Double nested Hilbert schemes and the local stable pairs theory of curves

We propose a variation of the classical Hilbert scheme of points - the double nested Hilbert scheme of points - which parametrizes flags of zero-dimensional subschemes whose nesting is dictated by a Young diagram. Over a smooth quasi-projective curve, we compute the generating series of topological Euler characteristic of these spaces, by exploiting the combinatorics of reversed plane partitions. Moreover, we realize this moduli space as the zero locus of a section of a vector bundle over a smooth ambient space, which therefore admits a virtual fundamental class. We apply this construction to the stable pair theory of a local curve, that is the total space of the direct sum of two line bundles over a curve. We show that the invariants localize to virtual intersection numbers on double nested Hilbert scheme of points on the curve, and that the localized contributions to the invariants are controlled by three universal series for every Young diagram, which can be explicitly determined after the anti-diagonal restriction of the equivariant parameters. Under the anti-diagonal restriction, the invariants are matched with the Gromov-Witten invariants of local curves of Bryan-Pandharipande, as predicted by the MNOP correspondence. Finally, we discuss $K$-theoretic refinements \`a la Nekrasov-Okounkov.

1. Introduction 1.1. Double nested Hilbert scheme of points. Let X be a quasi-projective scheme over C. We denote by X [n] the Hilbert scheme of n points on X, which parametrizes 0-dimensional closed subschemes Z ⊂ X of length n. Given a tuple of non-decreasing integers n = (n 0 ≤ · · · ≤ n d ), the nested Hilbert scheme of points X [n] parametrizes flags of zero-dimensional subschemes (Z 0 ⊂ · · · ⊂ Z d ) of X, where each Z i has length n i . The scheme structure of these moduli spaces has been intensively studied in the literature, see for example [13]. We propose a variation of this moduli space, by parametrizing flags of subschemes nesting in two directions. Let λ be a Young diagram and n λ = (n ) ∈λ a reversed plane partition, that is a labelling of λ by non-negative integers non-decreasing in rows and columns. We denote by X [n λ ] the double nested Hilbert scheme of points, the moduli space parametrizing flags of 0-dimensional closed subschemes (Z ) ∈λ  where each Z has length n . If λ is a horizontal or vertical Young diagram, the nesting is linear and we recover the usual nested Hilbert scheme of points. The scheme structure of these moduli spaces is interesting already in dimension one, for a smooth curve C. Cheah proved [13] that the nested Hilbert scheme C [n] is smooth, being isomorphic to 1 a product of symmetric powers of C via a Hilbert-Chow type morphism. However, as soon as we allow double nestings, C [n λ ] can have several irreducible components (see Example 2.6), therefore failing to be smooth. Our first result is a closed formula for the generating series of topological Euler characteristic of C [n λ ] in terms of the hook-lengths h( ) of λ. This is achieved by exploiting the power structure on the Grothendieck ring of varieties K 0 (Var C ), by which we reduce to the combinatorial problem of counting the number of reversed plane partitions of a given Young diagram, which was solved by Stanley and Hillman-Grassl [56,22]. Motivic analogues of this formula are studied in [34].
1.2. Virtual fundamental class. The double nested Hilbert scheme C [n λ ] is in general singular, making it hard to perform intersection theory. To remedy this, we show that C [n λ ] admits a perfect obstruction theory in the sense of Behrend-Fantechi and Li-Tian [5,28]. In fact, we can (globally!) realize C [n λ ] as the zero locus of a section of a vector bundle over a smooth ambient space. Theorem 1.2 (Theorem 2.8). Let C be an irreducible smooth quasi-projective curve. There exists a section s of a vector bundle E over a smooth scheme A C,n λ such that E C [n λ ] ∼ = Z(s) A C,n λ .
By this construction C [n λ ] naturally admits a perfect obstruction theory (see Example 2.7) and in particular carries a virtual fundamental class [C [n λ ] ] vir , which recovers the usual fundamental class in the case where the nesting is linear. We pause a moment to explain this construction in the easiest interesting example, that is for the reversed plane partition n 00 n 01 n 10 n 11 The embedding in the smooth ambient space is (Z 00 , Z 01 , Z 10 , Z 11 ) → (Z 00 , Z 10 − Z 00 , Z 11 − Z 01 , Z 01 − Z 00 , Z 11 − Z 10 ).
In other words, A C,n λ records the subscheme in position (0, 0) and all possible vertical and horizontal differences of subschemes, where sum and difference are well-defined by seeing the closed subschemes Z ij as divisors on C. At the level of closed points, the image of the embedding is given by all (Z 00 , X 1 , X 2 , Y 1 , Y 2 ) ∈ A C,n λ such that X 1 + Y 2 = Y 1 + X 2 -again, as divisors. Notice that X 1 + Y 2 and Y 1 + X 2 are effective divisors of the same degree, therefore they are equal if and only if one is contained into the other, say X 1 + Y 2 ⊂ Y 1 + X 2 . This relation is encoded into a section of a vector bundle E, as we now explain. Denote by X 1 , X 2 , Y 1 , Y 2 the universal divisors on A C,n λ × C and set Γ 1 = Y 1 + X 2 , The vector bundle E is defined as where π : A C,n λ × C → A C,n λ is the projection. The section s of E is the one induced -via π *by the section of O AC,n λ ×C (Γ 1 ) which vanishes on Γ 1 and then restricted to Γ 2 .
1.3. Stable pair invariants of local curves. Let C be a smooth projective curve and L 1 , L 2 two line bundles over C. We denote by local curve the total space X = Tot C (L 1 ⊕ L 2 ) with its natural T = (C * ) 2 -action on the fibers. For d > 0 and n ∈ Z, we denote by P X = P n (X, d[C]) the moduli space of stable pairs [O X s − → F ] ∈ D b (X) with curve class d[C] and χ(F ) = n. The moduli space P X has a perfect obstruction theory [50], but is in general non-proper. Still, the T-action on X induces one on P X with proper T-fixed locus P T X , therefore we can define invariants via Graber-Pandharipande virtual localization [19] PT d,n (X) := where s 1 , s 2 are the generators of the T-equivariant cohomology and N vir is the virtual normal bundle. We denote its generating series by PT d (X; q) := n∈Z q n · PT d,n (X) ∈ Q(s 1 , s 2 )((q)).
Pandharipande-Pixton extensively studied stable pair theory on local curves [48,47] using degeneration techniques and relative invariants, focusing on the rationality of the generating series, including the case of descendent insertions. The novelty of this paper is the different approach which only relies on the Graber-Pandharipande localization -without degenerating the curve C -and the virtual structure constructed on the double nested Hilbert schemes C [n λ ] . This is in particular useful to address the K-theoretic generalizations of stable pair invariants (cf. Section 1.8).
Our main result is that the generating series PT d (X; q) of such invariants is controlled by some universal series and determine them under the anti-diagonal restriction s 1 + s 2 = 0. There are universal series A λ (q), B λ (q), C λ (q) ∈ Q(s 1 , s 2 ) q such that where λ is the conjugate partition of λ, n(λ) = l(λ) i=0 i · λ i and g = g(C). Moreover, under the anti-diagonal restriction s 1 + s 2 = 0 We sketch now the main steps required in proving Theorem 1.3. The generating series of these invariants, for every fixed Young diagram λ, is controlled by three universal series (Theorem 5.1) This universal structure is proven by following the strategy of [15]. In fact, these invariants are multiplicative on triples of the form (C, L 1 , L 2 ) = (C ′ ⊔ C ′′ , L ′ 1 ⊕ L ′′ 1 , L ′ 2 ⊕ L ′′ 2 ) and are polynomial in the Chern numbers of (C, L 1 , L 2 ). The latter is obtained by pushing the virtual intersection number to C [n λ ] on the smooth ambient space A C,n λ -a product of symmetric powers of Cand later to a product of Jacobians Pic ni (C), where the integrand is a polynomial on well-behaved cohomology classes.
By the universal structure any computation is reduced to a basis of the three-dimensional Qvector space of Chern numbers of triples (C, L 1 , L 2 ). A simple basis consists of the Chern numbers of (P 1 , O, O) and any two (P 1 , L 1 , L 2 ) with L 1 ⊗ L 2 = K P 1 . In both cases, the invariants are explictly determined under the anti-diagonal restriction s 1 + s 2 = 0 by further applying the virtual localization formula.
1.5. Toric computations. The C * -action on P 1 canonically lifts to the double nested Hilbert scheme P 1 [n λ ] , with only finitely many C * -fixed points, therefore we can further C * -localize the invariants (1.1) to obtain [P 1[nλ] ] vir e T (−N vir where s 3 is the generator of the C * -equivariant cohomology and T vir Z is the virtual tangent bundle of P 1 [n λ ] at the fixed point Z. Under the anti-diagonal restriction s 1 + s 2 = 0, this translates the computation of the invariants into a purely combinatorial problem, which we explictly solve in the trivial vector bundle case L 1 = L 2 = O P 1 and in the Calabi-Yau case L 1 ⊗ L 2 = K P 1 . A few remarks are in order. In the trivial vector bundle case, the solution is equivalent to the vanishing for every reversed plane partition of positive size |n λ | > 0. This relies on the vanishing e T×C * (−T vir Z − N vir P 1 ,L1,L2,Z ) = 0, which comes from a simple vanishing property of the topological vertex in stable pair theory proved in [33]. In the Calabi-Yau case, the invariants turn out to be topological, under the anti-diagonal restriction. Theorem 1.4 (Theorem 7.3). Let X be Calabi-Yau. Then the generating series of the invariants (1.1) coincides, up to a sign, with the generating series of the topological Euler characteristic This happens as, under the anti-diagonal restriction, each C * -fixed point Z contributes with a sign which is independent of Z and the invariants amount to a (signed) count of the C * -fixed points. It is not a priori clear how to obtain the the same sign through the vertex formalism for stable pairs developed by Pandharipande-Thomas [51]. Nevertheless, the topological nature of the invariants in the Calabi-Yau case is not surprising also for a non-toric curve C. If X is Calabi-Yau and P n (X, d[C]) is proper -which happens only in rare cases -the anti-diagonal restriction would compute its virtual Euler characteristic and Behrend's weighted Euler characteristic, which is a purely topological invariant of a scheme with a symmetric perfect obstruction theory [4].
1.6. Gromov-Witten/stable pairs correspondence. In the seminal work [30], a conjectural correspondence -known as the MNOP conjecture -between Gromov-Witten invariants and Donaldson-Thomas invariants of projective threefolds is formulated, proven for toric varieties in [30,31,32] for primary insertions. By defining the GW/DT invariants via equivariant residues, the conjecture has been extended to local curves in [9] and proven by combining the results of [9,43]. Stable pair invariants were later introduced by Pandharipande-Thomas [50] to give a more natural geometric interpretation of the MNOP conjecture through the DT/PT correspondence proved by Toda and Bridgeland in [60,7] using wall-crossing and Hall algebra techniques. The Gromov-Witten/stable pairs correspondence has been subsequently extended to include descendent insertions and to quasi-projective varieties whenever invariants can be defined through virtual localization. The correspondence had been confirmed by Pandharipande-Pixton for Calabi-Yau and Fano complete intersections in product of projective spaces and toric varieties [49,52] and had been recently addressed in [41]. See [46] for a complete survey on the subject.
where the dependence is only on the genus g = g(C), the degrees of the line bundles and the degree d. The Gromov-Witten theory of local curves had been solved by Bryan-Pandharipande [9, Theorem 7.1] using a TQFT approach. Moreover they deduced an explicit closed formula for the partition function under the anti-diagonal restriction s 1 + s 2 = 0.
With this explicit expression it is immediate to check the Gromov-Witten/stable pairs correspondence under the anti-diagonal restriction. Corollary 1.6 (Corollary 8.3). Let X be a local curve. Under the anti-diagonal restriction s 1 +s 2 = 0 the GW/stable pair correspondence holds after the change of variable q = −e iu .
A scheme X with a perfect obstruction theory is endowed not only with a virtual fundamental class, but also with a virtual structure sheaf O vir X ∈ K 0 (X). If X is proper, K-theoretic invariants are simply of the form . If X is a local curve the moduli space of stable pairs P X is in general not proper and K-theoretic stable pair invariants are defined by virtual localization [16] on the proper T-fixed locus P T X , that is one set In Section 9 we show that, also in the K-theoretic setting, the invariants are controlled by universal series.
The naive generalization of cohomological invariants is for V = O X , that is no insertions. However, we learn from Nekrasov-Okounkov [38] that it is more natural to consider the twisted virtual structure sheaf vir is a square root 1 of the virtual canonical bundle. Denote by PT K d (X; q) the generating series of K-theoretic invariants with V = K 1/2 vir . Theorem 1.7 (Corollary 9.5). There exist universal series A K,λ (q), B K,λ (q), C K,λ (q) ∈ Q(t 1/2 Moreover, the universal series are explicitly computed under t 1 t 2 = 1.
We are not aware of a K-theoretic Gromov-Witten refinement for which a refined GW/stable pairs correspondence holds.
Acknowledgements. I am grateful to Martijn Kool for suggesting this problem and for many helpful discussions. I wish to thank Jan-Willem van Ittersum and Andrea Ricolfi for useful discussions. S.M. is supported by NWO grant TOP2.17.004.
Conventions. All schemes are separated and of finite type over C. We let K 0 (X) be the K-group of vector bundles on X. When X carries an action by an algebraic torus T, we let K 0 T (X) be the Kgroup of T-equivariant vector bundles on X. Similarly, we let K 0 (X) denote the K-group of coherent sheaves on X, and we let K T 0 (X) be the K-group of (the abelian category of) T-equivariant coherent sheaves on X. When X is smooth, the natural Z-linear map K 0 (X) → K 0 (X), resp. K 0 T (pt)-linear map K 0 T (X) → K T 0 (X), is an isomorphism. Chow groups A * (X) and cohomology groups H * (X) are taken with rational coefficients. We use (·) ∨ for the derived dual of complexes and (·) * for the (underived) dual of coherent sheaves. For clarity of exposition, we suppress various pullback maps, whenever they are clear from the context.

Double nested Hilbert schemes
The number of parts of λ is called the length of λ and is denoted by l(λ). A partition λ can be equivalently described by its associated Young diagram, which is the collection of d boxes in Z 2 located at (i, j) where 0 ≤ j < λ i . 2 Given a partition λ, a reversed plane partition n λ = (n ) ∈λ ∈ Z ≥0 is a collection of non-negative integers such that n ≤ n ′ for any , ′ ∈ λ such that ≤ ′ . In other words, a reversed plane partition is a Young diagram labelled with non-negative integers which are non-decreasing in rows and columns. The size of a reversed plane partition is The conjugate partition λ is obtained by reflecting the Young diagram of λ about the i = j line. In the paper we will require the following standard quantities. Given a box in the Young diagram λ, define the content c( ) = j − i and the hooklength h( ) = λ i + λ j − i − j − 1. The total content c λ = ∈λ c( ) satisfies the following identities (cf [29, pag. 11]): For any Young diagram λ there is an associated graph, where any box of λ corresponds to a vertex and any face common to two boxes correspond to an edge connecting the corresponding vertices. A square of this graph is a circuit made of four different edges.
Lemma 2.1. Let λ be a Young diagram and denote by V, E, Q respectively the number of vertices, edges and squares of the associated graph. Then Proof. We prove the claim by induction on the size of λ. If |λ| = 1, this is clear. Suppose it holds for all λ with |λ| ≤ n − 1. Then we construct λ of size n by adding a box with lattice coordinates (i, j) to a Young diagramλ of size n − 1. There are two possibilities: either one of i, j is zero, so we added one vertex and one edge, or both i, j are non-zero, so we added one vertex, one square and two edges. In both cases the claim is proved.

2.2.
Double nested Hilbert schemes. Let X be a projective scheme and O(1) a fixed ample line bundle. The Hilbert polynomial of a closed subscheme Y ⊂ X is defined by Given a polynomial p(m), the Hilbert scheme is the moduli space parametrizing closed subschemes Y ⊂ X with Hilbert polynomial p(m), which is representable by a projective scheme (e.g. by [20]). We consider here a more general situation, where we replace closed subschemes by flags of closed subschemes, satisfying certain nesting conditions dictated by Young diagrams.
Let λ be a Young diagram and p λ = (p ) ∈λ ∈ Z[x] be a collection of polynomials indexed by λ. If all p are non-negative integers which are non-decreasing in rows and columns, p λ = n λ is a reversed plane partition.
Definition 2.2. Let X be a projective scheme and p λ as above. The double nested Hilbert functor of X of type p λ is the moduli functor Hilb p λ (X) : Sch op → Sets, Thanks to representability, double nested Hilbert schemes are equipped with universal subschemes, for any ∈ λ, Z ⊂ X × Hilb p λ (X), such that the fiber over a point Z = (Z ) ∈λ ∈ Hilb p λ (X) is Remark 2.5. If p λ = n λ , Definition 2.2 generalizes to X quasi-projective. In fact, let X ⊂ X be any compactification of X. We define the double nested Hilbert scheme points as the open subscheme X [n λ ] := Hilb n λ (X) ⊂ Hilb n λ (X) consisting of the 0-dimensional subschemes supported on X ⊂ X.
In general, X [n λ ] is singular even for X a smooth quasi-projective curve.
Example 2.6. Let C be a smooth curve and consider the reversed plane partition n λ 0 1 1 2 There are two types of flags of divisors, of the form where P, Q ∈ C. Therefore its reduced scheme structure consists of two irreducible components C × C ∪ C × C, intersecting at the diagonals of C × C.
Singularities make it hard to perform intersection theory on X [n λ ] . To remedy this we construct, in special cases, virtual fundamental classes in A * (X [n λ ] ). We briefly recall the language of perfect obstruction theories of Behrend-Fantechi and Li-Tian [5,28].

Perfect obstruction theories.
A perfect obstruction theory on a scheme X is the datum of a morphism φ : . A perfect obstruction theory is called symmetric (see [6]) if there exists an isomorphism θ : A perfect obstruction theory determines a cone Letting ι : X ֒→ E 1 be the zero section of the vector bundle E 1 , the induced virtual fundamental class on X is the refined intersection By a result of Siebert [55,Thm. 4.6], the virtual fundamental class depends only on the K-theory class of E.  where e(·) denotes the Euler class.
2.4. Points on Curves. Let C be an irreducible smooth quasi-projective curve and n λ a reversed plane partition. In this section we show that C [n λ ] is the zero locus of a section of a vector bundle over a smooth ambient space, and therefore admits a perfect obstruction theory as in Example 2.7.
To ease the notation, we denote its elements by Z = ((Z 00 , X ij , Y lk )) ij,lk ∈ A C,n λ , where Z 00 ⊂ C is a divisor of length n 00 and X ij ⊂ C (resp. Y lk ⊂ C) is a divisor of length n ij − n i−1,j (resp. n lk − n l,k−1 ). A C,n λ comes equipped with universal divisors, which we denote by with fibers are For every (i, j) ∈ λ with i, j ≥ 1 define the universal effective divisors Theorem 2.8. Let C be an irreducible smooth quasi-projective curve, π : C × A C,n λ → A C,n λ be the natural projection and define the vector bundle Then there exists a section s of E whose zero set is isomorphic to Notice that E is a vector bundle, as by cohomology and base change all higher direct images vanish where s ′ ij is the section vanishing on Γ 1 ij while the second morphism is the restriction along j : We claim that To prove it, we follow the strategy of [10, ij is enough to have the equality, as all fibers are divisors in C of the same degree. On the other hand, we have that f factors through Z(s) if and only if f * s is the zero section of Therefore f * s ij is the zero section if and only if Γ 2,T ij ⊂ Γ 1,T ij as required.
Thanks to Theorem 2.8, C [n λ ] falls in the situation of Example 2.7 and we obtain a virtual fundamental class. Corollary 2.9. Let C be a smooth quasi-projective curve and n λ a reversed plane partition. Then C [n λ ] has a perfect obstruction theory In particular there exists a virtual fundamental class

Topological Euler characteristic.
Recall that we can view Euler characteristic weighted by a constructible function as a Lebesgue integral, where the measurable sets are constructible sets, measurable functions are constructible functions and the measure of a set is given by its Euler characteristic (cf. [8,Sec. 2]). In this language we have e(X) = X 1 · de, for any constructible set X. The following lemma is reminiscent of the existence of a power structure on the Grothendieck ring of varieties.
Lemma 2.10 ([8, Lemma 32]). Let B be a scheme of finite type over C and e(B) its topological Euler characteristic. Let g : Z ≥0 → Z((p)) be any function with g(0) = 1. Let G : Sym n B → Z((p)) be the constructible function defined by . Using Lemma 2.10 we compute the topological Euler characteristic of double nested Hilbert schemes of points of any quasi-projective smooth curve.
Theorem 2.11. Let C be a smooth quasi-projective curve and λ a Young diagram. Then Proof. Consider the constructible map ρ n : In other words, ρ just forgets the distribution and the nesting of the divisor ∈λ Z among all ∈ λ. Let ax = i a i x i ∈ Sym n C, with x i different to each other. The fiber ρ −1 n (ax) is clearly 0dimensional and satisfies In particular the Euler characteristic of the fiber ρ n (nx) does not depend on the point x ∈ C and counts the number of reversed plane partition of size n and underlying Young diagram λ where for any ax ∈ Sym n C with x i different to each other, using (2.3) and (2.4) Now, g(a) = |n λ |=a 1 and G(ax) = ρ n * 1(ax) satisfy the hypotheses of Lemma 2.10 and therefore ∞ n=0 |n λ |=n . A closed formula for the generating series of reversed plane partitions was given by Stanley [ by which we conclude the proof.
2.6. Double nesting of divisors. We conclude this section with a generalization of the zero-locus construction of Theorem 2.8.
Let X be a smooth projective variety of dimension d and β λ = (β ) ∈λ be a collection of homology classes β ∈ H n−2 (X, Z). Denote by H β λ the double nested Hilbert scheme of effective divisors on X, which parametrizes flags of divisors (Z ) ∈λ ⊂ X satisfying the nesting condition dictated by β λ . Denote by where H β is the usual Hilbert scheme of divisors on X of class β. Analogously to Section 2.4, A X,β λ comes equipped with universal (Cartier) divisors Z 00 , X ij , Y lk ⊂ X × A X,β λ and for every (i, j) ∈ λ with i, j ≥ 1 we define the universal effective divisors Under some extra assumptions on X and β λ Theorem 2.8 generalizes.
Proposition 2.12. Assume that A X,β λ is smooth and E is a vector bundle. Then there exists a section s of E such that E In particular, H β λ has a perfect obstruction theory.
Proof. The smoothness of A X,β λ follows by the smoothness of in particular, D 1 , D 2 are linearly equivalent. Combining the long exact sequence in cohomology of the short exact sequence which depends only on the degree [D 1 ] = [D 2 ] ∈ H 2 (X, Z) and implies that the dimension of the fibers of E is constant, thus E is a vector bundle.

Moduli space of stable pairs
3.1. Moduli space of stable pairs. Moduli spaces of stable pairs were introduced by Pandharipande-Thomas [50] in order to give a geometric interpretation of the MNOP conjectures [30], through the DT/PT correspondence proved by Toda (for Euler characteristic) and Bridgeland in [60,7] using wall-crossing and Hall algebra techniques. For a smooth quasi-projective threefold X, a curve class β ∈ H 2 (X, Z) and n ∈ Z, we define P n (X, β) to be the moduli space of pairs in the derived category of X where F is a pure 1-dimensional sheaf with proper support [supp(F )] = β with χ(F ) = n and s is a section with 0-dimensional cokernel.
By the work of Huybrechts-Thomas [24], the Atiyah class gives a perfect obstruction theory on P n (X, β) where (·) 0 denotes the trace-free part, π : X × P n (X, β) → P n (X, β) is the canonical projection and If X is projective, the perfect obstruction theory induces a virtual fundamental class [P n (X, β)] vir ∈ A * (P n (X, β)) and one defines stable pair (or PT ) invariants by integrating cohomology classes γ ∈ H * (P n (X, β), Z) against the virtual fundamental class We focus here in the case of X a local curve, i.e. X = Tot C (L 1 ⊕ L 2 ) the total space of the direct sum of two line bundles L 1 , L 2 on a smooth projective curve C and X is a smooth quasi-projective threefold, therefore the moduli space of stable pairs P n (X, β) is hardly ever a proper scheme and one cannot define invariants as in (3.2). Nevertheless, the algebraic torus T = (C * ) 2 acts on X by scaling the fibers and the action naturally lifts to P n (X, d[C]), making the perfect obstruction theory naturally T-equivariant by [53,Example 4.6]. Moreover, the T-fixed locus P n (X, d[C]) T is proper (cf. Prop. 3.1), therefore by Graber-Pandharipande [19] there is naturally an induced perfect obstruction theory on P n (X, d[C]) T and a virtual fundamental class where s 1 , s 2 are the generators of T-equivariant cohomology and the virtual normal bundle is the T-moving part of the restriction of the dual of the perfect obstruction theory. Stable pair invariants with descendent insertions on local curves have been studied in [47,48,40].

3.2.
Torus representations and their weights. Let T = (C * ) g be an algebraic torus, with character lattice T = Hom(T, C * ) ∼ = Z g . Let K 0 T (pt) be the K-group of the category of Trepresentations. Any finite dimensional T-representation V splits as a sum of 1-dimensional representations called the weights of V . Each weight corresponds to a character µ ∈ T, and in turn each character corresponds to a monomial t µ = t µ1 1 · · · t µg g in the coordinates of T. The map sending the class of a T-representation to its decomposition into weight spaces is a ring isomorphism, where tensor product on the left corresponds to the natural multiplication on the right. We will therefore sometimes identify a (virtual) T-representation with its character. If X is a scheme with a trivial T-action, every T-equivariant coherent sheaf on X decomposes as 3.3. The fixed locus. In this section we prove that the T-fixed locus P n (X, where F ij is a coherent sheaf on C. For example Since p is affine, the pushdown does not lose any information, and we recover the O X -module structure of F by the p * O X -action that p * F carries. This is generated by the action of the −1 pieces L −1 which commute with both the actions of O C and T. In other words, (3.5) are T-equivariant maps of O C -modules. By T-equivariance, they are sums of maps gives a graded map which commutes with the maps (3.6). Writing we find that the T-fixed stable pair (F, s) on X is equivalent to the following data of sheaves and commuting maps on C By the purity of F , each G ij is either zero or a pure 1-dimensional coherent sheaf on C, and the "vertical" maps are generically isomorphisms. In particular, for every (i, j) such that either i < 0 or j < 0, it follows that G ij is zero-dimensional and therefore vanishes by the purity assumption. Moreover, if G ij is non-zero, it is a rank 1 torsion-free sheaf on a smooth curve, that is a line bundle on C (with a section). Finally, any T-equivariant stable pair on X is set-theoretically supported on C, thus is properly supported on X and only finitely many G ij can be non-zero. This results in a diagram of the following shape (3.8) O where Z ij are divisors on C and all "horizontal" maps are injections of line bundles. Therefore, a T-fixed stable pair (F, s) is equivalent to a nesting of divisors where the nesting is dictated by a Young diagram λ. This results into a point of the double nested Hilbert scheme C [n λ ] , where by Riemann-Roch where for a Young diagram λ and g, k 1 , k 2 ∈ Z we define (3.10) Conversely, any nesting of divisors as in (3.9) corresponds to a diagram of sheaves as in (3.8), which corresponds to a T-fixed stable pair on X. Therefore we have a bijection of set: where the disjoint union is over all Young diagrams λ of size d and all reversed plane partitions n λ satisfying n = |n λ | + f λ,g (deg L 1 , deg L 2 ). We mimic [27,Prop. 4.1] to prove that the above bijection on sets is an isomorphism of schemes.
Proposition 3.1. There exists an isomorphism of schemes where the disjoint union is over all Young diagrams λ of size d and all reversed plane partitions n λ satisfying In particular, P n (X, d[C]) T is proper.
Proof. Let B be any (connected) scheme over C. We need to adapt the construction of this section to a T-fixed stable pair on X × B, flat over B. Pushing down by the affine map p : X × B → C × B gives a graded sheaf i,j F ij on C × B, flat over B (therefore so are all its weight spaces F ij ). The original sheaf F on X × B can be reconstructed from the maps (3.6). Therefore a T-fixed pair (F, s) on X × B, flat over B, is equivalent to the data (3.7), with each G ij on C × B, flat over B. If (F, s) is a stable pair, over each closed fiber C × {b}, where b ∈ B, we showed that each (non-zero) G ij is a line bundle. By [23,Lemma 2.1.7], this shows that each (non-zero) G ij is a line bundle on C × B. Together with its non-zero section, this defines divisors Z ij ⊂ C × B, flat over B, satisfying the nesting condition of (3.9), which yields a B-point B → n λ C [n λ ] . Conversely, any B-point B → n λ C [n λ ] defines a diagram (3.8), equivalent to a T-fixed stable pair (F, s) on X × B, flat over B.
As a corollary, we compute the generating series of the topological Euler characteristic of the moduli space of stable pairs on a local curve.
Corollary 3.2. Let C be a smooth projective curve of genus g, L 1 , L 2 line bundle on C and set X = Tot C (L 1 ⊕ L 2 ). Then for any d > 0 we have Proof. The topological Euler characteristic of a T-scheme is the same of its T-fixed locus, therefore n∈Z e(P n (X, d[C])) · q n = n∈Z e P n (X, where in the second line we applied Proposition 3.1 and in the last line Theorem 2.11. We start by describing the class in K-theory of (the restriction of) the perfect obstruction theory E ∈ K T 0 (C [n λ ] ). To ease readability we will omit various pullbacks whenever they are clear from the context. Recall the following identities in K-theory Again, to ease notation, we keep denoting i × id C [n λ ] , p × id C [n λ ] by i, p and by π the composition π • i. We compute where in the second equality we applied Grothendieck duality 3 and the projection formula and for a locally free sheaf V and extend it by linearity to any class in K 0 T (C). By [14,Lemma 5.4.9] , for every T-equivariant coherent sheaf F ∈ K T 0 (C), we have where in the second line we used adjunction in the derived category. To simplify the notation, for any (i, j), (l, k) ∈ λ set ∆ ij;lk = Z lk − Z ij , which is an effective divisor if (i, j) ≤ (l, k). Putting all together we have the following identity in Theorem 4.1. There is an identity of virtual fundamental classes where the class on the left-hand-side is induced by (3.1) by Graber-Pandharipande localization and the one on the right-hand-side is constructed in Corollary 2.9.
We explained in Section 2.1 how to associate a graph to any Young diagram λ. Notice that boxes (i, j) ∈ λ are in bijection with the vertices V , boxes (i, j) ∈ λ, such that i ≥ 1 (resp. j ≥ 1) are in bijection with vertical (resp. horizontal) edges E and boxes (i, j) ∈ λ, such that i, j ≥ 1 are in bijection with squares Q of the associated graph. By Lemma 2.1 Combining this identity with the universal exact sequences whenever ∆ ij,lk is an effective divisor, one gets Moreover, in the expression above, all higher direct images R k π * vanish for k > 0 by cohomology and base change, therefore We finally show that this is precisely the same class in K-theory as where E → A C,n λ is the vector bundle constructed in Theorem 2.8. In fact, in the notation of Section 2.4, we have by which we conclude that In virtue of Proposition 4.1, we denote now on by T vir C [n λ ] the class in K-theory of the dual of the perfect obstruction theory (2.2) and the one induced by the fixed part of (3.1), which we showed to agree. In order to compute stable pair invariants (3.3) one needs to express the virtual normal bundle (3.4) in terms of K-theoretic classes which are easier to handle. For instance, we could express N vir in terms of pullbacks of the line bundles L 1 , L 2 , K C and the universal divisors ∆ ij;lk , but that would lead to cumbersome expressions difficult to manipulate. Example 4.2. As a concrete example, we compute the weight space of E ∨ relative to the character t 1 t 2 , which we denote by E ∨ t1 t2 . An application of Grothendieck duality and projection formula on π : and some vanishing of higher direct images yields The situation notably simplifies if we impose X to be Calabi-Yau.
Proof. If X is Calabi-Yau, the perfect obstruction theory (3.1) satisfies ) are classes of T-equivariant vector bundles, (4.5) implies that We have that (E ∨ ) fix = T vir C [n λ ] , therefore which concludes the argument.

Remark 4.4.
A simple computation shows that we could take Ω to be of the form All other choicesΩ must be of the form for any A ∈ K 0 T (C [n λ ] ) having no weight spaces corresponding to the characters (t 1 t 2 ) 0 , t 1 t 2 . In particular, this implies that the parity of rk Ω is independent by the choice of Ω.

Universal expression.
In the previous sections, given a triple (C, L 1 , L 2 ) with C an irreducible smooth projective curve and L 1 , L 2 line bundles on C, we reduced stable pair invariants (with no insertions) of Tot C (L 1 ⊕ L 2 ) to the computation of where n λ is a reversed plane partition and the virtual normal bundle N vir C,L1,L2 is the T-moving part of the class in K-theory (4.1). We state our main results, describing the generating series of (5.1) in terms of three universal functions exploiting the universality techniques used in [15,Thm. 4.2] for surfaces. Furthermore, we find explicit expressions for these universal series under the anti-diagonal restriction s 1 + s 2 = 0.
Theorem 5.1. Let C be a genus g smooth irreducible projective curve and L 1 , L 2 line bundles over C. We have an identity where A λ , B λ , C λ ∈ Q(s 1 , s 2 ) q are fixed universal series for i = 1, 2, 3 which only depend on λ. Moreover Proof. The proof is similar to [15,Thm. 4.2]. Consider the map given by where C g,deg L1 deg L2 is the leading term of the generating series of the integrals (5.1). By Proposition 5.2 the integral (5.1) is multiplicative and by Corollary 5.4 it is a polynomial on g, deg L 1 , deg L 2 . This implies that Z factors through where γ(C, L 1 , L 2 ) = (g − 1, deg L 1 , deg L 2 ) and Z ′ is a linear map. A basis of Q 3 is given by the images and the image of a generic triple (C, L 1 , L 2 ) can be written as We conclude that which gives the universal series we were looking for. The second claim just follows by interchanging the role of L 1 and L 2 .
We devote the remainder of Section 5 to prove the multiplicativity and polynomiality of (5.1). In Section 6 we compute the leading term of the generating series of (5.1), while in Section 7 we explicitly compute the integral (5.1) in the toric case under the anti-diagonal restriction. These computations will lead to the proof of the second part of the main Theorem 1.3 (see Theorem 8.1).

Multiplicativity.
We show now that the integral (5.1) is multiplicative. First of all, notice that if C = C ′ ⊔ C ′′ is a smooth projective curve with two connected components, the construction of Theorem 2.8 does not directly work and we need to adjust it to define a virtual fundamental class. For any reversed plane partition n λ there is an induced stratification We set which is a smooth projective variety. Let E C ′ ,n ′ λ , E C ′′ ,n ′′ λ denote the vector bundles over A C ′ ,n ′ λ , A C ′′ ,n ′′ λ of Theorem 2.8. We define a vector bundle E C,n λ over A C,n λ by declaring its restriction to any con- λ ⊞ E C ′′ ,n ′′ λ . By Theorem 2.8, there exists a section s of E C,n λ such that

and therefore an induced virtual fundamental class [C
By iterating this construction, there exists a natural virtual fundamental class on C [n λ ] for any smooth projective curve C (with any number of connected components). Proof. Let n λ be a fixed reversed plane partition. We claim that the restriction of the virtual normal bundle to the connected component In fact, N vir C,L1,L2 is a linear combination of K-theoretic classes of the form , where ∆ is a Z-linear combination of the universal divisors Z ij on C × C [n λ ] , a, b ∈ Z and t µ is a T-character and notice that Consider the induced stratification Denote by ∆ ′ , ∆ ′′ the corresponding universal divisor on and consider the projection maps

On every component (C
Consider the cartesian diagram given by the natural projections Flat base change yields and analogously for C ′′ , which implies

Chern numbers dependence.
We now show that the integral 5.1 is a polynomial in the Chern numbers of the triple (C, L 1 , L 2 ). Our strategy is to express the integral on a product of Picard varieties Pic n (C) -via the Abel-Jacobi map -where the integrand is a polynomial expression on tautological classes. Through this section, we follow the notation as in [

Tautological integrals on
Pic n (C). Let C be a smooth curve of genus g. If n > 2g − 2, the Abel-Jacobi map is a projective bundle. In fact, consider the diagram AJ and the Poincaré line bundle P on Pic n (C) × C, normalized by fixing for a certain c ∈ C. Then C (n) ∼ = P(π * P).
Consider now the product of Abel-Jacobi maps AJ : C (n1) × · · · × C (ns) → Pic n1 (C) × · · · × Pic ns (C), where each n i > 2g −2. We denote by P i (the pullback of) the Poincaré line bundle on Pic ni (C)×C, each normalized at a point c i ∈ C, and by ω i the first Chern classes of the tautological bundles on C (ni) . Finally we denote by Z i ⊂ C (ni) × C the universal divisors and by I i (the pullback of) their ideal sheaves, which in this case are line bundles. We are interested in studying integrals of the form where f is a polynomial in the Chern classes of the K-theoretic classes . The Chern classes of the last expression are a linear combination of where ch denotes the Chern character for certain m i ∈ Z ≥0 . Integrating this class yields Using a standard identity [18,Sec. 3.1] we can express the pushforward AJ * ω mi i in terms of Segre classes (and therefore Chern characters) of π * P i . These Chern characters appearing in the integral are computed by Grothendieck-Riemann-Roch The Chern character of the Poincaré line bundle is (cf. [2, pag. 335]) Here, in the decomposition where ∆ ⊂ C × C is the diagonal and θ i is the theta divisor. All of this results in wheref is a polynomial in the classes

5.3.2.
Extension to all n. In the previous section we assumed that all n i > 2g − 2; we explain now how to remove this assumption, following closely [27,Sec. 10.1].
Let n ∈ Z ≥0 and N > 2g − 2. Then C (N ) ∼ = P(π * P), where P is the Poincaré line bundle on Pic N (C) × C normalized at c ∈ C. We can embed C (n) in C (N ) as the zero section of a vector bundle an Artinian thickened point. Moreover, the section is regular, therefore Recall that we are interested in the integrals (5.4). Choose N i > 2g−2, denote by I ′ i the universal ideal sheaves of the universal divisors W i on C (N1) × · · · × C (Ns) × C and by π ′ the projection map. By base change we can write where each G i = π * O(W i )| C (N i ) ×(Ni−ni)ci and f ′ is a polynomial in the Chern classes of The exact sequence yields the identity in K-theory by which we conclude that we can apply the construction in Section 5.3.1 to express wheref is a polynomial in the classes Smooth projective curves of the same genus are diffeomorphic to each other, therefore Pic n (C) is diffeomorphic to a g-dimensional complex torus. By the intersection theory on Pic n (C) developed in [2, Sec. VIII.2] we immediately obtain the following result.

Proposition 5.3. Let f be a polynomial in the Chern classes of the K-theory classes
where L k are line bundles on C and I, J are families of indices (possibly with repetitions). Then wheref is a polynomial in the classes (5.5). In particular, the integral is a polynomial in the genus g = g(C) and the degrees of the line bundles {L k } k .
As a corollary, we obtain that the localized contributions on C [n λ ] only depend on the Chern numbers of (C, L 1 , L 2 ). Proof. Let i : C [n λ ] ֒→ A C,n λ be the embedding of Theorem 2.8 and to ease the notation set A C,n λ = C (n1) × · · · × C (ns) . We claim that for a certain classÑ vir C,L1,L2 ∈ K 0 T (A C,n λ ). In fact, N vir C,L1,L2 is a linear combination of classes in K-theory of the form , where ∆ is a Z-linear combination of the universal divisors Z ij on C × C [n λ ] , a, b ∈ Z and t µ is a T-character. Each of such universal divisors ∆ can be expressed, in the Picard group of C [n λ ] , as a linear combination of the divisors i * Z 00 , i * X ij , i * Y ij , with notation as in Section 2.4; applying base change proves the claim. Therefore the integral (5.1) can be expressed as an intersection number on the product of symmetric powers of curves A C,n λ where E is the vector bundle of Theorem 2.8. In particular, the K-theory class of E −Ñ vir C,L1,L2 is a linear combination of classes of the form where L k are line bundles on C and I, J are families of indices (possibly with repetitions). By Proposition 5.3 this integral is a polynomial in g and the degrees of L k . We conclude the proof by noticing that all line bundles L k possibly occuring are a linear combination of L 1 , L 2 , K C .

The leading term
We compute the leading term of the generating series of the integrals (5.1), which is essential for the computation of the full generating series in Theorem 8.1. Proposition 6.1. Let C be a smooth projective curve of genus g and L 1 , L 2 line bundles on C. Then under the anti-diagonal restriction s 1 + s 2 = 0 we have where 0 λ is the unique reversed plane partition of size 0 and underlying Young diagram λ.

Proof.
We have that C [0 λ ] ∼ = pt and [C [0 λ ] ] vir = [pt] ∈ A * (pt), therefore E ∨ is completely T-movable and where by (4.1), (4.4) we express the class in K-theory of the virtual normal bundle as Applying Riemann-Roch, every line bundle L on C satisfies therefore we can write the virtual normal bundle as where the weights µ range among the weights ν range among (i − l, j − k) ∈ Z 2 such that ((i − j), (l − k)) ∈ λ, (i, j) = ((l, k), (l + 1, k + 1)), (6.3) and, for a weight µ = (µ 1 , µ 2 ), we set m µ = µ 1 deg L 1 + µ 2 deg L 2 . In particular, the weights 1, t 1 t 2 do not appear in (6.1), as the virtual tangent bundle of the T-fixed locus has rank 0 and by the explicit description of the weight space of t 1 t 2 in Example 4.2. For every weight µ, we compute where s = (s 1 , s 2 ) and µ · s is the standard inner product. With a simple manipulation we end up with Following the proof of Lemma A.1, we see that for any weight ν = (a, a) (with a ∈ Z) there is either a weight µ = (a, a) or µ = (1 − a, 1 − a) (and viceversa). This implies that In particular, the anti-diagonal restriction s 1 + s 2 = 0 is well-defined; we get where the product is over all µ, ν = (a, a). Moreover, it is immediate to check that #µ − #ν = |λ|. By Lemma A.2, we conclude that the last expression equals to where σ(λ) is defined in Lemma A.2. We are only left with a sign computation; we conclude by Lemma A.3 where ρ(λ) was defined is Lemma A.3 and we set k i = deg L i .

Toric computations
7.1. Torus action. Let U 0 , U ∞ be the standard open cover of P 1 . The torus C * acts on the coordinate functions of P 1 as t · x = tx (resp. t · x = t −1 x) in the chart U 0 (resp. U ∞ ). The C * -representation of the tangent space at the C * -fixed points of P 1 is We prove some identities of C * -representations that will be useful later in this section.
Lemma 7.1. Let Z = Z 0 ⊔ Z ∞ ⊂ P 1 be a closed subscheme, where Z 0 (resp. Z ∞ ) is a closed subscheme of length n 0 (resp. n ∞ ) supported on 0 (resp. ∞). For any a ∈ Z, we have the following identities in K 0 C * (pt) Proof. We have that where in the second line we applied the classical K-theoretic localization [59] on P 1 . Secondly, we have where in the second line we usedČech cohomology and in the third line we used [44,Ex. 3.4.5].
Thirdly, by applyingČech cohomology as before we have Finally, combining Serre duality and the previous result yields The C * -action on P 1 naturally lifts to a C * -action on the Hilbert scheme of point P 1 [n] , whose C * -fixed locus consists of length n closed subschemes Z ⊂ P 1 supported on 0, ∞. Therefore, there is an induced C * -action on A P 1 ,n λ , whose C * -fixed locus is 0-dimensional and reduced. This C * -action restricts to a C * -action on P 1 [n λ ] , whose C * -fixed locus is necessarily 0-dimensional and reduced. Moreover, the perfect obstruction theory (2.2) is naturally C * -equivariant, as all the ingredients of Theorem 2.8 are.
Proposition 7.2. Let Z ∈ P 1 [n λ ],C * be a C * -fixed point. Then T vir P 1[n λ ] ,Z is completely C * -movable. In particular, the induced perfect obstruction theory on P 1 [n λ ],C * is trivial.
Proof. We need to show that the class in K-theory does not have C * -fixed part. Recall that we have an identity in K 0 C * (pt) where for simplicity we denoted by X ij = Z ij − Z i−1,j and by Y ij = Z ij − Z i,j−1 . Moreover we have where for simplicity we denoted by W ij = Z ij − Z i−1,j−1 . Therefore the virtual tangent space is a sum of classes of the form RΓ(P 1 , O Z (Z)), with Z ⊂ P 1 a closed subscheme, which is entirely C * -movable by the description in Lemma 7.1.

7.2.
Case I: Calabi-Yau. We compute the integral (5.1) for C = P 1 in the case of L 1 ⊗ L 2 = K P 1 , showing that it coincides (up to a sign) with the topological Euler characteristic e P 1 [n λ ] .
Theorem 7.3. Let L 1 , L 2 be line bundles on P 1 such that L 1 ⊗ L 2 = K P 1 . For any reversed plane partition n λ , we have Proof. By Graber-Pandharipande [19], there is an induced perfect obstruction theory and virtual fundamental class on the C * -fixed locus P 1 [n λ ],C * , both trivial by Proposition 7.2. By Proposition 4.3 , and applying the virtual localization formula with respect to the C * -action yields where s 3 is the generator of the equivariant cohomology H * C * (pt) and we denoted by . Under the Calabi-Yau restriction s 1 + s 2 = 0, we have by Lemma 7.6 Moreover, by Lemma 7.5 rk V Z = deg L 1 (c λ + |λ|) + |λ| + |n λ | mod 2.
Therefore, we conclude that  as the Euler characteristic of a C * -scheme coincides with the Euler characteristic of its C * -fixed locus (in our case the number of fixed points).
Exploiting the close formula for the generating series of the topological Euler characteristic proved in Theorem 2.11, we derive the following close expression in the Calabi-Yau case.
Corollary 7.4. Let L 1 , L 2 be line bundles on P 1 such that L 1 ⊗ L 2 = K P 1 and λ be a Young diagram. We have where the sum is over all reversed plane partition n λ .
We devote the remainder of this section to prove the technical lemmas we used in Theorem 7.3.
Lemma 7.5. Let C be a smooth projective curve of genus g, L 1 , L 2 be line bundles on C such that L 1 ⊗ L 2 = K C and let Ω ∈ K 0 T (C [n λ ] ) be the K-theory class of Remark 4.4. Then Finally (i,j),(l,k)∈λ (i,j) =(l,k) (i,j) =(l+1,k+1) (n lk − n ij ) + (i,j),(l,k)∈λ (i,j) =(l−1,k) (i,j) =(l,k+1) (n lk − n ij ) Combining all these identities together, we conclude that Lemma 7.6. Let Z ∈ P 1 [n λ ],C * be a C * -fixed point and set where Proof. Denote by V CY Z the sub-representation of V Z consisting of weight spaces corresponding to the characters (t 1 t 2 ) a , for all a ∈ Z, with respect to the T × C * -action. We claim that V CY Z is of the form for a suitable A Z ∈ K 0 T×C * (pt).
Step I: Assuming the claim, setṼ where none of the characters t µ , t ν is a power of t 1 t 2 . Write e T×C * (t µ ) = µ · s, where s = (s 1 , s 2 , s 3 ). Then we conclude that where we used that no µ · s, ν · s is a multiple of s 1 + s 2 and that rk V Z = rkṼ Z mod 2.
Step II: We prove now our claim on V CY Z . Firstly, by Proposition 7.2 T vir P 1[n λ ] ,Z is C * -movable, which implies that there are no weight spaces corresponding to a power of t 1 t 2 . It is clear that the T × C * -weight spaces of Ω| Z relative to the characters (t 1 t 2 ) a are given by Ω ′ | Z , where as we just considered the weight spaces of (t 1 t 2 ) a in Ω of Remark 4.4. We notice that Ω ′ | Z is a sum of T × C * -representations of the form where Z a ⊂ P 1 are effective divisors. We have the following identities of C * -representations By Lemma 7.1, their C * -fixed part is Combining everything together, we conclude that where sgn is the usual sign function. Our claim follows by Lemma A.1. Proof. The leading term is computed in Proposition 6.1, therefore we just need to prove that the integral vanishes for |n λ | > 0. We apply Graber-Pandharipande virtual localization [19] with respect to the C * -action on P 1 [n λ ] where s 3 is the generator of H * C * (pt). The C * -action on P 1 [n λ ] is just the restriction of the natural T × C * -action on P n (P 1 × C 2 , d[P 1 ]), where d = |λ| and n = |λ| + |n λ |. This means that, as T × C * -representation, can be described via the topological vertex formalism of Pandharipande-Thomas [51,Thm. 3], which states where V 0,Z , V ∞,Z are the vertex terms corresponding to the two toric charts of P 1 × C 2 and E Z is the edge term. By [33,Lemma 22] we have that e T×C * (−V 0,Z − V ∞,Z ) is divisible by (s 1 + s 2 ) if |n λ | > 0, while e T×C * (−E Z ) is easily seen to be coprime with (s 1 + s 2 ). 4 This implies that the anti-diagonal restriction s 1 + s 2 = 0 is well-defined on every localized term and satisfies by which we conclude the required vanishing.
Remark 7.8. We could prove Proposition 7.7 without relying on the vertex formalism for stable pairs, by simply carrying out a detailed (but probably longer) analysis of the weight space of t 1 t 2 as in Lemma 7.6.
8. Summing the theory up 8.1. Anti-diagonal restriction. We combine the computations in Section 6, 7 to prove the second part of Theorem 1.3 from the Introduction.
Theorem 8.1. Under the anti-diagonal restriction s 1 + s 2 = 0 the three universal series are Proof. Let C g,deg L1 deg L2 ∈ Q(s 1 , s 2 ) be the leading term of the generating series of Theorem 5.1.
We can write for suitableÃ λ,i ∈ 1+Q(s 1 , s 2 ) q , where logÃ i are well-defined asÃ i are power series starting with 1. The claim is therefore reduced to the computation of the leading term (under the anti-diagonal restriction s 1 + s 2 = 0) and to the solution of a linear system. The leading term is computed in Proposition 6.1, which also shows that it is invertible in Q(s 1 , s 2 ). The linear system is solved by computing the generating series of the integrals (5.1), under the anti-diagonal restriction, on a basis of Q 3 . The classes are linearly independent and we computed their generating series in Proposition 7.7 and Corollary 7.4.

Degree 1.
In degree 1, we consider the Young diagram consisting of a single box and the corresponding double nested Hilbert scheme is C [n] ∼ = C (n) , the symmetric power of a smooth projective curve C, with universal subscheme Z ⊂ C × C (n) . Given line bundles L 1 , L 2 on C, the class (4.1) in K-theory of the virtual normal bundle is In fact, E Z is the T-representation of the tangent space at a T-fixed point of Hilb |λ| (C 2 ). The claim follows by noticing that the T-fixed locus coincides with the fixed locus of the subtorus { t 1 t 2 = 1 } ⊂ T preserving the Calabi-Yau form of C 2 , which is 0-dimensional and reduced.
where A λ,i are the universal series of Theorem 5.1. Moreover, under the anti-diagonal restriction Comparing with Bryan-Pandharipande's results -cf. Theorem 1.5 and [9, Sec. 8] for the fully equivariant result in degree 1 -we obtain a proof of the Gromov-Witten/stable pairs correspondence for local curves. Corollary 8.3. Let X be a local curve. Under the anti-diagonal restriction s 1 + s 2 = 0 the GW/stable pair correspondence holds after the change of variable q = −e iu . Moreover it holds fully equivariantly in degree 1.  [4]. Applying Riemann-Roch, this situation may appear only when H l (C, L i ) = 0 for l = 0, 1 and deg L i = g − 1 for i = 1, 2.
An interesting example is the resolved conifold X = Tot P 1 (O(−1) ⊕ O(−1)); in this case, the invariants can be further packaged into a generating series where we used some identities involving the Schur function s λ (see e.g. [29]). This last generating series agrees with the expression of the unrefined limit of the topological vertex of Iqbal-Kozçaz-Vafa [25] and can be seen as a specialization both of the motivic invariants of Morrison-Mozgovoy-Nagao-Szendrői [35] and of the K-theoretic invariants of Kononov-Okounkov-Osinenko [26].
9. K-theoretic refinement 9.1. K-theoretic invariants. Let X = Tot C (L 1 ⊕ L 2 ) be a local curve. The perfect obstruction theory on P X := P n (X, d[C]) induces a (T-equivariant) virtual structure sheaf O vir PX ∈ K T 0 (P X ) [16] which depends only on the K-theory class of the perfect obstruction theory [57,Cor. 4.5]. K-theoretic PT invariants are defined by virtual K-theoretic localization [16] for any V ∈ K T 0 (P X ) is defined for every locally free sheaf V and then extended by linearity to any class in K-theory.
Remark 9.1. Differently than the case of equivariant cohomology, χ(M, F ) ∈ K T 0 (pt) loc is welldefined for any T-equivariant coherent sheaf F on a non-proper scheme M , as long as the weight spaces of F are finite-dimensional; in this case, the virtual localization formula is an actual theorem, rather than an ad-hoc definition of the invariants.
Using the description of the T-fixed locus (P n (X, d[C)) T of Proposition 3.1, K-theoretic stable pair invariants on X (with no insertions) are reduced to intersection numbers on C [n λ ] . The same techniques of Section 5 can be applied in this setting, yielding the following result.
Proposition 9.2. Let C be a genus g smooth irreducible projective curve and L 1 , L 2 line bundles over C. We have an identity where A K,λ , B K,λ , C K,λ ∈ Q(t 1 , t 2 ) q are fixed universal series for i = 1, 2, 3, only depending on λ. Moreover t 1 ), Proof. The proof follows the same strategy as Theorem 5.1. We just need to notice that Λ • (·) is multiplicative and that, via virtual Hirzebruch-Riemann-Roch [16], we can express where f is a polynomial expression of classes of the same form as in Proposition 5.3.
Denote by PT K d (X; q) = n∈Z q n PT K d,n (X) the generating series of K-theoretic stable pair invariants.
Corollary 9.3. Let X = Tot C (L 1 ⊕ L 2 ) be a local curve. We have which are an algebro-geometric analogue of the A-genus of a spin manifold. On the T-fixed locus, we have an identity in K 0 where we define Λ • (·) = Λ • (·)⊗det(·) −1/2 . Again, the same techniques of Section 5 and Proposition 9.2 and can be applied in this setting, yielding the following result.
Theorem 9.4. Let C be a genus g smooth irreducible projective curve and L 1 , L 2 line bundles over C. We have an identity 2 ) q , 5 To take square roots equivariantly, we need to replace the torus T with the minimal cover where the characters t where A K,λ , B K,λ , C K,λ ∈ Q(t 1/2 2 ) q are fixed universal series for i = 1, 2, 3. Moreover Denote by PT K d (X; q) = n∈Z q n PT K d,n (X) the generating series of Nekrasov-Okounkov Ktheoretic stable pair invariants.
Corollary 9.5. Let X = Tot C (L 1 ⊕ L 2 ) be a local curve. We have .
The techniques of Section 6, 7 can be adapted to compute the generating series of Nekrasov-Okounkov K-theoretic invariants under the anti-diagonal restriction t 1 t 2 = 1. In fact, as in the proof of Theorem 8.1, we just need to compute the leading term of the generating series and the cases g = 0, L 1 ⊗ L 2 = K P 1 and L 1 = L 2 = O P 1 . Similarly to the proof of Theorem 7.3, as we work with C ∼ = P 1 , applying the K-theoretic virtual localization formula [16]

Explicit computations yields
A K,λ (q, t 1 , t −1 1 ) = (−1) |λ| · F −2 λ , B K,λ (−q, t 1 , t −1 1 ) = (−1) n(λ) · F λ · ∈λ (1 − q h( ) ), Appendix A. The combinatorial identities In this appendix we collect the proofs of some technical results on the combinatorics of Young diagrams we have used. Then we have Proof. We prove the first claim by induction on the size of λ. If |λ| = 1 this is clear. Suppose now the claim holds for all Young diagrams of size |λ| = n and consider a Young diagram of size |λ| = n + 1 obtained by adding to a Young diagram λ a box whose lattice coordinates are (i, j) ∈ Z 2 .
With an analogous analysis one proves the second claim as well.
Proof. The sign σ(λ) is easily determined, so we just need to compute the absolute value. To ease the notation, we adopt the following convention for the remainder of the proof: we set |0| = 1, which is merely a formal shortcut to include in the productory trivial factors we would have otherwise excluded. The claim therefore becomes (i,j)∈λ |i − j| · (i,j),(l,k)∈λ Denote the left-hand-side of (A.1) by H λ . We prove this claim on the induction on the size of λ. If |λ| = 1, the claim is trivially satisfied. Assume it holds for all Young diagrams of size n and consider a Young diagram λ ′ of size n + 1 obtained by a Young diagram of size λ by adding a box with lattice coordinates (i, j) ∈ Z 2 . We have To avoid confusion, we denote by h( ) (resp. h ′ ( )) the hooklength of ∈ λ (resp. ∈ λ ′ ). The strategy now is to divide the boxes of λ ′ in sub-collections and compute separately each contribution of the product on the right-hand-side.
We multiply now the last expression for all boxes (ĩ, j) withĩ = 0, . . . , i − 1 This is the contribution of all boxes (l, k) ∈ λ such that k ≥ j. By symmetry, we get that the contribution of all boxes (l, k) ∈ λ such that l ≥ i is given by Notice that the left-hand-side of (A.2) can be rewritten as (i,j)∈λ τ (−i, −j) (i,j),(l,k)∈λ τ (i − l, j − k)τ (1 + i − l, j − k), and denote this last expression by F λ . We prove the claim (A.2) on induction on the size of λ. If |λ| = 1, the result is clear. Assume it holds for all Young diagrams of size n and consider a Young diagram λ ′ of size n + 1 obtained from a Young diagram of size λ by adding a box with lattice coordinates (i, j) ∈ Z 2 ; we have We analyze the contribution of every box (l, k) ∈ λ in the product above. We say that a box (l, k) is in the same diagonal as (i, j) if it is of the form (l, k) = (i + a, j + a), a ∈ Z.
We say that a box (l, k) is in the lower diagonal of (i, j) if it is of the form (l, k) = (i + a, j + a − 1), a ∈ Z.
We say that a box (l, k) is in the upper diagonal of (i, j) if it is of the form (l, k) = (i + a − 1, j + a), a ∈ Z.
The contribution of the boxes in the upper diagonal of (i, j) is A completely analogous analysis shows that all other boxes (i, j) ∈ λ do not contribute to the product. Therefore the contribution to the sign in the product is just given by the boxes in the upper or lower diagonal of (i, j), as displayed in the picture 1 1 1 1 1 1 -1 1 -1 1 1 1 ij 1 1 1 Figure 3. The label ±1 in every box represents the contribution to the final sign, with (i, j) = (2, 3).
If we denote by δ + , δ − respectively the number of boxes in the upper diagonal and lower diagonal, we conclude that (l,k)∈λ where ⌈·⌉, ⌊·⌋ denote the ususal ceiling and floor functions. One readily proves that (δ + , δ − ) =        (j + 1, j) i > j, With a direct analysis we can show that in all the three cases, by which we conclude the inductive step.