JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 10, Number 2, April 1997, Pages 427{443 S 0894-0347(97)00230-0 MAXIMAL DEGENERACY POINTS OF GKZ SYSTEMS S. HOSONO, B. H. LIAN, AND S.-T. YAU 1. Introduction In this paper, we study the Gel0fand-Kapranov-Zelevinsky hypergeometric PDE systems [12]. We are mainly concerned with a case of the so-called resonant expo- nent β, though some of our results apply to generic β as well. This resonant case in fact plays an especially important role for applications in mirror symmetry. The problem we consider is motivated by mirror symmetry [25] as follows. Given a family of Calabi-Yau manifolds π : S, one would like to give a suitable com- pactification in which there is a boundaryX→ point, called a large radius limit, where all but one of period integrals of the manifolds become singular near that point. An astonishing discovery [6] in mirror symmetry is that, under some assumptions, the period integrals near this point combine to form a generating function which predicts the number of rational curves (a.k.a. instanton contributions) on another Calabi-Yau manifold. Thus the existence and construction of a large radius limit pose an important problem. When the Calabi-Yau manifolds are anticanonical divisors in a suitable toric variety, then their period integrals are natural solutions to a GKZ system [2]. Thus the GKZ theory becomes an important tool for studying the behavior of such period integrals ([15], [16], [4], [17]). Motivated by this, we study a GKZ system canonically associated with a complete regular fan, and a class of integral solutions. We prove a general existence theorem for the so-called maximal degeneracy points [19] for these integral representations. These special points can be geometrically interpreted as fixed points under a canonical torus action on the parameter space. Some complete regular fans (in most interesting examples we know) arise in mirror symmetry [1]. When they do, the maximal degeneracy points turn out to be large radius limits where the enumeration of instanton contributions is carried out ([6], [19], [18], [10], [7], [15], [8], [16], [4], [17]). 1.1. Conventions and notation. N : rank n lattice M : dual lattice N : N R R ⊗ MR : M R N : Z ⊗N M : Z × M × Received by the editors May 28, 1996 and, in revised form, November 13, 1996. 1991 Mathematics Subject Classification. Primary 14C30, 32G20. Key words and phrases. Mirror symmetry, hypergeometric systems, period integrals, Calabi- Yau manifolds, toric varieties, compactification, indicial ideal, Gr¨obner bases. c 1997 by the authors 427 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 428 S. HOSONO, B. H. LIAN, AND S.-T. YAU µ : a point in N µ :1 µ Σ : a× complete rational polyhedral fan in N : a finite set of integral points which spans the affine hyperplane 1 N A × R ⊂ N R L = L : the lattice of relations on A A PΣ : the toric variety associated to the fan Σ Σ(1) : the set of primitive elements on the edges of the fan Σ σ(1) : the set of primitive elements on the edges of the cone σ SΣ : the secondary fan of Σ GΣ:theGr¨obner fan cone(S) : the cone generated by a finite set S conv(S) : the convex hull of a finite set S An n-dimensional cone in NR is also called a large cone. A cone which contains no nontrivial linear subspace is called strongly convex. A cone generated by finitely many integral points is called rational. A rational cone is called regular if it is generated by a subset of an integral base. A fan is regular if it consists of regular cones. The dual of a cone σ is denoted σ∨. 2. GKZ -hypergeometric systems A Let N be a set of p + 1 integral points which span the affine plane 1 N . A⊂ × R We sometimes write = µ0,...,µp . The projection map N N is denoted A { p+1 } → µ µ.LetL=L := l Z µ lµµ =0, which we call the lattice of7→ relations on . ThisA is clearly{ ∈ a lattice| ∈A of rank p }n. The GKZ system with A P p+1 − exponent β N is the PDE system on C = CA given by ∈ R (2.1) Π(a)=0, u, µ θ u, β Π(a)=0 l h i aµ −h i for l L, u M [12]. Here X:= ( ∂ )lµ ( ∂ ) lµ and θ := R l lµ>0 ∂aµ lµ<0 ∂aµ − aµ ∂ ∈ ∈ − aµ . Equivalently we can write ∂aµ Q Q γ γ (2.2) a− a f(a)=0, u, µ θ f(a)=0, l h i aµ γ where f(a)=a− Π(a)andγlies in the (p Xn)-dimensional affine subspace Φ(β):= − γ γ µ=β . We shall always assume that µ := 1 ~0 . { | µ µ } 0 × ∈A P 3. Period integrals We shall consider a class of (multi-valued) analytic functions attached to the data as follows. Given a choice of basis e1,...,en of N, we have an isomorphism A n TM =Hom(N,C×) ∼= (C×) . On this algebraic torus, there is a family of rational functions defined by the Laurent polynomials: µ (3.1) f (X, a)= aµX . A µ X∈A The zero locus of this function is a divisor Ha in the torus TM .NowTMhas a canonical T -invariant holomorphic n-form given by dXi . Following [2], for M Xi every homology n-cycle α in the manifold TM Ha, we define the period integral − Q 1 dXi (3.2) Πα(a)= . α f (X, a) Xi Z A Y License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use MAXIMAL DEGENERACY POINTS OF GKZ SYSTEMS 429 This integral converges in some domain, and is singular in some divisor D CA. ⊂ Under analytic continuation in CA D, the integral will in general have monodromy. − Thus the period integrals are local sections of a locally constant sheaf over CA D. − As an example, we have the following. Let α0 be the real torus cycle X1 = = X =1 . The following two results are proved in [2]. {| | ··· | n| } Proposition 3.1. In the domain a0 a1,..., ap , we have the convergent power series representation | || | | | 1 n dX a i 0 f (X, a) X α0 i=1 i (3.3) Z A Y (l + +l )! al1 alp n l + +lp 1 p 1 p =(2πi) ( 1) 1 ··· ··· ··· . l + +lp − l1! lp! a 1 ··· l µ + +lpµp=0,l,...,lp 0 0 1 1 ··· X 1 ≥ ··· Proof. First we have the geometric series a 1 0 µ1 µp r = r ( a1X apX ) . f (X, a) a0 − −···− A r 0 X≥ Also we have n dX Xλ i =(2πi)nδ . X λ,0 α0 i=1 i Z Y Thus the terms in the geometric series which contribute to the integral are powers l1 lp l1+ +lp a1 ap of the form ( 1) ··· l +··· +lp ,wherel1µ1+ +lpµp =0,l1,...,lp 0. In 1 ··· − a0 ··· ≥ 1 µ1 µp r each term r ( a1X apX ) , each tuple (l1,...,lp) with r = l1 + +lp a0 − −···− ··· (l + +lp)! contributes exactly 1 ··· copies of such a power. Now summing over all r 0 l1! lp! ≥ and over all those tuples,··· we get the desired sum. The sum is convergent because it is a subseries of the standard series (with z = a /a ): i − i 0 (l + +l )! 1 p l1 lp (3.4) ··· z1 zp . l1! lp! ··· l ,...,lp 0 1 X≥ ··· Proposition 3.2 (see [13]). The period integrals Πα(a) are solutions to the GKZ system with the exponent β = 1 ~0. − × Proof. We claim that l Πα (a) = 0 for all l L.Infact,lG(f(a, X)) = 0 for any differentiable function G.Firstwehave∈ A (k) µj1 µjk (3.5) ∂aj ∂aj G(f (a, X)) = G (f)X X . 1 ··· k A ··· + Write l L as l = l l−,wherel± have only nonnegative entries and have disjoint ∈ − + support. Note that the sum of the entries of each l is the same as that of l−.Call this sum k.Thenwehave + (k) l µ l−µ (3.6) lG(f (a, X)) = G (f )(X µ X µ ). A A P − P + But l L implies that 0 = lµµ = (lµ lµ−)µ. It follows that lG(f (a, X)) = 0. ∈ − A P P Now we have to check that the periods Πα(a) are annihilated by the order 1 operators in the GKZ system. We consider the action of λ := (λ0,λ1,...,λn) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 430 S. HOSONO, B. H. LIAN, AND S.-T. YAU on the periods as follows: under Xi λiXi, the cycle α gets moved to αλ and dm := n dXi is invariant. Thus we7→ get i=1 Xi Q 1 1 1 (3.7) dm = dm. λ a Xµ a λµXµ 0 Zα µ Zαλ µ Now for λ closed enough toP the identity element,P it is clear that the cycle αλ is homologous to α. So we can replace αλ on the RHS by α. Differentiating the equation above by λ ∂ and then setting λ =id=(1,...,1), we get i ∂λi 1 1 a µ λµXµ δ dm = − µ i dm − λ i,0 a X µ ( a λµXµ)2 0 λ=id Zα µ Zα P µ λ=id e , µ a Xµ (3.8) P = P i µ dm − h i 2 α f (a, X) Z PA 1 = ei, µ θaµ dm, h i α f (a, X) Z A ∂ X where θa := aµ .