# Lecture 1. Toric Varieties: Basics

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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

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