Borel $(\alpha,\beta)$-multitransforms and Quantum Leray-Hirsch: integral representations of solutions of quantum differential equations for $\mathbb P^1$-bundles

In this paper, we address the integration problem of the isomonodromic system of quantum differential equations ($qDE$s) associated with the quantum cohomology of $\mathbb P^1$-bundles on Fano varieties. It is shown that bases of solutions of the $qDE$ of the total space of the $\mathbb P^1$-bundle can be reconstructed from the datum of bases of solutions of the corresponding $qDE$ associated with the base space. This represents a quantum analog of the classical Leray-Hirsch theorem in the context of the isomonodromic approach to quantum cohomology. The reconstruction procedure of the solutions can be performed in terms of some integral transforms, introduced in arXiv:2005.08262, called $Borel$ $(\alpha,\beta)$-$multitransf\!orms$. We emphasize the emergence, in the explicit integral formulas, of an interesting sequence of special functions (closely related to iterated partial derivatives of the B\"ohmer-Tricomi incomplete Gamma function) as integral kernels. Remarkably, these integral kernels have a universal feature, being independent of the specifically chosen $\mathbb P^1$-bundle. When applied to projective bundles on products of projective spaces, our results give Mellin-Barnes integral representations of solutions of $qDE$s. As an example, we show how to integrate the $qDE$ of blow-up of $\mathbb P^2$ at one point via Borel multitransforms of solutions of the $qDE$ of $\mathbb P^1$.

1. Introduction 1.1.Enumerative geometry is that branch of geometry that concerns the number of solutions to a given geometrical problem, rather than explicitly finding them all.In the last decades, ideas coming from physics brought innovation to enumerative geometry, with both new techniques and the emergence of new rich geometrical structures.As an example, Gromov-Witten theory, which focuses on counting numbers of curves on a target space, lead to the discovery of quantum cohomology and the closely related quantum differential equations.
Given a smooth complex projective variety X, its quantum cohomology QH • (X) is a family of commutative, associative, unital C-algebra structures on H • (X, C), obtained by deforming the classical cohomological cup product.Such deformation is performed by adding some "quantum correction terms", containing information on the number of rational curves on X. Namely, the structure constants of the quantum cohomology algebras can be expressed as third derivatives of a generating power series F X 0 (t), with t = (t1 , . . ., t n ) and n = dim C H • (X, C), of genus 0 Gromov-Witten invariants of X.Under the assumption of convergence of F X 0 (t) in some domain M ⊆ H • (X, C) ∼ = C n , the points t ∈ M can be used to label the quantum algebra structures on H • (X, C), the corresponding product being denoted by • t .This equips the quantum cohomology QH • (X) with an analytic Dubrovin-Frobenius structure, with M being the underlying complex manifold [Dub96,Man99,Her02,Sab08].
Points t ∈ M are parameters of isomonodromic deformations of the quantum differential equation (for short, qDE) of X.This is a system of linear differential equations of the form where ς is a z-dependent holomorphic vector field 1 on M , and U , µ are two endomorphisms of the holomorphic tangent bundle of M .The first operator U is the operator of •-multiplication by the Euler vector field, a distinguished vector field on M , obtained as perturbation of the constant vector field given by the first Chern class c 1 (X).The second operator µ, called grading operator, keeps track of the non-vanishing degrees of H • (X, C).
The qDE is a rich object associated with X.In the first instance, the Gromov-Witten theory of X can be reconstructed from the datum of the qDE (1.1) only.For details on a Riemann-Hilbert-Birkhoff approach to reconstruct the generating function F X 0 (t), and consequently the Dubrovin-Frobenius structure of QH • (X), see [Dub96,Dub99,Cot21a,Cot21b].In the second instance, the qDE of X encodes not only information about the enumerative (or symplectic) geometry of X, but also (conjecturally) about its topology and complex geometry.In order to disclose such a great amount of information is via the study of the asymptotics and the monodromy of its solutions, see [Dub98,GGI16,CDG18,Cot20]. The purpose of this paper is to construct new analytic tools, namely some integral representations of the solutions of qDE, which will be particularly convenient to the study of asymptotics, Stokes phenomena, and other analytical aspects.This will represent a continuation of the research direction started in [Cot22].
1.2.The projectivization of vector bundles is one of the most natural constructions of smooth projective varieties.Projective bundles have consequently been among the first varieties whose quantum cohomology algebras have been studied.See [QR98, CMR00, AM00].
The role played by the Gromov-Witten theory of projective bundles has recently been revealed as central not only in the context of open deep conjectures (such as the crepant transformation conjecture for ordinary flops [LLW16a,LLW16b,LLQW16]) but even for delicate foundational aspects of Gromov-Witten theory, such as its functoriality [LLW15].
The classical Leray-Hirsch theorem prescribes how to reconstruct the classical cohomology algebra H • (P, C) of a projective bundle P = P(V ) → X on a variety X, from the knowledge of the algebra H • (X, C) and the Chern roots c k (V ), k = 0, . . ., rk V .These data only, indeed, allow to write down an explicit presentation of H • (P, C).Several "quantum counterparts" of this theorem have been proved over the years.Many of the main results proved in [MP06, Ele05, Ele07, Bro14, LLW10, LLW15, LLW16a, LLW16b, Fan21] have a common thread: they allow to deduce information about the quantum cohomology (or, more generally, the Gromov-Witten theory) of a projective bundle P → X starting from information on the quantum cohomology of X. See Section 3.4 for a more detailed discussion.
Following the same philosophy, in this paper we address the following question: Q1.Is it possible to reconstruct a basis of solutions of the qDE of a projective bundle P → X from the datum of a basis of solutions of the qDE of X?
We obtain a positive result, under the assumption that P is a Fano split P 1 -bundle on X.In order to present the main results, we first briefly introduce some preliminary notions.
1.3.The first notion we want to introduce is that of master function.We call master function of X at p ∈ QH • (X) any C-valued function Φ ς , holomorphic on the universal cover C * of C * , of the form where ς is a solution of the qDE (1.1) specialized at t = p.
Rather than working directly with the space of solutions of the qDE, it is more convenient to focus on the space S p (X) of master functions of X at p ∈ QH • (X).More precisely, in addressing question Q1 above, we can split the problem into two parts: Q2(1).Is it possible to reconstruct the space of solutions of the qDE (specialized at a point p) from the datum of the space S p (X) of master functions only?Q2 (2).Is it possible to reconstruct the space of master functions S π * p (P ) from the datum of the space of master functions S p (X), where p ∈ QH • (X) and2 π : P → X? Question Q2(1) has been extensively studied in [Cot22].In loc.cit., it is shown that the answer is positive for generic p ∈ QH • (X).The problem of reducing a vector differential equation to a scalar one is well known in the theory of ordinary differential equations.Such a scalar reduction is equivalent to the choice of what is traditionally called a cyclic vector for the differential system [Del70, Lemma II.1.3].Moreover, several algorithmic reduction procedures have been developed, see e.g.[Bar93] and references therein.On any Dubrovin-Frobenius manifold M , we have a natural candidate for the cyclic vector, namely the unit vector of the Frobenius algebras at each point p ∈ M .It turns out that such a choice is working on the complement of an analytic subset of M , called A Λ -stratum [Cot22, Sec.2].More details will be given in Section 3.3.Consequently, question Q2(2) represents the main problem to be still addressed.
The second notion we want to recall is that of (analytic) Borel (α, β)-multitransforms, introduced in [Cot22].These are C-multilinear integral transforms of tuples of analytic functions.Given two h-tuples α, β ∈ (C * ) ×h , with h ⩾ 1, and an h-tuple of analytic functions where H is a Hankel-type contour of integration, originating from −∞, circling the origin once in the positive direction, and returning to −∞.See Figure 1.
The third and last object we need to introduce for presenting our main results is a sequence of special functions E k , with k ∈ N. Consider the function E(s, z), analytic and single valued on C × C * , defined by the integral Alternatively, the function E can be defined via the series expansion3 We define the functions E k ∈ O( C * ), k ∈ N, as the iterated partial derivatives For more explicit formulas for the E k -functions, see Section 3.6.
The function E is closely related to the upper and lower incomplete Gamma functions.These are the functions Γ(s, z) and γ(s, z) defined by the integrals The incomplete Gamma functions were first investigated, for real z, in 1811 by A.M. Legendre [Leg11, Vol. 1, pp. 339-343 and later works].The significance of the decomposition Γ(s) = γ(s, z) + Γ(s, z) was recognised by F.A. Prym in 1877 [Pry77], who seems to have been the first to investigate the functional behavior of these functions (that he denoted by P and Q, and which are often referred to as the Prym functions).The inconvenience of the function γ(s, z) is not only of having poles at s = 0, −1, −2, . . ., but even of being multivalued in z.Both inconveniences can be avoided by introducing, following P.E.Böhmer [Böh39] and F.G. Tricomi [Tri50], the normalized incomplete Gamma function γ * γ * (s, z) := z −s Γ(s) γ(s, z), which is entire on C 2 .
Our function E satisfies the identities  [Gau98].Curiously enough, however, in all these classical handbooks (and including [AS64], or the even more recent [OLBC10]) the higher order derivatives ∂ k s Γ(s, z), ∂ k s γ(s, z) and ∂ k s γ * (s, z) are not studied for k > 1.These exactly are those derivatives related to our functions E k .To the best of our knowledge, the first analysis of the functions ∂ k s Γ(s, z) was developed in [GGMS90, §4].
1.4.In this paper, we give explicit integral representations of master functions of P in terms of Borel (α, β)-multitransforms of master functions of the base space X under the following assumptions on X and P : (I) we assume that X is a product X = X 1 ×• • •×X h of smooth Fano projective varieties X i , (II) and that π : P = P(V ) → X is the projectivization of a split rank 2 vector bundle where L → X is the external tensor product of fractional powers of the determinant bundles det T X i .
Our first main result, Theorem 3.20, claims that any master function of P , at a point π * δ ∈ H 2 (P, C) of its small quantum cohomology, can be expressed in terms of Borel (α, β)multitransforms of master functions of X i at the point and det T X j = L ⊗ℓ j j , for ample line bundles L j → X j and 0 < d j < ℓ j , then any master function of P at π * δ is a C-linear combination of integral 4 According to W. Gautschi, Tricomi was fascinated by the incomplete Gamma functions, and he was fond of calling them affectionately the Cinderella functions, see [Gau98].
of the form where • α, β are the (h + 1)-tuples • Φ j is a master function of X j at the point δ j ∈ H 2 (X j , C), • and k = 0, . . ., dim C X + 1.
Remark 1.2.We emphasize a universal feature of the integral kernels E k , their emergence in formula (1.2) being independent of the specifically chosen P 1 -bundle P = P(V ) on X.
Our second main result concerns the more specific case of projective bundles on products of projective spaces.Consider the bundle P . ., h.Our Theorem 3.21 claims that the space S 0 (P ) of master functions of P at 0 ∈ H • (P, C) is a linear combination of integrals of the form where k = 0, 1, . . ., 1−h+ h i=1 n i , γ i are parabolas encircling the poles of the factors Γ(s i ) n i , the functions φ i j i (s i ) are defined by for any h-tuple j = (j 1 , . . ., j h ) with 0 ⩽ j i ⩽ n i − 1.
1.5.The paper is organized as follows.
In Section 2, we recall basic notions and results in Gromov-Witten theory, Frobenius manifolds and quantum cohomology.
In Section 3, we first introduce the quantum differential equation of a smooth projective variety and recall several results on the cyclic stratum and master functions from [Cot22].Subsequently, we overview several results in the literature on the quantum cohomology of projective bundles, focusing on quantum analogs of the classical Leray-Hirsch theorem.We also recall the definition of Borel (α, β)-multitransform in the analytic setting.We introduce the sequence of E k -functions and describe their series expansions and integral representations.Then, we formulate the main results of the paper.
In Section 4, after introducing the notion of topological-enumerative solution of the qDE and of the closely related J-function, we recall the statement of the Elezi-Brown theorem [Ele05,Ele07,Bro14].Consequently, we prove the main result of the paper, Theorem 3.20.
Finally, in Section 5 we exemplify our results on the specific case of the qDE of the blow-up of the projective plane at a point.We make explicit a base of solutions, obtained from a base of solutions for the qDE of P 1 via the Borel multitransforms.
In all the paper, unless otherwise stated, the Einstein summation rule over repeated Greek indices will be used.

Gromov
Given β ∈ H 2 (X, Z) tf , denote by M 0,k (X, β) the Kontsevich-Manin moduli stack of stable k-pointed rational maps of degree β and with target X.This stack parametrizes isomorphism classes of triples (C, x, f ) where (1) C is an algebraic curve of genus 0 with at most nodal singularities, (2) x = (x 1 , . . ., x k ) is a k-tuple of pairwise distinct points of the smooth locus of C, the group of automorphisms of (C, x, f ) is finite.
The moduli space M 0,k (X, β) is a proper Deligne-Mumford stack of virtual dimension The (genus 0) Gromov-Witten invariants of X, and their descendants, are rational numbers ⟨τ d 1 γ 1 , . . ., τ d k γ k ⟩ X 0,k,β ∈ Q defined via the intersection numbers of cycles on M 0,k (X, β).More precisely, they are defined by the integrals where, for each i = 1, . . ., k, we have See [BF97] for its construction.

2.3.
Mori and Kähler cones.The stack M 0,k (X, β) is non-empty only if β is an element of the Mori cone of effective curves.This is the semigroup NE(X) of classes in the lattice H 2 (X, Z) tf representable by algebraic curves, i.e. the classes we have a distinguished open convex cone, the Kähler cone of X, denoted by K X .This is defined as the set of all cohomology classes [ω] ∈ H 1,1 (X, C) associated with any Kähler form ω on X.
The Mori and Kähler cones are dual to each other, in the following sense.

By extension of scalars, we have an inclusion
We will identify H 2 (X, Z) tf with its image along i.In particular, we have It is standard to denote by Nef(X) the dual cone of the closure NE(X) R , and to call it the cone of nef divisors.In this notation, the statement of Proposition 2.2 becomes Nef(X) = K X .We recommend the monograph [Laz04] for a comprehensive account of the general theory of ample, nef, effective cones on smooth projective varieties.
2.4.Novikov ring, and Gromov-Witten potentials.Fix a Kähler form ω on X.We define the Novikov ring Λ X,ω to be the ring of formal power series, in an indeterminate Q, of the form It is easy to see that the product of two such a series is well defined.In simple terms, Λ X,ω can be intended as an "upward" completion of the group ring C[H 2 (X, Z) tf ], by allowing sums with infinite terms in the direction of increasing values of the functional β → β ω.
In what follows, it is convenient to associate with X also a big phase space P X .This consists of an infinite product of countably many copies of H • (X, C), that is P X := H • (X, C) N .We will identify H • (X, C) with the 0-th factor of P X , called the small phase space.Correspondingly to the choice of the base (T 0 , . . ., T n ), we denote by6 (τ k T 0 , . . ., τ k T n ) the basis of the k-th copy of H • (X, C) in P X .The coordinates of a point γ ∈ P X with respect to (τ k T α ) α,k will be denoted by t • = t α,k α,k .Hence, we will identify t α ≡ t α,0 for α = 0, . . ., n.Instead of denoting by γ = (t α,k τ k T α ) α,k a generic point of P X we will write this as a formal series The restriction of F X 0 to the small phase space (i.e. by setting t α,0 = t α and t α,p = 0 for p > 0) defines the (genus 0) Gromov-Witten potential of X, Remark 2.3.The above formulas define two elements of 2.5.Quantum cohomology as Λ X,ω -formal Frobenius manifold.Consider • the finite rank free Λ X,ω -module By extension of scalars, the C-basis (T 0 , . . ., T n ) of H • (X, C) induces a basis of the module H.The algebra K can be then identified with the formal power series ring In what follows we set The formal spectrum M := Spf(K) is the formal scheme structure supported on a point (the "origin") with structure sheaf of formal functions Γ(M, O M ) ∼ = K, and with space of formal vector fields The Gromov-Witten potential F X 0 can be thus interpreted as a formal function on M .Theorem 2.4.[KM94,Man99] The function F X 0 ∈ O M satisfies the following properties.
(1) It is a quasi-homogeneous function.More precisely, if E ∈ T M denotes the formal vector field on The vector field E of equation ( 2.3) will be called Euler vector field.
Define the K-linear product • of formal vector fields on M via the formulas The product • is associative, by WDVV equations (2.4), commutative, and compatible with the Poincaré pairing, as follows Moreover, the vector field e := T 0 is the unit for the •-product.This makes (H K , •, η, e) a Frobenius algebra.
The datum of (M, F X 0 , η, e, E) defines a formal Frobenius manifold structure over the Novikov ring Λ X,ω , called quantum cohomology of X. See [Man99].
2.6.Quantum cohomology as C-formal Frobenius manifold.If we fix an arbitrary basis (β 1 , . . ., β r ) of H 2 (X, Z) tf , we can identify the Novikov ring Λ X,ω with the ring of formal power series, in r indeterminates Q 1 , . . ., Q r (where Series as such are called generalized Laurent series in [HS95]. Denote by B X the set of points q ∈ (C * ) r such that for any k ⩾ 3, and any α 1 , . . ., α k ∈ {0, . . ., n}, the generalized Laurent series Remark 2.5.By multi-linearity of Gromov-Witten invariants, the sum (2.6) is convergent if and only if all the series are convergent at Q = q for any k ⩾ 3, any α 1 , . . ., α k ∈ {0, . . ., n}, and any choice of Assumption A: The set B X is non-empty.
Proposition 2.6.Assumption A holds if and only if By the divisor axiom of Gromov-Witten invariants, we have7 Hence, q 1 ∈ B X if and only if q 0 ∈ B X .□ Proposition 2.7.If X is Fano, each series (2.6) is finite.Hence, Assumption A holds true. Proof.
2).As a consequence of Kleiman ampleness criterion [KM98, Cor.1.19], there exist only finitely many tuples d ∈ Z r for which the dimensional constraint holds.□ Whenever Assumption A holds, the specialization F X 0 | Q=q , with q ∈ B X , is a well-defined formal power series with complex coefficients.A formal Frobenius manifold over C, E , is defined for each q ∈ B X , by specializing the Gromov-Witten potential F X 0 at Q = q.We call such a formal Frobenius manifold quantum cohomology of X specialized at Q = q.2.7.Quantum cohomology as Dubrovin-Frobenius manifold.In this paper we will consider the case of convergent Gromov-Witten potentials only.This will allow us to promote the formal Frobenius manifold structures to analytic ones.Consider a variety X for which Assumption A holds, so that Proof.Let q 0 , q 1 ∈ B X , and set F 0 , F 1 ∈ C[[t]] be the formal power series defined by the specializations The same computations in the proof of Proposition 2.6, invoking the divisor axiom, show that F (1) is a derivative of F (0) .More precisely, if V i denotes the vector field corresponding to the cohomology class log The following result can be useful to deduce the validity of Assumption B. Theorem 2.9.[Cot21a,Cot21b] Let Assumption A hold.Assume that the quantum cohomology of X specialized at some point Under the validity Assumption B, the series F X 0 | Q=q have the same domain of convergence M ⊆ C n+1 for any q ∈ (C * ) r .Without loss of generality, in what follows we will consider the specialization of F X 0 at Q = 1 = (1, 1, . . ., 1).
Let T M (resp.T * M ) the holomorphic tangent (resp.cotangent) bundles of M .At each p ∈ M , we have a canonical identification of vector spaces T p M ∼ = H • (X, C), via the map ∂ ∂t α → T α , for α = 0, . . ., n. Via this identification, the Poincaré metric defines a holomorphic section η ∈ Γ 2 T * M .This is a symmetric non-degenerate O M -bilinear 2-form, for simplicity called metric, with flat Levi-Civita connection ∇.
together with the two holomorphic vector fields e, E ∈ Γ(T M ) defined by the constant field e = T 0 ∈ H • (X, C) and equation (2.3), respectively.
The product defined in equation (2.5) is now convergent: each tangent space T p M is equipped with a Frobenius algebra structure, holomorphically depending on the point p.The tangent vector e p equals the unit of the Frobenius algebra at p ∈ M .For this reason e is called unit vector field.The Euler vector field E satisfies the following identities: The datum of (M, η, c, e, E) defines a Dubrovin-Frobenius manifold structure [Dub96,Dub99].
The operator U ∈ Γ(End T M ) is the tensor defined by We will denote by µ and U, respectively, the matrices of components of the tensors µ and U in the ∇-flat coordinates t.
Lemma 3.2.For any vector fields v 1 , v 2 ∈ Γ(T M ), we have Proof.The result directly follows from the definitions.□ Consider a punctured complex line C * , with global coordinate z, and introduce the extended space M := C * × M .Given the canonical projection π : M → M , we can consider the pull-back bundle π * T M .All the tensors η, c, E, µ, U can be lifted to π * T M , and their lifts will be denoted by the same symbols.Moreover, the Levi-Civita connection ∇ can be uniquely lifted on π * T M in such a way that where π −1 T M the sheaf of sections of π * T M constant on the fibers of π.
3.3.Cyclic stratum, A Λ -stratum, master functions.For any k ∈ N, introduce a vector field e k ∈ Γ(π * T M ), defined by the iterated covariant derivative (3.9) Definition 3.6.The cyclic stratum M cyc is defined to be the maximal open subset U of M such that the bundle π * T M | U is trivial and the collection of sections (e k | U ) n k=0 defines a basis of each fiber.On M cyc we will also introduce the dual coframe (ω j ) n j=0 , by imposing ⟨ω j , e k ⟩ = δ jk . (3.10) The frame (e k ) n k=0 will be called cyclic frame, and its dual (ω j ) n j=0 cyclic coframe.Definition 3.7.The Λ-matrix is the matrix-valued function Λ = (Λ iα (z, p)), holomorphic on M cyc , defined by the equation Λ iα e i , α = 0, . . ., n. (3.11) , where A 0 , . . ., A ( n 2 ) are holomorphic functions on M .Moreover, if n > 1 and if the eigenvalues of the grading operator µ are not pairwise distinct, then the function A ( n 2 ) is identically zero. □ the function det Λ takes the form Definition 3.9.We define the set A Λ ⊆ M to be the set Consider now the system (3.7),(3.8).We have the following result.
Theorem 3.10.[Cot22, Th. 2.29] The matrix differential equation (3.8), specialized at a point p ∈ M \ A Λ , can be reduced to a single scalar differential equation of order n + 1 in the unknown function ϖ 0 .The scalar differential equation admits at most n 2 apparent singularities.□ The scalar differential equation above will be called the master differential equation of X.
We can more explicitly describe the master differential equation.At points (z, p) ∈ M cyc , introduce the column vector ϖ defined by where Λ is the matrix defined as in (3.11).The entries of ϖ are the components ϖ j of the ∇-flat covector ϖ with respect to the cyclic coframe (ω j ) n j=0 .The vector ϖ satisfies the differential system Theorem 3.11.[Cot22, Cor.2.27] The system of differential equations (3.13) is the companion system of the master differential equation of X. □ Remark 3.12.Since e 0 = e, we have ϖ 0 = ϖ 0 .
Definition 3.13.Fix p ∈ M , consider the system of differential equations (3.8) specialized at p, and let X p be the C-vector space of its solutions.Let ν p : X p → O C * be the morphism defined by 3.4.Projective bundles: classical and quantum aspects.Let X be a smooth projective variety over C, and V → X a holomorphic vector bundle on X of rank r + 1.For short denote by P the total space of the projective bundle10 π : P(V ) → X, and set ξ := c 1 (O P (1)) , where O P (−1) is the tautological line bundle on P.
The classical cohomology H • (P, C) is an algebra over H • (X, C) via pullback.More precisely, by the classical Leray-Hirsch Theorem, the pull-back map π * : H • (X, C) → H • (P, C) is a monomorphism of rings, and via this map the algebra H • (P, C) admits the following presentation Moreover, the C-linear map H • (X, C) ⊕(r+1) → H • (P, C) defined by (α 0 , . . ., α r ) → i ξ i π * α i is an isomorphism of C-vector spaces, so that Remark 3.14.Two projective bundles P(V ) and P(V ′ ) are isomorphic as X-schemes if and only if V ′ ∼ = V ⊗ L for some line bundle L → X.In such a case, we have O P ′ (−1) ∼ = π * L ⊗ O P (−1), so that ξ ′ = ξ − c 1 (L).See [EH16, Prop.11.3].Consequently, in the case of a split bundle V = r j=0 L j , without loss of generality we can assume that L 0 = O X .
We say that V → X is a Fano bundle if the projectivization P = P(V ) is a Fano manifold.The existence of a Fano bundle of X automatically implies that X itself is Fano [SW90a, Th. 1.6].Fano bundles have been extensively studied, and even completely classified in several cases: (1) In [SW90a, SW90b, SW90c, SSW91] a complete classification of rank two Fano bundles up to dimension 3 is given.(2) In [APW94] it is shown that a rank two Fano bundle on P n , with n ⩾ 4, and a quadric Q n , with n ⩾ 6, splits into a direct sum of line bundles.On Q 4 and Q 5 there are some exceptions: two spinor bundles and a 7-dimensional family of stable bundles on Q 4 , and Cayley bundles on Q 5 [Ott90].
(3) In [MOS12a,MOS12b], rank two Fano bundles on projective Grassmannians G(1, n), with n ⩾ 4, have been classified.It is shown that the only non-split rank two Fano bundles are, up to twists, the universal quotient bundles Q → G(1, n).Subsequently, in [MOS14], a classification of rank two Fano bundles on manifolds X with If E → X is a complex vector bundle, let c(E) be the total Chern class c(E) = j⩾0 c j (E).
The next result shows how to compute the whole Chern class c(P ) in terms of c(X), c(V ), and ξ.
Lemma 3.15.We have c(P ) = c(π * V ⊗ O P (1))π * c(X).In particular, we have Proof.The first claim follows from the relative Euler exact sequence and the exact sequence 0 → T P/X → T P → π * T X → 0. The k-th Chern class of π * V ⊗ O P (1) can be computed by the formula In [MP06], D. Maulik and R. Pandharipande developed a quantum Leray-Hirsch theorem for P 1 -bundles P = P(O X ⊕ L) over a smooth base X.In particular, they showed that the Gromov-Witten theory of P can be uniquely and effectively reconstructed from the Gromov-Witten theory of X and the Chern class c 1 (L).

Contemporarily, in [Ele05]
A. Elezi formulated a conjectural quantum Leray-Hirsch result, which allows the reconstruction of the small J-function of P starting from the J-function of X, in the case P is the projectivization of a split Fano bundle on X.The validity of this conjecture was first proved in several cases in [Ele07].Subsequently, in [Bro14] J. Brown proved Elezi conjecture as a special case of a more general result, allowing the reconstruction of the J-function of the total space of a toric fibration F → X (constructed from a split vector bundle E → X) from the J-function of the base X.
The result of Brown was the starting point for a big project of Y.-P.Lee, H.-W. Lin and C.-L. Wang, developed in the series of papers [LLW10,LLW16a,LLW16b].The original Brown's theorem was formulated in terms of Givental's Lagrangian cones formalism [Giv04,CG07].Lee, Lin and Wang first showed how to explicitly reconstruct the J-function of a toric fibration in terms of generalized mirror transformations and Birkhoff factorizations.Then, they used these results to show the invariance of quantum cohomology under ordinary flops.Moreover, in a subsequent paper [LLQW16] joint with W. Qu, they showed how it is possible to remove any splitting assumption in the quantum Leray-Hirsch theorem (quantum splitting principle).
In [Fan21], H. Fan recently proved that the Gromov-Witten theory of a projective bundle P(V ) → X is uniquely determined by the Gromov-Witten theory of X and the Chern class c(V ).
In the very recent preprint [Kot22], it is shown that if Assumption B holds for the quantum cohomology of X, then the same holds for the quantum cohomology of P .In particular, its quantum cohomology can be equipped with a Dubrovin-Frobenius manifold structure.

3.6.
The functions E k .The Böhmer-Tricomi normalized Gamma function γ * is the entire function on C 2 , defined by the integral This function was originally introduced by P.E.Böhmer in [Böh39, pp.124-125], and subsequently studied in [Tri50].It is closely related to the upper and lower incomplete Gamma functions Γ and γ, introduced in 1877 by F.E. Prym [Pry77].These are the holomorphic functions defined on C × C * whose general values are respectively defined by without restrictions on the integration paths.We have For general properties of the incomplete Gamma functions, see e. Let E be the holomorphic function on C × C * defined by Proposition 3.17.For any (s, z) ∈ C × C * , we have • (β n ) n∈N is the sequence of real numbers defined by • G m,n p,q z a 1 , . . ., a p b 1 , . . ., b q denotes the Meijer G-function defined by the integral for a suitable integration path L defined in [Luk69, pag.144 (3)] and separating the poles of n i=1 Γ(1 − a i + s) from the poles of m i=1 Γ(b i − s), • T (m, z), with m ∈ N * , is the specialization where L surrounds the multipole at t = −1 and the simple poles at We have the following explicit formula for E k .
Proposition 3.19.For any j ∈ N, we have . By results of [GGMS90, §4], we also have Hence, we have and the result follows.□ Example.The first elements of the sequence 3.7.Main theorems.Let X 1 , . . ., X h , with h ⩾ 1, be Fano smooth projective complex varieties.Assume that det T X j = L ⊗ℓ j j , with ℓ j ∈ N * , for ample line bundles L j → X.
Let δ j ∈ H 2 (X j , C), with j = 1, . . ., h, and let be the corresponding point of H 2 (X, C), under Künneth isomorphism.Denote by S δ j (X j ) and S π * δ (P ) the corresponding space of master functions.
Theorem 3.20.The space S π * δ (P ) is contained in the finite dimensional C-vector space generated by the images of the maps B α,β,k : where and k = 0, . . ., dim C X + 1.In other words, every element of S π * δ (P ) is a finite sum of integrals of the form and Φ j ∈ S δ j (X j ).
We can make the result more explicit, in the case of projective bundles over product of projective spaces [QR98, CMR00, AM00, Str15].
Theorem 3.21.Let h ⩾ 1, and P be the projective bundle with 0 < d i < n i for any i = 1, . . ., h.Any master function in S 0 (P ) is a C-linear combination of integrals of the form for k = 0, 1, . . ., 1 − h + h i=1 n i , and j = (j 1 , . . ., j h ) ∈ h i=1 {0, . . ., n i − 1}.The paths γ i , with i = 1, . . ., h, are parabolas of the form Re s i = −ρ 1,i (Im s i ) 2 + ρ 2,i , for suitable ρ 1,i , ρ 2,i ∈ R + , so that they encircle the poles of the factors Γ(s i ) n i .The function φ i j i is defined as follows • for n i even: • for n i odd: Proof.The result follows by applying Theorem 3.20 to the case X i = P n i −1 , ℓ i = n i .For each factor P n i −1 the space S 0 (P n i −1 ) of master functions equals the space of solutions g of the differential equation So, a basis of the space S 0 (P n i −1 ) is given by the integrals where k = 0, . . ., dim C X + 1, and j i = 0, . . ., n i − 1 with i = 1, . . ., h. □ 4. Proof of the main theorem 4.1.Topological-enumerative solution, and J-function.
Definition 4.1.Define the functions θ β,p (z, t), θ β (z, t), with β = 0, . . ., n and p ∈ N, by Define the matrix Θ(z, t) by Let R be the matrix associated with the C-linear morphism with respect to the basis (T 0 , . . ., T n ), and define the matrix Z top (z, t) by ]).The matrix Z top (z, t) is a fundamental system of solutions of the joint system (3.5)-(3.6).□ Definition 4.3.The solution Z top (z, t) is called topological-enumerative solution of the joint system (3.5),(3.6).
Let ℏ be an indeterminate. .
The restriction of the J-function to the small quantum locus, i.e. to points τ ∈ H 2 (X, C), has a simpler expansion.
Set P := P(V ).Let s i : X → P , with i = 0, . . ., r, be the section of π : P → X determined by the i-th summand of V .The following lemma describes the Mori cone of P .Lemma 4.7 ([Ele05, Lem.1.0.1]).Assume that each line bundle (2) The Mori cone of X and P are related by where [ℓ] is the class of a line in the fiber of π.Here NE(X) is embedded in NE(P ) via the section s 0 .□ Consider the small J-function of X: by Lemma 4.5, it is of the form for suitable coefficients J X β (δ).For each β ∈ H 2 (X, Z) and each ν ∈ N, introduce the twisting factor and define the small I-function (defined on H 2 (P, C) ∼ = H 2 (X, C) ⊕ Cξ) as the hypergeometric modification where the sum β + ν denotes the element β + ν[ℓ] of NE(P ), in the notations of Lemma 4.7.
The following result was first conjectured, and proved in several cases, by A. Elezi [Ele05,Ele07], and finally proved in full generality by J. Brown [Bro14].
(4.5) □ Remark 4.9.The result proved by J. Brown in [Bro14] is actually more general, since it concerns the more general case of a toric fiber bundle on X rather than a projective bundle.Its formulation, however, is less explicit than the equality (4.5).Namely, Brown's result is stated only in terms of the A. Givental's formalism of Lagrangian cones [Giv04,CG07].If x ∈ M A,κ , denote by x ′ its "nilpotent part", i.e. the projection of x onto Nil(A).
We have two maps ν κ : M A,κ → N h and ι κ : M A,κ → A defined by By universal property of the direct sums of monoids, the natural inclusions M A,κ i → M A,κ induce a unique morphism On M A,κ we can define the partial order x ⩽ y iff x ′ = y ′ and ν κ (x) ⩽ ν κ (y), the order on N h being the lexicographical one.This order makes (M A,κ , ⩽) a strictly ordered monoid, that is if a, b ∈ M A,κ are such that a < b, then a + c < b + c for all c ∈ M A,κ .
Define F κ (A) to be the set of all functions f : M A,κ → A whose support is (1) Artinian, i.e. every subset of supp(f ) admits a minimal element, (2) and narrow, i.e. every subset of supp(f ) of pairwise incomparable elements is finite.
The set F κ (A) is an A-module with respect to pointwise addition and multiplication of A-scalars.We will denote the element f ∈ F κ (A) by where Z is an indeterminate.Given f 1 , f 2 ∈ F κ (A), define where we set The following result is a consequence of P. Ribenboim's theory of generalized power series [Rib92,Rib94].
Theorem 4.10.The product above is well-defined.The set F κ (A) is equipped with an A-algebra structure with respect to the operations above.□ Definition 4.11.Let r o ∈ Nil(A).We say that an element In what follows we will usually take F (x) = Γ(λ+x) with λ ∈ C\Z ⩽0 , or F (x) = 1/Γ(λ+x) with11 λ ∈ C, where Γ denotes the Euler Gamma function.
Definition 4.13.Let α, β, κ ∈ (C * ) h .We define the Borel (α, β)-multitransform as the A-linear morphism which is defined, on decomposable elements, by Let s = ((κ i n i 1 A ) h i=1 , r) ∈ M A,κ .We define the analytification Z s of the monomial Z s ∈ F κ (A) to be the A-valued holomorphic function Notice that the sum is finite, since r ∈ Nil(A).
Let f ∈ F κ (A) be a series The analytification f of f is the A-valued holomorphic function defined, if the series absolutely converges, by • the functions f i are well defined on C \ R <0 .
We have provided that both sides are well-defined.□ In the remaining part of the paper, we will consider the Ribenboim algebras F κ (X) := F κ (H • (X, C)) where X is a smooth projective variety such that H odd (X, C) = 0.This is necessary in order to work with commutative cohomological algebras.4.5.The Ribenboim E-series of a projective bundle.Given a projective bundle P → X as in Section 3.4, we associate to P a distinguished element of F 1 (P ) as follows.
By Künneth isomorphism, and by the universal property of coproduct of algebras (i.e.tensor product), we have injective12 maps H • (X i , C) → H • (X, C).In order to ease the computations, in the next formulas we will not distinguish an element of H • (X i , C) with its image in H • (X, C).The same will be applied to elements of H 2 (X, Z).So, for example, we have where the last identity follows from Lemma 3.15.
The space of master functions S δ (X) is generated by the components (with respect an arbitrary basis of H • (X, C)) of the small J-function , by the R. Kaufmann's quantum Künneth formula [Kau96].By Lemma 4.5, for each j = 1, . . ., h, we have where (T 0,j , . . ., T n j ,j ) is a fixed basis of H • (X j , C), T α j := n j λ=0 η αλ j T α,j , and η j is the Poincaré metric on H • (X j , C).So, we can rewrite the J-function J where the coefficient J j k j (δ j ) equals Analogously, the space of master functions S π * δ (P ) is spanned by the components of the H • (P, C)-valued function J P (π * δ + c 1 (P ) log z)| Q=1
By Elezi-Brown Theorem 4.8, we have On the one hand, each factor in the tensor product (4.6)equals the analytification J X j of the series J X j ∈ F ℓ j (X) defined by On the other hand, consider the Borel multitransform B α,β [π * J X 1 , . . ., π * J X h , E P ] for arbitrary tuples α = (α 1 , . . ., α h+1 ) and β = (β 1 , . . ., β h+1 ).We have , by definitions of the formal Borel multitransform and of the E-series.Then, for the choice of weights we have Then, the space S π * δ (P ) is spanned by the components, with respect to an arbitrary ba- . This follows from the invertibility of the morphism The statement of Theorem 3.20 then follows from Theorem 4.14 and Proposition 4.16.
5. An example: blowing-up a point in P 2 5.1.Classical and quantum cohomology.Let P be the blow-up of a point in P 2 .The variety P admits a P 1 -bundle structure on P 1 , namely .
By looking at P as the compactification of the total space of O(−1) → P 1 , we can introduce two cycles ε, ν ∈ H 2 (P, Z) defined as follows: • ε represents the homology class of the section "at infinity" of O(−1), • ν represents the homology class of a fiber of O(−1).
The classical cohomology algebra admits the following presentation , and T 1 T 2 = T 3 .
For shortening the following formulas, set For j = 0, we have where in the last equality we used the identity Similarly, we have Hence, after some computations, one can explicitly check that the functions in (5.4) solve the differential equation (5.2).This can be done by expressing the coefficients C k z in terms of values of polygamma functions ψ (k) , namely B k −ψ (0) (z), −ψ (1) (z), −ψ (2) (z), . . ., −ψ (k−1) (z) , where B k is the k-th Bell polynomial as in (3.16), and by invoking the following well-known identities (see [OLBC10, Ch. 5]): The explicit formulas of Theorem 5.1 open the possibility to the study of analytical properties of the solutions of the qDE (5.1), such as their asymptotics, Stokes phenomenon, etc.As shown in [Cot22], these analytical properties remarkably encode information not only about the topology, but even about the algebraic geometry of the surface P : the entries of the connection and Stokes matrices can be related to explicit characteristic classes of P and of objects of exceptional collections in the derived category D b (P ), see [Cot22,Th. 11.8.3].
We underline that the basis of solutions of the qDE (5.1) considered in [Cot22] was constructed via Laplace (α, β)-multitransforms, by realizing P as an hypersurface in P 1 × P 2 .Notice that the computations involved in the proof of Theorem 5.1 are much faster then those performed in [Cot22, Ch. 11, App.B].

3.
QDEs, Borel multitransforms, E k functions, and main results 3.1.Extended deformed connection.Consider the Dubrovin-Frobenius manifold M defined in the previous section.Let us introduce two distinguished holomorphic (1, 1)-tensors on M .Definition 3.1.The grading operator µ ∈ Γ(End T M ) is the tensor defined by

2
z) = −γ EM − log z + (z) + 12γ EM log(z) + 6γ 2 EM + π 2 + ∞ ℓ=1 (−1) ℓ ℓ!ℓ 2 z ℓ ,we deduce The classical theory of the incomplete Gamma functions (including series expansions of various kinds, asymptotics expansions, differentiation and recurrence relations, continued fractions, integral representations, etc.) can be found in [Nie06, Kap.II, XV,XXI] [Böh39, Kap.V].This research was boostered after the late 1940s, when Tricomi recognised the importance of these functions 4 , revitalized their study, and gave important contributions of his own.Tricomi himself summarized the knowledge as of 1950s in the second volume of the Bateman Project [Erd53, Ch.IX, pp.133-151], and gave an even more detailed exposition in his monograph [Tri54, § § 4.1-4.6].For a beautiful overview of recent results on the incomplete Gamma functions since Tricomi see