PPERIOD INTEGRALS AND MUTATION
KETIL TVEITEN
Abstract.
Let f be a Laurent polynomial in two variables, whose New-ton polygon strictly contains the origin and whose vertices are primitivelattice points, and let L f be the minimal-order differential operator thatannihilates the period integral of f . We prove several results about f and L f in terms of the Newton polygon of f and the combinatorial operationof mutation , in particular we give an in principle complete descriptionof the monodromy of L f around the origin. Special attention is givento the class of maximally mutable Laurent polynomials, which has ap-plications to the conjectured classification of Fano manifolds via mirrorsymmetry. Introduction
Let N (cid:39) Z d be a lattice, and C [ N ] the ring of Laurent polynomials in d variables. Let P ⊂ N be a lattice polytope, and let f ( a, x ) = (cid:80) m ∈ P a m x m be a generic Laurent polynomial with Newton polytope N ewt ( f ) = P . Forsuch an f , and C an d -cycle in H d ( { x ∈ ( C ∗ ) d | f ( x ) (cid:54) = 0 } , C ) , consider theintegral φ f ( a ) = (cid:90) C f ( a, x ) dx x · · · dx n x n . As a function of the coefficients a m of f , this integral satisfies a system ofdifferential equations of the GKZ type, as follows: Let P (cid:48) be the image of P under the embedding N (cid:44) → Z × N “at height 1” given by m (cid:55)→ (1 , m ) . Then φ ( f ) is a solution to the GKZ system H γ ( P (cid:48) ) , where γ = (0 , . . . , , − (see[SST00, 5.4.2]).An interesting variant of this is the classical period integral of f , π f ( t ) = ( 12 πi ) n (cid:90) | x | = ··· = | x n | = ε − tf ( a, x ) dx x · · · dx n x n which is a (possibly multivalued) holomorphic function of t in a punctureddisk around t = 0 . The GKZ system annihilating φ f specializes to a Picard-Fuchs operator L f = (cid:80) p i ( t ) ∇ i ∈ C [ t, ∇ t ] (here ∇ t = t∂ t ). This operator L f plays an important role in the conjectured classification of Fano manifoldsvia mirror symmetry, for a certain class of Laurent polynomials f called maximally mutable [CCG +
13, ACC + L f around the origin by using toolsfrom toric geometry, and in particular the relation between properties of L f and the combinatorial data of the Newton polytope P , in particularthrough the operation of mutation [ACGK12]; in addition, we try to extract a r X i v : . [ m a t h . AG ] J a n KETIL TVEITEN some information about the global behaviour of L f from the combinatoricsof P . The former problem we can in principle solve completely (althoughwe only do it explicitly for simple cases), for the latter the best we cando is make plausible conjectures backed up by empirical data. We also proveseveral results about the relationship between Laurent polynomials and theirNewton polygons. 2. Preliminaries
We can of course explicitly compute L f , by several different methods, themost efficient of which are recently developed by Lairez in [Lai14]. The com-putation is expensive, and for larger examples in practice undoable withoutfixing values for the coefficients. We hope to work around this problem byapplying the Riemann-Hilbert correspondence; the operator L f is equivalentto the monodromy data of the solution sheaf Sol ( L f , O ) , which is a localsystem away from the singular points of L f . This lets us work with general f , unfortunately we pay the price of only being able to talk about the singlesingular point t = 0 . The problem of finding the remaining singular pointsand their local monodromy data is again a prohibitively expensive computa-tional problem, though we will conjecture ways to extract some informationin the final section.As we will discuss the operation of mutation of polygons and Laurent poly-nomials, we must restrict to the class of Fano polygons , where this operationis well-behaved.
Definition 2.1.
Let N be a two-dimensional lattice and let P ⊂ N ⊗ R bea convex lattice polygon such that(1) dim P = 2 ;(2) ∈ int ( P ) , that is, the origin is a strict interior point of P ; and(3) the vertices of P are primitive lattice points.Such a polygon is called a Fano polygon (see [KN12]).In the remainder, all polygons are assumed to be Fano polygons, and allLaurent polynomials are such that
N ewt ( f ) is Fano. We consider two poly-gons to be equal if they differ by an element of GL ( N ) , and similarly considertwo Laurent polynomials to be equal if they are related by an automorphismof C [ N ] induced by an element of GL ( N ) . The one-dimensional faces of apolygon are called edges and the zero-dimensional faces are called vertices .Let P ⊂ N be a Fano polygon, let M = Hom ( N, Z ) , and let Y P be thetoric del Pezzo surface defined by the normal fan of P in M . Recall thata variety Y is Fano if the anticanonical divisor − K Y is very ample (two-dimensional Fano varieties are usually called del Pezzo surfaces for historicalreasons). The rays u i generating the normal fan of P are the inward normalsto the edges E i of P , so for each edge E i of P , let D E i = D i denote thecorresponding divisor on Y P . There is a distinguished divisor on Y P , D P = (cid:88) i h i D i , here h i = −(cid:104) u i | E i (cid:105) is the lattice height of the edge E i . D P is very ample, andits global sections Γ( Y P , D P ) can be identified with the set of Laurent poly-nomials with Newton polygon N ewt ( f ) = P (see [CLS11, 4.3.3/4.3.7]). A ERIOD INTEGRALS AND MUTATION 3 section f (or generally a linear system δ ⊂ Γ( Y P , D P ) of sections) determinesa rational map τ := 1 f : Y P (cid:57)(cid:57)(cid:75) P . Let Γ τ = { ( y, τ ( y )) ∈ Y P × P } be the graph of τ . Via the embedding Y P ⊂ Γ τ there is an induced rational map Γ τ (cid:57)(cid:57)(cid:75) P , and if we resolve all singularitiesof Γ τ (and generally any base points of δ ) we get a smooth surface (cid:102) Y P suchthat the induced map (cid:101) τ : (cid:102) Y P → P is a morphism. Let D be the pullback of D P to (cid:102) Y P . The fiber X = (cid:101) τ − (0) is equal to the support of D , and the fiber X t over a general point t ∈ P is smooth.Now by general D -module theory the solution sheaf Sol ( L f ) on P isisomorphic to the constructible sheaf with fiber H ( X t , C ) at t ∈ P ; here H denotes homology with closed support (also known as Borel-Moore homology ),and not the usual singular homology. Indeed, L f is a D -module theoreticdirect image of the GKZ module, and from [BGK +
87] (VII.9.6, VIII.13.4 andVIII.14.5.1) we have that the solution complex
Sol ( L f ) • is the exceptionaldirect image of the cohomology complex of that module, which works out togive the described constructible sheaf. In other words, we wish to find themonodromy of H ( X t , C ) around t = 0 .3. Mutation
We introduce some notation and terminology which will remain in forcefor the remainder of the paper.Let P ⊂ N be as before, with vertices p i and edges E i with inward normalvectors u i ∈ M , we number these so that E i is the edge between p i and p i +1 .The lattice height of an edge E i is −(cid:104) u i | E i (cid:105) , and the lattice width of E i is (cid:104) u i − | p i − p i +1 (cid:105) = (cid:104) u i +1 | p i +1 − p i (cid:105) ; the lattice width is equal to the numberof lattice points on E i minus one. The following definition is due to Akhtarand Kasprzyk (see [AK14]). Definition 3.1.
Let C ⊂ N be a primitive lattice cone of lattice height h and lattice width w . If h = w , we say that C is a primitive T -cone . If w is apositive multiple of h , we say that C is a T -cone . If w is strictly less than h ,we say that C is an R -cone .Let E be an edge of P of height h and width w = hk + r . The cone over E from the origin may be subdivided into k primitive T -cones and an R -coneof width r ; we say that these cones are on the edge E . There are k + 1 waysto do this, so saying e.g. “the R -cone on the edge E ” is strictly speaking notwell-defined, but this observation is not relevant anywhere in what follows(i.e. any subdivision gives the same result), so we permit ourselves to abuselanguage in this way. The most important quantity here is the number ofinternal lattice points of P lying in R -cones (or later, f -rigid cones, see 3.8),which does not depend on the subdivision, by [AK14, 2.3]. Whenever we saye.g. “internal points of an R -cone”, it should always be understood to meanlattice points in P . Definition 3.2.
Let C ⊂ N be a primitive lattice cone, with primitivespanning vectors u and v . If { u, v } is a lattice basis for N , we say that C is KETIL TVEITEN smooth . If C is not smooth, there is a point p ∈ N such that p = r u + ar v ,and { u, p } and { v, p } are lattice bases for N ; in this case we say that C is oftype r (1 , a ) . Remark . The type of cones parallels the classification of cyclic quo-tient singularities; a cone of type r (1 , a ) defines a toric variety isomor-phic to the cyclic quotient singularity C /µ r , where µ r acts with weight (1 , a ) . See [AK14] for further details on R - and T -cones, and the correspond-ing singularities, called R - and T -singularities. The operation of mutation ,which we will now describe, (conjecturally, see [ACC + Q -Gorenstein deformation of toric varieties; roughly speaking T -cones (resp. T -singularities) are mutable (resp. Q -Gorenstein smoothable), while R -conesand R -singularities are rigid under mutation/deformation. Definition 3.4 ([ACGK12]) . Let P ∈ N be a Fano polygon, and let E be anedge of P , with inward normal vector u , lattice height h , and lattice width w = kh + e , where k > , e ≥ are integers (in other words, E supports kT -cones and an R -cone of width e ), let h (cid:48) be the minimal lattice height (withrespect to u ) of the points in P , and let F ∈ u ⊥ ⊂ N be a primitive vector.For any r ∈ Z let P r = { p ∈ P | u ( p ) = r } be the points of P at height r withrespect to u (in particular, P h = E ). Notice that we can for each < r ≤ h write P r as a Minkowski sum kr · F + Q r , where Q r is some (possibly empty)polygon. The mutation of P with respect to the mutation data u, F is thepolygon P (cid:48) = mut u ( P ) defined by P (cid:48) r = (cid:40) ( k − rF + Q r ≤ r ≤ hP r + rF h (cid:48) ≤ r < . Intuitively, we are removing slices rF from each positive height r > , andadding slices r (cid:48) F at each negative height r (cid:48) < . Equivalently, we are con-tracting a T -cone from E , and putting in a T -cone on the opposite side of P .Any polygons P , Q related by a chain of mutations are said to be mutation-equivalent .We observe that R -cones are rigid under mutation: they do not have suf-ficient lattice width to permit mutation. Example 3.5.
Let P be the Fano polygon with vertices ( − , , ( − , , (3 , , (3 , − and ( − , − . It has one R -cone of type (1 , (shaded dark grey), and nine T -cones. We will perform a mutation with factor F = { ( − , , (0 , } (indi-cated by an arrow) and height function h (( x, y )) = − y , which will contractaway the lightly shaded T -cone, and add a new T -cone on the other side ofthe polygon. • • • • •• · · · · •• · · · •• • • • • • F ERIOD INTEGRALS AND MUTATION 5
After the mutation, we have this picture; the lightly shaded T -cone has beencontracted, a new T -cone has been added to the opposite side, and the R -cone and the T -cone beneath it have been skewed to fit. • • •• · · · •• · · · •• • • • • • • Observe that the numbers of T - and R -cones are unchanged, and the typeof the R -cone is preserved. Definition 3.6 ([AK14]) . Let P be a Fano polygon, let k be the numberof T -cones in P and B the list of types of R -cones in P , ordered cyclically.The set B is called the singularity basket of P . The singularity content of P is the pair ( k, B ) .Any mutation removes one T -cone and adds another, so the total numberof T -cones is unchanged. The R -cones and their relative order is unchangedby mutation, so the singularity content is an invariant under mutation (see[AK14]). If the cyclical order isn’t important, it may be useful to think ofthe singularity basket as a mere multiset; we will do this in Theorem 4.17. Example 3.7.
The polygons in example Example 3.5 have singularity con-tent (9 , { (1 , } ) . Definition 3.8.
Let f be a Laurent polynomial with Newton polygon P ,and let P (cid:48) be the mutation of P with mutation data u, F . The map µ : x a (cid:55)→ x a ( γ + ηx F ) (cid:104) u | a (cid:105) (where a ∈ N, γ, η ∈ C ) defines an automorphism of C ( N ) ,the rational functions in two variables, and is called a cluster transformation .We say that f is mutable with respect to u, ( γ + ηx F ) if f (cid:48) = f ◦ µ is in C [ N ] (notice N ewt ( γ + ηx F ) = [0 , F ] ), i.e. is a Laurent polynomial, and in thiscase that f (cid:48) is a mutation of f ; the Newton polygon of f (cid:48) is P (cid:48) . Any twoLaurent polynomials related by a chain of mutations are said to be mutationequivalent .We also say that f is mutable over the T -cone contracted by the mutation P (cid:55)→ P (cid:48) ; we say that a cone over which f is mutable is an f -mutable cone, anda cone over which f is not mutable is an f -rigid cone, or if f is understoodsimply call these mutable and rigid cones respectively.Let us make explicit what’s going on. For f (cid:48) to be in C [ N ] , we require thefollowing: if P r are the points of P at height r as in Definition 3.4, let f r be the terms of f corresponding to the points of P r . To perform a mutationwith factor ( γ + ηx F ) , it is neccessary that ( γ + ηx F ) r is a factor of f r for < r ≤ h . The mutated polynomial f (cid:48) = mut ( f ) can be described by f (cid:48) r = f r ( γ + ηx F ) − r . It is clear that
N ewt ( mut ( f )) = mut ( N ewt ( f )) . In particular, any mutationof f gives an underlying mutation of N ewt ( f ) . KETIL TVEITEN
The relevant fact for us is that mutation of f preserves the classical periodintegral π f ( t ) , and thus the Picard-Fuchs operator L f . A proof of this factcan be found in [ACGK12]. The analysis of π f ( t ) and L f is then indepen-dent of which f in the mutation class we use, which allows for a great dealof flexibility. In particular, we consider the class of maximally mutable poly-nomials. We note that the following definition is a special case only valid forthe two-dimensional case; for general dimension a somewhat more involvedformulation must be used (see [KT15]). The problems that occur in higherdimensions are not relevant to us, so we keep it simple here. Definition 3.9 ([KT15]) . A Laurent polynomial f is called maximally mu-table if whenever there is a sequence of mutations N ewt ( f ) = P → P →· · · → P n , there are Laurent polynomials f i with f = f and N ewt ( f i ) = P i ,such that f i is mutable over the mutation P i → P i +1 , and f i +1 is the resultingmutation of f i .If in addition f has zero constant term, and for every edge E of N ewt ( f ) oflattice height h E and lattice width w E , the polynomial f E is (up to GL ( N ) )equal to x h E (1 + x ) w E (i.e. f has “binomial edge coefficients”), f is called standard maximally mutable .We may for simplicity refer to maximally mutable Laurent polynomials assimply MMLP’s .The standard MMLP’s are of particular importance for the mirror symme-try classification of Fano manifolds. In that literature one usually considerswhat we call “standard maximally mutable” polynomials, with the additionalcondition that for lattice points on an edge internal to an R -cone, the cor-responding coefficient is zero—this is called having T -binomial edge coeffi-cients )—for these, the mutations will always be with factor (1 + x F ) (e.g. in[ACC +
15, CCG +
13, OP15]). In the remainder, we will work as generally aspossible, but we will return to the specializations of coefficients in the finalsection.Requiring that we can mutate f across the whole graph of mutations of N ewt ( f ) means that we must ensure that for every T -cone of height h , theslices f r at height < r ≤ h must be divisible by some factor ( γ + ηx F ) r ,or in other words, the maximally mutable Laurent polynomials are those forwhom the f -mutable cones are the T -cones, and the f -rigid cones are the R -cones. The process of finding the MMLP’s for a given polygon P is bestillustrated with an example. Example 3.10.
Let P be the polygon with vertices ( − , , (1 , , (2 , , (2 , − , ( − , − and ( − , ; this has two R -cones of type (1 , and seven T -cones. It is easiest to show the process of finding the maximally mutableLaurent polynomials by labelling the vertices of P by the associated coef-ficients. The cones are indicated; the R -cones are shaded grey, the T -cones ERIOD INTEGRALS AND MUTATION 7 are white. We begin with generic coefficients: a − , a , a , a − , a − , a , a , a − , a − , a , a , a − , − a − , − a , a , a , − a , − a , − First impose the factorization conditions along the edges, with a linearfactor ( γ + ηx ) for each T -cone. This will determine the “internal” coefficientson the edges with T -cones of height 2, e.g. a − , + a , x + a , x = ( γ + ηx ) (for some γ, η ) implies that a , = 2( a − , a , ) . In the same way, a , = 2( a , a , − ) and a − , = 2( a − , a − , − ) . To reduce visual clutter,we rename the free parameters on the edges by a, b, c, . . . . a ab ) bi a − , a , a , hi ) a − , a , a , h g c cd ) df e We now require the polynomial i yx + a − , yx + a , y + a , xy + cx y alongthe y = 1 row to be divisible by a + b x , the polynomial a y x + a − , yx + a − , x + g xy along the x = − line to be divisible by h + i x , and thepolynomial bxy + a , xy + a , x + e xy along the x = 1 line to be divisible by c + d x . Solving the equations this imposes, we get • a − , = h i a − , − ahi + gi h , • a , = b a a − , + a b a , − acb − bia , • a , = d c a , − bdc + c ed .If for simplicity we let f be standard maximally mutable, setting theconstant term to zero and imposing binomial edge coefficients, we get thefollowing picture (where we set a − , = p, a , = q to reduce visual clutter): KETIL TVEITEN p p + q − q p + 3 q + 3 a, b, c, d, e, f, g, h, i, a − , , a , a , in the coefficients; there is one for each vertex, one for the origin, and onefor each point of P not internal to a T -cone; in the standard case there areonly free parameters corresponding to the internal points in the R -cones.This last observation is true in general, and a proof is given in [KT15]. Proposition 3.11.
The number of free parameters in a Laurent polynomialwith
N ewt ( f ) = P is equal to the number of lattice points in P not inter-nal to an f -mutable cone. In particular, the number of free parameters ina maximally mutable Laurent polynomial with N ewt ( f ) = P is equal to thenumber of lattice points in P not internal to a T -cone, and the number offree parameters of a standard MMLP is equal to the number of lattice pointsin P internal to an R -cone. Proposition 3.12.
Let P be a Fano polygon, and let f be a generic Laurentpolynomial with N ewt ( f ) = P . Let P (cid:48) be a mutation of P that contractsa T -cone of height h on an edge E , and suppose f is mutable over thiscone. Then f has an ordinary multiple point of multiplicity h on supp ( E ) .In particular, a generic maximally mutable Laurent polynomial has a multiplepoint of multiplicity h i for each T -cone of P of height h i .Proof. Recall from Definition 3.8 the conditions for mutability: choose localcoordinates x, y so the edge E is contained in the hyperplane y = h , andlet f r be the polynomial made up of terms of f corresponding to points atheight r (using the same height function). Then in these coordinates, we canwrite f r = ( γ + ηx ) r y r h r , where h r = h r ( x ) is some Laurent polynomial in x . Examining in localcoordinates, e.g. in the toric chart corresponding to one of the vertices of E , where f becomes an honest polynomial, we can easily see that f has anordinary h -uple point here (at the point corresponding to x = − γ/η ). (cid:3) Theorem 3.13.
Let f be a generic Laurent polynomial with N ewt ( f ) = P .The general fiber X t ⊂ (cid:102) Y P , which is the desingularization of the curve f = 0 in Y P , has genus equal to the number of internal lattice points of the f -rigidcones of P , counting the origin. In particular, if f is a generic maximallymutable Laurent polynomial, the genus is the number of internal lattice pointsof the R -cones of P , counting the origin, and we call this number the mutablegenus of Y P and denote it by g mut ( Y P ) ; it is mutation-invariant. ERIOD INTEGRALS AND MUTATION 9
Proof.
Recall that the genus g ( D P ) of the desingularization of D P , calledthe sectional genus of Y P , is equal to the number of internal lattice points of P [CLS11, 10.5.8]. This is the genus of a generic curve in the complete linearsystem of curves linearly equivalent to D P . The curves defined by Laurentpolynomials mutable over a given collection of T -cones form a base point-free linear subspace of this linear system in the obvious way, and to find thegenus of such a curve, we need to examine how a general Laurent polynomial f with the appropriate mutability differs from a generic section of D P .It follows from Proposition 3.12 that a Laurent polynomial has an h -uplepoint for every T -cone of height h over which it is mutable; we have imposedno other conditions, so there are no other special points that affect the genus.The effect of an ordinary h -uple point on the genus of a curve is well known(see e.g. [GH94, pp.500-508]): the genus drops by h ( h − for every suchpoint. Thus, the genus of the curve defined by f is g ( D P ) − (cid:80) h i ( h i − ,where the sum runs over the T -cones of N ewt ( f ) over which f is mutableand the i ’th cone has lattice height h i .Now observe that h ( h − is exactly the number of internal lattice pointsin a T -cone of height h (this follows directly from Pick’s formula [CLS11,Ex. 9.4.4]), so the genus of f is equal to g ( D P ) − (cid:80) h i ( h i −
1) = | int ( P ) ∩ N | − | int ( P ) ∩ N ∩ f -mutable cones | , that is, the number of internal latticepoints in P that are in f -rigid cones, counting the origin. In particular if f is a generic MMLP, it is mutable over all the T -cones, so the genus is thenumber of lattice points in P that are in R -cones, counting the origin.To see that this genus g is mutation-invariant, it is enough to recall that thesingularity content of P , in particular the set of R -cones, is invariant undermutation (3.6, also see [AK14]), which of course implies that the number ofinternal lattice points in the R -cones is invariant; in particular it is preservedby those mutations of P over which f is mutable. (cid:3) Remark . We remark that the genus is unchanged even if some themultiple points on the T -cones on the same edge over which the polynomialis mutable are allowed to coincide, i.e. if f is mutable with the same factoron all the cones. This is because when we deform k ordinary h -uple pointsto coincide, the result is not an ordinary kh -uple point, but an h -uple pointwhere the branches meet with an order k tangency (the case k = 2 , h = 2 isthe familiar tacnode); an order k tangency will drop the genus by k for everybranch, so the total defect is still k · h ( h − [GH94, pp.500-508]. Note alsothat the standard MMLP’s may have genus lower than g mut ( Y P ) , as fixing acoefficient 0 at the origin may cause problems. An example is the picturedpolygon (the internal point is the origin): • • ••• Fixing binomial edge coefficients we get a polynomial f = y x + 2 y + xy + y + a ; the curve f = 0 has genus 1 unless a = ± or a = 0 , in which case ithas genus 0.Recall that we can write L f = (cid:80) r p r ( t ) ∇ r . The order of L f is the maximal r occurring in the sum, and the degree of L f is the maximal degree (in t )of the p r ’s. The degree is hard to say anything about (but see Section 5),however the order is now available to us: Corollary 3.15.
Let f be a Laurent polynomial with N ewt ( f ) = P . Thenthe order of the Picard-Fuchs operator L f is 2 times the genus of X t ; inparticular if f is a generic maximally mutable Laurent polynomial, the orderof L f is g mut ( Y P ) .Proof. It follows from the Cauchy-Kovalevski theorem ([Hör90, 9.4.5]) thatthe order of L f is equal to the rank of its solution space, and it is a well-known fact that H ( X, C ) (cid:39) C g if X is a compact Riemann surface of genus g . (cid:3) Monodromy at t = 0 To compute the monodromy of H ( X t , C ) , we need to find a suitable basisof cycles, and a description of the monodromy automorphism. We will dothis by explicitly constructing a model for X t by means of local calculations,explicitly carrying out the resolution (cid:102) Y P → Y P .Let us recap what we know so far: The general fiber X t ⊂ (cid:102) Y P is a genus g mut curve, which degenerates as t → to the support of the divisor D , thepullback of D P to (cid:102) Y P . This divisor is in any case a collection of P ’s, topo-logically a necklace of spheres, with some chains of spheres attached (eachsphere corresponds to an edge of P or an exceptional curve of the resolution (cid:102) Y P → Y P ). Recall from Theorem 3.13 that g mut is equal to the number ofinternal lattice points of P that are not internal to an f -mutable cone, whichalways includes the origin as P by assumption is Fano. A necklace of spheresis a degeneration of a topological surface of genus at least one, which wouldaccount for the contribution to the genus from the lattice point at the origin.By 3.13 the rest of the genus comes from the internal points of the R -conesof P , so there must be some singularities on the P ’s corresponding to theedges that resolve to give a higher-genus surface.We may thus reduce to a series of local considerations, which we will referto as the contributions from the vertices and edges, and f -rigid cones respec-tively. The contribution from the vertices is this: intersection points betweenthe components are degenerations of the form { x m y n = t } → { x m y n = 0 } ,and we must describe which of these occur and what the monodromy does tothem (see Figure 1). The contributions from the R -cones is this: on the com-ponents of D P corresponding to edges with f -rigid cones, we must identifywhat singular points occur and resolve them to get a positive-genus curve (cid:101) C → P ; then find an appropriate automorphism of (cid:101) C that fixes the inverseimages of all the singular points and intersection points with the adjoiningcomponents of D P (see Figure 2). ERIOD INTEGRALS AND MUTATION 11
Figure 1.
Local picture of the degeneration over an inter-section between components of D P ; the vanishing cycle isindicated in red, and the relative cycle in blue. Figure 2.
The component of D corresponding to an f -rigidcone, showing the exceptional curves of some resolved singu-lar points; this is a degeneration of a higher-genus surface,vanishing and relative cycles indicated. Notice how the mon-odromy automorphism of X t must fix these vanishing cyclesto degenerate correctly to the special fiber.To compute the whole monodromy action on X t , we will then cut thecurve into pieces and consider each piece by itself, and then assemble theresults afterwards. Some of the basis cycles of H ( X t , C ) will exist entirelywithin these pieces (that is, they are homologous to cycles contained in thelocal piece), these will be cycles that degenerate to a point in the specialfiber, and are as such called vanishing cycles . The remaining cycles will inthe local pictures enter and exit the local piece through the cuts, these willbe called relative cycles in the local pictures.We fix some notation: There is in fact only a single such global cyclethat locally becomes a relative cycle, from here on we will call this cycle(and its local images) α . The vanishing cycles over the intersections betweencomponents of D P are all homologous, and we will call this cycle β .Observe that the local monodromy action on the relative cycles need notbe integral, as long as these globally add up to something integral (indeed,exploiting this fact will be crucial in some of the local calculations).4.1. The singularities of Y P , and intersections between the com-ponents. After we have resolved the singularities of Y P , we may look atthe monodromy action over the intersections between the components of Figure 3.
The cycle α marked in blue and the vanishingcycle β marked in red, and their images in the local pieces. D . Locally at the intersection between two components of D , of multi-plicities m and n respectively, in the local coordinates given by the toricchart corresponding to the vertex of intersection, we can write supp ( D ) as { x m y n = 0 } . In these local coordinates, the global sections and f be-come x m y n and higher-order terms ) respectively, and we can write − tf as x m y n − t (1 + ( higher-order terms )) , locally analytically equiv-alent to x m y n − t . The degeneration when t → is now equivalent to { x m y n = t } → { x m y n = 0 } . The monodromy action is then locally themonodromy of the curve x m y n = t as t goes around zero. Lemma 4.1.
Let β be the vanishing cycle of x m y n = t when t → (withpositive orientation), and let α be the relative cycle. The monodromy actionon α, β in x m y n = t as t goes around t = 0 in the positive direction is givenby β (cid:55)→ β, α (cid:55)→ α − mn β .Proof. Consider the Riemann surface of y = n (cid:113) tx m , for fixed t . This is an n -sheeted covering of the punctured complex plane with a singularity at x = 0 ,where as you trace along the surface around the singularity, y will alternatebetween approaching + ∞ and −∞ as x approaches zero, alternating a totalof m times (see Figure 4 for a picture of what this looks like). Notice the m -fold rotational symmetry of the surface.Write t = e iθ and x = e iτ (we may ignore the magnitude as only theargument is relevant to the monodromy action); we may now express thesurface as y = ( tx − m ) n = ( e i ( θ − mτ ) ) n . In other words, when t moves around the origin, the resulting surface satisfiesan equation y = ( x − mθ ) n , where x θ = e iτ ( θ ) , and the argument satisfies − mτ ( θ ) = θ − mτ . From this, τ ( θ ) = − θ/m + τ , we see that the surface ERIOD INTEGRALS AND MUTATION 13
Figure 4.
The Riemann surface of y = Re ( (cid:113) x ) ; the solidcurve is the relative cycle α , and the vanishing cycle β can beidentified with the outer boundary of the displayed surface.The dashed curve is the cycle α − β .will rotate in the same direction as t , with m ’th the speed. Thus, when t hascompleted a full revolution, the surface will have rotated by an angle of πm ,or one step along the m -fold rotational symmetry.To find the effect of this on the cycles α and β , we give an explicit model foreach. The vanishing cycle β is homologous to the curve { ( e iθ , e − iθn ) | ≤ θ ≤ nπ } that winds around the singularity n times, following the sheets untilit meets itself. This curve is preserved under the rotational symmetry of thesurface, so the monodromy action on β is the identity. The relative cycle α can be modelled by a curve going along the topmost sheet of the surfacefrom ( ε, ε − mn ) to ( K, K − mn ) , where ε (cid:28) and K (cid:29) are real numbers (notethe orientation). The monodromy action can be modelled by pinning theinitial point ( ε, ε − mn ) in place (i.e. letting it rotate along with the surface)while holding the other fixed over x = K . After the monodromy action, theinital point has been moved to ( ε · e πi/m , ε − mn e πi/m ) , while the final point,fixed to lie over x = K , will be on the sheet immediately below the topmostone. The resulting curve is homologous to α − mn β , as the m -fold rotationalsymmetry moves a point mn ’th of the length of β . (cid:3) We now find the pullback of D P when resolving the singularities of Y P ,this will give us all the points on X that locally are of the form x m y n = 0 ,and now 4.1 tells us what the local monodromy action is. Notice that wecan combine the local actions without a problem, as the vanishing cycles β appearing in all of them are homologous, so the local actions commute.Recall that D P = (cid:80) h i D i , where h i are the lattice heights of the edges E i of P corresponding to the divisors D i . The resolved divisor D can be written D = D P + (cid:80) m j F j , where F j are some exceptional curves and m j are theirmultiplicities. An interesting fact is that the numbers m j are such that itmakes sense to think of the F j as corresponding to “edges” of P of widthzero and height m j ; we will however not need this. Suppose now v is a vertex of P , corresponding to a cone in the normalfan where Y P has a singularity of type r (1 , a ) , and that v is joining edges E and E (cid:48) , of heights h and h (cid:48) . The singularity is resolved according to [CLS11,Chapter 10]; recall in particular the notion of Hirzebruch-Jung continuedfractions, denoted as follows: [ b , b , . . . , b k ] = b − b − ... − bk . We introduce some notation: suppose the Hirzebruch-Jung continued fractionexpansion of r/a is [ b , . . . , b k ] ; let s = t k = 1 , and define positive integers s i , t i by s i /s i − := [ b i − , . . . , b ] , ≤ i ≤ kt i /t i +1 := [ b i +1 , . . . , b k ] , ≤ i ≤ k − . Note that we may extend this to letting s = t k +1 = 0 and s k +1 = t = r .When resolving the singularity at v , we get k exceptional curves F , . . . F k ,with self-intersections F i = − b i . Let m i denote the multiplicity of E i in D ;these multiplicities are determined by the criterion that E i .D = 0 . Lemma 4.2. (1) s i +1 + s i − = b i s i and t i +1 + t i − = b i t i . (2) m i = r ( t i m + s i m k +1 ) . (3) m = s i +1 m i − s i m i +1 .Proof. 1. By definition, s i +1 /s i = [ b i , . . . , b ] = b i − / [ b i − , . . . , b ] = b i − s i − /s i and it follows that s i +1 + s i − = b i s i , a similar rearrangement showsthe other identity. Recall that the m i are defined by the system of equations E i .D = 0 . Asthe only components of D that are involved are D F , D F (cid:48) and the E i ’s, and theintersection numbers are 1 for adjacent components and 0 for non-adjacentcomponents, we get equations m i − − b i m i + m i +1 = 0 (for ≤ i ≤ k ).Successive elimination, applying item 1 at each step, now yields the desiredconclusion. We show this by induction. The base case is the equation m i − − b i m i + m i +1 = 0 for i = 1 , using that s = 1 and s = b . The induction step isto show that s i +1 m i − s i m i +1 = s i +2 m i +1 − s i +1 m i +2 ; rearranging we have s i +1 m i + s i +1 m i +2 = s i +2 m i +1 + s i m i +1 , and applying the identity s i +2 + s i = b i +1 s i +1 on the right-hand side and the equation m i + m i +2 = b i +1 m i +1 onthe left-hand side we see that both sides equal b i +1 s i +1 m i +1 . (cid:3) Lemma 4.3. (cid:80) ki =0 1 m i m i +1 = rm m k +1 .Proof. Observe first that m m + m m = m m + m m m , and by 4.2(3) m + m = s m + m = s m , so we get m m + m m = s m m . In similar fashion wesee that s i m m i + m i m i +1 = m i s i m i +1 + m m m i +1 = s i +1 m m i +1 , so by induction we have (cid:80) ki =0 1 m i m i +1 = s k +1 m m k +1 = rm m k +1 . (cid:3) Proposition 4.4.
The contribution to the global monodromy of the cycles α, β ∈ H ( X t ) from the vertices of P is α (cid:55)→ α − ( K ) β and β (cid:55)→ β , where Π is the toric variety defined by the spanning fan of P and K Π is its canonicaldivisor. ERIOD INTEGRALS AND MUTATION 15
Proof.
Combining 4.1, 4.2 and 4.3 tells us that the contribution from a vertexof P is α (cid:55)→ α − rmn β , where m, n are the lattice heights of the adjoiningedges and the singularity of Y P in the corresponding chart is of type r (1 , a ) (or if Y P is smooth here, take r = 1 ).It is well-known that K is equal to the lattice volume of the dual polytope P ◦ ⊂ M R of P (see [CLS11, 13.4.1]). To show the claim, it is enough to showthat the volume of the cone C v in P ◦ corresponding to the vertex v of P isequal to rmn . Let u, w be primitive lattice generators of C v . By Definition3.2, C v is of type r (1 , a ) when { r u + ar w, w } is a lattice basis for M . As { r u + ar w, w } is a lattice basis, we have det( r u + ar w, w ) = 1 , and it followsthat det( u, v ) = r . Observing now that C v is spanned by m u and n w , weare done as det( m u, n w ) = rmn . (cid:3) Monodromy over an f -rigid cone. We know from Theorem 3.13that the f -mutable cones do not contribute to the genus of X t , and so wemay ignore them for purposes of the monodromy computation. Assume wehave an edge E of P supporting a single f -rigid cone, of height h and width w , and denote by X E the inverse image under the map X t → X of thestrict transform of D E under the resolution (cid:102) Y P → Y P . The strict transformof D E is a P , and it intersects the adjacent components of the pullback of D P , as well as the exceptional curves coming from resolving the singularitiesof f on D E . The inverse image X E is then the part of X t bounded by thevanishing cycles over these points of intersection. From this and 3.13 we seethat topologically, X E is a surface with a number of punctures, one for eachof these vanishing cycles, with genus equal to ( h − w , the number ofinternal points in the f -rigid cone (this follows directly from Pick’s formula).The terms of f along the edge E can be written as y h (cid:81) wi =1 ( x − η i ) insuitable coordinates. Now locally at each η i it is not hard to see that − tf is analytically equivalent to y h − t ( x + y ) , an A h − -singularity. Indeed,the “suitable coordinates” just referred to is the chart of Y P correspondingto one of the vertices of E ; in these coordinates, 1 becomes x e y h (where e is some positive integer, for now it isn’t important which) and f is anordinary polynomial in x, y . The variable change x (cid:55)→ x − η i is best describedpictorially by looking at the effect of the Newton polytope of − tf ; in thepicture circles represent points at height 0 (relative to t ), while everythingelse is at height 1. Note that x e y h (cid:55)→ ( x − η i ) e y h under this variable change,which changes the one point at height zero (the 1 in − tf ) to several points.The points now visible from the origin are (0 , h, , (0 , , and (1 , , , so − tf is analytically equivalent to y h − t ( x + y ) at the singular point. •• • h w •· ••• (cid:110) Thus, we have on D E the w singular points of f , each of type A h − , andthe two points of intersection with the adjacent components of D P . As D E has multiplicity h , we can now model our X E as a ramified degree h coverof P , with two ramification points of ramification index h (corresponding tothe intersection points with the adjacent divisors), and w ramification pointscorresponding to the singular points of f . More precisely, X E is homotopic tosuch a surface, punctured at the two ramification points over the intersectionswith the adjacent components. The ramification index e p of these points isfound by the Riemann-Hurwitz formula: setting g = ( h − w in g − − h + 2( h −
1) + w ( e p − gives e p = h , so we have a degree h map, ramified at w + 2 points of ramifi-cation index h .The local monodromy action on H ( X E ) must then be induced by anautomorphism of X E with w + 2 fixed points, near which the automorphismhas order h (to be compatible with the ramification index); this implies theautomorphism has order h everywhere ( a priori it has order a factor of h ).We have shown: Lemma 4.5.
Let E be a edge of P with an f -rigid cone of height h and width w . Then the local monodromy action on H ( X E ) , where X E is as above, isgiven by an order h automorphism of a genus ( h − w surface with w + 2 fixpoints. There are of course many such automorphisms, but we can limit the pos-sibilities to some extent, by taking advantage of the fact that a Riemannsurface is a quotient of a plane (the Euclidean plane if g = 1 , the hyperbolicplane if g ≥ ) by a suitable lattice; any order h automorphism of the surfacedescends from an order h automorphism of the plane. The two cases mustbe treated separately, but are ultimately quite similar. The fact that theglobal monodromy of H ( X t ) is integral (see [Żoł06, 5.4.32]) gives anotherrestriction; we already know (Proposition 4.4) the total contribution fromthe vertices of P , and whatever numbers we get from the contribution of the f -rigid cones must fit with this to produce something integral.Recall from 4.4 that the total monodromy from the vertices of P is equalto the degree K of the toric variety defined by the spanning fan of P . Wemay reformulate this to a count of contributions from the edges by using aresult of Akhtar and Kasprzyk ([AK14, Prop. 3.3]). Recall also from section4.1, for a singularity σ of type r (1 , a ) , the numbers b , . . . , b k σ making up ERIOD INTEGRALS AND MUTATION 17 the Hirzebruch-Jung continued fraction expansion of r/a , and the numbers s i , t i ( ≤ i ≤ k σ ) defined in terms of the b i . Let also d i = ( s i + t i ) /r − , andlet A ( σ ) = k σ + 1 − (cid:80) k σ i =1 d i b i + 2 (cid:80) k σ − i =1 d i d i +1 . Note that if σ is a primitive T -cone, A ( σ ) = 1 . Proposition 4.6 (Akhtar-Kasprzyk,[AK14]) . Let Π be a complete toric sur-face with singularity content ( n, B ) . Then K = 12 − n − (cid:88) σ ∈B A ( σ ) . This and 4.4 together give that for each R -cone of type σ , the contributionto the total monodromy of the relative cycle α from the vertices is α (cid:55)→ α + A ( σ ) β . The number A ( σ ) is in general not an integer, and so the actionon α in the local monodromy of the corresponding H ( X E ) must be of theform α (cid:55)→ ( something ) + B ( σ ) β , where A ( σ ) + B ( σ ) is an integer.We must separate the cases of g = 1 and g ≥ ; in the first case the surfaceis a quotient of the Euclidean plane by a lattice, and so its automorphismgroup is isomorphic to GL ( Z ) , in the second case the surface is a quotient ofthe hyperbolic plane, so its automorphism group is isomorphic to a subgroupof P SL ( C ) . In either case, to have an order h automorphism, the eigenvaluesmust be h ’th roots of unity.4.2.1. The case of genus 1.
For genus 1, note that requiring g = ( h − w =1 implies that either h = 3 and w = 1 , or h = w = 2 . Thus, the only possible f -rigid cones giving a genus one surface are the cones of type (1 , and (1 , ; we should have w + 2 fixpoints, so we want an order 3 automorphismwith 3 fixpoints, or an order 2 automorphism with 4 fixpoints. Lemma 4.7.
Let A ∈ GL ( Z ) . Then (1) if A = I and A has exactly 4 fixpoints modulo Z , then A = − I (here I is the × identity matrix), and the fixpoints are of the form ( n , m ) with n, m = 0 or . (2) if A = I and A has exactly 3 fixpoints modulo Z , then the fixpointsare of the form ( n , m ) with n, m = 0 , or 2, and A is similar to thematrix (cid:18) − − (cid:19) .Proof. For (1), it is easy to compute that a × integer matrix with order2 is one of the following: • ± I , or • (cid:18) a b − a b − a (cid:19) for a, b ∈ Z such that b | − a .The reader can now easily verify that of these, only − I has exactly 4 fixpointsmodulo Z .For (2), observe that order three implies that the eigenvalues must the twoprimitive third roots of unity, and such a matrix by necessity has determinant1 and trace -1. Imposing on a general integer matrix (cid:18) a bc d (cid:19) these conditions gives a matrix of the form (cid:18) a b − a + a +1 b − ( a + 1) (cid:19) (we may safely assume b (cid:54) = 0 , as assuming either b = 0 or c = 0 from thebeginning yields a noninteger matrix, because the only solutions to a + d = − , ad = 1 are the two primitive third roots of unity). We now find conditionson possible fixpoints: if ( x, y ) is a fixpoint modulo Z , that requires x ≡ Z ax + byy ≡ Z − a + a + 1 b x − ( a + 1) y which after some simplification gives x ≡ Z and y ≡ Z . Any fixpoints arethus of the form ( n , m ) , for n, m = 0 , , . Now, if ( n , m ) is fixed, so too is ( n , m ) , as 2 and 3 are coprime, so if we want exactly three fixpoints, theyare either (0 , , ( , ) , ( , ) , or (0 , , ( , ) , ( , ) . These configurations aremirror images of each other, so they are congruent.By conjugating with elementary matrices, we see that the matrix of theabove form with parameters ( a, b ) is similar to those with parameters ( a ± b, b ) , ( a, − b ) , ( − a − , − ( a + a + 1) /b ) , ( a + ( a + a + 1) /b, b + ( a + a +1) /b + 2 a + 1) or ( a − ( a + a + 1) /b, b + ( a + a + 1) /b − a − . Iteratingapplication of these similarities we can eventually arrive at (0 , − ; this isbecause a, b and ( a + a + 1) /b are necessarily coprime. (cid:3) Proposition 4.8.
Let X be a Riemann surface of genus one, possessing anorder two automorphism ω with exactly four fixpoints p , p , q and q , andlet X (cid:48) be X punctured at p and p . On X (cid:48) , let α be a relative cycle passingfrom p to p , and let β be a cycle going once around p . Then there is achoice of cycles a , a such that { α, β, a , a } is a basis for H ( X (cid:48) ) , and theautomorphism ω of X (cid:48) has an induced action on homology given (in thisbasis) by the matrix − − − − . Proof.
From 4.7 we have that the automorphism must be − I (modulo Z ),and the fixpoints must be (0 , , ( , , (0 , ) and ( , ) . Changing basis in Z if necessary, we may assume p = (0 , , p = ( , ) , q = ( , and q =(0 , ) . Now take the basis cycles a , a to be the edges of the fundamentaldomain. It is now clear that ω ( a i ) = − a i and ω ( β ) = β , and observe (seeFigure 5) that ω ( α ) − α + β is homologous to − a − a . (cid:3) Proposition 4.9.
Let X be a Riemann surface of genus one, possessing anorder three automorphism ω (cid:48) with three fixpoints p , p and q , and let X (cid:48) be X with p , p removed. On X (cid:48) , let α be a relative cycle passing from p to p ,and let β be a cycle going once around p . Then there is a choice of cycles a , a such that { α, β, a , a } is a basis for H ( X ) , and the automorphism ω (cid:48) of X gives an induced action on homology given (in this basis) either by the ERIOD INTEGRALS AND MUTATION 19 p p p p p q q a a a a α ω ( α ) β β Figure 5.
The automorphism from Proposition 4.8. matrix −
10 0 0 − − − or by its inverse.Proof. From 4.7 we have that the automorphism is similar to the one inducedby (cid:18) − − (cid:19) ; let a , a be the basis cycles in which ω (cid:48) has this representation;we may choose q = (0 , , p = ( , ) and p = ( , ) .The action on the basis { α, β, a , a } can be visualized by constructinga model for X as follows: divide the fundamental domain in C along onediagonal to form two triangles; label the sides of the fundamental domain by a , a , and label the diagonal by a , with orientations as indicated in Figure 6.Let p , p be the centroids of the triangles, and let q be the origin. We choosetwo of the cycles a i as basis cycles, e.g. a and a ; we have β = a + a + a and α passes from p to p , we may choose it to cross a . So, in the basis { α, β, a , a } , we may write a = β − a − a .The effect of ω (cid:48) is now to rotate these triangles by one step; we can seethat a (cid:55)→ a , a (cid:55)→ a = β − a − a , and β (cid:55)→ β , the only nontrivial thingis ω (cid:48) ( α ) . The cycle ω (cid:48) ( α ) passes from p to p crossing a rather than a ,so by adding β near each of the p i we can form the cycle ω (cid:48) ( α ) − α + β ,which is homologous to a + a . In other words, we have ω (cid:48) ( α ) − α + β = a + a = β − a , or ω (cid:48) ( α ) = α + 13 β − a and we are done. (cid:3) Now combining Propositions 4.4 and 4.9 gives us a main result of thispaper: • • • • p p a a a a a αω ( α ) ω ( α ) β β Figure 6.
The automorphism ω (cid:48) from Proposition 4.9 Theorem 4.10.
Let P be a Fano polygon with singularity content ( k, { n × (1 , } , and let X t be defined using a generic maximally mutable Laurentpolynomial f with N ewt ( f ) = P . Then there is a basis { α, β, a , a , . . . , a n , a n } of cycles in H ( X t , Z ) such that in terms of this basis, the monodromy auto-morphism ω of H ( X t , Z ) is given by • ω ( α ) = α + ( k + 2 n − β − (cid:80) nj =1 a j , • ω ( β ) = β , • ω ( a j ) = a j for ≤ j ≤ n , and • ω ( a j ) = β − a j − a j for ≤ j ≤ n .Proof. For each edge E with a (1 , -cone, 4.9 gives us two candidates forthe local monodromy automorphism of H ( X E ) : the automorphism ω (cid:48) , andits inverse; the only thing we have to do is find out which one gives integralglobal monodromy. Now, ω (cid:48) ( α ) = α + 13 β − a while ω (cid:48)− ( α ) = α − β + a , and from 4.6 (from which we compute A ( (1 , ) and 4.4 we have thatonly ω (cid:48) gives integral global monodromy. (cid:3) The case of g ≥ . For general genus, we instead have a quotient ofthe hyperbolic plane, by some hyperbolic lattice. Here we are unable to pindown the fixpoints as we did in the previous case, but we can confine thepossibilities very strongly (the following result is surely well known, but weare unable to locate an exact reference):
Lemma 4.11.
Up to conjugation, there are only n order n elements of P SL ( R ) .Proof. Elements of
P SL ( R ) are represented by × real matrices withdeterminant one. A matrix of order n must have as eigenvalues n ’th roots ERIOD INTEGRALS AND MUTATION 21 of unity, and as the determinant is one, the two eigenvalues must be mutu-ally inverse, and the trace must be πn ) . A matrix with this trace anddeterminant is of the form (cid:18) a b b ( − a + 2 a cos(2 π/n ) −
1) cos(2 π/n ) − a (cid:19) where b (cid:54) = 0 . All such matrices are similar, as the matrix with parameters a, b can be transformed to the matrix with parameters c, d by conjugatingwith the real matrix (cid:32) d ( a − π/n ) a +1) b ( c − π/n ) c +1) ( a − c ) dc − π/n ) c +1 (cid:33) (this is ok as x − π/n ) x + 1 has no real roots). There are n possible n ’th roots of unity, so from this the result follows. (cid:3) The automorphism group of a genus g ≥ surface is a subgroup of P SL ( R ) , so this implies that if we can find one primitive order h auto-morphism of our surface with the required number of fixpoints, its powerswill give us all the other possibilities, up to choice of basis.To give a complete description valid for any P would require computingthe number A ( σ ) of Proposition 4.6 for all singularity types σ = r (1 , a ) cor-responding to an R -singularity (it is always 1 for a primitive T -singularity),and finding an automorphism of a curve of the appropriate genus (with ap-propriate order and number of fixed points) that compensates for the non-integer part of A ( σ ) . We will do this by giving a model for the surface insimilar fashion to Figure 6 and finding a description in terms of cycles of theresulting automorphism. We can not give a concise general formula for whatpower of this automorphism is the right one, but it reduces case-by-case tomodular arithmetic and is easily doable by hand; we compute some simplecases in Proposition 4.13.Let us first restrict attention to the simplest case of an R -cone with latticewidth ; here by necessity the lattice height r ≥ must be odd (otherwisethe cone can’t be spanned by primitive lattice vectors).Suppose now we have an edge E with an R -cone of height r and widthone, and as in the genus one case let X E be the inverse image under X t → X of the proper transform of D E under the resolution (cid:102) Y P → Y P . By 4.5the local monodromy automorphism of H ( X E ) is induced by an order r automorphism of a genus ( r − / Riemann surface with three fixed points(two punctures, and one other distinguished point), and by 4.11 it is enoughto find one such automorphism; one of its powers will be the one that inducesthe local monodromy automorphism on H ( X E ) . Proposition 4.12.
Let X be a Riemann surface of genus ( r − / , withtwo removed points p , p and a third distinguished point q , and an order r automorphism that fixes these points. Let α be a relative cycle passing from p to p , and let β be a cycle going once around p . Then there is a choice ofcycles c , . . . , c r − such that { α, β, c , . . . , c r − } is a basis for H ( X ) , and theautomorphism is a power of the automorphism ω r given on homology cyclesby • α (cid:55)→ α + (1 − r ) β + c − (cid:80) r − i =3 c i p ••• • • p •• • c c c c c c c c ω ( α ) ω ( α ) c α Figure 7.
The automorphism ω in the model of a genus 2 surface. • β (cid:55)→ β , • c i (cid:55)→ c i +1 for ≤ i ≤ r − , and • c r − (cid:55)→ β − (cid:80) r − i =1 c i .Proof. Recall from the genus one case where we constructed a model for thesurface by gluing together two triangles, and an order 3 automorphism withthree fixpoints by rotating the triangles. For the w = 1 , r ≥ odd, case wewill construct a model in a similar way; gluing together two r -gons to makea genus ( r − / surface, and getting an order r automorphism with threefixpoints by rotating the r -gons.More precisely: Take two regular r -gons, with edges labelled c , . . . , c r going around in the positive direction. On one r -gon choose an orientationfor each edge, and one the other give the corresponding edges the oppositeorientation. Finally, identify edges with the same label according to theirorientation; this gluing is easily seen to give a surface of the desired genus.Let the punctures p i be the centroids of the r -gons, and let q be a vertex of an r -gon (these all are identified by the gluing). Let α be the relative cycle goingfrom p to p across c r , and let β be the cycle going once around p . Choosing g = r − of the c i ’s, e.g. c , . . . , c r − , we have a basis { α, β, c , . . . , c r − } of H ( X ) ; the cycle c r can be expressed as β − (cid:80) r − i =1 c i (see Figure 7 for anillustration).Now define the automorphism ω r by c (cid:55)→ c (cid:55)→ · · · (cid:55)→ c r (cid:55)→ c , which we may visualize as rotating the r -gons in the positive direction. Itis clear that ω r has order r and fixes p , p and q . The cycle β , which canbe identified with (cid:80) ri =1 c r , is clearly fixed. For the cycle α , its image ω r ( α ) goes from p to p crossing c , and similar to the genus one case, the cycle ω r ( α ) − α + r β is homologous to c + c r = β − (cid:80) r − i =2 c i . From this we find ω r ( α ) = α + r − r β − r − (cid:88) i =2 c i . (cid:3) Now as above let E be an edge of P with an R -cone of height r and widthone. By 4.5, 4.11 and 4.12 the local monodromy automorphism of H ( X E ) is induced by some power of the map ω r defined in 4.12. To find which it is ERIOD INTEGRALS AND MUTATION 23 necessary to compute the number A ( σ ) for the singularity type σ = r (1 , a ) of the R -cone, and find a power ω kr of ω r such that ω kr ( α ) + A ( σ ) β is integralin the basis from 4.12. We will do this explicitly for the simplest cases r (1 , a ) with a = 1 , or 3; it is not a hard computation, but we do not have a generalexpression yet, and must do it case-by-case.It is straightforward to verify that ω kr ( α ) = α + (1 − kr ) β + k − (cid:88) i =1 c i − r − (cid:88) i = k +1 c i , so for each σ = r (1 , a ) , we want to find a k such that A ( σ ) + (1 − kr ) is aninteger. Proposition 4.13.
Let A ( σ ) be the number defined in 4.6, and let m ( σ ) bean integer such that A ( σ ) + 1 − m ( σ ) r is an integer. Then (1) m ( r (1 , ≡ r − , (2) m ( r (1 , ≡ r − l , where r = 2 l + 1 , and (3) m ( r (1 , ≡ r (cid:40) l − if r = 3 l + 1 − l + 1) if r = 3 l + 2 . Proof.
Recall that r is odd in all cases. We compute A ( r (1 , − r − r ,so A ( σ ) + 1 − mr ≡ Z reduces to − − m ≡ r , which as r is odd gives m ( r (1 , ≡ r − .If now r = 2 l + 1 , we have that A ( r (1 , − l + l − r ; we now solve thecongruence l − − m ≡ r and find the solution m ≡ r − l .For the r (1 , case we must separate into the cases r = 3 l + 1 (here l is even) and r = 3 l + 2 (here l is odd). In the former case, A ( r (1 , − l + r ( l − , and the conguence l − − m ≡ r has the solution m ≡ r l − .For the latter case, we have A ( r (1 , − l − ( l + 6) /r , and the congruence − l − − m ≡ r has the solution m ≡ r − l + 1) . (cid:3) For the general g ≥ case, with width w ≥ , we can do essentially thesame thing as above; in the general case there are however several choices tobe made during the process, and no apparent means of identifying the “rightone”. The ambiguity in choices can be isolated to the image of the relativecycle α . Recall that above we had ω ( α ) = α + h − h β + (cid:80) ± c i , where the c i ’swere basis elements constructed from the edges of the polygons we glued toobtain our surface; in the general case we will have a similar formula, butseveral possible choices for what should go in the sum. As the most interestingpart of ω ( α ) is the coefficient of β , this can be considered a minor loss. Proposition 4.14.
Let w ≥ , and let X be a Riemann surface of genus ( h − w with two removed points p , p and w distinguished points q , . . . , q w ,and an order h automorphism fixing these points. Let α be a relative cyclepassing from p to p , and let β be a cycle going once around p . Then thereis a choice of cycles e , . . . , e ( h − w such that { α, β, e , . . . , e ( h − w } is a basisfor H ( X ) , and the automorphism is a power of the map ω given on homologycycles by • ω ( β ) = β , • ω ( e j ) = (cid:40) e j + w j ≤ ( h − wβ − (cid:80) k (cid:54) =1 e j + kw j > ( h − w, and • ω ( α ) = α + (1 − h ) β − (cid:80) h − k = w e kw + ( ∗ ) ,here ( ∗ ) is a Z -linear combination of e j ’s, and the indices in the second itemmust be taken modulo ( h − w .Proof. As in Propositions 4.12 and 4.9, we make a model for X by gluing twopolygons as follows: Take two regular hw -gons, and for the first label the cen-troid by p and the vertices in anticlockwise order by q , q , . . . , q w , q , q , . . . q w ,and label the edges by c , . . . c hw , oriented anticlockwise. On the second hw -gon, label the vertices in clockwise order q , q , . . . , q w , q , q , . . . q w , and labelthe edges with labels c , . . . c hw in such a way that for each i , c i + w is w stepsfrom c i going anticlockwise, each c i is connecting the same points q j , q j +1 asin the first polygon, and such that c i +1 does not follow c i going clockwise.There are a total of ( h − w − ( h − ways of doing this: at an edge connect-ing q j and q j +1 , there are h possible labels c j , c j + w , . . . , c j + hw (reading theindices modulo hw ), and there are w such sets; within each such set the orderis fixed by the requirement that c i + w and c i are separated by w steps. As c i +1 cannot follow c i , each choice rules out a choice on the adjacent edges, so thereare h − choices on each edge once one choice has been made, except thefinal one where there are only h − as it gets constraints imposed from bothsides. By rotation of the polygon, we can regard one set c j , c j + w , . . . , c j + hw as fixed; this gives ( h − ( w − ( h − possible arrangements.The demands that c i not be followed by c i +1 and that each c i connectsthe same points q j , q j +1 in both polygons ensure that when gluing accordingto the labels, we get a surface of genus ( h − w . The demand that c i isseparated by w steps from c i + w preserves the order h automorphism givenby “rotation by h ”: c i (cid:55)→ c i + w ; this is our ω .Now define the homology basis by e j := (cid:80) w − k =0 c j + k , β is equivalent to (cid:80) hwi =0 c i , and we may take α to pass from p to p crossing c hw . Applying c hw = β − (cid:80) hw − i =0 c i , it is easy to check that ω ( e j ) = (cid:40) e j + w j ≤ ( h − wβ − (cid:80) k (cid:54) =1 e j + kw j > ( h − w, and it is obvious that ω ( β ) = β . For the relative cycle α , as in 4.12 we seethat ω ( α ) − α + h β is homologous to c w + c hw + (cid:80) w − i =1 c i + (cid:80) w − i =1 c [ i ] , here c [ i ] = c i + kw for some k (this is where the ambiguity in labelling edges comesin, we can only fix these mod w ). Rearranging using (cid:80) hw − i = w c i = (cid:80) h − k =1 e kw ,we have ω ( α ) = α + (1 − h ) β + w − (cid:88) i =1 c [ i ] + c w − h − (cid:88) k =1 e kw , and for each choice of c [ i ] ’s, we can of course express (cid:80) w − i =1 c [ i ] + c w in termsof the e j ’s, but there is no concise general formula. (cid:3) Example 4.15.
The only choice of h, w with w ≥ that gives an unambigu-ous labelling is h = 3 , w = 2 , the (1 , R -cone. A picture of the labellingis given in Figure 8. ERIOD INTEGRALS AND MUTATION 25 p q q q q q q p q q q q c c c c c c c c c c c Figure 8.
Labels for the local model for X E , for h = 3 , w = 2 . Example 4.16.
For h = w = 3 (the primitive T -cone of height 3), there are (3 − (3 − (3 −
2) = 2 possible labellings. If the first -gon is labelled c , . . . , c (in anticlockwise order), the second must be labelled (in anticlockwise order)either c , c , c , c , c , c , c , c , c , or c , c , c , c , c , c , c , c , c . Theorem 4.17.
Let P be a Fano polygon, let f be a maximally mutableLaurent polynomial with N ewt ( f ) = P , and let L f be the associated Picard-Fuchs operator. The monodromy at zero of L f determines and is determinedby the singularity content of P (thought of as a multiset).Proof. It is clear by 4.4, 4.6, 4.9, 4.12 and 4.14 that the singularity contentdetermines the monodromy.Suppose now that the singularity content of P is ( k, B ) , and that we aregiven the monodromy matrix in the bases we have described. By 4.9, 4.12and 4.14 this matrix is of the form B rc M M . . . M n here r and c are some vectors and M i are block matrices of size g i × g i where g i is the genus of the local piece X E i , and B = 12 − k − (cid:80) σ ∈B m ( σ ) ,where m ( σ ) = A ( σ ) − (1 − ph ) and p is the power of the local monodromymap required to make A ( σ ) − (1 − ph ) an integer (as in Proposition 4.13).Each block M i is associated to an R -cone of type r i (1 , a i ) . The sizes g i of the blocks M i give us the r in r (1 , a ) , as g i = w i ( h i − ( h i and w i arethe height and width of the R -cone, respectively) and necessarily the matrix M i has order h i , so we can solve for w i and get r i = h i w i ; the a i can bededuced by finding the correct power p i (as done in 4.13) for each R -coneof this height and width (the list is finite) and selecting the one that equals M i .Now having identified the singularity basket B , we deduce the number of T -cones as k = 12 − B − (cid:80) σ ∈B m ( σ ) . That we cannot recover the cyclical order of the singularity basket followsfrom the easily verified fact that the local monodromy automorphisms overthe f -rigid cones commute; also we may reorder the blocks M i as desired byreordering the basis. (cid:3) Example 4.18.
The numbers m ( σ ) for the cases r (1 , a ) for a = 1 , , are • m ( r (1 , − r , • m ( r (1 , , and • m ( r (1 , (cid:40) if r ≡ − ( r + 1) if r ≡ ,as can be easily computed from 4.12 and 4.13. Corollary 4.19.
With the assumptions of Theorem 4.17, suppose the singu-larity basket contains n i R -cones of height h i . Then the monodromy of L f atzero has eigenvalues 1 with multiplicity 2, and each h i ’th root of unity (otherthan 1) with multiplicity n i . Ramification and degree of L f We now have a good description of the monodromy of L f around theorigin. From this we can deduce some information about L f , for instance wealready know from 3.15 that the order of L f (i.e. the highest degree in thedifferential variable ∇ t ) is twice the mutational genus. It is more difficultto prove anything about the degree (i.e. the degree in the variable t of theleading term of L f ). We do have some observations and conjectures, however.A local system V on P \ S (where S is a finite set) has monodromy T s around each point s ∈ S , and we can gather up some information aboutthe total monodromy group in a quantity called the ramification index of V ,defined by rf ( V ) = (cid:88) s ∈ S dim ( V x /V T s x ) − rk ( V ) , where x ∈ P \ S is some point (it doesn’t matter which, as T s is only definedup to conjugation, i.e. up to choice of base point). It is a general fact that rf ( V ) ≥ , in particular local systems with rf ( V ) = 0 seem interesting intheir own right (also see [CCG + V = Sol ( L f ) at the singularpoint t = 0 , we see that eigenspace has dimension either one or two, depend-ing on whether the number B defined in the proof of Theorem 4.17 is oneor zero, respectively. Both cases occur, for instance there are 26 mutationclasses of polygons with singularity content ( k, { n × (1 , } ) (see [KNP15]),and of these 6 have B = 0 and the rest have B > .The origin thus contributes either dim ( Sol ( L f )) − or dim ( Sol ( L f )) − to the ramification. We can re-express the ramification defect as rf ( Sol ( L f )) = (cid:88) s ∈ S \{ } dim ( Sol ( L f ) x /Sol ( L f ) T s x ) − g mut ( Y P ) − δ where S is the singular locus of L f , δ is 1 or 2, and rk ( Sol ( L f )) = 2 g mut .From now on, we write rf ( L f ) for rf ( Sol ( L f )) for simplicity. Now let E i be ERIOD INTEGRALS AND MUTATION 27 the dimension of the eigenspace of 1 at the singular point s i ∈ S \ { } , thenusing | S \ { }| = deg ( L f ) , we have after some rearrangement that (writing d = deg ( L f ) and g = g mut ) rf ( Sol ( L f )) = 2 g ( d − − δ − (cid:88) E i . We have some empirical evidence of some further information: for thoseinstances of L f that have been explicitly computed, which are the smoothFano polygons and several of those with singularity content ( k, { n × (1 , } ) ,a pattern emerges: Conjecture 5.1.
Let P be a Fano polygon with singularity content ( k, { n × (1 , } ) , let f be a maximally mutable Laurent polynomial with N ewt ( f ) = P , and let L f be the associated Picard-Fuchs operator. Then (1) the degree of L f is equal to g + 3 g − g · rf ( L f ) , and (2) the ramification index rf ( L f ) is equal to n + k eff − , where k eff isthe number of multiple points on the curve f = 0 .Remark . The number k eff here is equal to k for the generic MMLP’s,and drops by one whenever two T -cones on the same edge of P have the sameassociated factor ( γ + ηx ) in f . Thus, the minimal possible value for rf ( L f ) occurs when all the T -cones have the same factor (e.g. in the standard MMLPcase), and is equal to the minimal number of vertices of polygons mutation-equivalent to P , minus three. We should point out that 5.1 only applies topolygons with singularity basket { n × (1 , } ; it is not entirely clear how togeneralize it. As an example, the polygon with vertices ( − , , (2 , , (3 , − has singularity content (2 , { × (1 , } ) ; the conjecture would predict adegree of 17 for the standard MMLP, but the actual value is 19. Example 5.3.
The computations are expensive, as noted at the start ofSection 2, and the output is very large and not particularly enlightening,so we’ll show only the simplest few examples here. The simplest smoothFano polygon polygon is the one with vertices (0 , , (1 , and ( − , − ,with singularity content (3 , ∅ ) : the standard MMLP is x + y + xy and L f is ∇ − t ( ∇ + 1)( ∇ + 2) (as before, ∇ = t∂ t ); this has ramification indexzero, degree 3 and order 2. The second simplest smooth Fano polygon is theone with vertices (0 , , (1 , , ( − , − and (1 , , with singularity content (4 , ∅ ) ; here the standard MMLP is x + y + xy + xy and L f is ∇ + t ∇ (17 ∇ − − t (5 ∇ + 8)(11 ∇ + 8) − t (30 ∇ + 78 ∇ + 47) − t ( ∇ + 1)(103 ∇ +147) − t ( ∇ + 1)( ∇ + 2) , this has ramification index 1, order 2 and degree5. We may observe (though we don’t know how to prove this) that there isno way to mutate this polygon into one with three vertices.There are nonequivalent polygons with the same singularity content, butgiving different ramification index for the associated L f ’s. The simplest ex-ample is for singularity content (5 , { × (1 , } : for the polygon with vertices ( − , , (3 , and (0 , − , we have order 4, degree 9, and ramification indexzero; for the polygon with vertices ( − , − , ( − , , (1 , and (2 , − wehave order 4, degree 13, and ramification index one. Remark . Conjecture 5.1 suggests a way to distinguish nonequivalent mu-tation classes of Fano polygons with the same singularity content; in the minimal case the value for k eff is the minimal number of edges (or vertices)of polygons mutation-equivalent to P . This number together with the singu-larity content could be an invariant that completely classifies Fano polygonsup to mutation.Conjecture 5.1, if true, gives us the ramification index and degree directlyfrom N ewt ( f ) , and so gives some bounds on the E i . We have (cid:88) E i = 2 g ( d − − δ − rf ( L f ) , and as there are d singular points, we can write each E i = 2 g − ε i , where (cid:80) i ε i = 2 g + δ + rf ( L f ) . This number is always smaller than the degree(assuming 5.1), so there are guaranteed to be at least d − (cid:80) i ε i = g + g − g − δ ) rf ( L f ) points with trivial monodromy and possibly more. Thisis quite special, as a generic local system has nontrivial monodromy at everysingular point. Acknowledgements
This paper came out of the project to classify Fano manifolds via mirrorsymmetry, to which I was introduced during the PRAGMATIC 2013 Re-search school in Algebraic Geometry and Commutative Algebra “Topics inHigher Dimensional Algebraic Geometry” held in Catania, Italy, in Septem-ber 2013. I am very grateful to Alfio Ragusa, Francesco Russo, and GiuseppeZappalá, the organizers of the PRAGMATIC school, for making all that hap-pen. I am also grateful to Alessio Corti and Al Kasprzyk for introducing meto this topic and for the fruitful collaboration that has followed, and to TomCoates and Rikard Bøgvad for helpful comments.
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Ketil Tveiten, Department of Mathematics, Stockholm University, 106 91Stockholm.
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