Minicourse on Toric Varieties University of Buenos Aires July, 2001 David A. Cox Amherst College Lecture I: Toric Varieties and Their Constructions 1. Varieties ........................................ .........................................1 2. Characters and 1-Parameter Subgroups . ......................................1 3. ToricVarieties ................................... .........................................2 4. First Construction: Cones and Fans . ......................................3 5. Properties of Toric Varieties . .........................................7 6. Second Construction: Homogeneous Coordinates . ....................................8 7. Third Construction: Toric Ideals . .......................................10 Lecture II: Toric Varieties and Polytopes 1. The Toric Variety of a Polytope . .....................................12 2. The Dehn-Sommerville Equations . ....................................14 3. TheEhrhartPolynomial............................. ......................................15 4. TheBKKTheorem.................................... ...................................18 5. Reflexive Polytopes and Fano Toric Varieties . ....................................19 Lecture III: Toric Varieties and Mirror Symmetry 1. TheQuinticThreefold.............................. ......................................22 2. InstantonNumbers................................. ......................................23 3. TheQuinticMirror................................. ......................................24 4. Superconformal Field Theory. ......................................25 5. The Batyrev Mirror Construction . .....................................27 6. Other Aspects fo Mirror Symmetry. ....................................28 Acknowledgements Bibliography Lecture I: Toric Varieties and Their Constructions 1 Varieties We will work over the complex numbers C . Basic examples of varieties are: n C ¯ Affine space and affine varieties n ´ ;:::; µ C V V f1 fs ¡¡¡ defined by the polynomial equations f1 fs 0. n È ¯ Projective space and projective varieties n ´ ;:::; µ È V V F1 Fs ¡¡¡ defined by the homogeneous equations F1 Fs 0. £ £ n n C Ò ´C µ C ´ ;:::; µ Example 1.1 Let C 0 . Then is an affine variety since the map t1 tn ´ ; = ¡¡¡ µ t1 ;:::; tn 1 t1 tn gives a bijection · £ n n 1 C µ ³ ´ ¡¡¡ µ C : ´ V x x x 1 1 2 n·1 £ We call C the n-dimensional complex torus. Also recall that given varieties V and W, we can form the product variety V ¢ W. Then a ¢ morphism ϕ : V W is a function whose graph is a subvariety of V W. 2 Characters and 1-Parameter Subgroups £ n ´C µ The torus T has: ¯ The character group £ χ C χ : M : T is a morphism and a group homomorphism ¯ The group of 1-parameter subgroups £ λ C λ : N : T is a morphism and a group homomorphism Note that: n n ³ Z ´ ;:::; µ ¾ Z ¯ M where m m1 mn gives m m1 mn ;:::; µ ¡¡¡ : χ ´t t t t 1 n 1 n n n ³ Z ´ ;:::; µ ¾ Z ¯ N where u u1 un gives u u1 un µ ´ ;:::; µ: λ ´t t t 1 £ £ k λ ¾ χ Æ λ C C ¾ Z Given χ ¾ M and N, the composition : is of the form t t for some k . χ ; λ If we set k , then χ ;λ : λ ´ µ χ Æ t t One can easily check that: ¢ Z ´ χ ; λ µ χ ; λ ¯ The map M N given by is a perfect pairing. ´ ;:::; µ ´ ;:::; µ ¯ Given m m1 mn and u u1 un , then m u χ ; λ · ¡¡¡ · : m1u1 mnun Furthermore: n Z ¾ ¯ We will usually identify M with and write m M. However, when we think of m as a £ n m ´C µ χ function on T , we continue to write . n u Z ¾ λ ¯ Similarly, we identity N with and write u N, though we write when thinking of u as a 1-parameter subgroup. m u ; χ ; λ ¯ Finally, we will usually write m u instead of . 3 Toric Varieties £ n ´C µ The torus T can be regarded as a Zariski open subset of a larger variety X in many ways: £ n n ´C µ C ¯ under the natural inclusion. £ n n ´C µ È ´ ;:::; µ ´ ;:::; ; µ ¯ under the map t1 tn t1 tn 1 . 4 £ 4 ´ µ C ´C µ ´ ; ; µ ´ ; ; ; = µ ¯ V V xy zw contains the Zariski open set V . The map r s t r s t rs t £ £ £ 3 4 3 C µ ³ ´C µ ´C µ induces a bijection ´ V . Thus V contains a copy of as a Zariski open set. Definition 3.1 A toric variety is a normal variety X of dimension n which contains a torus T £ n C µ ´ as a Zariski open set in such a way that the natural action of T on itself given by the group structure extends to an action of T on X. All of the above examples are toric varieties. The main goal of Lecture I is to explain three constructions of toric varieties. The common thread of these constructions is the rich combinatorial structure which underlies a toric variety. Here is an example, £ 2 2 2 C µ È Ê ª Ê Example 3.2 Let’s show that ´ gives the following picture in N : Z (1.1) 2 u £ 2 2 λ C È È A 1-parameter subgroup u ¾ N gives a map : . is complete, so that u µ limλ ´t t 0 2 2 u ´ ; µ ¾ Z λ exists in È . If u a b N, then the description of given on page 1 implies that u a b µ ´ ; ; µ: λ ´t t t 1 It is then straightforward to compute that 8 > ´ ; ; µ ; > > 0 0 1 a b 0 > > > > ´ ; ; µ > ; > 0 1 1 a 0 b 0 > > > > ; ; µ ; > ´1 0 1 a 0 b 0 < u a b λ ´ µ ´ ; ; µ ; ; µ (1.2) lim t lim t t 1 ´1 1 1 a b 0 > t 0 t 0 > > ´ ; ; µ > ; < > 0 1 0 a b b 0 > > > > ´ ; ; µ < ; < > 1 0 0 a 0 a b > > : ; ; µ < ; : ´1 1 0 a 0 a b The first four cases are trivial. To see how the fifth case works, note that a b a b b ; ; µ ´ ; ; µ lim´t t 1 lim t 1 t t 0 t 0 < ´ ; ; µ since these are homogeneous coordinates. Then a > b and b 0 imply that the limit is 0 1 0 , as claimed. The last two cases are similar. Now observe that (1.1) decomposes the plan into 7 disjoint regions: ; > ; > > ; < ¯ The open sets 1: a b 0, 2: a 0 b 0, 3: a b b 0. > ; ; > < ; ¯ The open rays 1: a 0 b 0, 2: a 0 b 0, 3: a 0 a b. ¯ The point a b 0. The corresponds perfectly with (1.2). In the next section, we will see that (1.1) is the fan corre- 2 sponding to the toric variety È . 4 First Construction: Cones and Fans £ n ´C µ Let X be a toric variety with T , M and N as above. We first explain how the character m £ χ C group M leads to pictures generalizing (1.1). The idea is that m ¾ M gives : T . Since χm T X, we can regard as a rational function on X. The divisorof this rational function has some nice properties. It is supported on the complement of T in X. This complement will be a union of irreducible divisors, which we denote ¡ ¡ ¡ : X Ò T D1 Dr Then the divisor of χm can be written r m χ µ ; div´ ∑ aiDi i=1 3 χm where ai is the order of vanishing (or the negative of the order of the pole) of along Di. This is one of the reasons we require that X be normal—it ensures that the ai are well-defined. The key observation is that the map m ai is a homormorphism. (Exercise: Prove this.) Using ; the duality between M and N, we get ni ¾ N such that ai m ni . This implies r m χ µ ; : (1.3) div´ ∑ m ni Di i=1 ¾ It follows that the toric structure of X uniquely determines a unique set of elements n1 ;:::; nr N. The ni are part of the fan determined by X. Ê However, before we can define fans, we must consider cones. We will let M M ª and Ê Z Ê N N ª denote the real vector spaces obtained from M and N. Ê Z A rational polyhedral cone σ N is a cone generated by finitely many elements of N: Ê © ¨ ; λ · ¡¡¡ · λ ¾ λ ;:::; λ σ u sus N s 0 1 1 Ê 1 ¾ where u1 ;:::; us N. Then: σ σ ´ σ µ ¯ is strongly convex if 0 . σ σ ¯ The dimension of is the dimension of the smallest subspace containing . σ ` σ ` ¯ A face of is the intersection 0 , where is a linear form which is nonnegative on µ σ. The set of faces of σ of dimension r is denoted σ ´r . σ ρ ¾ σ ´ µ ρ ¾ σ ´ µ ¯ The edges of are the 1-dimensional faces 1 . The primitive element nρ of 1 is ; ρ ¾ σ ´ µ σ the unique generator of ρ N. The primitive elements nρ 1 , generate the . σ σ ¯ The facets of are the codimension-1 faces. When dim n, each facet has an inward pointing normal which is naturally an element of M . We get a unique inward normal by Ê requiring that it is in M and has minimal length. σ If σ N be a strongly convex rational polyhedral cone, then its dual cone M is Ê Ê ¨ © : ; ¾ σ σ m ¾ M m u 0 for all u Ê This is a rational polyhedral cone of dimension n. Then consider the semigroup algebra σ ℄ C M ¼ ¼ m m m m·m χ χ ¡ χ χ consisting of linear combinations of characters , with multiplication given by . σ ℄ C Gordan’s Lemma implies that C M is a finitely generated algebra over . σ ;:::; Example 4.1 First consider an n-dimensional cone generated by a basis e1 en of N. The n Z σ basis gives an isomorphism N ³ which takes to the “first quadrant” where all coordinates £ σ are nonnegative. In terms of the dual basis ei of M, has the same description.