On the motive of the nested Quot scheme of points on a curve

Let $C$ be a smooth curve over an algebraically closed field $\mathbf{k}$, and let $E$ be a locally free sheaf of rank $r$. We compute, for every $d>0$, the generating function of the motives $[\mathrm{Quot}_C(E,\boldsymbol{n} )] \in K_0(\mathrm{Var}_{\mathbf{k}})$, varying $\boldsymbol{n} = (0\leq n_1\leq\cdots\leq n_d)$, where $\mathrm{Quot}_C(E,\boldsymbol{n} )$ is the nested Quot scheme of points, parametrising $0$-dimensional subsequent quotients $E \twoheadrightarrow T_d \twoheadrightarrow \cdots \twoheadrightarrow T_1$ of fixed length $n_i = \chi(T_i)$. The resulting series, obtained by exploiting the Bialynicki-Birula decomposition, factors into a product of shifted motivic zeta functions of $C$. In particular, it is a rational function.


INTRODUCTION
Let K 0 (Var k ) be the Grothendieck ring of varieties over an algebraically closed field k. If Y is a kvariety, its motivic zeta function Sym n Y q n ∈ K 0 (Var k ) q is a generating series introduced by Kapranov in [23], where he proved that for smooth curves it is a rational function in q .
In this paper we compute the motive of the nested Quot scheme of points Quot C (E , n ) on a smooth curve C , entirely in terms of ζ C (q ). Here, E is a locally free sheaf on C , and n = (0 ≤ n 1 ≤ · · · ≤ n d ) is a non-decreasing tuple of integers, for some fixed d > 0. The scheme Quot C (E , n ) generalises the classical Quot scheme of Grothendieck (recovered when d = 1): it parametrises flags of quotients E ։ T d ։ · · · ։ T 1 where T i is a 0-dimensional sheaf of length n i .
Our main result, proved in Theorem 4.2 in the main body, is the following.
Theorem A. Let C be a smooth curve over k, let E be a locally free sheaf of rank r on C . Then ζ C α−1 q i q i +1 · · ·q d ∈ K 0 (Var k ) q 1 , . . ., q d , where = [ 1 k ] is the Lefschetz motive. In particular, this generating function is rational in q 1 , . . ., q d .
The statement taken with d = 1, thus regarding the motive [Quot C (E , n)] of the usual Quot scheme of points, was proved in [1]. Our result is a natural generalisation, which was inspired by Mochizuki's paper on "Filt schemes" [24].
Our formula fits nicely in the philosophical path according to which "rank r theories factorise in r rank 1 theories".
There are to date a number of examples of this phenomenon in Donaldson-Thomas theory, exhibiting a generating series of rank r invariants as a product of r (suitably shifted) generating series of rank 1 invariants: see for instance [2,28] for enumerative DT invariants, [15] for K-theoretic DT invariants, [6,7] for motivic DT invariants and [26,14] for the parallel pictures in string theory. MATHEMATICAL  The paper is organised as follows. In Section 1 we introduce the nested Quot scheme and prove its connectedness. In Section 2 we describe its tangent space and prove that, for a smooth quasiprojective curve, the nested Quot scheme is smooth. Under the assumption that the locally free sheaf is split, in Section 3 we describe a torus framing action and its associated Białynicki-Birula decomposition. In Section 4 we prove that the motive of the nested Quot scheme is independent of the locally free sheaf, and exploit the Białynicki-Birula decomposition to prove Theorem A. Our result readily implies closed formulae for the generating series of Hodge-Deligne polynomials, χ y -genera, Poincaré polynomials, Euler characteristics, since these are all motivic measures; we provide some explicit formulae in Section 4.4.
After our paper was written, we were informed that our formula for the motive of the nested Quot scheme on a projective curve can be alternatively obtained, after some manipulations, from general results on the stack of iterated Hecke correspondences [17,Corollary 4.10] (see also [20,Section 3] for a related computation of the Voevodsky motive with rational coefficients). Our paper provides a direct and self-contained argument for this formula, exploiting the geometry of the nested Quot scheme.
where each T i ∈ Coh(X ) is a 0-dimensional quotient of E of length n i . The nested Quot scheme comes with a closed immersion In particular, it is projective as soon as X is projective. If C is a smooth proper curve over and E ∈ Coh(C ) is a locally free sheaf, the cohomology of Quot C (E , n ) was studied by Mochizuki [24].
Example 1.1. The classical Quot scheme Quot X (E , n) of length n quotients of E is obtained by setting n = (n), i.e. taking d = 1 and n d = n. If we set n = (1 ≤ 2 ≤ · · · ≤ d ), we obtain Mochizuki's complete Filt scheme Filt(E , d ), which for d = 1 reduce to Filt(E , 1) = (E ) [24]. When E = X , we use the notation Hilb n (X ) to denote Quot X ( X , n ). This space is the nested Hilbert scheme of points, studied extensively by Cheah [9,8,10].
1.2. Support map and nested punctual Quot scheme. Fix a variety X , a coherent sheaf E and a dtuple of non-negative integers n = (n 1 ≤ · · · ≤ n d ) for some d > 0. Composing the embedding (1.1) with the usual Quot-to-Chow morphisms yields the support map We make the following definition. Definition 1.2 (Nested punctual Quot scheme). Let X be a variety, x ∈ X a point, E ∈ Coh(X ) a coherent sheaf, n = (n 1 ≤ · · · ≤ n d ) a tuple of non-negative integers. The nested punctual Quot scheme attached to (X , E , n , x ) is the closed subscheme defined as the preimage of the cycle (n 1 x , . . ., n d x ) along the support map h E ,n .
The name 'punctual' refers, as for the classical Quot schemes, to the fact that all quotients are entirely supported at a single point. We will not need the following result. Lemma 1.3. Let X be a smooth quasiprojective variety of dimension m , and let E be a locally free sheaf of rank r on X . For every d -tuple n = (n 1 ≤ · · · ≤ n d ), and for every x ∈ X , one has a non-canonical isomorphism Proof. The result follows from the isomorphism Quot X (E , k ) x → Quot m ( ⊕r , k ) 0 relating the classical punctual Quot schemes, which is proved in full detail in [27, Section 2.1] exploiting a choice of étale coordinates around x (which exist by the smoothness assumption, and which explain the non-canonical nature of the isomorphism). It remains to observe that the induced isomorphism Quot m ( ⊕r , n i ) 0 ← → ∼ maps the subscheme Quot X (E , n ) x isomorphically onto Quot m ( ⊕r , n ) 0 .

Connectedness.
We prove the following connectedness result for the nested Quot scheme. A proof in the case (r, d , n ) = (1, 1, n) of the classical Hilbert scheme was first given by Hartshorne [19], and by Fogarty in the surface case [16]. We shall also exploit Cheah's connectedness result for Hilb n (X ), see [9, Sec. 0.4].
Theorem 1.4. If X is an irreducible quasiprojective k-variety and E is a locally free sheaf on X , then Quot X (E , n ) is connected for every n = (n 1 ≤ · · · ≤ n d ). In particular, the classical Quot scheme Quot X (E , n) is connected for every n ≥ 0.
Proof. The proof consists of several steps. STEP 1: We reduce to proving the statement when E = ⊕r x and x ′ are connected in Quot X (E , n ) by any point in this intersection. STEP 2: The scheme Quot X ( ⊕r X , n ) has a framing T-action with non-empty fixed locus, where T = r m (see Proposition 3.1 for an explicit description of this fixed locus: we shall exploit it in the next step). Let x ∈ Quot X ( ⊕r X , n ) be an arbitrary point. Then the closure of its orbit contains a T-fixed point -this will be explained in Section 3. Therefore it is enough to prove that any two T-fixed points where 1≤α≤r n α = n = 1≤α≤r n ′ α . But since each nested Hilbert scheme Hilb m (X ) is connected (cf. [9, Sec. 0.4]), we can in fact choose a pair of convenient x and x ′ . We fix them satisfying the condition that Supp(T d ), Supp(T ′ d ) consist of n d distinct points. When viewed in the full space Quot X ( ⊕r X , n ), the points x and x ′ both belong to the open subset where ∆ big ⊂ Sym n i X is the big diagonal and the bottom map is the support map (1.2). In other words, where the quotients T ′ i are supported on n i − n i −1 distinct points (and we set n 0 = 0). The scheme V i is the image On the other hand, is irreducible, hence A i is irreducible, and in particular V i is irreducible, being the image of an irreducible space along a continuous map. Therefore U → i V i is also irreducible, in particular connected, which completes the proof.

TANGENT SPACE AND SMOOTHNESS IN THE CASE OF CURVES
Fix (X , E , n ) as in the previous section. For any point We have a flag of subsheaves The embedding (1.1) induces a k-linear inclusion of tangent spaces which can be described as follows: a d -tuple of maps (δ 1 , . . ., commutes. This is formalised in terms of the map ∆ x in the next proposition.
In particular, if E is locally free of rank r on a smooth curve C , we have that Proof. Along the same lines of [29, Prop. 4.5.3(i)] it is easy to see that the tangent space is given by the maps making Diagram (2.1) commute, which is equivalent to belonging to the kernel of ∆ x . Let Q i be the 0-dimensional sheaf fitting in the exact sequences If X = C is a smooth curve, we have that each K i is a locally free sheaf of rank r (because torsion free is equivalent to locally free on smooth curves); since Q i is a 0-dimensional sheaf, we obtain the vanishings Therefore each ψ i is a surjective map, which implies that ∆ x is surjective and that the dimension of the tangent space is computed as Since the tangent space dimension is constant and Quot C (E , n ) is connected by Theorem 1.4, it is enough to find a smooth open subset U ⊂ Quot C (E , n ) of dimension r n d . We shall exploit the fact that the classical Quot scheme Quot C (E , m ) is smooth of dimension r m , which follows from standard deformation theory and the vanishing Ext 1  [25] for the classification of smoothness of Quot X (E , n ) when X has arbitrary dimension.

BIAŁYNICKI-BIRULA DECOMPOSITION
Let E be a locally free sheaf of rank r on a variety X . Assume that E = r α=1 L α splits into a sum of line bundles on X . Then Quot X (E , n ) admits the action of the algebraic torus T = r m as in [4]. Indeed, T acts diagonally on the product d i =1 Quot X (E , n i ) and the closed subscheme Quot X (E , n ) is T-invariant. Its fixed locus is determined by a straightforward generalisation of the main result of [4]. Quot X (L α , n α ).
Proof. We construct a bijection on k-valued points, which is straightforward to verify in families. Fix tuples n α = (n α,1 ≤ · · · ≤ n α,d ) such that n i = 1≤α≤r n α,i for every i = 1, . . ., d . An element of the connected component corresponding to (n 1 , . . ., n r ) in the right hand side is a tuple of nested quotients where each T (α) i is the structure sheaf of a finite subscheme of X of length n α,i . By Bifet's theorem on the T-fixed locus of ordinary Quot schemes [4], we have that for each i = 1, . . ., d , and since each of the original tuples of quotients was nested according to n , it follows that also the tuples (3.1) are nested according to n , and this proves that (3.1) defines a point in The reverse inclusion follows by an analogous reasoning relying once more on Bifet's result [4].
Assume now X = C is a smooth quasiprojective curve and let x ∈ Quot C (E , n ) T be a T-fixed point, corresponding to the tuple The tangent space at x can be written as Denote by w 1 , . . ., w r the coordinates of the algebraic torus T, which we see as irreducible T-characters. As a T-representation, the tangent space admits the following weight decomposition We recall the classical result of Białynicki-Birula (see [3,Section 4]), by which we obtain a decomposition of Quot X (E , n ) in the case when E is completely decomposable.

Theorem 3.3 (Białynicki-Birula)
. Let X be a smooth projective scheme with a m -action and let { X i } i be the connected components of the m -fixed locus X m ⊂ X . Then there exists a locally closed stratification X = i X + i , such that each X + i → X i is an affine fibre bundle. Moreover, for every closed point x ∈ X i , the tangent space is given by T where T x (X ) fix (resp. T x (X ) + ) denotes the m -fixed (resp. positive) part of T x (X ). In particular, the relative dimension of X + i → X i is equal to dim T x (X ) + for x ∈ X i . The Białynicki-Birula "strata" are constructed as follows. If t denotes the coordinate of m , we have In particular, the properness assumption assures that the closure of each m -orbit in X contains the m -fixed point lim t →0 t · x . Recently Jelisiejew-Sienkiewicz [22] generalised Theorem 3.3, proving the the X + i always exists even when X is not projective (or even not smooth). However, in the smooth case they cover X as long as the closure of every m -orbit contains a fixed point.
We now determine a Białynicki-Birula decomposition for Quot C (E , n ), where C is a smooth quasiprojective curve. See Mochizuki's paper [24,Section 2.3.4] for an equivalent construction and tangent space calculation (in the projective case), using a slightly different, but technically equivalent, tangent complex. 1 Let m → T be the generic 1-parameter subtorus given by w → (w, w 2 , . . ., w r ); it is clear that be the connected component of the fixed locus corresponding to the r -tuple n = (n α ) 1≤α≤r decomposing n 1 + · · · + n r = n . Proposition 3.4. Let C be a smooth quasiprojective curve and E = r α=1 L α . Then the nested Quot scheme admits a locally closed stratification where n = (n α ) 1≤α≤r are such that n 1 + · · · + n r = n and Q + n → Q n is an affine fibre bundle of relative dimension 1≤α≤r (α − 1)n α,d .
Proof. The strata Q + n are induced by Theorem 3.3 -we just need to show that the closure of every orbit contains a fixed point. Choose a compactification C → C , an extension L α of each line bundle L α and consider the induced open immersion The closure of every orbit must contain a fixed point in Quot C r α=1 L α , n , but the m -action does not move the support of a nested quotient, by which we conclude that such a fixed point had to be already contained in Quot C r α=1 L α , n . Let x ∈ Q n be a fixed point as in (3.3). The positive part of the tangent space (3.4) is x is the restriction of the map ∆ x . Thanks to the vanishings (2.2), ∆ + x is surjective, therefore the relative dimension is computed as i ) has rank 1. The proof is complete.

THE MOTIVE OF THE NESTED QUOT SCHEME ON A CURVE
4.1. Grothendieck ring of varieties. Let B be a scheme locally of finite type over k. The Grothendieck group of B -varieties, denoted K 0 (Var B ), is defined to be the free abelian group generated by isomorphism classes [X → B ] of finite type B -varieties, modulo the scissor relations, namely the identities The neutral element for the addition operation is the class of the empty variety. The operation The main rules for calculations in K 0 (Var k ) are the following: (1) If X → Y is a geometric bijection, i.e. a bijective morphism, then These are, indeed, the only properties that we will use. The Lefschetz motive is the class = [ 1 k ] ∈ K 0 (Var k ). It can be used to express, for instance, the class of the projective space, namely [ n k ] = 1 + + · · · + n ∈ K 0 (Var k ).

Independence of the vector bundle.
The following result generalises [27, Corollary 2.5], which in turn generalises the main theorem of [1] extending it from proper smooth curves to arbitrary smooth varieties.
Proposition 4.1. Let E be a locally free sheaf of rank r on a k-variety X . For every n , the motivic class of Quot X (E , n ) is independent of E , that is Proof. Let (U k ) 1≤k ≤e be a Zariski open cover trivialising E . We can refine it to a locally closed stratification X = W 1 ∐ · · ·∐ W e such that W k ⊂ U k , so that in particular E | W k = ⊕r W k for every k . Each W k is taken with the reduced induced scheme structure.
Let Quot X ,W k (E , n ) ⊂ Quot X (E , n ) be the preimage of Sym n d (W k ) ⊂ Sym n d (X ) along the projection where h E ,n is the support map (1.2). We endow Quot X ,W k (E , n ) with the reduced scheme structure. We have a geometric bijection It is enough to prove that these are independent of E . In the cartesian diagram the open immersion j is in fact surjective, hence an isomorphism. But we can repeat this process with ⊕r X in the place of E . It follows that Quot X ,W k (E , n k ) ∼ = Quot U k ,W k ( ⊕r U k , n k ) ∼ = Quot X ,W k ( ⊕r X , n k ), which yields the result.

Proof of the main theorem.
Let X be a smooth quasiprojective variety and E a locally free sheaf of rank r . Define Z X ,r,d (q ) = n Quot X (E , n ) q n ∈ K 0 (Var k ) q 1 , . . ., q d , where n = (n 1 ≤ · · · ≤ n d ) and we use the multivariable notation q = (q 1 , . . ., q d ) and q n = d i =1 q n i . The notation Z X ,r,d reflects the independence on E that we proved in Proposition 4.1. If X = C is a smooth quasiprojective curve and r = d = 1, then Z C ,1,1 (q ) is simply the Kapranov motivic zeta function (4.1) Sym n (C ) q n .
We can now prove our main theorem, first stated in Theorem A in the Introduction.
In particular, Z C ,r,d (q ) is a rational function in q 1 , . . ., q d .
Proof. By Proposition 4.1 the motive [Quot C (E , n )] is independent on the vector bundle E , so we may assume E = ⊕r C . In this case, we may compute the motive exploiting the decomposition of Quot C ( ⊕r C , n ) given by Proposition 3.4. Every stratum is a Zariski locally trivial fibration over a connected component of the fixed locus, with fibre an affine space whose dimension we computed in Proposition 3.4. In what follows, we denote by n α = (n α,1 ≤ · · · ≤ n α,d ) a nested tuple of non-negative integers and by l α = (l α,1 , . . ., l α,d ) a tuple of non-negative integers. Clearly the two collections of tuples are in bijection, by means of the correspondence (4.2) (n α,1 ≤ · · · ≤ n α,d ) ←→ (n α,1 , n α,2 − n α,1 , . . ., n α,d − n α,d −1 ).

Remark 4.3.
We can reformulate our main theorem in terms of the motivic exponential, for which a minimal background is provided in Appendix A. The case r = d = 1 yields the classical expression The general case becomes This is actually the formula defining the power structure on [u, v ]. The motivic measure E can be proved to be a morphism of rings with power structure, see [18] for full details.
Let C be a smooth projective curve of genus g . We have For E a locally free sheaf of rank r over C , define As a direct consequence of Theorem 4.2, we obtain the following corollary.

Corollary 4.5.
There is an identity Proof. This follows by combining Theorem 4.2 and Equation (4.3) with one another, after observing that E is multiplicative (being a ring homomorphism) and sends to u v .
The generating function of the signed Poincaré polynomials is obtained from E C ,r,d (q ) by setting u = v . The result confirms a result of L. Chen [11] obtained in the case C = 1 . The generating series of topological Euler characteristics is obtained from E C ,r,d (q ) by setting u = v = 1, also in the quasiprojective case. So we obtain n e top (Quot C (E , n ))q n = d i =1 1 − q i q i +1 · · ·q d −r ·e top (C ) .