ON THE LOG-LOCAL PRINCIPLE FOR THE TORIC BOUNDARY

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ON THE LOG-LOCAL PRINCIPLE FOR THE TORIC BOUNDARY

Pierrick Bousseau, Andrea Brini, Michel van Garrel

To cite this version:

Pierrick Bousseau, Andrea Brini, Michel van Garrel. ON THE LOG-LOCAL PRINCIPLE FOR THE TORIC BOUNDARY. 2021. �hal-03277609�

ON THE LOG-LOCAL PRINCIPLE FOR THE TORIC BOUNDARY

PIERRICK BOUSSEAU, ANDREA BRINI, AND MICHEL VAN GARREL

Abstract. LetXbe a smooth projective complex variety and letD=D1+· · ·+Dlbe a reduced normal crossing divisor onX with each componentDj smooth, irreducible, and nef. The log-local principle put forward in [17] conjectures that the genus 0 log Gromov–Witten theory of maximal tangency of (X, D) is equivalent to the genus 0 local Gromov–Witten theory of X twisted by Ll

j=1O(−Dj). We prove that an extension of the log-local principle holds forXa (not necessarily smooth)Q-factorial projective toric variety,Dthe toric boundary, and descendent point insertions.

1. Introduction

Let X be a smooth projective complex variety of dimension n and let D = D1 +· · ·+Dl be an effective reduced normal crossing divisor with each componentDj smooth, irreducible and nef. We can then consider two, a priori very different, geometries associated to the pair (X, D):

• the n-dimensionallog geometry of the pair (X, D),

• the (n+l)-dimensional local geometry of the total space Tot Ll

j=1O_X(−D_j) . The genus zero log Gromov–Witten invariants of (X, D) virtually count rational curves

f :P¹ →X

of a fixed degree f∗[P¹]∈ H₂(X,Z), with insertions, such as passing through a number of general points, and with prescribed intersections with D. Such an f is said to be of maximal tangency if f(P¹) meets each Dj in only one point of full tangency. On the other hand, the local Gromov- Witten theory of Tot

L_l

j=1O_X(−D_j)

is a way to study the local contribution of X to the enumerative geometry of a compact (n+l)-dimensional varietyY containingXwith normal bundle Ll

j=1O_X(−D_j).

The existence of a relation between the log and the local theory of (X, D) was introduced by the log-local principle of [17, Conjecture 1.4]:

Conjecture 1.1. Let d be an effective curve class such that d ·Dj > 0 for all 1 ≤ j ≤ l.

After dividing by Ql

j=1(−1)^d·Dj+1d·D_j, the genus 0 log Gromov-Witten invariants of maximal tangency and class d of (X, D) equal the genus 0 local Gromov-Witten invariants of class d of Tot

Ll

j=1O_X(−D_j)

(with the same insertions).

This project has been supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 746554 (M. vG.), the Engineering and Physical Sciences Research Council under grant agreement ref. EP/S003657/2 (A. B.) and by Dr. Max R¨ossler, the Walter Haefner Foundation and the ETH Z¨urich Foundation (P. B. and M. vG.).

1

Theorem 1.2 ([17]). The log-local principle holds if X is a smooth projective variety and D is smooth and nef.

There are two natural directions to generalise the log-local principle further. The first is to in- vestigate extensions to correspondences with other invariants. At the level of BPS invariants [12–14, 19, 30] this is proven for the pair of P² and smooth cubic in [5, 6] and in higher genus in [9]. In [7, 8], we extend the correspondences to the non-toric and higher genus/refined setting and include open Gromov–Witten invariants, their underlying open BPS counts, as well as quiver Donaldson–Thomas invariants to the set of correspondences. Another direction is the relationship between local and orbifold invariants [3, 31].

The second natural question is to what extent the log-local principle generalises to the case whenX andDare not smooth: log Gromov-Witten theory is indeed well-defined for any pair (X, D) which is log smooth, but it is unclear how to define a local geometry in such generality. In the present paper, we consider a situation that goes beyond the smoothness assumptions of Conjecture 1.1 and where both log and local sides can be defined: we take for X a Q-factorial projective toric variety and for Dthe toric boundary divisor of X. AsX isQ-factorial, it makes sense to require that the componentsDj ofDare nef. We show in Proposition 2.1 that requiring eachDj to be nef forcesX to be a product of fake weighted projective spaces. While such anX is not necessarily smooth, and Dis typically not normal crossing, (X, D) can naturally be viewed as a log smooth variety, and so log Gromov-Witten invariants of (X, D) are well-defined. On the other hand,X can be naturally viewed as a smooth Deligne–Mumford stack, and the local geometry Tot

Ll

j=1O_X(−D_j)

makes sense in the category of orbifolds. The local Gromov-Witten invariants can be defined using orbifold Gromov–Witten theory [2], and it thus makes sense to ask if the genus 0 log invariants of maximal tangency of such a pair (X, D) are related in the sense of Conjecture 1.1 to the corresponding local invariants. Our main result is the following Theorem 1.3; we refer to Theorems 3.1–3.4 for precise statements.

Theorem 1.3. Let X be a Q-factorial projective toric variety and let D be the toric boundary divisor of X. Assume that all the components Dj of D are nef. Then the genus 0 log Gromov- Witten invariants of maximal tangency of (X, D), and the genus 0local Gromov-Witten invariants of (X, D), both with descendent point insertions, can be computed in closed form for all degrees.

As a corollary, the log-local principle holds for the resulting invariants.

Except for the well-studied case when X = P¹, the log and local Gromov–Witten invariants of (X, D) are non-zero only for one, two or, providedX= (P¹)ⁿ, three point insertions. ForX= (P¹)ⁿ we prove an equality of virtual fundamental classes and refer to well-known techniques to compute the invariants. For the other cases, our proof proceeds by calculating both sides to obtain explicit closed formulas for these invariants for all (X, D) (Theorems 3.2 and 3.3). To compute the log invariants we use the tropical correspondence result [25] and an algorithm of [24] for the tropical multiplicity. The log-local principle of Conjecture 1.1 then predicts an explicit formula in all degrees

2

for the local invariants, which we verify using local mirror symmetry techniques and a reconstruction result from small to big quantum cohomology.

Relation to[27]and[7,8]. After this paper was finished, we received the manuscript [27] where the log-local principle is considered for simple normal crossings divisors. The respective strategies have different flavours in the proof and complementary virtues in the outcome: [27] consider the log/local correspondence for X smooth and Dj a hyperplane section, with a beautiful geometric argument reducing the simple normal crossings case to the case of smooth pairs, and with no restrictions on X. The combinatorial pathway we pursued in the toric setting allows on the other hand to relax the hypotheses on the smoothness ofX, the normal crossings nature ofD, and the very ampleness of Dj, and it lends itself to a wider application to the case when D is not the toric boundary and the refinement to include all-genus invariants. We consider this specifically in the follow-up papers [7, 8], where we prove the log-local principle for log Calabi–Yau surfaces with the components of the anticanonical divisor smooth and nef and suitably reformulate it to, and verify it for, the higher genus theory in these cases. In addition, we extend the correspondences to include open Gromov–Witten invariants, the various underlying BPS counts, and quiver Donaldson–Thomas invariants.

Remark 1.4. In its most recent version, [27] gives a counter-example in principle to Conjecture 1.1. It is proven that there is a choice of (unspecified) insertion leading to a counter-example.

The geometry however is not log Calabi–Yau and the insertion is not formed of point insertions.

It remains open whether the conjecture holds in the more restrictive setting of a log Calabi–Yau variety with only point insertions. The present paper as well as [7, 8] provide evidence for it.

Acknowledgements. We thank Helge Ruddat for getting this project started by asking us to compute some local Gromov–Witten invariants of P⁴ and for helpful discussions all along. We thank Travis Mandel for detailed explanations on [24, 25]. We are grateful to Miles Reid and Andrea Petracci for some discussions on the classification of toric varieties with nef divisors. We thank Makoto Miura for a discussion on Hibi varieties. Finally, it is a pleasure to thank Navid Nabijou and Dhruv Ranganathan for discussions on their parallel work [27].

2. Setup

2.1. Notation. LetX be aQ-factorial projective toric variety of dimensionnX and letD=D1+

· · ·+D_lDbe the toric boundary divisor ofX. In the foregoing discussion, we writer_X := rank Pic(X) for the rank of the Picard group of X, so thatl_D =n_X +r_X, and χ_X =χ(X) := dim_CH(X,C) for the dimension of the cohomology of X. The variety X has a natural presentation as a GIT quotientC^nX+rX//t((C^?)^rX ×G_X) for G_X a finite abelian group; for every 1≤j≤l_D, we writeDj

for the divisor corresponding to the (C^?)^rX×G_X reduction toX of thej^th coordinate hyperplane inC^nX+rX. Note in particular thatPlD

j=1Dj =−K_X.

We also fix a further piece of notation, which will turn out to be convenient when dealing with the book-keeping of indices for products of fake weighted projective spaces. Let m ∈ N0. If

3

v= (v1, . . . , vm)∈N^m is a lattice point in the non-negativem-orthant, we write |v|=Pm

i=1vi for its 1-norm; in the following we will consistently use serif fonts for orthant points and italic fonts for their Cartesian coordinates. For R a finitely generated commutative monoid with generators α1, . . . , αm, x = α^j₁¹. . . α^jm^m ∈ R a reduced word in αi, and v ∈ N^m, we write x^v for the product Q

iα^j_i^ivi ∈ R. We introduce partial orders on the m-orthant by saying that v ≺w (resp. v w) if v_i < w_i (resp. v_i ≤ w_i) for all i = 1, . . . , m. Also, we will write Q^X_ij ∈ Z, i = 1, . . . , r_X, j = 1, . . . , n_X +r_X, for the weight of the i^th factor of the (C^?)^rX torus action on the j^th affine factor of C^nX+rX.

Definition 2.1. A nef toric pair (X, D)is a pair given by X a Q-factorial complex projective toric variety with toric boundary divisor D=D1+· · ·+Dl_D, such that all the componentsDj are nef.

Nefness of all the components D_j of the toric bundary divisor imposes strong conditions onX, as the Proposition 2.1 below shows.

Definition 2.2. LetX be aQ-factorial projective toric variety, and letC^nX+rX//_t((C^?)^rX ×G_X)be its natural GIT description. We say thatX is a fake weighted projective space if C^nX+rX//t(C^?)^rX is a weighted projective space.

Proposition 2.1. Let X be a Q-factorial projective variety such that every effective divisor on X is nef. Then X is a product of fake weighted projective spaces.

Proof. By [16, Proposition 5.3], X admits a finite surjective toric morphism Q

Pⁿⁱ → X. Let Σ⊂N⊗Rbe the fan ofQ

Pⁿⁱ and Σ⁰ ⊂N⁰⊗Rthe fan ofX. Then we have an injective morphism of lattices N →N⁰ of finite index. Identifying N with its image in N⁰, Σ = Σ⁰. It follows that X is the quotient of Q

Pⁿⁱ byN⁰/N. HenceX is a product of fake weighted projective spaces.

By Proposition 2.1, there isn_X ∈N^rX such thatn_X =|n_X|andX is a product of r_X,n_i:= (n_X)_i- dimensional fake weighted projective spaces,

X =

rX

Y

i=1

P^Gi

w⁽ⁱ⁾_X ,

with w⁽ⁱ⁾_X = ((wX)⁽ⁱ⁾₁ , . . . ,(wX)⁽ⁱ⁾_n

i+1) ∈ Nⁿⁱ⁺¹, which we may assume not to have any common factors, and

P^Gi

w⁽ⁱ⁾_X

:=P

w⁽ⁱ⁾_X

//tGi,

forGia finite abelian group. Notice that, for fixediand definingεi :=Pi−1

k=1(nk+1), we have Q^X_i,j+εi =

( (w_X)⁽ⁱ⁾_j 1≤j≤n_i+ 1,

0 else, (2.1)

independent of the G_i. Let H_i := pr^∗_ic₁(O

P^Gi(w⁽ⁱ⁾)(1)) denote the pull-back to X of the (orbi-) hyperplane class of the i^th factor of X and let H := H1. . . HrX. These generate the classical

4

cohomology ring,

H^•(X,C) = C[H1, . . . , HrX] Dn

H_iⁿⁱ⁺¹orX

i=1

E, (2.2)

which is independent of the Gi, and we can take a homogeneous linear basis for H^•(X,C) in the form {H^l}_li≤ni. Notice, in particular, that

[pt] =

rX

Y

i=1

Gi

Y

i,j

(wX)⁽ⁱ⁾_j H^nX.

Indeed, if G_i is trivial, this follows from applying [22, Theorem 1] to each component in the product; and ifGiis non-trivial, then the extra factor comes from the component-wise identification H^• P^Gi(w⁽ⁱ⁾),C

= H^•(P(w⁽ⁱ⁾),C)^Gi. We will also write d = (d1, . . . , dr_X) for the curve class d₁H₁+· · ·+d_rH_r and

d^nX :=

rX

Y

i=1

dⁿ_iⁱ. (2.3)

We order the toric divisorsDj ofX,j= 1, . . . ,|n_X|+rX, in such a way that Q^X_ij =

i^th

(0, . . . ,0,1,0, . . . ,0)·D_j, where the 1 is in the i^th position. Finally, we define

e^X_j (d) :=X

i

Q^X_ijd_i =d·D_j, e^X(d) :=

|n_X|+r_X

X

j=1

e^X_j (d) =−d·K_X. (2.4) 2.2. Log Gromov-Witten invariants. Let (X, D) be a nef toric pair and let d be an effective curve class on X.¹ For the definition of log Gromov-Witten invariants, we endow² X with the divisorial log structure coming fromD, and view (X, D) as a log smooth variety. The log structure is used to impose tangency conditions along the components Dj of D: in this paper we consider genus 0 stable maps into X of class d that meet each component D_j in one point of maximal tangency d·Dj. The appropriate moduli space M^log_0,m(X, D,d) of genus 0 m-marked maximally tangent stable log maps was constructed (in all generality) in [1, 10, 20]. In this description, we have m marked points that have tangency 0 with the boundary (interior marked points), and l_D marked points with maximal tangency with eachD_j respectively. In cased·D_j = 0 for somej, this means that the corresponding maximal tangency marked point is an interior marked point. There is a virtual fundamental class

[M^log_0,m(X, D,d)]^vir ∈H_2vdim(X,D,d) log

(M^log_0,m(X, D,d)), where

vdim^(X,D,d)_log = −d·K_X + dimX−3 +m−

l_D

X

j=1

(d·D_j−1)

= nX +m+lD −3 = 2nX +rX+m−3.

1Note that unlike in Conjecture 1.1 we do not require thatd·Dj>0 for all 1≤j≤lD.

2We refer to [18] for an introduction to log geometry.

5

Evaluating at the marked pointspi yields the evaluation maps ev_i: M^log_0,m(X, D,d)−→X.

ForLi thei^thtautological line bundle on M^log_0,m(X, D,d), whose fiber at [f: (C, p1, . . . , pm)→X] is the cotangent line ofC at p_i, there are tautological classesψ_i := c₁(L_i). We are interested in the calculation of the genus 0 log Gromov–Witten invariants of maximal tangency of (X, D) with 1, 2 or 3 point insertions andψ-class insertions at one point, defined as follows:

Rp^X_d :=

Z

[M^log_0,1(X,D,d)]^vir

ev^∗₁([pt])∪ψⁿ₁^X+rX−2, (2.5) Rq^X_d :=

Z

[M^log_0,2(X,D,d)]^vir

ev^∗₁([pt])∪ev^∗₂([pt])∪ψ^r₂^X−1. (2.6)

The invariantRp^X_d (resp.Rq^X_d) is a virtual count of rational curves inXof degreed= (d₁, . . . , d_rX) that meet each toric divisor Dj in one point of maximal tangency d·Dj = Pr

i=1diQ^X_ij = e^X_j (d) and that pass through one point in the interior with ψ^nX+rX−2 condition (resp. two points in the interior, one of which with a ψ^rX−1 condition).

Remark 2.2. Having a point condition onX cuts down the dimension of the moduli space bynX. Thus (2.5) and (2.6) cover all possible invariants with descendent point insertions except for two families of cases. For the first, one distributes the descendent insertions along both points in (2.6).

Adapting the log calculations of Section 5 to that case is left as an exercise to the reader, see also Remark 6.5 for the local side. The second family of cases concerns the invariants of (P¹)ⁿ with any number of marked points ifn= 1 and up to 3 marked points if n≥2. We treat (P¹)ⁿ separately in Theorem 3.1.

2.3. Local Gromov-Witten invariants. Let (X, D) be a nef toric pair as in Definition 2.1 and write X_D^loc := Tot(L

iO_X(−D_i)) for the target space of the local theory. By Proposition 2.1, we can viewX andX_D^loc as the coarse moduli schemes of smooth Deligne–Mumford stacksX andX_D^loc overC, where

X :=

rX

×

h

C^(nX)i\ {0}

/(C^?×G_i)i ,

X_D^loc :=

rX

×

h

C^(nX)i \ {0}

×C^(nX)i+1

/(C^?×G_i)i

. (2.7)

Even thoughX_D^locis not proper and may be singular, the locution “Gromov–Witten theory ofX_D^loc” receives a meaning in terms of the orbifold Gromov–Witten theory of X twisted byL

iO_X(−D_i) [2, 11] and restricted over its non-stacky part, and we refer the reader in particular to [2] for the relevant background on the Gromov–Witten theory of Deligne–Mumford stacks. Let M_0,m(X,d) be the moduli stack of twisted genus 0m-marked stable maps [f :C → X] withf∗([C]) =d∈H2(X,Q),

6

whereCis anm-pointed twisted curve³ [2], and write M0,m(X,d) for the substack of twisted stable maps such that the image of all evaluation maps is contained in the zero-age component of the (rigidified, cyclotomic) inertia stack of X. The stack M_0,m(X,d) can be equipped with a virtual fundamental class [2, Section 4.5], which induces a virtual fundamental class of pure homological degree over the stack M0,m(X,d) of stable maps to the coarse moduli space,

M0,m(X,d)vir

∈H_2vdim(X,D,d)(M0,m(X,d),Q), where

vdim^(X,D,d):=−K_X ·d+ dimX+m−3 =vdim^(X,D,d)_log +e_X(d)−l_D. Let nowdbe such thatd·Dj >0 for all 1≤j ≤lD. Then H⁰(C, f^∗LlD

j=1O_X(−D_j)) = 0 for every twisted stable map [f :C → X] withf∗([C]) =d, and so Ob_D :=R¹π∗f^∗

LlD

j=1O_X(−D_j)

is a vec- tor bundle on M_0,m(X,d), which is of rankPlD

j=1(d·D_j−1) and has fibre H¹(C, f^∗LlD

j=1O_X(−D_j)) at a stable map [f :C → X]. Restricting to the zero-age component defines the virtual fundamental class

[M_0,m(X_D^loc,d)]^vir:= [M_0,m(X,d)]^vir∩c_top(Ob_D)∈H_2(vdim(X,D,d)+lD−e_X(d))(M_0,m(X,d),Q), (2.8) and we have

vdim M0,m(X_D^loc,d) =vdim^(X,D,d)−eX(d) +lD =vdim^(X,D,d)_log .

The restriction to the untwisted sector gives well-defined evaluation maps evi: M0,m(X,d) −→X, and there are tautological classes ψi := c1(Li), where the fibre of Li at a stable map [f :C → X] is given by the cotangent line to the coarse moduli space of C at the i^th point. The (untwisted) local Gromov–Witten invariants of (X, D) are then caps of pull-backs of classes inH^•(X,C) via the evaluation maps against the virtual fundamental class (2.8). In particular, the local counterparts of (2.5) and (2.6) are defined by

p^X_d :=

Z

[M0,1(X_D^loc,d)]^vir

ev^∗₁([pt])∪ψ₁^nX+rX−2, (2.9) q^X_d :=

Z

[M0,2(X_D^loc,d)]^vir

ev^∗₁([pt])∪ev^∗₂([pt])∪ψ^r₂^X−1. (2.10) 3. Main results

We first consider the case of (P¹)ⁿ and treat the general case thereafter.

Theorem 3.1. Conjecture 1.1 holds forX = (P¹)ⁿ with its toric boundary.

Proof. For X = P¹, the log-local principle (at the level of the virtual fundamental classes) is a direct consequence of [17] since the toric divisors are disjoint. ForX = (P¹)ⁿwithn≥2, we apply

3This means that the coarse moduli space of C is a pre-stable curve in the ordinary sense, with cyclic-quotient

stackiness allowed at special points, and satisfying kissing (balancing) conditions for the stacky structures at the nodes. See [2, Section 4] for more details.

7

the log product formula [21, 29] on the log side and the product formula [4] on the local side to

obtain an equality of virtual fundamental classes.

Note that computational techniques to compute the invariants of X = (P¹)ⁿ (with arbitrary numbers of point insertions if n = 1) are well-developed. For example, using tropical corre- spondence results one may show that the maximal tangency 3-pointed invariants of (P¹)ⁿ are Q2n

j=1d·Dj =Qn i=1d²_i.

The following Theorems 3.2 and 3.3 compute the log and local Gromov-Witten invariants defined in Sections 2.2 and 2.3 in all degrees for a nef toric pair (X, D).

Theorem 3.2. Let (X, D) be a nef toric pair and let d be an effective curve class on X. If there is j such that d·Dj = 0, then Rp^X_d =Rq^X_d = 0. If d·Dj >0 for all1≤j ≤lD, then we have

Rp^X_d = 1, (3.1)

Rq^X_d =

rX

Y

i=1

G_i



 Y

i,j

w_X(i) j



d^nX. (3.2)

We write

◦

Q

je^X_j (d) to mean the product ofe^X_j (d) over j∈ {1, . . . ,|n_X|+r_X

e^X_j (d)6= 0}.

Theorem 3.3. Let (X, D) be a nef toric pair and letd be an effective curve class on X. Then p^X_d = (−1)^{eX(d)−nX−rX}

◦

Q

je^X_j (d)

, (3.3)

q^X_d =

rX

Y

i=1

G_i



 Y

i,j

w_X(i) j



d^nXp^X_d

=

rX

Y

i=1

Gi



 Y

i,j

wX

(i) j



d^nX (−1)^{eX(d)−nX−rX}

◦

Q

je^X_j (d)

. (3.4)

We deduce from these the log-local principle proved in the present paper.

Theorem 3.4. The log-local principle holds for nef toric pairs(X, D) with descendent point inser- tions and with no assumptions ond·Dj. That is, for every effective curve classd, the log and local invariants are equal up to the factor

lD

Y

j=1

(−1)^d·Dj+1d·Dj = (−1)^{eX(d)−|nX|−rX}

|n_X|+r_X

Y

j=1

e^X_j (d).

Theorem 3.4 is a direct corollary of the combination of Theorems 3.1–3.3. We will prove Theorem 3.2 using a tropical correspondence principle, and Theorem 3.3 using an equivariant mirror theorem.

We review these technical tools in Section 4, and explain how to apply them to the proofs of Theorems 3.2 and 3.3 in Sections 5 and 6 respectively.

8

4. Computational methods 4.1. The log side: tropical curve counts. Let X = QrX

i=1P^Gi(w⁽ⁱ⁾) as in Section 2.1 and let Σ ⊂ N_R be the fan of X = X_Σ; here N ' Z^|nX| and N_R := N ⊗_Z R. Define furthermore N_Q :=N⊗_ZQand letM := Hom(N,Z) be the dual ofN. Denote by [D₁], . . . ,[D_|nX|+rX] the rays of Σ corresponding to the irreducible effective toric divisors of X. We use correspondence results with tropical curve counts as developed in [25, 26, 28] (see [18] for an introduction) and state them in the generality needed for our purposes.

Denote by Γ the topological realisation of a finite connected graph and by Γ the complement of a subset of 1-valent vertices. We require that Γ has no univalent and no bivalent vertices. The set of its vertices, edges, non-compact edges and compact edges is denoted by Γ^[0], Γ^[1], Γ^[1]∞ and Γ^[1]c

respectively. Γ comes with a weight function w : Γ^[1] → Z≥0. The non-compact edges come with markings. Weight 0, resp. positive weight, non-compact edges areinterior, resp.exterior, markings.

There will be 1 or 2 interior point markings, which we denote byP1 and P2, and|n_X|+rX exterior markings corresponding to the toric divisors, which we denote by [D1], . . . ,[D_|nX|+rX] as well.

Definition 4.1. A genus 0 degree d maximally tangent parametrised marked tropical curvein X consists of Γ as above and a continuous map h: Γ→N_R satisfying

(1) ForE ∈Γ^[1],h|_E is constant if and only ifw(E) = 0. Otherwise, h|_E is a proper embedding into an affine line with rational slope.

(2) Let V ∈ Γ^[0] with h(V) ∈ N_Q. For edges E 3 V, denote by u_(V,E) the primitive integral vector at h(V) into the direction h(E) (and set u_(V,E) = 0 if w(E) = 0). The balancing condition holds:

X

E3V

w(E)u_(V,E)= 0.

(3) For each exterior marking D_j, h|_Dj is parallel to the ray[D_j]and w(D_j) =d·D_j. (4) The first Betti number b₁(Γ) = 0.

If(Γ⁰, h⁰)is another such parametrised tropical curve, then an isomorphismbetween the two is given by a homeomorphism Φ : Γ→ Γ⁰ respecting the discrete data and such that h=h⁰◦Φ. A genus 0 degree d maximally tangent marked tropical curve then is an isomorphism class of such.

Moreover, we say that an interior marking E satisfies a ψ^k-condition if h(E) is a k+ 2-valent vertex.

Denote by T(p)^X_d the (moduli) space of genus 0 degree d maximally tangent tropical curves in X with the interior marking equipped with a ψ^|nX|+rX−2-condition passing through a fixed general point in R^|nX|+rX. Denote by T(q)^X_d the moduli space of genus 0 degree d maximally tangent tropical curves inX with the two interior markingsP₁ andP₂ mapping to two fixed general points inR^|nX|+rX and such that P2 has a ψ^rX−1-condition. We will see in Propositions 5.1 and 5.4 that

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each of T(p)^X_d and T(q)^X_d consist of one element. Since T(p)^X_d and T(q)^X_d are finite hence, their elements arerigid [24, Definition 2.5].

Counts of tropical curves are weighted with appropriate multiplicities. There are a number of ways of defining the multiplicity Mult(Γ) of Γ. The version we use was formulated (forX smooth) in [24, Theorem 1.2]. We state it for our setting. Set A := Z[N]⊗_Z Λ^•M. For n ∈ N and α ∈ Λ^•M, write zⁿα for zⁿ⊗α and ι_nα for the contraction of α by n. Recall that if α ∈ Λ^sM, thenιnα∈Λ^s−1M. Fork≥1, define`k:A^⊗k→A via

`_k(zⁿ¹α₁⊗ · · · ⊗z^nkα_k) :=z^n1+···+nkι_n1+···+nk(α₁∧ · · · ∧α_k).

Let now h : Γ→N_R be in T(p)^X_d or T(q)^X_d and choose a vertex V∞ of Γ. Consider the flow on Γ with sink vertex V∞. To each edge E of Γ, we inductively associate an element ζ_E =z^nEα_E ∈A, well-defined up to sign:

• For the exterior markings, set ζ_Dj = z^w(Dj)∆(j), where ∆(j) is the primitive generator of [D_j].

• For an interior marking P, set ζP to be one of the two generators of Λ^|nX|M.

• If E₁, . . . , E_k are the edges flowing into a vertex V 6=V∞ and E_out is the edge flowing out, set ζEout=`k(ζE1 ⊗ · · · ⊗ζEk).

By [24, Theorem 1.2],ζ_Γ:=Q

E3V∞ζ_E ∈z⁰⊗Λ^|nX|M and Mult(Γ) is the index ofζ_Γ in Λ^|nX|M. It then follows from [25, Theorem 1.1] thatRp^X_d is the number of Γ in T(p)^X_d counted with multiplicity Mult(Γ), and Rq^X_d is the weighted cardinality of{Γ∈T(q)^X_d}, each weighted by Mult(Γ).

Remark 4.1. Note that a priori [25, Theorem 1.1] is stated for smooth varieties; in the cases of interest to us, however, the curves never meet the deeper toric strata and the arguments of [25]

carry through.

4.2. The local side: mirror symmetry for toric stacks. The second technical result we will use for the calculation of local Gromov-Witten invariants is Theorem 4.2 below. Consider a torus T ' C^? acting on X_D^loc := Tot (L

iO_X(−D_i)) transitively on the fibres and covering the trivial action on the image of the zero section. We will denote byλ:=c1(O_P^∞(1)) the polynomial generator of the T-equivariant cohomology of a point, H_T(pt) = H(BT) ' C[λ]. The basis elements H^l of Section 2.1 for the cohomology of X have canonical T-equivariant lifts, which by a slight abuse of notation we denote with the same symbol, to cohomology classes in X_D^loc forming a C(λ) basis of H_T(X_D^loc), where as usual C(λ) is the field of fractions of H_T(pt). The T-equivariant cohomology HT(X_D^loc) is furthermore endowed with a non-degenerate, symmetric bilinear form given by the restriction of the T-equivariant Chen–Ruan [15, Section 2.1] pairing on the untwisted component of the inertia stack ofX_D^loc,

η_lm:= (H^l, H^m)_Xloc D :=

Z

X

H^l∪H^m

∪_ie_T(O_X(−D_i)), (4.1) where eT denotes theT-equivariant Euler class.

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Let nowτ ∈HT(X_D^loc). The equivariant big J-function of X_D^loc is the formal power series J^X

loc D

big (τ, z) :=z+τ+ X

d∈NE(X)

X

n∈Z⁺

X

l,mn_X

1 n!

τ, . . . , τ, H^l z−ψ

X_D^loc 0,n+1,d

H^mη^lm, (4.2) where we employed the usual correlator notation for Gromov-Witten invariants,

D

τ1ψ^k₁¹, . . . , τnψ^k_nⁿ EX^loc

0,n,d:=

Z

[M0,m(X_D^loc,d)]^vir

Y

i

ev^∗_i(τi)ψ_i^ki, (4.3) and η^lm:= (η⁻¹)_lm. Restriction tot=t₀1_H(X)+PrX

i=1t_iH_i and use of the Divisor Axiom leads to theequivariant small J-function ofX_D^loc,

J_small^XDloc(t, z) :=ze^Ptiφi/z



1 + X

d∈NE(X)

X

l,mn_X

e^Ptidi

H^l z(z−ψ1)

X_D^loc 0,1,d

H^mη^lm



. (4.4) The n-pointed genus zero Gromov–Witten invariants with one marked descendant insertion (re- spectively, the 1-pointed genus zero descendant invariants) of X_D^loc, and no twisted insertions, can thus be read off from the formal Taylor series expansion ofJ_big (resp., J_small) at z=∞.

The following theorem provides an explicit hypergeometric presentation of J^X

loc D

small(t, z). Let κ_j :=

c₁(O(−D_j)) be the T-equivariant first Chern class of O(D_j) andy_i ∈ SpecC[[t]],i= 1, . . . , r_X be variables in a formal disk around the origin. Writing (x)n := Γ(x+n)/Γ(x) for the Pochhammer symbol of (x, n) withn∈Z, theT-equivariantI-functionsofXand X_D^loc are defined as the H_T(X) and H_T(X_D^loc) valued Laurent series

I^X(y, z) := z1_H(X)+Y

i

y^H_i ^i/z X

d∈NE(X)

Y

i

y^d_iⁱz^d·KX 1 Q

j κj

z + 1

d·D_j

, (4.5)

I^XDloc(y, z) := z1_H(X)+Y

i

y^H_i ^i/z X

d∈NE(X)

Y

i

y^d_iⁱz^{d·(KX+D)−lD} Q

jκj κj

z + 1

d·Dj−1

Q

j κj

z + 1

d·D_j

(4.6) and their mirror maps as their formalO(z⁰) coefficient,

t˜ⁱ_X(y) :=

z⁰H_i

I^X(y, z),

˜tⁱ_Xloc

D (y) :=

z⁰Hi

I^XDloc(y, z). (4.7)

Note that X and X_D^loc are smooth toric Deligne–Mumford stacks with coarse moduli schemes X and X_D^loc that are projective over their affinisation, and at this level of generality a result of [15]

can be applied to provide a Givental-style equivariant mirror statement for them, as follows. In the language of [15], the I-functions (4.5) and (4.6) are the stackyI-functions of [15, Definition 28 and 29] for X and X_D^loc respectively, restricted to insertions in the zero-age sector of their inertia stack. The main result of [15] identifies the small J-function of a semi-projective toric Deligne–

Mumford stack to its stacky I-function, up to a change-of-variables given by itsO(z⁰) term as in (4.7). In particular, the following statement is a projection to the untwisted sector of X and X_D^loc of [15, Theorem 31 and Corollary 32].

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Theorem 4.2 ([15]). We have

J_small^X (˜t_X(y), z) =I^X(y, z), J_small^XDloc

˜t_X(y) + ˜t_Xloc

D (y)−logy, z

=I^XDloc(y, z). (4.8)

5. The log side: proof of Theorem 3.2 Assume first that there isj such that d·Dj = 0. Given that

X=

rX

Y

i=1

P^Gi

w⁽ⁱ⁾_X

is given its toric boundary, each Dj is of the form (up to reordering of the factors) D^j×Y

i6=k

P^Gi

w_X⁽ⁱ⁾

for some k and with D^j a prime toric divisor in P^Gk

w^(k)_X

. As d= (di)i, d·Dj = 0 implies by ampleness ofD^j thatd_k·D^j = 0 and thusd_k= 0. This means that each genus 0 degreedmaximally tangent stable log map factors through

P^Gk

w^(k)_X

× Y

i6=k

P^Gi

w_X⁽ⁱ⁾ ,

and is trivial on the first component. By the log product formula [21, 29], the invariant reduces to the corresponding invariant ofQ

i6=kP^Gi

w⁽ⁱ⁾_X

. This moduli problem however is in positive virtual dimension and thus Rp^X_d =Rq^X_d = 0. For the remainder of this section, we therefore assume that d·D_j >0 for all 1≤j≤l_D.

Moving to the general case with 1 or 2 point insertions, recall that X = QrX

i=1P^Gi(w⁽ⁱ⁾) is given by the fan Σ ⊂ N_R where N ' Z^|nX| and N_R = N ⊗_Z R. Then Σ is the product fan of the fans Σ_i ⊂ (N_i)_R of P^Gi(w⁽ⁱ⁾), where N_i ' Zⁿⁱ. Writing ε_i := Pi−1

k=1(n_k + 1), the rays of Σ_i are [Dεi+1], . . . ,[Dεi+ni+1] and have primitive generators ∆(εi+ 1), . . . ,∆(εi+ni+ 1), which sat- isfy

w⁽ⁱ⁾_ε

i+1∆(ε_i+ 1) +· · ·+w⁽ⁱ⁾_ε

i+ni+1∆(ε_i+n_i+ 1) = 0.

Write L_i for the sublattice of N_i generated by the [∆(ε_i+j)] and writeB_i for the change of basis matrix from a Z-basis of N_i to a Z-basis ofL_i. Then

|detB_i|= N_i/L_i

= G_i

.

LetL be the sublattice of N generated by the Li and let B be the change of basis matrix fromN toL given by theB_i. We have that|detB|=Qr_X

i=1|detBⁱ|=Qr_X i=1|G_i|.

Proposition 5.1. The set T(p)^X_d has an unique element Γ of multiplicity 1.

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Proof. Each element Γ of T(p)^X_d has |n_X|+rX exterior markings (=rays) parallel to the rays [D1], . . . ,[D_|nX|+rX] and one vertex (=unique interior marking) with valency |n|+rX. Thus the only possibility is that Γ is the translate of the rays of the fan of X. Write ζ for one of the two generators of Λ^|nX|M. Then Mult(Γ) is given by the index of

|n_X|+r_X

Y

j=1

z^eXj(d)∆(j)=ζ ∈Λ^|nX|M

in Λ^|nX|M, which equals 1.

It follows from Proposition 5.1 and the correspondence result of [25] that Rp^X_d = 1.

We calculate the multiplicity of the element ofT(q)^X_d in three steps of increasing generality.

Proposition 5.2. Assume that X is the fake weighted projective plane P^G(w1,w2,w3), where we assumed thatgcd(w1,w2,w3) = 1. Then T(q)^X_d has an unique element of multiplicity|G|w1w2w3d². Proof. Fromw₁∆(1) +w₂∆(2) +w₃∆(3) = 0, it follows that |∆(1)∧∆(2)|= w₃|detB|. Choose the basis {∆(1),∆(2)} of N_R. In this basis, choose P₁ to be (1,0) and P₂ to be (0,1). Then the unique genus 0 degree d maximally tangent tropical curve passing through P1 and P2 consists of the rays [D1],[D2],[D3], meeting at 0 = (0,0), and with weights wjdon [Dj].

Choose 0 to be the sink vertex and let E1, resp. E2, be the edge connecting 0 withP1, resp. P2. Choose moreover{e₁, e2} to be a Z-basis ofM with dual basis {e^∗₁, e^∗₂}. Then

ζ_E1 =`<sub>2</sub>(ζ<sub>D1</sub>⊗ζ<sub>P1</sub>) =`₂(z^w1d∆(1)⊗(e^∗₁∧e^∗₂)) =z^w1d∆(1)ι_w1d∆(1)(e^∗₁∧e^∗₂)

=z^w1d∆(1) (ι_w1d∆(1)e^∗₁)∧e^∗₂−e^∗₁∧ι_w1d∆(1)(e^∗₂)

=z^w1d∆(1)(e^∗₁(w₁d∆(1))e^∗₂−e^∗₂(w₁d∆(1))e^∗₁). Similarly

ζ_E2 =z^w2d∆(2)(e^∗₁(w₂d∆(2))e^∗₂−e^∗₂(w₂d∆(2))e^∗₁) and

ζΓ=ζD3ζE1ζE2 =−e^∗₁(w1d∆(1))e^∗₂(w2d∆(2))e^∗₂∧e^∗₁−e^∗₂(w1d∆(1))e^∗₁(w2d∆(2))e^∗₁∧e^∗₂

=w₁w₂d²(e^∗₁(∆(1))e^∗₂(∆(2))−e^∗₂(∆(1))e^∗₁(∆(2)))e^∗₁∧e^∗₂

=w1w2d²|∆(1)∧∆(2)|e^∗₁∧e^∗₂=w1w2w3d²|detB|e^∗₁∧e^∗₂,

which is indeed of index |G|w₁w₂w₃d² in Λ²M.

Proposition 5.3. Assume that rX = 1, i.e. X=P^G(w1, . . . ,wn+1) and that n≥3. Then, for an appropriate choice of marked pointsP₁ andP₂, the set T(q)^X_d has a unique elementΓof multiplicity

|G|Qn+1 j=1wjdⁿ.

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Proof. We choose as basis ofN_Rthe basis{∆(1), . . . ,∆(n)}. We choose our second point (interior marking) P2 to have coordinate (a1, . . . , an) for ai < 0 and general. We choose our first marked point P₁ to have coordinate (b,0, . . . ,0) for b >0 large enough so that restricted to the halfspace {(x₁, . . . , xn)|x₁ > b}, anyh∈T(q)^X_d is affine linear with image (b,0, . . . ,0) +R>0∆(1) and weight e1(d).

For 1< j≤n, writew1j∆(1j) :=−w₁∆(1)−wj∆(j) with ∆(1j) primitive andw1j ∈N. Consider the finite abelian group

G^j := (h∆(1),∆(2)i_R∩N)/h∆(1),∆(2),∆(1j)i.

Given Γ ∈ T(q)^X_d, projecting to the plane h∆(1),∆(2)i_R leads to a genus 0 maximally tangent tropical curve in P^G

j(w1,wj,w1j) passing through 2 general points. By Proposition 5.2, there is only one such curve (and it has multiplicity|G^j|w₁w_jw_1jd²). These curves lift to a unique maximally tangent curveh: Γ→N_R.

Choose P₂ to be the sink vertex and consider the associated flow. Since the a_i are chosen to be general, on the set {(x_i)|x_i< ai},h is affine linear with slope parallel to ∆(n+ 1). We reorder the

∆(j) such that following the flow fromP2, the rays that are added to Γ are successively translates of [D_n],[Dn−1], . . . ,[D₂]. Note that all vertices are 3-valent sinceP₁ andP₂ are in general position.

Starting atP1 and following the flow, we label the compact edges successively E1, . . . , En. Choose a Z-basise1, . . . , en ofN. Then

ζE1 =`2(z^w1d∆(1)⊗e^∗₁∧ · · · ∧e^∗_n) =z^w1d∆(1)ι_w1d∆(1)(e^∗₁∧ · · · ∧e^∗_n).

At the next step,

ζ_E2 =z^{w1d∆(1)+w2d∆(2)}ι_{w1d∆(1)+w2d∆(2)}◦ι_w1d∆(1)(e^∗₁∧ · · · ∧e^∗_n)

=z^{w1d∆(1)+w2d∆(2)}ι_{(w1d∆(1)+w2}d∆(2))∧w₁d∆(1)(e^∗₁∧ · · · ∧e^∗_n)

=z^{w1d∆(1)+w2d∆(2)}ι_(w1w2d²∆(1)∧∆(2))(e^∗₁∧ · · · ∧e^∗_n).

Iterating this process, we obtain that

ζ_En=z^w1d∆(1)+···+wnd∆(n)ι_(w1···wndⁿ∆(1)∧···∧∆(n))(e^∗₁∧ · · · ∧e^∗_n).

Since w1∆(1) +· · ·+wn+1∆(n+ 1) = 0, |∆(1)∧ · · · ∧∆(n)|=wn+1|detB|and hence ζ_Γ=ι_(w1···wndⁿ∆(1)∧···∧∆(n))(e^∗₁∧ · · · ∧e^∗_n)e^∗₁∧ · · · ∧e^∗_n

=w₁· · ·w_ndⁿ|∆(1)∧ · · · ∧∆(n)|e^∗₁∧ · · · ∧e^∗_n

=w1· · ·wnwn+1dⁿ|detB|e^∗₁∧ · · · ∧e^∗_n.

which is indeed of index |G|w₁· · ·w_nw_n+1dⁿ in ΛⁿM. Proposition 5.4. Let X =QrX

i=1P^Gi(w⁽ⁱ⁾) be the product of fake weighted projective spaces. Then the set T(q)^X_d has a unique element Γ of multiplicityQr_X

i=1

G_i

Q

i,j w_X(i) j

d^nX.

14