Journal of Mathematical Sciences, Vol. 186, No. 3, October, 2012
ALGORITHM FOR CONSTRUCTING AN INTEGRAL REPRESENTATION FROM THE FAN OF A TORIC VARIETY A. A. Kytmanov Institute of Space and Information Technologies of Siberian Federal University 26, ul. Kirenskogo, Krasnoyarsk 660074, Russia
[email protected] UDC 517.55 + 004.94
We describe algorithms for constructing a toric variety from a fan and an integral representation in Cd associated with the toric variety. Bibliography: 7 titles.
Introduction As is known, the kernel of the Bochner–Martinelli integral representation in Cn+1 and the Fubini–Study form for the projective space Pn = CPn are closely connected by the formula ω(z) =
1 dλ ∧ ω0 ([ξ]) 2πi λ
(1)
(cf., for example, [1, 2]), where ω is the Bochner–Martinelli form n+1 zk n! (−1)k−1 2n+2 dz[k] ∧ dz, ω(z) = (2πi)n+1 |z| k=1
dz = dz1 ∧ . . . ∧ dzn+1 , dz[k] is obtained from dz by eliminating the differential dz k , and ω0 ([ξ]) is the volume form for the Fubini–Study metric in Pn (cf. [3]) ω0 ([ξ]) = where E(ξ) =
n! E(ξ) ∧ E(ξ) , (2πi)n |ξ|2(n+1) n+1
(−1)k−1 ξk dξ[k]
k=1
is the Euler form, ξ = (ξ1 , . . . , ξn+1 ) are the homogeneous coordinates of the point [ξ] ∈ Pn , and ξ, z ∈ Cn+1 are connected with λ ∈ C by the relation z = λ ξ. The Bochner–Martinelli form is a “standard” form of degree 2n + 1 in the set Cn+1 {0} which is a bundle over Pn with the one-dimensional torus C∗ for a fibre. In other words, Translated from Vestnik Novosibirskogo Gosudarstvennogo Universiteta: Seriya Matematika, Mekhanika, Informatika 10, No. 2, 2010, pp. 61-70. c 2012 Springer Science+Business Media, Inc. 1072-3374/12/1863-0453
453
Pn = [Cn+1 {0}]/G, where G = { (λ, . . . , λ) ∈ Cn+1 : λ ∈ C∗ } is the group of transformations generated by diagonal matrices. The projective space is a particular case of a toric variety. In the general case, an n-dimensional toric variety is the quotient space (cf. [4]) (2) X = Cd Z(Σ) /G. Here, Z(Σ) is the union of some coordinate subspaces in Cd constructed from a fan Σ ⊂ Rn with d generators, whereas G is a group that is isomorphic to the torus (C∗ )r , r = d − n, and is also constructed from the fan Σ. An arbitrary compact toric variety X = Xn of complex dimension n is defined from a complete simplicial primitive fan Σ ⊂ Rn generated by a collection of vectors v1 , . . . , vd ∈ Zn (onedimensional generators of the fan). In the construction of X, an important role is played by relations between vk . Suppose that a11 v1 + . . . + a1d vd = 0, .....................
(3)
ar1 v1 + . . . + ard vd = 0, where r = d−n, are all independent linear relations over Zn between vk . With each vector vk one can associate the complex variable zk so that z = (z1 , . . . , zd ) plays a role of the homogeneous coordinates in the toric variety X. The author [4] constructed an integral representation for holomorphic functions in a d-circular polyhedron W = Wρ defined by the inequalities a11 |z1 |2 + . . . + a1d |zd |2 < ρ1 , (4)
..................... 2
2
ar1 |z1 | + . . . + ard |zd | < ρr , where the integral is taken over the skeleton of the polyhedron W and ω(z) denotes the kernel of this representation. The skeleton is obtained from (4) by replacing all the inequalities with equalities. The integral representation holds if the radius-vector ρ = (ρ1 , . . . , ρr ) is taken from the K¨ahler cone for X (cf. [4]) and the fan is convex. The kernel ω(z) of the integral representation is a generalization of the Bochner–Martinelli differential form; it looks like a G-invariant differential (d, n)-form: ω(z) =
h(z) ∧ dz , g(z, z)
(5)
where h(z) is a generalization of the Euler form and g(z, z) is a polynomial whose set of zeros coincides with Z(Σ). The main result of [4, Theorem 1] asserts that, in a neighborhood of ζ = 0, any holomorphic function f (ζ) in the polyhedron Wρ admits the integral representation 1 f (z)ω(z − ζ), (6) f (ζ) = C Γ(ρ)
where the normalization coefficient C is independent of f . 454
The goal of this paper is to present an algorithm for constructing a toric variety and an integral representation connected with the variety from a given fan (one-dimensional generators of the fan and cones of maximal dimension). To describe the algorithm, we consider the construction of a toric variety from a fun and then the construction of an integral representation in the space Cd .
1
Construction of Toric Variety
We recall the main notions used in this paper. By a cone generated by vectors v1 , . . . , vk ∈ k Zn {0} we mean the set σ = bj vj : bj 0 in Rn . The vectors v1 , . . . , vk are called the j=1
generators (of the cone). A face of a cone σ is a subset of σ such that bj = 0 for some j. The dimension of a cone is the dimension of the minimal subspace containing this cone. A fan Σ is a collection of cones such that 1) Σ is closed under taking intersections; moreover if a cone is the intersection of some two cones, then it is a face of each of them. 2) Σ is closed under taking faces. The dimension of a fan is the maximal dimension of cones in the fan. Definition 1. A fan cone is simplicial if its dimension coincides with the number of the fan generators. A fan is said to be simplicial if all its cones are simplicial. Definition 2. An n-dimensional fan Σ ⊂ Rn generated by v1 , . . . , vd is primitive if the matrix of column-vectors of any collection of the vectors vi1 , . . . , vin forming the cone of maximal dimension is unimodular. Definition 3. An n-dimensional fan Σ ⊂ Rn is complete if the union of all its cones coincides with the entire space Rn . Definition 4. An n-dimensional fan Σ ⊂ Rn generated by v1 , . . . , vd is convex if the ends of all the vectors v1 , . . . , vd lie on the boundary of their convex hull. The convexity of a fan is an important notion in toric geometry. Any convex fan is the dual of a reflexive polyhedron. As was already mentioned, an important role in the construction of a toric variety X is played by the relations (3). As was proved in [4], for any complete n-dimensional fan Σ ⊂ Rn one can write the system (3) of independent linear relations connecting vk with nonnegative coefficients aij . Therefore, without loss of generality we can assume that all the coefficients aij in (3) are nonnegative. To each vector vk we associate the variable zk . Definition 5. A collection of vectors vk1 , . . . , vkm of a fan Σ is primitive if this collection generates no cone in Σ, but any proper subcollection generates a cone in Σ. Any primitive collection of vectors vk1 , . . . , vkm of a fan generates the coordinate plane in Z(Σ) (cf. [4]) so that Z(Σ) =
{zk1 = . . . = zkm = 0} .
vk1 ,...,vkm −prim.
455
Moreover, Z(Σ) can be represented as the intersection (cf [4]):
zj = 0 , Z(Σ) = σ∈Σ
vj ∈σ /
which is equivalent to the previous construction. The group G is defined by the relations (3). Thereby the basis for the lattice of relations is formed by the vectors ai = (ai1 , . . . , aid ), i = 1, . . . , r. The group G is an r-parameter surface {(λa111 · . . . · λar r1 , . . . , λa11d · . . . · λar rd ) : λj ∈ C∗ } ⊂ (C∗ )d , so that z ∼ η ⇔ ∃(λ1 , . . . , λr ) : z = (z1 , . . . , zd ) = (λa111 · . . . · λar r1 η1 , . . . , λa11d · . . . · λar rd ηd ). The moment mapping (cf. [4]) μ : Cd → Rd /Rn Rr is defined as μ(z1 , . . . , zd ) = (ρ1 , . . . , ρr ), where
a11 |z1 |2 + . . . + a1d |zd |2 = ρ1 (7)
..................... 2
2
ar1 |z1 | + . . . + ard |zd | = ρr . For a fixed ρ = (ρ1 , . . . , ρr ) ∈ Rr the relations (7) define the set Γ(ρ) = μ−1 (ρ). The K¨ ahler cone [4] for a toric variety X is defined as follows. To each vector vi in Σ we associate a number bi ∈ R, i.e., we consider a function ψ : {v1 , . . . , vd } → R on a finite set of vectors vi . This function can be extended to a piecewise linear function in the entire space Rn by setting it to be linear on each cone σ ∈ Σ. We are interested in those vectors (b1 , . . . , bd ) ∈ Rd , for which these piecewise linear functions are strictly convex. The set of such vectors forms a in Rd (i.e., a cone bounded by finitely many hyperplanes). polyhedral cone K We denote by μ : Rd → Rr the linear mapping with matrix (aij ) in (7). in the space Rr is called the K¨ Definition 6. The interior K of the image μ (K) ahler cone. is described as follows [4]. For every primitive collection of vectors In fact, the cone K vk1 , . . . , vkm we represent their sum in the form vk1 + . . . + vkm = ci1 vi1 + . . . + cin vin ,
c i1 , . . . , c in ∈ Q + ,
(8)
where vi1 , . . . , vin form a cone containing the sum. Then such a collection vk1 , . . . , vkm yields the inequalities for b = (b1 , . . . , bd ) ∈ K: bk1 + . . . + bkm ci1 bi1 + . . . + cin bin .
(9)
is given by the system of inequalities (9) for vectors vk , . . . , vk running over all Thereby K m 1 the collections and belonging no cone of the fan, where i1 , . . . , in and ci1 , . . . , cin depend on the choice of vk1 , . . . , vkm . Identifying bj with |zj |2 and considering the images μ(z1 , . . . , zd ) = (ρ1 , . . . , ρr ), we conclude that the K¨ahler cone K is given by the inequalities for ρj : |zk1 (ρ)|2 + . . . + |zkm (ρ)|2 > ci1 |zi1 (ρ)|2 + . . . + cin |zin (ρ)|2 ;
(10)
moreover, the number of inequalities is the same as the number of primitive collections [4]. For ρ ∈ K the cycle Γ(ρ) = μ−1 (ρ) does not intersect Z(Σ) (cf. [4]). 456
2
Construction of Integral Representation
In [4], the author constructed a form (an analog of the Bochner–Martinelli form) in Cd Z(Σ) that is the kernel of an integral representation. This form has bidegree (d, n), is closed, and is represented by formula (5), where the numerator is a form of type (d, n), dz = dz1 ∧ . . . ∧ dzd , and vJ z[J]dzJ (11) h(z) = J
is an analog of the Euler form. Here, J = (j1 , . . . , jn ) is the multiindex and the prime means the summation relative to strictly increasing indices 1 j1 < . . . < jn d, |J| = = j1 +. . .+jn , vJ = vj1 ...jn = det(vj1 , . . . , vjn ); z[J] = z[j1 , . . . , jn ] is the product of zl , l = 1, . . . , d, l = j1 , . . . , jn ; dzJ = dzj1 ∧ . . . ∧ dzjn . (n)
To construct the denominator g of the form (5), we introduce the notation. Let σm , m = 1, . . . , M , be all cones of the fun of maximal dimension n. For the sake of brevity, we write σm and, unless the contrary is allowed, assume that the dimension of the cone is equal to n. Let the cone σm be generated by vectors vm1 , . . . , vmn . Since the fan is primitive, we have det(vm1 , . . . , vmn ) = ±1. We assume that for every n-dimensional cone σm the order of vectors vm1 , . . . , vmn is fixed, so that det(vm1 , . . . , vmn ) = 1. We set νlσm := −
n
det(vm1 , . . . , vmi−1 , vl , vmi+1 , . . . , vmn ).
(12)
i=1
If the fan is convex, then for any cone σm of maximal dimension we have νlσm −1 ∀ 1 l d,
(13)
where the equality holds if l ∈ {m1 , . . . , mn } since, in this case, all but one term vanish because of the linear dependence of columns in the determinants. Now, we can define g as a polynomial function in z and z: g(z, z) =
M
cm (zz)ν
σm +1
,
m=1
where (zz)ν
σm +1
=
σm
(zl z l )νl
+1
.
1ld
We define a domain D = Dρ by the formula
D := z ∈ Cd : |zk1 |2 + . . . + |zkm |2 < |zk1 (ρ)|2 + . . . + |zkm (ρ)|2
− ci1 |zi1 (ρ)|2 − . . . − cin |zin (ρ)|2 ∀k1 , . . . , km : vk1 , . . . , vkm are primitive .
(14)
The domain D is nonempty if (10) holds. We formulate the main result of [4]. Theorem 1. Suppose that a function f (z) is holomorphic in the domain W defined by the inequalities (4) and is continuous in the closure of W . Then, in D ∩ W , where the domain D is defined by the inequalities (14), the integral representation (6) holds, where Γ is defined by (7) and C is the normalization constant: C = ω > 0. Γ
457
3
Algorithm
The main procedure ToricVariety realizes the construction of a toric variety and the associated integral representation. The input data of the procedure are a collection of vectors (one-dimensional generators) of a fan Σ and a collection of the numbers of vectors that form cones of maximal dimension in Σ. The procedure sequentially computes the dimension d and constructs the set Z(Σ) and group G involved in the construction of a toric variety X of the form (2). For constructing the set Z(Σ) the procedure PrimColl is used. This procedure for a given set of the numbers of vectors cone list, forming cones of maximal dimension in the fan, computes all primitive collections of vectors of the fan. For constructing the group G the procedure LinRel is used. For a given collection of vectors vec list this procedure computes all independent linear relations on these vectors. The output of the procedure is the list of row-vectors of the matrix of linear relations. Further the main procedure deals with construction of the kernel ω of an integral representation. To construct the denominator of the form, the procedure nu sigma is used. This procedure computes the exponents νlσm by formula (12). It uses the procedure Permut with a vector and a number for the input data, whereas the output is a list of vectors obtained from a given one by substituting a given number into the first, second, etc. positions. The input data of the procedure nu sigma is a list of vectors of the fan and the result of substituting the number l into the vector (m1 , . . . , mn ) in (12). To construct the domain D, the main procedure involves an algorithm for computing the K¨ahler cone of a given toric variety X. The procedures SubsetsMinus1 and SetOfAllSubsets which are also involved in the algorithm, can be regarded as auxiliary. The input data of SubsetsMinus1 is a set, and its output is the set of all subsets of a given set with the number of elements less by 1 than that of the input set. The input data of SetOfAllSubsets are a set and a nonnegative integer K, and the output of this procedure is the set of all subsets of a given set if K = 0 and the set of all K-element subsets of a given set if K > 0. We refer to [5] for details.
4
Examples
This section contains two tests of the above algorithm for constructing a toric variety from a fan and then the integral representation associated with the toric variety. The algorithm was realized by using Maple 12 and computers with Intel Core 2 Duo E6550 (2,33 GHz) with 1 Gb RAM, Windows XP SP 3. The computation time is 0,0 seconds in both examples. Example 1. We consider the complete convex fan in R2 generated by v1 = (1, 0), v2 = (0, 1), v3 = (−1, 0), v4 = (−1, −1), v5 = (0, −1). The cone σ generated by vi1 , . . . , vin is denoted by σ = vi1 , . . . , vin . We write the cones of maximal dimension 2: σ1 = v1 , v2 ,
σ2 = v2 , v3 ,
σ3 = v3 , v4 ,
σ4 = v4 , v5 ,
σ5 = v5 , v1 .
Thus, the input data of the main procedure ToricVariety is the list of vectors [[1, 0], [0, 1], [−1, 0], [−1, −1], [0, −1]] and the list of cones [[1, 2], [2, 3], [3, 4], [4, 5], [5, 1]]. The toric variety is the quotient space (2) with d = 5 and 458
Z(Σ) = {z1 = z3 = 0} ∪ {z1 = z4 = 0} ∪ {z2 = z4 = 0} ∪ {z2 = z5 = 0} ∪ {z3 = z5 = 0}. The independent relations between vi are v1 + v2 + v4 = 0, v1 + v3 = 0, and v2 + v5 = 0. Hence the basis for the lattice of relations is formed by μ1 = (1, 1, 0, 1, 0), μ2 = = (1, 0, 1, 0, 0), and μ3 = (0, 1, 0, 0, 1). The group of action G is the 3-parameter surface {(λ1 λ2 , λ1 λ3 , λ2 , λ1 , λ3 ) : λj ∈ C∗ } ⊂ (C∗ )5 which is isomorphic to the torus (C∗ )3 . The integration cycle Γ(ρ) = μ−1 (ρ) is defined by ρ1 = |z1 |2 + |z2 |2 + |z4 |2 ,
ρ2 = |z1 |2 + |z3 |2 ,
ρ3 = |z2 |2 + |z5 |2 .
(15)
The K¨ ahler cone for the toric variety is described by the inequalities ρ2 > 0,
ρ1 > ρ3 ,
ρ1 > ρ2 ,
ρ3 > 0,
ρ2 + ρ 3 > ρ1 .
The standard form ω in C5 Z(Σ) has bidegree (5, 2) and is represented by formula (5) with dz = dz1 ∧ dz2 ∧ dz3 ∧ dz4 ∧ dz5 and the following analog of the Euler form: h(z) = −z3 z4 z5 dz1 ∧ dz2 + z2 z3 z5 dz1 ∧ dz4 + z2 z3 z4 dz1 ∧ dz5 − z1 z4 z5 dz2 ∧ dz3 − z1 z3 z5 dz2 ∧ dz4 − z1 z2 z5 dz3 ∧ dz4 − z1 z2 z4 dz3 ∧ dz5 − z1 z2 z3 dz4 ∧ dz5 . The denominator g is a polynomial: g(z, z) = |z1 |2 |z2 |4 |z3 |2 + |z1 |4 |z2 |2 |z5 |2 + |z1 |4 |z4 |2 |z5 |4 + |z2 |4 |z3 |4 |z4 |2 + |z3 |4 |z4 |6 |z5 |4 . The domain D and polyhedron W in Theorem 1 are described by the inequalities ⎧ ⎪ |z1 |2 + |z3 |2 < ρ2 , ⎪ ⎪ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ |z1 |2 + |z4 |2 < ρ1 − ρ3 , |z |2 + |z2 |2 + |z4 |2 < ρ1 , ⎪ ⎪ ⎨ 1 ⎨ W : D: |z2 |2 + |z4 |2 < ρ1 − ρ2 , |z |2 + |z3 |2 < ρ2 , ⎪ 1 ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ |z2 |2 + |z5 |2 < ρ3 , |z2 |2 + |z5 |2 < ρ3 . ⎪ ⎪ ⎪ ⎪ ⎩ |z |2 + |z |2 < ρ + ρ − ρ , 3 5 2 3 1 The obtained results coincide with the results of [6]. Example 2. We consider the complete nonconvex fan in R2 generated by v1 = (1, 0), v2 = (−2, 1), v3 = (−1, 0), v4 = (−2, −1). We write the cones of maximal dimension 2: σ1 = v1 , v2 ,
σ2 = v2 , v3 ,
σ3 = v3 , v4 ,
σ4 = v4 , v1 .
Thus, the input data of the main procedure ToricVariety are the list of vectors [[1, 0], [−2, 1], [−1, 0], [−2, −1]] and the list of cones [[1, 2], [2, 3], [3, 4], [4, 1]]. The toric variety is the quotient space (2) with d = 4 and Z(Σ) = {z1 = z3 = 0} ∪ {z2 = z4 = 0}. The independent relations between vi are 4v1 + v2 + v4 = 0 and v1 + v3 = 0. The basis for the lattice of relations is formed by the vectors μ1 = (4, 1, 0, 1) and μ2 = (1, 0, 1, 0). The group of action G is the 2-parameter surface {(λ41 λ2 , λ1 , λ2 , λ1 ) : λj ∈ C∗ } ⊂ (C∗ )4 which is isomorphic to the torus (C∗ )2 . The integration cycle Γ(ρ) = μ−1 (ρ) is defined by ρ1 = 4|z1 |2 + |z2 |2 + |z4 |2 ,
ρ2 = |z1 |2 + |z3 |2 .
(16) 459
The K¨ ahler cone for the toric variety is defined by the inequalities ρ2 > 0, ρ1 > 4ρ2 . The standard form ω in C4 Z(Σ) has bidegree (4, 2) and is represented by formula (5) with dz = dz1 ∧ dz2 ∧ dz3 ∧ dz4 and the following analog of the Euler form: h(z) = −z3 z4 dz1 ∧ dz2 + z2 z3 dz1 ∧ dz4 − z1 z4 dz2 ∧ dz3 − 4z1 z3 dz2 ∧ dz4 − z1 z2 dz3 ∧ dz4 . The denominator g is the rational function: g(z, z) = |z3 |4 |z4 |12 + |z1 |4 |z4 |−4 + |z1 |4 |z2 |−4 + |z2 |12 |z4 |4 . The domain D and polyhedron W in Theorem 1 are defined by the inequalities 4|z1 |2 + |z2 |2 + |z4 |2 < ρ1 , |z1 |2 + |z3 |2 < ρ2 , W : D: |z2 |2 + |z4 |2 < ρ1 − 4ρ2 , |z1 |2 + |z3 |2 < ρ2 . The obtained results coincide with the results of [7]. Note that, in [7], the denominator g(z, z) was written as a single fraction since it contains negative powers.
Acknowledgments The work is supported by a grant of the President of the Russian Federation for leading scientific schools (project No. NSh–7347.2010.1) and the target program “Development of Scientific Potential of Higher School” (project No. 2.1.1/4620).
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2.
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3.
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4.
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5.
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6.
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7.
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Submitted on October 8, 2010 460
