Cobordism Applications Complex Bordism and Cobordism Applications V. M. Buchstaber Mini-course in Fudan University, April-May 2017 Main goals: --- To describe the main notions and constructions of bordism and cobordism; ---

More information

TOPOLOGY OF LINE ARRANGEMENTS. Alex Suciu. Northeastern University. Workshop on Configuration Spaces Il Palazzone di Cortona September 1, 2014

TOPOLOGY OF LINE ARRANGEMENTS. Alex Suciu. Northeastern University. Workshop on Configuration Spaces Il Palazzone di Cortona September 1, 2014 TOPOLOGY OF LINE ARRANGEMENTS Alex Suciu Northeastern University Workshop on Configuration Spaces Il Palazzone di Cortona September 1, 2014 ALEX SUCIU (NORTHEASTERN) TOPOLOGY OF LINE ARRANGEMENTS CORTONA,

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

GROUPS DEFINABLE IN O-MINIMAL STRUCTURES

GROUPS DEFINABLE IN O-MINIMAL STRUCTURES GROUPS DEFINABLE IN O-MINIMAL STRUCTURES PANTELIS E. ELEFTHERIOU Abstract. In this series of lectures, we will a) introduce the basics of o- minimality, b) describe the manifold topology of groups definable

More information

EQUIVARIANT COHOMOLOGY IN ALGEBRAIC GEOMETRY LECTURE TWO: DEFINITIONS AND BASIC PROPERTIES

EQUIVARIANT COHOMOLOGY IN ALGEBRAIC GEOMETRY LECTURE TWO: DEFINITIONS AND BASIC PROPERTIES EQUIVARIANT COHOMOLOGY IN ALGEBRAIC GEOMETRY LECTURE TWO: DEFINITIONS AND BASIC PROPERTIES WILLIAM FULTON NOTES BY DAVE ANDERSON 1 For a Lie group G, we are looking for a right principal G-bundle EG BG,

More information

Free Loop Cohomology of Complete Flag Manifolds

Free Loop Cohomology of Complete Flag Manifolds June 12, 2015 Lie Groups Recall that a Lie group is a space with a group structure where inversion and group multiplication are smooth. Lie Groups Recall that a Lie group is a space with a group structure

More information

As is well known, the correspondence (2) above is generalized to compact symplectic manifolds with torus actions admitting moment maps. A symplectic m

As is well known, the correspondence (2) above is generalized to compact symplectic manifolds with torus actions admitting moment maps. A symplectic m TORUS ACTIONS ON MANIFOLDS Mikiya Masuda An overview. In this series of lectures I intend to develope the theory of toric varieties, which is a bridge between algebraic geometry and combinatorics, from

More information