Abstract. We develop a theory of reduced Gromov-Witten and stable pair invariants of surfaces and their canonical bundles.We show that classical Severi degrees are special cases of these invariants. This proves a special case of the MNOP conjecture, and allows us to generalise the Göttsche conjecture to the non-ample case. In a sequel we prove this generalisation.We prove a remarkable property of the moduli space of stable pairs on a surface. It is the zero locus of a section of a bundle on a smooth compact ambient space, making calculation with the reduced virtual cycle possible. IntroductionMotivation. Fix a nonsingular projective surface S and a homology class β ∈ H 2 (S, Z). There are various ways of counting holomorphic curves in S in class β; in this paper we focus on Gromov-Witten invariants [Beh, LT] and stable pairs [PT1,Ott]. Since these are deformation invariant they must vanish in class β if there exists a deformation of S for which the Hodge type of β is not (1, 1). We can see the origin of this vanishing without deforming S as follows. For simplicity work in the simplest case of an embedded curve C ⊂ S with normal bundle N C = O C (C). As a Cartier divisor, C is the zero locus of a section s C of a line bundle L := O S (C), giving the exact sequenceThe resulting long exact sequence describes the relationship between first order deformations and obstructions H 0 (N C ), H 1 (N C ) of C ⊂ S, and the deformations and obstructionsThe resulting "semi-regularity map"S) takes obstructions to deforming C to the "cohomological part" of these obstructions. Roughly speaking, if we deform S, we get an associated obstruction in H 1 (N C ) to deforming C with it; its image in H 0,2 (S) is the (0, 2)-part of the cohomology class β ∈ H 2 (S) in the deformed complex structure. Thus it gives the obvious cohomological obstruction to deforming C: that β must remain of type (1, 1) in the deformed complex structure on S.In particular, when S is fixed, obstructions lie in the kernel ofMore generally, if we only consider deformations of S for which β remains (1, 1) then the same is true. And when h 0,2 (S) > 0 but H 2 (L) = 0, the existence of this trivial H 0,2 (S) piece of the obstruction sheaf guarantees that the virtual class vanishes.So it would be nice to restrict attention to surfaces and classes (S, β) inside the Noether-Lefschetz locus, 1 defining a new obstruction theory using only the kernel of the semi-regularity map.2 Checking that this kernel really defines an obstruction theory in the generality needed to define a virtual cycle -i.e. for deformations to all orders, over an arbitrary base, of possibly non-embedded 1 The locus of surfaces S for which β ∈ H 2 (S) has type (1, 1); for more details see [Voi, MP]. 2For embedded curves this means we use the obstruction space H 1 (L) to deforming sections of L. We have been able to remove the obstructions H 2 (O S ) to deforming L since the space of line bundles is smooth over the Noether-Lefschetz locus. Here we give quite a general construction using a mixture ...

We conjecture a formula for the generating function of virtual χ y -genera of moduli spaces of rank 2 sheaves on arbitrary surfaces with holomorphic 2-form. Specializing the conjecture to minimal surfaces of general type and to virtual Euler characteristics, we recover (part of) a formula of C. Vafa and E. Witten.These virtual χ y -genera can be written in terms of descendent Donaldson invariants. Using T. Mochizuki's formula, the latter can be expressed in terms of Seiberg-Witten invariants and certain explicit integrals over Hilbert schemes of points. These integrals are governed by seven universal functions, which are determined by their values on P 2 and P 1 × P 1 . Using localization we calculate these functions up to some order, which allows us to check our conjecture in many cases.In an appendix by H. Nakajima and the first named author, the virtual Euler characteristic specialization of our conjecture is extended to include µ-classes, thereby interpolating between Vafa-Witten's formula and Witten's conjecture for Donaldson invariants.

We study Hilbert schemes of points on a smooth projective Calabi-Yau 4-fold X. We define DT 4 invariants by integrating the Euler class of a tautological vector bundle L [n] against the virtual class. We conjecture a formula for their generating series, which we prove in certain cases when L corresponds to a smooth divisor on X. A parallel equivariant conjecture for toric Calabi-Yau 4-folds is proposed. This conjecture is proved for smooth toric divisors and verified for more general toric divisors in many examples.Combining the equivariant conjecture with a vertex calculation, we find explicit positive rational weights, which can be assigned to solid partitions. The weighted generating function of solid partitions is given by exp(M (q) − 1), where M (q) denotes the MacMahon function.The virtual class (1.1) depends on the choice of orientation o(L). On each connected component of Hilb n (X), there are two choices of orientations, which affects the corresponding contribution to the class (1.1) by a sign. We review facts about the DT 4 virtual class in Section 2.1.In order to define the invariants, we require insertions. Let L be a line bundle on X and denote by L [n] the tautological (rank n) vector bundle over Hilb n (X) with fibre H 0 (L| Z ) over Z ∈ Hilb n (X). Then it makes sense to define the following: Definition 1.1. Let X be a smooth projective Calabi-Yau 4-fold and let L be a line bundle on X. Let L be the determinant line bundle of Hilb n (X) with quadratic form Q induced from Serre duality. Suppose L is given an orientation o(L). We define DT 4 (X, L, n ; o(L)) := [Hilb n (X)] vir o(L) e(L [n] ) ∈ Z, if n 1,where e(−) denotes the Euler class. We also set DT 4 (X, L, 0 ; o(L)) := 1.We make the following conjecture for the generating series of these invariants: 1 2 YALONG CAO AND MARTIJN KOOL Conjecture 1.2 (Conjecture 2.2). Let X be a smooth projective Calabi-Yau 4-fold and L be a line bundle on X. There exist choices of orientation such that ∞ n=0 DT 4 (X, L, n ; o(L)) q n = M (−q) X c1(L)·c3(X) , where

A short proof of the Göttsche conjecture MARTIJN KOOL VIVEK SHENDE RICHARD P THOMASWe prove that for a sufficiently ample line bundle L on a surface S , the number of ı -nodal curves in a general ı -dimensional linear system is given by a universal polynomial of degree ı in the four numbers L 2 ; L : K S ; K 2 S and c 2 .S /. The technique is a study of Hilbert schemes of points on curves on a surface, using the BPS calculus of Pandharipande and the third author [22] and the computation of tautological integrals on Hilbert schemes by Ellingsrud, Göttsche and Lehn [8].We are also able to weaken the ampleness required, from Göttsche's .5ı 1/-very ample to ı -very ample.

Recently, Cao-Maulik-Toda defined stable pair invariants of a compact Calabi-Yau 4-fold X. Their invariants are conjecturally related to the Gopakumar-Vafa type invariants of X defined using Gromov-Witten theory by Klemm-Pandharipande. In this paper, we consider curve counting invariants of X using Hilbert schemes of curves and conjecture a DT/PT correspondence which relates these to stable pair invariants of X. After providing evidence in the compact case, we define analogous invariants for toric Calabi-Yau 4-folds. We formulate a vertex formalism for both theories and conjecture a relation between the (fully equivariant) DT/PT vertex, which we check in several cases. This relation implies a DT/PT correspondence for toric Calabi-Yau 4-folds with primary insertions.