We suggest a construction of virtual fundamental classes of certain types of moduli spaces.Comment: LaTeX, Postscript file available at http://www.math.ubc.ca/people/faculty/behrend/inc.p

We prove that Donaldson-Thomas type invariants are equal to weighted Euler characteristics of their moduli spaces. In particular, such invariants depend only on the scheme structure of the moduli space, not the symmetric obstruction theory used to define them. We also introduce new invariants generalizing Donaldson-Thomas type invariants to moduli problems with open moduli space. These are useful for computing Donaldson-Thomas type invariants over stratifications.

Gromov-Witten invariants for arbitrary projective varieties and arbitrary genus are constructed using the techniques from K. Behrend, B. Fantechi: The intrinsic normal cone.Comment: LaTeX, Postscript file available at http://www.math.ubc.ca/people/faculty/behrend/gwag.p

Given a smooth complex threefold X, we define the virtual motive [Hilb n (X)] vir of the Hilbert scheme of n points on X. In the case when X is Calabi-Yau, [Hilb n (X)] vir gives a motivic refinement of the n-point degree zero Donaldson-Thomas invariant of X. The key example is X = C 3 , where the Hilbert scheme can be expressed as the critical locus of a regular function on a smooth variety, and its virtual motive is defined in terms of the Denef-Loeser motivic nearby fiber. A crucial technical result asserts that if a function is equivariant with respect to a suitable torus action, its motivic nearby fiber is simply given by the motivic class of a general fiber. This allows us to compute the generating function of the virtual motives [Hilb n (C 3 )] vir via a direct computation involving the motivic class of the commuting variety. We then give a formula for the generating function for arbitrary X as a motivic exponential, generalizing known results in lower dimensions. The weight polynomial specialization leads to a product formula in terms of deformed MacMahon functions, analogous to Göttsche's formula for the Poincaré polynomials of the Hilbert schemes of points on surfaces..See §1.5 for the definition of Exp. While the above formula only applies when dim(X) = 3, it fits well with corresponding formulas for dim(X) < 3. In these cases, the Hilbert schemes are smooth and thus have canonical virtual motives which are easily expressed in terms of the ordinary classes [Hilb n (X)] in the Grothendieck group. The resulting partition functions have been computed for curves [17] and surfaces [15], and all these results can be expressed (Corollary 3.4) in the single formula Z X (T ) = Exp T [X] vir Exp T [P d−2 ] vir valid when d = dim(X) is 0, 1, 2, or 3. Hereand the class of a negative dimensional projective space is defined by (3.3). In particular, [P −1 ] vir = 0 and [P −2 ] vir = −1. There are some indications that the above formula has significance for dim X > 3; see Remarks 3.5 and 3.6. The weight polynomial specialization of the class of a projective manifold gives its Poincaré polynomial. For example, if X is a smooth 1 Note that the variable "t" has special meaning in the definition of "Exp"; in particular, one cannot simply substitute t for T in the above equation for Z X .1.3. Relative motivic weights. Given a reduced (but not necessarily irreducible) variety S, let K 0 (Var S ) be the Z-module generated by isomorphism classes of (reduced) S-varieties, under the scissor relation for S-varieties, and ring structure whose multiplication is given by fiber product over S. Elements of this ring will be denoted [X] S . A morphism f : S → T induces a ring homomorphism f * : K 0 (Var T ) → K 0 (Var S ) given by fiber product. In particular, K 0 (Var S ) is always a K 0 (Var C )-module. Thus we can let

We introduce differentiable stacks and explain the relationship with Lie groupoids. Then we study S 1 -bundles and S 1 -gerbes over differentiable stacks. In particular, we establish the relationship between S 1 -gerbes and groupoid S 1 -central extensions. We define connections and curvings for groupoid S 1 -central extensions extending the corresponding notions of Brylinski, Hitchin and Murray for S 1 -gerbes over manifolds. We develop a Chern-Weil theory of characteristic classes in this general setting by presenting a construction of Chern classes and Dixmier-Douady classes in terms of analogues of connections and curvatures. We also describe a prequantization result for both S 1 -bundles and S 1 -gerbes extending the well-known result of Weil and Kostant. In particular, we give an explicit construction of S 1 -central extensions with prescribed curvature-like data.2 folklore (see [15,34,39,40]). However, we feel that it is useful to spell it out in detail in the differentiable geometry setting, which is of ultimate interest for our purpose.Our main goal of this paper is to develop the theory of S 1 -gerbes over differentiable stacks. Motivation comes from string theory in which "gerbes with connections" appear naturally [13,16,23,46]. For S 1 -gerbes over manifolds, there has been extensive work on this subject pioneered by Brylinski [5], Chatterjee [8], Hitchin [21], Murray [32] and many others. Also, there is interesting work on equivariant S 1 -gerbes, e.g., by Brylinski [6], Meinrenken [29], Gawedzki-Neis [17], Stienon [41] and others, as well as on gerbes over orbifolds [27]. These endeavors make the foundations of gerbes over differentiable stacks a very important subject. An important step is to geometrically realize a class H 2 (X, S 1 ) (or H 3 (X, Z) when X is Hausdorff). Such a geometrical realization is crucial in applications to twisted K-theory [43,44,45].Our method is to use the dictionary mentioned above, under which we show that S 1 -gerbes are in one-to-one correspondence with Morita equivalence classes of groupoid S 1 -central extensions. Thus it follows from a well-known theorem of Giraud [19] that there is a bijection between H 2 (X, S 1 ) and Morita equivalence classes of Lie groupoid S 1 -central extensions. We note that there are several independent investigations of similar topics; see [7,36,37,42,50].An S 1 -central extension of a Lie groupoid X 1 ⇉ X 0 is a Lie groupoid R 1 ⇉ X 0 with a groupoid morphism π : R 1 → X 1 such that ker π ∼ = X 0 × S 1 lies in the center of R 1 . It is easy to see that π : R 1 → X 1 is then naturally an S 1 -principal bundle. A standard example is an S 1 -central extension of aČech groupoid: Let N be a manifold and α ∈ H 3 (N, Z), and let {U i } be a good covering of N . Then the groupoidwhich is called theČech groupoid, is Morita equivalent to the manifold N . Then the S 1 -gerbe corresponding to the class α can be realized as anare the same point x in the three-intersection U ijk considered as elements in the two-intersections, and c ijk : U ijk → S ...

We construct the motivic tree-level system of Gromov-Witten invariants for convex varieties.

Recall that in an earlier paper by one of the authors Donaldson-Thomas type invariants were expressed as certain weighted Euler characteristics of the moduli space. The Euler characteristic is weighted by a certain canonical Z-valued constructible function on the moduli space. This constructible function associates to any point of the moduli space a certain invariant of the singularity of the space at the point.In the present paper, we evaluate this invariant for the case of a singularity which is an isolated point of a C * -action and which admits a symmetric obstruction theory compatible with the C * -action. The answer is (−1) d , where d is the dimension of the Zariski tangent space.We use this result to prove that for any threefold, proper or not, the weighted Euler characteristic of the Hilbert scheme of n points on the threefold is, up to sign, equal to the usual Euler characteristic. For the case of a projective Calabi-Yau threefold, we deduce that the Donaldson-Thomas invariant of the Hilbert scheme of n points is, up to sign, equal to the Euler characteristic. This proves a conjecture of Maulik-Nekrasov-Okounkov-Pandharipande.

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