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 József u. 1, H Ép., H-1111 Budapest, Hungary E-mail : szenes@math.bme.hu Abstract. In this note, I review a result obtained in joint work with Michèle Vergne on a duality principle in toric geometry. I will demonstrate the essential points of our work on a concrete example. 1. Toric varieties and quotients The basic data is a real vector space g endowed with a basis [ω 1, ω 2,..., ω n ], and an exact sequence 0 a ι g π t 0 of real vector spaces of dimensions r, n and d, such that the three lattices g Z = n i=1zω i, t Z = π(g Z ), a Z = ker(π gz ). have maximal rank in the corresponding vector spaces. Clearly, we have n d = r. This construction gives rise to two sequences of vectors where ω i is the dual basis in g, and A = [α 1,..., α n ] a Z, α i = ι (ω i ) B = [β 1,..., β n ] t Z, β i = π(ω i ) One says that A and B are Gale dual to each other in this situation. November 4, 2004.
110 Mathematisches Institut, Seminars, 2004-05 We will assume that A is projective, i.e. there is γ a such that α i, γ > 0 for i = 1,..., n. This is equivalent to the condition that the origin is contained in the convex hull of B. We can write down our maps as integer matrices [ι] and [π] if we choose bases of a Z and t Z. Then the set of α i s are the column vectors of [ι] and the β i s are the row vectors of [π]. It is a pleasant exercise to check that a pair of matrices [ι] and [π] with integer coefficients represents an exact sequence as described above if the g.c.d of the r-by-r minors of [ι] is 1, the same condition holds for [π], and [ι] [π] = 0 We will consider the following example of this setup: [ι 0 ] = ( 0 1 0 ) 1 1 1 1 0 0 1 [π 0 ] = 1 0 1 1 1 0 Denoting the coordinates on a by x and y, we can list the α i s as follows: A 0 = [α 1 = y, α 2 = x + y, α 3 = y, α 4 = x] The set B 0 may be read off the matrix [π 0 ]. We obtain the following pictures: 1 3 2 1 4 2 4 A 0 B 0 3
A. Szenes: Toric reduction and tropical geometry 111 Next we introduce the set of Bases(A) as those subsets σ [1, n] for which {α i ; i σ} is a basis of a ; we used the notation [1, n] = {1,..., n} here. The set of chambers C(A), by definition, is the set of connected components the open subset ( ) n R >0 α i R >0 α i, i=1 σ Bases(A) which is thus the complement of the boundaries of r-dimensional simplicial cones spanned by the vectors from A. In our example Bases(A 0 ) = 5 and C(A 0 ) = 2. The following statement describes the Gale dual picture: Proposition 1.1. There is a one-to-one correspondence between the chambers C(A) and complete simplicial fans in t whose one-dimensional faces form a subset of B. One can associate a projective orbifold toric variety to each complete simplicial simplicial fan ([Ful93]). One can give a quotient construction of this variety using the Gale dual A-data as follows. Let the coordinates on g C = C n be x = (x 1,..., x n ), and define a diagonal action of the complexified torus T a = a C /a Z on C n by the formula i σ s a C, x C n (exp(2π 1 α i, s )x i, i = 1,..., n). Given a chamber c C(A), define the open subset { U c = x C n ; i σ x i 0 for some σ for which c i σ R >0 α i } of C n. Then U c is clearly T a -invariant, and we can define the d-dimensional compact variety V c = U c /T a. As an exercise, one may find the two fans corresponding to the two chambers of our example A 0, and one may check that the corresponding two surfaces are P 2 and the blow-up of P 2 at a point. Remark 1.2. Another data often used to describe toric varieties are polytopes. In our setup they appear as follows. Let θ c a Z and define the partition polytope { } n Π θ = (γ 1,..., γ n ) (R 0 ) n ; γ i α i = θ. i=1 This polytope is polar to the corresponding fan in t.,
112 Mathematisches Institut, Seminars, 2004-05 Now we introduce the vector κ = n i=1 α i a Z. In the simplest case, which we will consider here, κ will be in one of the chambers of A. The fan corresponding to this special chamber is the one induced by the faces of the convex hull of B. The general case is more complicated [BM02, SV04]. 2. Intersection numbers of toric varieties One can go one step further with the quotient construction of the previous section: given θ a Z one can define the (orbi)-line-bundle L θ over V c as L θ = U c Ta C θ, where C θ is the character of T a corresponding to θ. Taking the Chern class of this bundle, we obtain a map χ : a Z H 2 (V c, Q), which may be extended multiplicatively to a map χ : C[a] H (V c, R), i.e. to polynomials on a. We are interested in the intersection numbers of the classes χ(θ), θ a Z ; this means that we are looking for a formula for the map Q χ(q) V c for homogeneous degree d polynomials Q on a. First we write this intersection number as ( ) Q (2.1) χ(q) = JK c n i=1 α, i V A (c) where JK c is a functional on the space C A [a] of rational functions which are regular on the complement of the hyperplane arrangement U(A) = {u a C ; α i (u) 0, i [1, n]}. This functional is defined implicitly as follows: for any r-element subset σ [1, n] we set ( ) { 1 vol a (σ) 1, if c i σ (2.2) JK c = R>0 α i, α i σ i 0, if c i σ R>0 α i =, where vol a (σ) is the volume of the parallelepiped i σ [0, 1]α i as measured by the lattice α Z. We also declare that JK c vanishes on fuctions of degree different
A. Szenes: Toric reduction and tropical geometry 113 from r; using the multidimensional partial fraction expansion, it is easy to show that this defines a unique functional JK c on C A [a]. Let us turn to our example. Consider the chamber c(κ) containing κ = 2x + 3y. Then we obtain V c χ(x 2 ) = JK c(κ) ( x 2 x(x + y)y 2 ) ( ) 1 = JK c(κ) y 2 1 = 0 1 = 1. (x + y)y Similarly, we find that V c χ(y 2 ) = 0 and V c χ(xy) = 1. This functional distinctly has the flavor of a residue. From this point of view, one would like to find a cycle Z[c] for every c C(A), which has the property that (2.3) JK c (f) = f d r ϕ, where d r ϕ is the translation invariant holomorphic r form normalized using the lattice a Z and an orientation of a. In our example, we can let d 2 ϕ = dx dy. Our main result is an explicit description of such a cycle as a solution set of r polynomial equations. A priori, it is not at all clear that such a presentation exists, but here it is. Theorem 2.1. Let c C(A) be such that κ c. Then for a sufficiently generic ξ c and small ε > 0, the real algebraic subvariety of U(A) { } n Z ε (ξ) = u U(A); α i (u) αi,λ = ε ξ,λ for every λ a Z, i=1 when appropriately oriented, is a compact cycle which satisfies (2.3). Remark 2.2. 1. Formally, it looks like we imposed infinitely many conditions on Z ε (ξ), but, in fact, because of the multiplicativity of both sides of the equations, one can reduce the number of equations to r. This results in an r-dimensional real cycle. 2. The condition κ c is crucial. Otherwise Z ε (ξ) might be non-compact! 3. Nevertheless, by repeating some of the αs sufficiently many times we can achieve the condition κ c, thus we can write down a cycle for any of the chambers this way. Returning to our example, pick ξ = x + 2y c(κ) choose a basis (λ 1, λ 2 ) dual to (x, y). Our equations then have the following form Z[c] (2.4) x(x + y) = ε, y 2 (x + y) = ε 2.
114 Mathematisches Institut, Seminars, 2004-05 Naturally, the cycle for the other chamber may be realized by the system of equations where x and y are exchanged. The proof of this theorem is somewhat technical, but the central idea is that of the tropical Ansatz : x = ε a, y = ε b, x + y = ε c. Then the equations (2.4) imply the equalities { a + c = 1, (2.5) 2b + c = 2. These, naturally, do not determine a, b and c, but we observe that for small enough ε, if x + y is very small compared to x, then x and y should be rather close. This argument gives us the following three possibilities: 1. c a, b, which implies a b 2. b a, c, which implies a c 3. a c, b, which implies c b. Here by we mean significantly greater, and by we mean rather close. Now we can go back to our system (2.5), and solve it under the three possible scenarios a = b, a = c, or b = c. We obtain the following three solutions (1, 1, 0), (1/2, 3/4, 1/2), (1/3, 2/3, 2/3). However, only the second of these equations satisfies the corresponding inequalities! Thus we, informally, conclude, that for ξ = x+2y the cycle Z ε (ξ) consists of a torus which is very close to the torus { y = ε 3/4, x = ε 1/2 } C 2. Integration over such a torus is equivalent to an algebraic operation called iterated residue, which means ordering an r-tuple of αs and then taking the usual onedimensional residue with respect to each variable, one by one, while assuming all subsequent (from right to left) variables to be nonzero constants. Thus one check of our result is the equality JK c(κ) (f) = Res x Res f dx dy, y which is easy to see. All is not as simple as it appears, however. Take another ξ, say, let ξ = 3x + 4y. Completing the above computation, we obtain two solutions: (1, 1, 2) and (5/3, 4/3, 4/3) instead of one!!! It looks like there is an error but no: the union of the two tori we obtain this way is, in fact, homologous in {(x, y) C 2 ; x, y, x + y 0} to the single torus we computed above.
A. Szenes: Toric reduction and tropical geometry 115 If you are intrigued, check out [SV04, SV] for details and proofs! In [BM02, BM03] you can learn what all this has to do with mirror symmetry, and [Stu02] will explain why this method is called tropical. References [BM02] V. V. Batyrev & E. N. Materov Toric residues and mirror symmetry, Mosc. Math. J. 2 (2002), no. 3, p. 435 475, Dedicated to Yuri I. Manin on the occasion of his 65th birthday. [BM03], Mixed toric residues and Calabi-Yau complete intersections, in Calabi-Yau varieties and mirror symmetry (Toronto, ON, 2001), Fields Inst. Commun., vol. 38, Amer. Math. Soc., Providence, RI, 2003, p. 3 26. [Ful93] W. Fulton Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993, The William H. Roever Lectures in Geometry. [Stu02] B. Sturmfels Solving systems of polynomial equations, CBMS Regional Conference Series in Mathematics, vol. 97, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2002. [SV] A. Szenes & M. Vergne Mixed Toric Residues and tropical degenerations, math.ag/0410064. [SV04] A. Szenes & M. Vergne Toric reduction and a conjecture of Batyrev and Materov, Invent. Math. 158 (2004), no. 3, p. 453 495.