Motivic degree zero Donaldson–Thomas invariants

Abstract

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

An Introduction to (Motivic) Donaldson-Thomas Theory

“…-Instead of classifying objects up to isomorphism, we could allow weaker equivalences. For example, we could identify to families V (1) and V (2) (over S) of vector spaces or representations of an algebra A if there is a line bundle L on S such that…”

“…Every vector bundle V of rank r provides such a bundle by taking P := P(V ), the bundle of lines or hyperplanes in V . Two vector bundles V (1) , V (2) define isomorphic bundles P(V (1) ) ∼ = P(V (2) ) if and only if…”

“…Hint: Relate the slope of V / ker(f ) = im(f ) to µ(V (1) ) and to µ(V (2) ) by drawing the central charges of all objects involved. (2) Using the notation of the first part, let us assume µ(V (1) ) = µ(V (2) ) for the semistable representations V (1) , V (2) . Show that ker(f ) and coker(f ) are also semistable of the same slope µ(V (1) ).…”

“…(2) Use Exercise 3.12(1) to construct a Harder-Narasimhan filtration. Hint: Let V (1) be the subrepresentation constructed in the first step, and let V (2) be the preimage of a maximal subrepresentation in V /V (1) of maximal slope. Proceed in this way, and use the previous exercise to estimate the slopes.…”

“…The case of zero potential has been intensively studied by M. Reineke in a series of papers [30,31,32]. Despite some computations of motivic or even numerical Donaldson-Thomas invariants for quivers with potential (see [2,6,4,27]), the true nature of Donaldson-Thomas invariants for quiver with potential remained mysterious for quite some time. A full understanding has been obtained recently and is the content of a series of papers [7,5,26,24].…”