2.1 Mixed t-categories

For a subcategory $\mathcal{S}\subseteq \mathcal{D}$ (or a collection of subcategories) of a triangulated category $\mathcal{D}$ , let $\left \langle \mathcal{S}\right \rangle \subseteq \mathcal{D}$ be the minimal full subcategory that contains $\mathcal{S}\cup \left \{0\right \}$ and is closed under extensions. A t-structure $(\mathcal{D}^{\le 0},\mathcal{D}^{\geqslant 0})$ [Reference Beilinson, Bernstein and DeligneBBD82] on a triangulated category $\mathcal{D}$ is called bounded if $\mathcal{D}=\left \langle \mathcal{A}[n]:n\in \mathbb{Z}\right \rangle$ for the heart $\mathcal{A}=\mathcal{D}^{\le 0}\cap \mathcal{D}^{\geqslant 0}$ . For $M\in \mathcal{D}$ , we define the cohomology object $H^iM=\tau _{\le 0}\tau _{\geqslant 0}(M[i])\in \mathcal{A}$ . A (bounded) t-category is a triangulated category equipped with a (bounded) t-structure.

For $M\in \mathcal{A}$ , we write $w(M)\leqslant n$ if ${\rm Gr}^W_kM=0$ for $k\gt n$ and we write $w(M)\geqslant n$ if ${\rm Gr}^W_kM=0$ for $k\lt n$ . We define

\begin{align*}w(M)=\sup \left \{n\in \mathbb{Z}\,\middle |\,{\rm Gr}^W_nM\ne 0\right \}\in \mathbb{Z}\sqcup \left \{-\infty \right \}.\end{align*}

For $M\in \mathcal{D}$ , we write $w(M)\leqslant n$ if $w(H^iM)\leqslant n+i$ for all $i\in \mathbb{Z}$ and we write $w(M)\geqslant n$ if $w(H^iM)\geqslant n+i$ for all $i\in \mathbb{Z}$ . We define

\begin{align*}w(M)=\max \left \{w(H^iM)-i\,\middle |\,i\in \mathbb{Z}\right \}\in \mathbb{Z}\sqcup \left \{-\infty \right \}.\end{align*}

Let

\begin{gather*} \mathcal{D}_{\leqslant n}=\left \{M\in \mathcal{D}\,\middle |\,w(M)\leqslant n\right \}=\left \langle \mathcal{A}_m[i]:m+i\leqslant n\right \rangle ,\\[5pt] \mathcal{D}_{\geqslant n}=\left \{M\in \mathcal{D}\,\middle |\,w(M)\geqslant n\right \}=\left \langle \mathcal{A}_m[i]:m+i\geqslant n\right \rangle . \end{gather*}

Then $\mathcal{D}_{\leqslant n}[1]=\mathcal{D}_{\leqslant n+1}$ , $\mathcal{D}_{\geqslant n}[1]=\mathcal{D}_{\geqslant n+1}$ and

\begin{align*}{\rm Hom}(\mathcal{D}_{\leqslant m},\mathcal{D}_{\geqslant n})=0\quad \forall m\lt n.\end{align*}

2.2 Exactness

An additive functor $F:\mathcal{D}_1\to \mathcal{D}_2$ between triangulated categories is called exact (or triangulated) if it commutes with translations and preserves distinguished triangles. For an exact functor $F:\mathcal{D}_1\to \mathcal{D}_2$ between t-categories with hearts $\mathcal{A}_1,\mathcal{A}_2$ , we define

(6) \begin{align} {}^{{\rm p}}{F}:\mathcal{A}_1\to \mathcal{A}_2,\quad M\mapsto H^0FM. \end{align}

An exact functor $F:\mathcal{D}_1\to \mathcal{D}_2$ between t-categories is called right (respectively left) t-exact if $F(\mathcal{D}_1^{\le 0})\subseteq \mathcal{D}_2^{\le 0}$ (respectively $F(\mathcal{D}_1^{\geqslant 0})\subseteq \mathcal{D}_2^{\geqslant 0}$ ). We say that $F$ has t-amplitude $[a,b]$ if $F(\mathcal{D}_1^{\le 0})\subseteq \mathcal{D}_2^{\leqslant b}$ and $F(\mathcal{D}_1^{\geqslant 0})\subseteq \mathcal{D}_2^{\geqslant a}$ .

An exact functor $F:\mathcal{D}_1\to \mathcal{D}_2$ between mixed t-categories is called right (respectively left) w-exact if $F(\mathcal{D}_{1,\le 0})\subseteq \mathcal{D}_{2,\le 0}$ (respectively $F(\mathcal{D}_{1,\geqslant 0})\subseteq \mathcal{D}_{2,\geqslant 0}$ ). The following result is Deligne’s generalization of Weil’s conjectures in the $\ell$ -adic context [Reference DeligneDel80] and Saito’s theorem in the ${\rm MHM}$ context [Reference SaitoSai90].

Remark 2.10. The external tensor product [Reference SaitoSai90]

\begin{align*}\boxtimes :{\rm MHM}(X)\times {\rm MHM}(Y)\to {\rm MHM}(X\times Y)\end{align*}

is w-exact, meaning that if $M\in {\rm MHM}(X)$ and $N\in {\rm MHM}(Y)$ are pure, then $M\boxtimes N$ is pure of weight $w(M)+w(N)$ . The induced external tensor product on the derived categories is automatically t-exact (and w-exact). On the other hand, the tensor product $\otimes ^*$ on $D^b{\rm MHM}(X)$ , defined by $M\otimes ^* N=\Delta ^*(M\boxtimes N)$ for the diagonal $\Delta :X\to X\times X$ , is right t-exact and right w-exact (cf. [Reference Beilinson, Bernstein and DeligneBBD82, 5.1.14]). The duality functor $\mathbb{D}:{\rm MHM}(X)\to {\rm MHM}(X)^{{\rm op}}$ maps objects of weight $n$ to objects of weight $-n$ (actually $\mathbb{D} M\simeq M(n)$ for any pure object $M$ of weight $n$ ).

2.3 Mixed Hodge modules

For an algebraic variety $X$ over $\mathbb{C}$ , let $d_X=\dim X$ denote the maximal dimension of its irreducible components. Let ${\rm MHM}(X)$ be the category of mixed Hodge modules over $X$ . Let $D^b_c(\mathbb{Q}_X)$ be the derived category of bounded $\mathbb{Q}$ -complexes with constructible cohomology, and let ${\rm Perv}(\mathbb{Q}_X)\subset D^b_c(\mathbb{Q}_X)$ be the subcategory of perverse sheaves. The exact functor ${\rm rat}:D^b{\rm MHM}(X)\to D^b_c(\mathbb{Q}_X)$ maps ${\rm MHM}(X)$ to ${\rm Perv}(\mathbb{Q}_X)$ and is compatible with Grothendieck’s six operations on both sides. We have ${\rm MHM}({\bf pt})={\rm MHS}^p(\mathbb{Q})$ , the category of (graded-)polarizable mixed $\mathbb{Q}$ -Hodge structures.

Let $\mathbb{Q}(n)\in {\rm MHM}({\bf pt})$ be the Tate object of weight $-2n$ for $n\in \mathbb{Z}$ . It induces the Tate twist functor ${\rm MHM}(X)\to {\rm MHM}(X)$ , $M\mapsto M(n)=M\otimes \mathbb{Q}(n)$ . We consider the Lefschetz object

(7) \begin{align} \mathbb{L}=H^*_c(\mathbb{A}^1,\mathbb{Q})=\mathbb{Q}(-1)[-2]\in D^b{\rm MHM}({\bf pt}) \end{align}

of weight $0$ and the corresponding functor

(8) \begin{align} \mathbb{L}:D^b{\rm MHM}(X)\to D^b{\rm MHM}(X),\quad M\mapsto M(-1)[-2]. \end{align}

Note that $H^*_c(\mathbb{P}^n,\mathbb{Q})=\mathbb{Q}\oplus \mathbb{L}\oplus \dots \oplus \mathbb{L}^n$ for $n\geqslant 0$ .

In what follows, we will require the class $\mathbb{L}^{1/2}$ at the level of Grothendieck groups (see Section 2.4), but it will also be convenient to have a related object at the level of categories. We have two approaches to this problem. The first one is to embed ${\rm MHM}(X)$ into the category of monodromic mixed Hodge modules ${\rm MMHM}(X)$ (see [Reference Kontsevich and SoibelmanKS11, Reference Davison and MeinhardtDM20]), so that one has the objects $\mathbb{Q}(1/2)\in {\rm MMHM}({\bf pt})$ and $\mathbb{L}^{1/2}=\mathbb{Q}(-1/2)[-1]\in D^b{\rm MMHM}({\bf pt})$ and the corresponding twist functors. The second approach is to construct the root category $\mathcal{A}^{1/2}$ and the functor $\mathbb{T}^{1/2}=\mathbb{Q}(1/2):\mathcal{A}^{1/2}\to \mathcal{A}^{1/2}$ for the category $\mathcal{A}={\rm MHM}(X)$ and the functor $\mathbb{T}=\mathbb{Q}(1):\mathcal{A}\to \mathcal{A}$ . More generally, we construct the root category $\mathcal{A}^{1/r}$ and the functor $\mathbb{T}^{1/r}:\mathcal{A}^{1/r}\to \mathcal{A}^{1/r}$ for any $r\geqslant 1$ as follows. We define $\mathcal{A}^{1/r}=\bigoplus _{i=0}^{r-1}\mathcal{A}^{(i)}$ with $\mathcal{A}^{{(i)}}=\mathcal{A}$ and the functor $\mathbb{T}^{1/r}$ sending $\mathcal{A}^{(i)}\ni M\mapsto M\in \mathcal{A}^{(i+1)}$ for $0\leqslant i\lt r-1$ and $\mathcal{A}^{(r-1)}\ni M\mapsto \mathbb{T} M\in \mathcal{A}^{(0)}$ . We embed $\mathcal{A}=\mathcal{A}^{(0)}\hookrightarrow \mathcal{A}^{1/r}$ so that $(\mathbb{T}^{1/r})^r=\mathbb{T}$ . The weight of $M\in \mathcal{A}^{(i)}\subset \mathcal{A}^{1/2}$ is defined to be $w(M)-i$ (if $M\in \mathcal{A}$ is pure) so that $\mathbb{T}^{1/2}$ has weight $-1$ and $\mathbb{T}=\mathbb{Q}(1)$ has weight $-2$ as before. We can also define the functor $\mathbb{Q}(n/2)$ for any $n\in \mathbb{Z}$ . The same construction can be applied to $D^b{\rm MHM}(X)$ and we define $\mathbb{L}^{1/2}=\mathbb{Q}(-1/2)[-1]$ . By abuse of notation we will continue to write ${\rm MHM}(X)$ instead of ${\rm MHM}(X)^{1/2}$ . We define

(9) \begin{align} M\left \langle n\right \rangle =M(n/2)[n]=\mathbb{L}^{-n/2}M,\qquad M\in D^b{\rm MHM}(X). \end{align}

Note that the weight of $M\left \langle n\right \rangle$ is equal to the weight of $M$ .

Let $X$ be an irreducible algebraic variety of dimension $d$ . The object

\begin{align*}\mathbb{Q}_X=a_X^*\mathbb{Q}\in D^b{\rm MHM}(X),\quad a_X:X\to {\bf pt},\end{align*}

has weight $\le 0$ . If $X$ is smooth, then $\mathbb{Q}_X$ has weight 0 and $\mathbb{Q}_X[d]\in {\rm MHM}(X)$ . Generally, we define the intersection complex

(10) \begin{align} {\bf IC}_X =j_{!*}(\mathbb{Q}_U\left \langle d\right \rangle )\in {\rm MHM}(X), \end{align}

where $j:U\hookrightarrow X$ is an embedding of a nonempty, open, smooth subvariety and

(11) \begin{align} j_{!*}(M)=\Im ({}^{{\rm p}}j_! M\to {}^{{\rm p}}j_* M)\in {\rm MHM}(X),\quad M\in {\rm MHM}(U). \end{align}

The object ${\bf IC}_X$ is self-dual (meaning that $\mathbb{D}{\bf IC}_X\simeq {\bf IC}_X$ ) and has weight $0$ . The object ${\rm IC}_X={\rm rat}({\bf IC}_X)$ is the perverse intersection complex in ${\rm Perv}(\mathbb{Q}_X)$ . We define the intersection cohomology

(12) \begin{align} {\rm IH}^*(X,\mathbb{Q}) =H^*(X,{\bf IC}_X\left \langle -d\right \rangle ) \end{align}

so that ${\rm IH}^*(X,\mathbb{Q})=H^*(X,\mathbb{Q})$ for smooth $X$ .

2.4 Degrees

For a graded abelian group $V=\bigoplus _{i\in \mathbb{Z}}V_i$ and $x=\sum _{i\in \mathbb{Z}} x_i\in V$ with $x_i\in V_i$ , let

(13) \begin{align} \deg (x)=\sup \left \{i\,\middle |\,x_i\ne 0\right \}\in \mathbb{Z}\sqcup \left \{-\infty \right \}. \end{align}

If $n=\deg (x)$ is finite, we call $x_n$ the leading term of $x$ . If $A$ is a graded integral domain and $V$ is a graded torsion-free $A$ -module, then

(14) \begin{align} \deg (ax)=\deg (a)+\deg (x)\quad \forall a\in A,\, x\in V. \end{align}

We extend the degree map to $\mathcal{F}(A)\otimes _A V$ , where $\mathcal{F}(A)$ is the fraction field of $A$ , by the formula

(15) \begin{align} \deg (x/a)=\deg (x)-\deg (a)\quad \forall a\in A\backslash \left \{0\right \},\, x\in V. \end{align}

For an abelian category $\mathcal{A}$ (respectively a triangulated category $\mathcal{D}$ ), let $K(\mathcal{A})$ (respectively $K(\mathcal{D})$ ) denote its Grothendieck group. Let $[M]\in K(\mathcal{D})$ denote the class of $M\in \mathcal{D}$ . If $\mathcal{D}$ is a bounded t-category with the heart $\mathcal{A}$ , then the canonical map $K(\mathcal{A})\to K(\mathcal{D})$ is an isomorphism. If $\mathcal{D}$ is a mixed t-category with the heart $\mathcal{A}$ , then the Grothendieck group $K(\mathcal{A})\simeq K(\mathcal{D})$ is graded, where we define $\deg [M]=n$ for a pure object $M\in \mathcal{A}$ of weight $n$ . If $\mathcal{A}$ is of finite length and $C_n$ denotes the set of isomorphism classes of simple objects in $\mathcal{A}_n$ , then $K(\mathcal{A})\simeq \bigoplus _{n\in \mathbb{Z}}\mathbb{Z} C_n$ .

In particular, the Grothendieck group $K({\rm MHM}(X))\simeq K(D^b{\rm MHM}(X))$ is a graded module over the graded ring $K({\rm MHM}({\bf pt}))$ (cf. Section 3.3). Let

(16) \begin{align} \mathbb{L}=[H_c^*(\mathbb{A}^1,\mathbb{Q})]=[\mathbb{Q}(-1)]\in K({\rm MHM}({\bf pt})){\rm and} \end{align}

(17) \begin{align} \mathbf K({\rm MHM}(X))=K({\rm MHM}(X))\otimes _{\mathbb{Z}[\mathbb{L}]}\mathbb{Z}[\mathbb{L}^{\pm 1/2},(1-\mathbb{L}^n)^{-1}\mid n\geqslant 1]. \end{align}

Note that $\mathbb{L}$ has degree $2$ . We extend the degree function to $\mathbf K({\rm MHM}(X))$ so that $\mathbb{L}^{1/2}$ has degree $1$ . For $M\in D^b({\rm MHM}(X))$ , let $\deg (M)=\deg [M]$ . For example, $\deg (\mathbb{Q}[n])=0$ , while $w(\mathbb{Q}[n])=n$ for $n\in \mathbb{Z}$ . The duality functor $\mathbb{D}$ on ${\rm MHM}(X)$ induces the group homomorphism $\mathbb{D}:\mathbf K({\rm MHM}(X))\to \mathbf K({\rm MHM}(X))$ such that $\mathbb{D}(\mathbb{L}^n a)=\mathbb{L}^{-n}\mathbb{D}(a)$ for $n\in {\frac 12}\mathbb{Z}$ and $a\in \mathbf K({\rm MHM}(X))$ . The class $[{\bf IC}_X]\in \mathbf K({\rm MHM}(X))$ is self-dual (meaning that $\mathbb{D}[{\bf IC}_X]=[{\bf IC}_X]$ ).

2.5 Decompositions

We say that an object $M\in {\rm MHM}(X)$ is supported on a locally closed subvariety $Z\subseteq X$ if $M=j_{!*}{}^{{\rm p}}j^*M$ for the embedding $j:Z\hookrightarrow X$ , meaning that the canonical map $M\to {}^{{\rm p}}j_*{}^{{\rm p}}j^*M$ induces an isomorphism $M\xrightarrow {\sim } j_{!*}{}^{{\rm p}}j^*M\hookrightarrow {}^{{\rm p}}j_*{}^{{\rm p}}j^*M$ . For a simple object $M$ , this means that $Z$ contains a nonempty open subset of $\textrm {supp} M$ . We say that $M\in D^b{\rm MHM}(X)$ is supported on $Z$ if every $H^iM$ is supported on $Z$ . In what follows, a partition of $X$ means a finite collection $\mathcal{S}$ of locally closed subsets of $X$ such that $X=\bigsqcup _{S\in \mathcal{S}}S$ .

2.6 Defects

Let $\pi :X\to Y$ be a projective morphism between algebraic varieties. For $i\in \mathbb{N}\sqcup \left \{-\infty \right \}$ , let $Y_i=\left \{y\in Y\,\middle |\,\dim \pi ^{-1}(y)=i\right \}$ (note that $\dim \varnothing =-\infty$ ). The subsets $Y_{\leqslant k}=\bigsqcup _{i\leqslant k}Y_i$ are open in $Y$ . For a subvariety $Z\subseteq Y$ , we define the (fibre) defect (cf. [Reference Goresky and MacPhersonGM88, Reference de Cataldo and MigliorinidCM03])

(18) \begin{align} \delta (\pi ,Z)=\max \left \{2i+d_{Z\cap Y_i}-d_X\,\middle |\,i\geqslant 0\right \}. \end{align}

If $\mathcal{S}$ is a finite partition of $Y$ such that the fibres of $\pi$ over $S\in \mathcal{S}$ have dimension $c_S$ , then

(19) \begin{align} \delta (\pi ,Z)=\max _{S\in \mathcal{S}}{\delta (\pi ,Z\cap S)},\quad \delta (\pi ,Z\cap S)=2c_S+d_{Z\cap S}-d_X. \end{align}

Note that $\delta (\pi )=\delta (\pi ,Y)=d_{X\times _YX}-d_X\geqslant 0$ . A (surjective) morphism $\pi$ is called semismall if $\delta (\pi )=0$ and small if $\delta (\pi )=0$ and $\delta (\pi ,Y_i)=0$ only for $i=0$ .

In what follows, let $\mathcal{D}$ denote the t-category $D^b{\rm MHM}(X)$ with the standard t-structure or the t-category $D^b_c(\mathbb{Q}_X)$ with the perverse t-structure. These t-structures are compatible under the functor ${\rm rat}:D^b{\rm MHM}(X)\to D^b_c(\mathbb{Q}_X)$ . For $K\in D^b_c(\mathbb{Q}_X)$ , let $\mathsf{H}^iK\in {\rm Sh}(\mathbb{Q}_X)$ denote the $i$ th cohomology sheaf and ${}^{{\rm p}}H^iK\in {\rm Perv}(\mathbb{Q}_X)$ denote the $i$ th cohomology object with respect to the perverse t-structure.

Proof. By Lemma 2.17 we can assume that $Z\subset Y_k$ for some $k\geqslant 0$ . Consider the following Cartesian square.

An object $K\in D^b_c(\mathbb{Q}_X)$ is contained in ${}^{{\rm p}}D^{\leqslant n}_c(\mathbb{Q}_X) \iff \dim (\textrm {supp} \mathsf{H}^iK)\leqslant -i+n$ for all $i\in \mathbb{Z}$ . We have $j^*\pi _*{\rm IC}_X=\pi '_*j'^*\mathbb{Q}_X[d_X]$ . The object $K=j'^*\mathbb{Q}_X[d_X]$ satisfies $\dim (\textrm {supp} \mathsf{H}^iK)=d_{X'}=-i+(d_{X'}-d_X)$ for $i=-d_X$ and $\mathsf{H}^i K=0$ otherwise. This implies that $K\in {}^{{\rm p}}D^{\leqslant d_{X'}-d_X}_c(\mathbb{Q}_X)$ . The functor $\pi '_*$ has amplitude $[-k,k]$ . Therefore $j^*\pi _*{\rm IC}_X =\pi '_*K \in {}^{{\rm p}}D^{\leqslant k+d_{X'}-d_X} ={}^{{\rm p}}D^{\le \delta (\pi ,Z)}(\mathbb{Q}_X).$ The functor ${\rm rat}:D^b{\rm MHM}(X)\to D^b_c(\mathbb{Q}_X)$ sends $j^*\pi _*{\bf IC}_X$ to $j^*\pi _*{\rm IC}_X$ and preserves t-structures, and hence $j^*\pi _*{\bf IC}_X\in D^{\leqslant \delta (\pi ,Z)} {\rm MHM}(X)$ .

2.7 The leading term

If $\pi :X\to Y$ is a projective morphism and is a (topological) fibre bundle with all fibres irreducible and of dimension $n$ , then (cf. [Reference Göttsche and SoergelGS93])

(20) \begin{align} H^{n}\pi _*(\mathbb{Q}_X[d_X])\simeq \mathbb{Q}_Y[d_Y](-n). \end{align}

If $X$ and $Y$ are smooth (and irreducible), we obtain (cf. Appendix A)

(21) \begin{align} H^n \pi _*{\bf IC}_X\simeq {\bf IC}_Y(-n/2). \end{align}

We prove the following well-known result (cf. [Reference Göttsche and SoergelGS93, Theorem 5]) for completeness.

Proof. We have $M=\pi _*{\bf IC}_X\in \mathcal{D}^{\le 0}$ by Lemma 2.18. As $M$ is self-dual, we conclude that $M\in {\rm MHM}(X)$ . By Lemma 2.12 we have $M=\bigoplus _S {}^{{\rm p}}j_{S!*}{}^{{\rm p}}j^*_S M$ . If $\delta (\pi ,S)\lt 0$ , then $j_S^*M\in \mathcal{D}^{\lt 0}$ , and hence ${}^{{\rm p}}j_S^*M=0$ . Let us assume that $\delta (\pi ,S)=0$ and let $n=d_X-d_{X'}$ be the fibre dimension, where $X'=\pi ^{-1}(S)$ . Then

\begin{align*} {}^{{\rm p}}j_S^*M =H^0(\pi '_{*}\mathbb{Q}_{X'}[d_X](d_X/2)) =H^{n}(\pi '_{*}\mathbb{Q}_{X'}[d_{X'}])(d_X/2) \simeq \mathbb{Q}_S[d_S](d_X/2-n)={\bf IC}_S, \end{align*}

where we applied (20) to $\pi ':X'\to S$ .

3.1 Pre- $\lambda$ -rings of motivic functions

Let ${\rm Sch}={\rm Sch}_K$ be the category of algebraic schemes over a field $K$ of characteristic zero, by which we mean schemes $X$ equipped with a finite type morphism $a_X:X\to {\bf pt}={\bf Spec}(K)$ . For a scheme (or an Artin stack) $S$ locally of finite type over $K$ , let ${\rm Sch}/ S$ be the category of morphisms $X\to S$ over $K$ , where $X\in {\rm Sch}$ . For a morphism $f:S\to T$ we have the functor

\begin{align*}f_!:{\rm Sch}/ S\to {\rm Sch}/ T,\quad [X\to S]\mapsto [X\to S\to T].\end{align*}

If $f:S\to T$ is of finite type, then $f_!$ has the right adjoint functor

\begin{align*}f^*:{\rm Sch}/ T\to {\rm Sch}/ S,\quad [X\to T]\mapsto [X\times _T S\to S].\end{align*}

We also have the exterior product

\begin{align*} \boxtimes :{\rm Sch}/ S\times {\rm Sch}/ T\to {\rm Sch}/ (S\times T),\qquad [X\to S]\boxtimes [Y\to T]=[X\times Y\to S\times T]. \end{align*}

Let $K({\rm Sch}/ S)$ be the Grothendieck group of algebraic schemes over $S$ (see e.g. [Reference JoyceJoy07, Reference BridgelandBri12]). The above functors induce morphisms between Grothendieck groups. The Grothendieck group $K({\rm Sch})=K({\rm Sch}/{\bf pt})$ has a commutative ring structure defined by $[X]\cdot [Y]=[X\times Y]$ . The Grothendieck group $K({\rm Sch}/ S)$ is a module over $K({\rm Sch})$ . We define (cf. (17))

(22) \begin{align} \mathbb{L}=[\mathbb{A}^1]\in K({\rm Sch}),{\rm and} \end{align}

(23) \begin{align} \mathbf K({\rm Sch}/ S) =K({\rm Sch}/ S)\otimes _{\mathbb{Z}[\mathbb{L}]}\mathbb{Z}[\mathbb{L}^{\pm 1/2},(1-\mathbb{L}^n)^{-1}:n\geqslant 1]. \end{align}

In particular, the elements

(24) \begin{align} [\mathbb{P}^n]=\frac {1-\mathbb{L}^{n+1}}{1-\mathbb{L}},\quad [{\bf GL}_n]=\prod _{i=0}^{n-1}(\mathbb{L}^n-\mathbb{L}^i) \end{align}

are invertible in $\mathbf K({\rm Sch}/ {\bf pt})$ . Consider a morphism $f:\mathcal{X}\to S$ , where $\mathcal{X}$ is a finite type Artin stack with affine stabilizers. Then there exists an algebraic variety $X$ with an action of the group ${\bf GL}_n$ and a geometric bijection $X/{\bf GL}_n\to \mathcal{X}$ (see e.g. [Reference BridgelandBri12]). The class

(25) \begin{align} [\mathcal{X}\to S]=[X\to \mathcal{X}\to S]/[{\bf GL}_n]\in \mathbf K({\rm Sch}/ S) \end{align}

depends only on the morphism $f$ (see e.g. [Reference BridgelandBri12]).

Let $(S,\mu ,\eta )$ be a commutative monoid in the category of schemes (or Artin stacks) over $K$ . We equip ${\rm Sch}/ S$ with a symmetric monoidal structure having the tensor product

(26) \begin{align} [X\xrightarrow fS]\otimes [Y\xrightarrow gS]=\mu _!(f\boxtimes g) =[X\times Y\to S\times S\xrightarrow \mu S] \end{align}

and the unit object $\unicode {x1D7D9}=[{\bf pt}\xrightarrow \eta S]$ . This tensor product induces the ring structure on $K({\rm Sch}/ S)$ . We equip $K({\rm Sch}/ S)$ with a pre- $\lambda$ -ring structure having $\sigma$ -operations

(27) \begin{align} \sigma ^n[X\to S]=[S^n(X)\to S^n(S)\xrightarrow \mu S],\quad n\geqslant 1, \end{align}

where $S^n(X)=X^n/S_n$ . We extend the pre- $\lambda$ -ring structure to $\mathbf K({\rm Sch}/ S)$ by the formula

(28) \begin{align} \sigma ^n(y^m a)=y^{mn}\sigma ^n(a),\quad y=-\mathbb{L}^{1/2},\ a\in \mathbf K({\rm Sch}/ S). \end{align}

In particular, ${\bf pt}$ is a commutative monoid, and hence $K({\rm Sch})$ and $\mathbf K({\rm Sch})$ are pre- $\lambda$ -rings. The maps $\eta _!:K({\rm Sch})\to K({\rm Sch}/ S)$ and $a_{S!}:K({\rm Sch}/ S)\to K({\rm Sch})$ (induced by the unit $\eta :{\bf pt}\to S$ and the projection $a_S:S\to {\bf pt}$ ) are homomorphisms of pre- $\lambda$ -rings.

3.2 Graded pre- $\lambda$ -rings and completions

More generally, let $\Lambda$ be a commutative monoid and $S=\bigsqcup _{\gamma \in \Lambda }S_{\gamma}$ be a $\Lambda$ -graded commutative monoid in the category of schemes over $K$ . The pre- $\lambda$ -rings $K({\rm Sch}/ S)=\bigoplus _{\gamma \in \Lambda }K({\rm Sch}/ S_{\gamma} )$ and $\mathbf K({\rm Sch}/ S)=\bigoplus _{\gamma \in \Lambda }\mathbf K({\rm Sch}/ S_{\gamma} )$ are $\Lambda$ -graded pre- $\lambda$ -rings, meaning that

(29) \begin{align} \deg (ab)=\deg (a)+\deg (b),\quad \deg \sigma ^n(a)=n\deg a. \end{align}

In particular, we consider the commutative monoid $\boldsymbol{\Lambda }=\bigsqcup _{\gamma \in \Lambda }{\bf pt}_{\gamma}$ with ${\bf pt}_{\gamma} ={\bf Spec}(K)$ in the category of schemes over $K$ . Then $K({\rm Sch}/\boldsymbol{\Lambda })=K({\rm Sch})[\Lambda ]=\bigoplus _{\gamma \in \Lambda }K({\rm Sch})t^\gamma$ is a $\Lambda$ -graded pre- $\lambda$ -ring with the product and $\sigma$ -operations

\begin{align*}[X]t^\gamma \cdot [Y]t^{\gamma '}=[X\times Y]t^{\gamma +\gamma '},\quad \sigma ^n([X]t^\gamma )=[S^nX]t^{n\gamma }.\end{align*}

The degree map

(30) \begin{align} \deg :S\to \boldsymbol{\Lambda },\quad S_{\gamma} \ni x\mapsto {\bf pt}_{\gamma} \end{align}

is a homomorphism of commutative monoids and induces a homomorphism of pre- $\lambda$ -rings $\deg _!:K({\rm Sch}/ S)\to K({\rm Sch}/ \boldsymbol{\Lambda })=K({\rm Sch})[\Lambda ]$ .

Assume that there exists a filtration $\Lambda =\Lambda _0\supseteq \Lambda _1\supseteq \dots$ such that

(31) \begin{align} \Lambda _i+\Lambda _j\subseteq \Lambda _{i+j},\quad \left \lvert \Lambda \backslash \Lambda _i\right \rvert \lt \infty ,\quad \bigcap _{i\geqslant 0}\Lambda _i=\varnothing . \end{align}

For example, if $\Lambda \subseteq \mathbb{N}^n$ for some $n\geqslant 1$ , we can define $\Lambda _k=\left \{\gamma \in \Lambda \,\middle |\,\sum _{i=1}^n\gamma _i\geqslant k\right \}$ . For a $\Lambda$ -graded ring $A=\bigoplus _{\gamma \in \Lambda }A_{\gamma}$ , we define its completion

(32) \begin{align} \widehat A=\varprojlim _k \left (A/\bigoplus \nolimits _{\gamma \in \Lambda _k}A_{\gamma} \right ) \simeq \prod \nolimits _{\gamma \in \Lambda }A_{\gamma} , \end{align}

where the last isomorphism is an isomorphism of abelian groups. In particular, the completion $\widehat {\mathbf K}({\rm Sch}/ S)\simeq \prod _{\gamma \in \Lambda }\mathbf K({\rm Sch}/ S_{\gamma} )$ inherits the structure of a pre- $\lambda$ -ring. Let us assume that $\Lambda _1=\Lambda _+=\Lambda \backslash \left \{0\right \}$ , and consider the ideal $\widehat {\mathbf K_+}({\rm Sch}/ S)=\prod _{\gamma \in \Lambda _+}\mathbf K({\rm Sch}/ S_{\gamma} )$ . The plethystic exponential

(33) \begin{align} {\rm Exp}:\widehat {\mathbf K_+}({\rm Sch}/ S)\to 1+\widehat {\mathbf K_+}({\rm Sch}/ S),\quad a\mapsto \sum _{n\geqslant 0}\sigma ^n(a), \end{align}

where $\sigma^0(a)=1$ , is a group isomorphism (between the additive and the multiplicative groups).

3.3 $\lambda$ -rings of mixed Hodge modules

Let $(S,\mu ,\eta )$ be a commutative monoid in the category of complex algebraic varieties such that the product $\mu :S\times S\to S$ is a finite map. We will equip the category ${\rm MHM}(S)$ with a symmetric monoidal structure in a similar way to the case of ${\rm Sch}/ S$ considered earlier (see (26)).

Theorem 3.1. The category ${\rm MHM}(S)$ with the tensor product

\begin{align*}\otimes :{\rm MHM}(S)\times {\rm MHM}(S)\to {\rm MHM}(S),\qquad E\otimes F=\mu _!(E\boxtimes F),\end{align*}

and the unit object $\unicode {x1D7D9}=\eta _!\mathbb{Q}$ is a symmetric monoidal category. The tensor product is exact and w-exact.

Proof. Let $\sigma :S\times S\to S\times S$ be the permutation of factors. By [Reference Maxim, Saito and SchürmannMSS11, Theorem 1.9] (applied to the variety $X=S\sqcup S$ ), there exists a canonical isomorphism

\begin{align*}\sigma ^\#:E\boxtimes F\to \sigma _*(F\boxtimes E)\end{align*}

satisfying $\sigma _*\sigma ^\#\circ \sigma ^\#=id$ . Applying the pushforward $\mu _!$ and using commutativity of the monoid, we obtain an isomorphism $\sigma _{EF}:E\otimes F\to F\otimes E$ satisfying $\sigma _{EF}\circ \sigma _{FE}=id$ . This is the required braiding for the tensor product. We have $\unicode {x1D7D9}\otimes F =\mu _!(\eta _!\mathbb{Q}\boxtimes F) =\mu _!(\eta \times {\rm id})_!(\mathbb{Q}\boxtimes F)=F$ . The tensor product is exact (respectively w-exact) as both $\mu _!$ and $\boxtimes$ are exact (respectively w-exact).

The above tensor product $\otimes$ on ${\rm MHM}(S)$ should not be confused with the tensor product $\otimes ^*$ on $D^b{\rm MHM}(S)$ defined in Remark 2.10, which is only right w-exact.

Remark 3.2. Without the assumption that $\mu$ is finite, we can similarly equip the category $D^b{\rm MHM}(S)$ with a symmetric monoidal structure. There is a monoidal functor

(34) \begin{align} \chi _c:{\rm Sch}/ S\to D^b{\rm MHM}(S),\quad [X\xrightarrow fS]\mapsto f_!\mathbb{Q}_X. \end{align}

Let $\mathcal{A}={\rm MHM}(S)$ (or any other Karoubian symmetric monoidal category linear over $\mathbb{Q}$ ). By [Reference GetzlerGet96, Reference HeinlothHei07], the split Grothendieck group $\overline K(\mathcal{A})$ (with relations induced by direct sums) is a (special) $\lambda$ -ring, with the product and $\sigma$ -operations defined by

(35) \begin{align} [E]\cdot [F]=[E\otimes F], \end{align}

(36) \begin{align} \sigma ^n[E]=[S^n(E)],\quad S^n(E)=\Im \bigg (\frac 1{n!}\sum _{\sigma \in S_n}\sigma \bigg )\subseteq E^{\otimes n}, \end{align}

where $e=\frac 1{n!}\sum _{\sigma \in S_n}\sigma \in {\rm End}(E^{\otimes n})$ is an idempotent and $\Im (e)$ is obtained by its splitting.

Remark 3.3. More generally, for any partition $\lambda \vdash n$ , let $V_\lambda$ be the corresponding simple representation of the group $S_n$ . We define the Schur functor (cf. [Reference DeligneDel02])

(37) \begin{align} S_\lambda :\mathcal{A}\to \mathcal{A},\quad E\mapsto {\rm Hom}_{S_n}(V_\lambda ,E^{\otimes n}), \end{align}

where we first construct ${\rm Hom}(V_\lambda ,E^{\otimes n})$ as a direct sum of $\dim V_\lambda$ copies of $E^{\otimes n}$ and then take the $S_n$ -invariant subobject of ${\rm Hom}(V_\lambda ,E^{\otimes n})$ by splitting the idempotent $e=\frac 1{n!}\sum _{\sigma \in S_n}\sigma$ . In particular, for the trivial representation we obtain $S^n(E)=S_{(n)}(E)$ and for the alternating representation we obtain

(38) \begin{align} \Lambda ^n(E)=S_{(1^n)}(E)=\Im \bigg (\frac 1{n!}\sum _{\sigma \in S_n}(-1)^{\sigma }\sigma \bigg )\subseteq E^{\otimes n}. \end{align}

We have $s_\lambda [E]=[S_\lambda (E)]$ , where $s_\lambda$ is the symmetric Schur function. In particular, $\lambda ^n[E]=[\Lambda ^n(E)]$ .

The monoidal functor defined in (34) induces a morphism of pre- $\lambda$ -rings (cf. [Reference SchuermannSch09, Reference Maxim and SchürmannMS12])

(39) \begin{align} \chi _c:K({\rm Sch}/ S)\to K({\rm MHM}(S)),\quad [X\xrightarrow fS]\mapsto [f_!\mathbb{Q}_X]. \end{align}

We extend the $\lambda$ -ring structure to $\mathbf K({\rm MHM}(S))$ using (28). Then $\chi _c$ extends to a morphism of pre- $\lambda$ -rings $\chi _c:\mathbf K({\rm Sch}/ S)\to \mathbf K({\rm MHM}(S))$ .

Note that we use the notation $\mathbb{L}$ both for the class $\chi _c[\mathbb{A}^1]=[H_c^*(\mathbb{A}^1,\mathbb{Q})]\in K({\rm MHM}({\bf pt}))$ (16) and for the class $[\mathbb{A}^1]\in K({\rm Sch})$ (22), depending on the context. For a smooth (equidimensional) algebraic variety $X$ of dimension $d$ (or a smooth Artin stack with affine stabilizers), we define its virtual class

(40) \begin{align} [X]_{{\rm vir}}=\mathbb{L}^{-d/2}[X]\in \mathbf K({\rm Sch}). \end{align}

For example,

(41) \begin{align} [\mathbb{P}^n]_{{\rm vir}}&=\mathbb{L}^{-n/2}(1+\mathbb{L}+\dots +\mathbb{L}^n) =\mathbb{L}^{-n/2}+\mathbb{L}^{-n/2+1}+\dots +\mathbb{L}^{n/2}, {\rm and}\end{align}

(42) \begin{align} [{\bf GL}_n]_{{\rm vir}}&=\mathbb{L}^{-n^2/2}\prod _{i=0}^{n-1}(\mathbb{L}^n-\mathbb{L}^i)=\mathbb{L}^{-n/2}\prod _{i=1}^n(\mathbb{L}^i-1). \end{align}

Note that ${\bf IC}_X=\mathbb{Q}_X\left \langle d\right \rangle =\mathbb{L}^{-d/2}\mathbb{Q}_X$ , and hence the class of

\begin{align*} H_c(X,{\bf IC}_X) =\mathbb{L}^{-d/2}H_c(X,\mathbb{Q})\end{align*}

is equal to $\chi _c[X]_{{\rm vir}}$ , which we will also denote by $[X]_{{\rm vir}}$ by abuse of notation.

Let $\Lambda$ be a commutative monoid admitting a filtration satisfying (31), and $\Lambda _1=\Lambda _+=\Lambda \backslash \left \{0\right \}$ . Let $S=\bigsqcup _{\gamma \in \Lambda }S_{\gamma}$ be a $\Lambda$ -graded commutative monoid in the category of complex algebraic varieties (so that the $S_{\gamma}$ are of finite type) such that the product $\mu :S_{\gamma} \times S_{\gamma '}\to S_{\gamma +\gamma '}$ is finite. We define

\begin{align*}K({\rm MHM}(S))=\bigoplus \nolimits _{\gamma \in \Lambda }K({\rm MHM}(S_{\gamma} ))\end{align*}

and equip it with the $\mathbb{Z}$ -graded (and $\Lambda$ -graded) $\lambda$ -ring structure using the same formulas as before. We extend this $\lambda$ -ring structure to $\mathbf K({\rm MHM}(S))=\bigoplus \nolimits _{\gamma \in \Lambda }\mathbf K({\rm MHM}(S_{\gamma} ))$ using (28). There is a commutative diagram

(43)

where $\chi _c$ is defined in (39), $\deg :S\to \boldsymbol{\Lambda }$ is the degree map (30), and all arrows are homomorphisms of $\Lambda$ -graded pre- $\lambda$ -rings. These maps induce homomorphisms between completions (32).

As in (33), we have the plethystic exponential

(44) \begin{align} {\rm Exp}:\widehat {\mathbf K_+}({\rm MHM}(S))\to 1+\widehat {\mathbf K_+}({\rm MHM}(S)),\quad a\mapsto \sum _{n\geqslant 0}\sigma ^n(a). \end{align}

In terms of the Adams operations, ${\rm Exp}$ and its inverse (called the plethystic logarithm) are

(45) \begin{align} {\rm Exp}(a)=\exp \bigg (\sum _{n\geqslant 1}\frac 1n\psi ^n(a)\bigg ),\quad {\rm Log}(1+a)=\sum _{n\geqslant 1}\frac {\mu (n)}n\psi ^n(\log (1+a)), \end{align}

where $\mu$ is the Möbius function.

4.1 Stability of framed objects

Let ${\rm Vec}_{\mathrm f}$ be the category of finite-dimensional vector spaces over a field $K$ . Let $\mathcal{A}$ be an abelian $K$ -linear category and $\Phi :\mathcal{A}\to {\rm Vec}_{\mathrm f}$ be a left exact functor, to be called a framing functor.

Let $Z=-\mathsf d+\mathsf r{\bf i}:\Gamma =K(\mathcal{A})\to \mathbb{C}$ be a stability function on an abelian category $\mathcal{A}$ (see e.g. [Reference BridgelandBri07]), where $\mathsf d,\mathsf r\in {\rm Hom}_{\mathbb{Z}}(\Gamma ,\mathbb{R})$ . For every $0\ne E\in \mathcal{A}$ , we have either $\mathsf r(E)\gt 0$ or $\mathsf r(E)=0$ and $\mathsf d(E)\gt 0$ . We define the slope of $E$

\begin{align*}\mu _Z(E)=\frac {\mathsf d(E)}{\mathsf r(E)}\in \mathbb{R}\sqcup \left \{+\infty \right \}\end{align*}

and we say that an object $E$ is $Z$ -semistable (respectively $Z$ -stable) if, for every proper $0\ne F\subset E$ , we have $\mu _Z(F)\le \mu _Z(E)$ (respectively $\mu _Z(F)\lt \mu _Z(E)$ ). For $\mu \in \mathbb{R}$ , let $\mathcal{A}^\mu \subseteq \mathcal{A}$ be the subcategory of $Z$ -semistable objects having slope $\mu$ (including the zero object). This is a weakly Serre subcategory of $\mathcal{A}$ , meaning an abelian subcategory closed under extensions, kernels and cokernels.

It will be convenient to extend the notion of framed objects $(E,s)$ as follows. Let $\mathcal{A}_{\mathsf{f}}$ be the category consisting of triples $(E,V,s)$ , where $E\in \mathcal{A}$ , $V\in {\rm Vec}_{\mathrm f}$ and $s:V\to \Phi (E)$ is linear. The category $\mathcal{A}_{\mathsf{f}}$ is an abelian category, called the category of framed objects. A pair $(E,s)$ with $s\in \Phi (E)$ can be identified with the triple $(E,K,s)\in \mathcal{A}_{\mathsf{f}}$ . For $\varepsilon \gt 0$ , we define the stability function on $\mathcal{A}_{\mathsf{f}}$

\begin{align*}Z_\varepsilon (E,V,s)=Z(E)-\varepsilon \cdot \dim V\end{align*}

and the corresponding notion of stability in $\mathcal{A}_{\mathsf{f}}$ . A framed object $(E,s)$ is $Z_\varepsilon$ -stable for $0\lt \varepsilon \ll \left \lvert Z(E)\right \rvert$ (we will say that it is $Z_+$ -stable) if and only if:

1. (i) $E$ is $Z$ -semistable; and

2. (ii) for every proper $F\subset E$ with $s\in \Phi (F)$ , we have $\mu _Z(F)\lt \mu _Z(E)$ .

The above result implies that the notion of $Z_+$ -stability of framed objects in $\mathcal{A}^\mu$ is equivalent to the notion of stability of framed objects from Definition 4.1. To define such a framed object, it is enough to have a left exact functor $\Phi :\mathcal{A}^\mu \to {\rm Vec}_{\mathrm f}$ . Usually, this functor will be exact.

Example 4.3. Let $Q$ be a quiver and $\mathcal{A}={\rm Rep} Q$ be the category of $Q$ -representations over a field $K$ . For a fixed (framing) vector ${\bf w}\in \mathbb{N}^{Q_0}$ , we define the exact functor

(46) \begin{align} \Phi :\mathcal{A}\to {\rm Vec}_{\mathrm f},\quad M\mapsto \bigoplus _{i\in Q_0}{\rm Hom}(K^{{\bf w}_i},M_i) \simeq \bigoplus _{i\in Q_0}M_i^{\oplus {\bf w}_i}. \end{align}

The category $\mathcal{A}_{\mathsf{f}}$ of framed objects can be identified with the category of representations of the new quiver $Q'$ obtained from $Q$ by adding a new vertex $*$ and ${\bf w}_i$ arrows $*\to i$ for all $i\in Q_0$ . An object $(M,V,s)\in \mathcal{A}_{\mathsf{f}}$ is identified with the representation $M'\in {\rm Rep} Q'$ such that $M'_i=M_i$ for $i\in Q_0$ and $M'_*=V$ . The linear map $s:V\to \Phi (M)=\bigoplus _{i\in Q_0}M_i^{\oplus {\bf w}_i}$ induces linear maps $M'_*\to M'_i$ corresponding to the arrows $*\to i$ in $Q'$ . A framed object $(M,s)$ corresponds to $M'\in {\rm Rep} Q'$ with $M'_*=K$ . It is stable if and only if $M'_*$ generates the whole representation.

Example 4.4. Let $C$ be a smooth projective curve of genus $g$ and $\mathcal{A}={\rm Coh} C$ be the category of coherent sheaves over $C$ . We consider the stability function $Z(E)=-\deg E+{\rm rk} E\cdot {\bf i}$ on $\mathcal{A}$ . For $\mu \in \mathbb{Q}$ , let $\mathcal{A}^\mu ={\rm Coh}^\mu C\subset {\rm Coh}C$ be the category of semistable vector bundles of slope $\mu$ . For a fixed line bundle $L$ (or any coherent sheaf) over $C$ , we consider the left exact functor

(47) \begin{align} \Phi :\mathcal{A}\to {\rm Vec}_{\mathrm f},\quad E\mapsto {\rm Hom}(L,E). \end{align}

If the line bundle $L$ has degree $\ell \lt \mu -(2g-2)$ , then the functor $\Phi :\mathcal{A}^\mu \to {\rm Vec}_{\mathrm f}$ is exact. Indeed, ${\rm Ext}^1(L,E)\simeq {\rm Hom}(E,L\otimes \omega _C)^*=0$ for $E\in \mathcal{A}^\mu$ since $\mu \gt \ell+2g-2$ .

On the other hand, for a fixed point $p\in C$ , the fibre functor

(48) \begin{align} \Phi :\mathcal{A}^\mu \to {\rm Vec}_{\mathrm f},\quad E\mapsto E(p)=E_p\otimes _{\mathcal{O}_{C,p}}K, \end{align}

is also exact. Framed objects in this situation can be identified with parabolic vector bundles.

As before, let $\mathcal{A}^\mu \subseteq \mathcal{A}$ be the category of $Z$ -semistable objects of $\mathcal{A}$ having slope $\mu \in \mathbb{R}$ . The stable objects of $\mathcal{A}$ having slope $\mu$ are exactly the simple objects of $\mathcal{A}^\mu$ . Let

\begin{align*}E_1,\dots ,E_n\in \mathcal{A}^\mu \end{align*}

be a collection of (pairwise non-isomorphic) simple objects, and let $\mathcal{C}=\left \langle E_1,\dots ,E_n\right \rangle \subset \mathcal{A}^\mu$ be the abelian category generated by them. This is a Serre subcategory of $\mathcal{A}^\mu$ , meaning an abelian subcategory closed under taking extensions, subobjects and quotients. Given a left exact functor $\Phi :\mathcal{A}^\mu \to {\rm Vec}_{\mathrm f}$ , we define the category $\mathcal{C}_{\mathsf{f}}$ of framed objects in $\mathcal{C}$ in the same way as before.

We are going to describe the categories $\mathcal{C}$ and $\mathcal{C}_{\mathsf{f}}$ as categories of quiver representations. Let $Q$ be the quiver with vertices $1,\dots ,n$ and the number of arrows from $i$ to $j$ equal to

(49) \begin{align} a_{ij}=\dim {\rm Ext}^1(E_i,E_j). \end{align}

Let $Q'$ be the quiver obtained from $Q$ by adding a new vertex $*$ and ${\bf w}_i=\dim \Phi (E_i)$ arrows $*\to i$ for all $i\in Q_0$ . A representation of $Q$ is called nilpotent if it has a filtration by sub-representations such that the arrows of $Q$ act trivially on the factors.

Proof. For the first statement, see [Reference Deng and XiaoDX98, Section 1.5]. For the second statement we can assume that $\mathcal{C}={\rm Rep}^{{\rm nil}}(Q)$ and we will show that an exact functor $\Phi :\mathcal{C}\to {\rm Vec}_{\mathrm f}$ is uniquely determined (up to a non-unique natural transformation) by its values on the simple objects. This will imply the second statement, as ${\rm Rep}^{{\rm nil}}(Q')$ can be identified with the category of framed objects in ${\rm Rep}^{{\rm nil}}(Q)$ (see Example 4.3).

Let $A=\Bbbk Q$ be the path algebra, $\mathfrak{r}\subset A$ be the radical, and $e_i\in A$ be the idempotent corresponding to the trivial path at a vertex $i\in Q_0$ . It is enough to show that $\Phi$ is uniquely determined on the subcategory $\mathcal{C}_t={\rm Rep} A_t\subset \mathcal{C}$ (which is not an exact subcategory), where $A_t=A/\mathfrak{r}^{t+1}$ for $t\geqslant 0$ . Let $S_i$ be the simple modules and $P_i=A_te_i$ their projective covers as $A_t$ -modules. For the projective module $P=\bigoplus _i \Phi (S_i)^*\otimes _\Bbbk P_i$ , where $\Phi (S_i)^*$ denotes the dual vector space of $\Phi (S_i)$ , we consider the functor $h_P={\rm Hom}(P,-):\mathcal{C}\to {\rm Vec}_{\mathrm f}$ . We will construct an isomorphism $a:h_P\to \Phi$ of functors over $\mathcal{C}_t$ . The map

\begin{align*}{\rm Fun}(h_P,\Phi )\simeq \Phi (P) =\bigoplus _i \Phi (S_i)^*\otimes _\Bbbk \Phi (P_i) \to \bigoplus _i \Phi (S_i)^*\otimes _\Bbbk \Phi (S_i) \end{align*}

is surjective, as $P_i\to S_i$ is surjective and $\Phi$ is exact. Consider a preimage $a:h_P\to \Phi$ of the sum of identity maps. This corresponds to $a_i:\Phi (S_i)\to \Phi (P_i)$ such that the composition $\Phi (S_i)\to \Phi (P_i)\to \Phi (S_i)$ is the identity for all $i$ . This induces a chain of maps

\begin{align*} h_P(M)&={\rm Hom}(P,M)=\bigoplus _i \Phi (S_i)\otimes _\Bbbk {\rm Hom}(P_i,M)\\[5pt] &\to \bigoplus _i \Phi (P_i)\otimes _\Bbbk {\rm Hom}(P_i,M)\to \Phi (M). \end{align*}

For $M=S_j$ , the composition of the last two arrows is an isomorphism. This implies that $a:h_P\to \Phi$ induces an isomorphism on simple objects. By the exactness of $h_P$ and $\Phi$ on $\mathcal{C}_t$ , we conclude that $a:h_P\to \Phi$ is an isomorphism of functors on the whole category $\mathcal{C}_t$ .

The above theorem implies that the moduli space of stable pairs $(E,s)$ with $E\in \mathcal{C}$ can be identified with the moduli spaces of stable nilpotent representations of $Q'$ having dimension one at the vertex $*$ (see Example 4.3). Stability of a representation $M\in {\rm Rep} Q'$ means that $M_*$ generates the whole representation.

Example 4.7. Let $S$ be a smooth projective surface and $H$ be an ample divisor such that $H\cdot K_S\lt 0$ , where $K_S$ is the canonical divisor of $S$ . Define the slope function

\begin{align*}\mu (E)=\frac {H\cdot c_1(E)}{{\rm rk} E},\quad E\in {\rm Coh} S.\end{align*}

For $\mu \in \mathbb{R}$ , let $\mathcal{A}^\mu ={\rm Coh}^\mu S\subset {\rm Coh} S$ be the category of semistable vector bundles (also called Mumford semistable) having slope $\mu$ . This category is hereditary. Indeed, for any $E,F\in \mathcal{A}^\mu$ , we have ${\rm Ext}^2(E,F)\simeq {\rm Hom}(F,E\otimes \omega _S)^*=0$ as both $F$ and $E\otimes \omega _S$ are semistable and

\begin{align*}\mu (E\otimes \omega _S)=\frac {H\cdot c_1(E)+{\rm rk} E\cdot H\cdot K_S}{{\rm rk} E}=\mu (E)+H\cdot K_S\lt \mu (F).\end{align*}

For $n\in \mathbb{Z}$ , we consider the left exact functor

(50) \begin{align} \Phi _n:\mathcal{A}^\mu \to {\rm Vec}_{\mathrm f},\qquad E\mapsto {\rm Hom}(\mathcal{O}(-nH),E). \end{align}

This functor is not exact in general, but one can prove the following (cf. [Reference Huybrechts and LehnHL10, Sections 1.7, 3.3]). For $E\in {\rm Coh} S$ , let ${\rm cl} (E)=({\rm rk} E,H\cdot c_1(E),c_2(E))\in \mathbb{Z}^3$ . For any finite subset $C\subset \mathbb{Z}^3$ , there exists $n\gt 0$ such that $R^i\Phi _n(E)=0$ for all $i\gt 0$ and $E\in \mathcal{A}^\mu$ with ${\rm cl}(E)\in C$ .

4.2 Moduli spaces

Let $\mathcal{A}$ be an abelian ( $\mathbb{C}$ -linear) category and ${\rm cl}:K(\mathcal{A})\to \Gamma$ be a homomorphism to an abelian group $\Gamma$ equipped with a bilinear form $\chi$ compatible with the Euler form on $\mathcal{A}$

(51) \begin{align} \chi ({\rm cl} E,{\rm cl} E')=\chi (E,E')=\sum _{i\geqslant 0}(-1)^i\dim {\rm Ext}^i(E,E'). \end{align}

Let $Z=-\mathsf d+\mathsf r{\bf i}:\Gamma \to \mathbb{C}$ be a homomorphism inducing a stability function $K(\mathcal{A})\xrightarrow {{\rm cl}}\Gamma \xrightarrow Z\mathbb{C}$ (also denoted by $Z$ ). For $\mu \in \mathbb{R}$ , let $\mathcal{A}^\mu \subseteq \mathcal{A}$ be the category of $Z$ -semistable objects having slope $\mu$ (including the zero object). We define the semigroups

(52) \begin{align} \Lambda =\left \{{\rm cl}(E)\in \Gamma \,\middle |\,E\in \mathcal{A}^\mu \right \},\quad \Lambda _+=\Lambda \backslash \left \{0\right \}. \end{align}

We assume that $\mathcal{A}^\mu$ is hereditary and we have an exact functor $\Phi :\mathcal{A}^\mu \to {\rm Vec}_{\mathrm f}$ and a homomorphism $\phi :\Lambda \to \mathbb{Z}$ such that

(53) \begin{align} \dim \Phi (E)=\phi ({\rm cl} E)\quad \forall E\in \mathcal{A}^\mu . \end{align}

Example 4.8. As in Example 4.4, let $C$ be a smooth projective curve of genus $g$ over $\mathbb{C}$ , and let $\mathcal{A}={\rm Coh} C$ and $\mathcal{A}^\mu ={\rm Coh}^\mu C\subset {\rm Coh} C$ for a fixed $\mu \in \mathbb{Q}$ . We consider the Chern character map ${\rm ch}:K(\mathcal{A})\to \Gamma =\mathbb{Z}^2$ , $[E]\mapsto ({\rm rk} E,\deg E)$ . Let $\Phi ={\rm Hom}(L,-):\mathcal{A}^\mu \to {\rm Vec}_{\mathrm f}$ for a line bundle $L$ of degree $\ell \lt \mu -(2g-2)$ . For $\gamma ={\rm ch}(E)=(r,d)$ and $\gamma '={\rm ch} E'=(r',d')$ , we have

(54) \begin{align} \chi (\gamma ,\gamma ')=\chi (E,E')=(1-g)rr'+(rd'-r'd), \end{align}

(55) \begin{align} \phi (\gamma )=\chi (L,E)=(1-g-\ell )r+d. \end{align}

Note that the Euler form is symmetric on $\Lambda$ . Indeed, if $\gamma ,\gamma '\in \Lambda$ , then $rd'=r'd$ , hence $\chi (\gamma ,\gamma ')=(1-g)rr'=\chi (\gamma ',\gamma )$ .

We assume that for every $\gamma \in \Lambda$ there exists a moduli space $\mathcal{M}_{\gamma}$ of $Z$ -semistable objects in $\mathcal{A}$ having class $\gamma$ , and a moduli space ${\mathcal{M}}^{\mathsf{f}}_{\gamma}$ of stable framed objects $(E,s)$ with $E\in \mathcal{A}^\mu$ having class $\gamma$ (see e.g. [Reference Huybrechts and LehnHL95] in the case of framed objects on a curve). The moduli space ${\mathcal{M}}^{\mathsf{f}}_{\gamma}$ is smooth as all objects are stable and the second ${\rm Ext}$ -group vanishes (see e.g. [Reference ThaddeusTha94]). The moduli space $\mathcal{M}_{\gamma}$ parametrizes poly-stable objects of the form

\begin{align*}E=\bigoplus _{i=1}^n E_i^{m_i},\end{align*}

where $E_i$ are pairwise non-isomorphic stable objects in $\mathcal{A}$ with $\gamma _i={\rm cl} E_i\in \Lambda _+$ such that $\gamma =\sum _i m_i\gamma _i$ . Note that the objects $E_i$ are simple objects in $\mathcal{A}^\mu$ . We define the type of $E$ to be the pair $(\boldsymbol{\gamma },{\bf m})$ with

\begin{align*}\boldsymbol{\gamma }=(\gamma _i)_{i=1}^n\in (\Lambda _+)^n,\quad {\bf m}=(m_i)_{i=1}^n\in \mathbb{N}^n.\end{align*}

The set $S_{\boldsymbol{\gamma },{\bf m}}\subset \mathcal{M}_{\gamma}$ of all objects having type $(\boldsymbol{\gamma },{\bf m})$ is called the Luna stratum of type $(\boldsymbol{\gamma },{\bf m})$ . In our applications they form a finite partition of $\mathcal{M}_{\gamma}$ into locally closed subsets. If $n=1$ and $m_1=1$ , then $S_{\boldsymbol{\gamma },{\bf m}}$ is equal to the stable locus ${\mathcal{M}}^{\mathsf{s}}_{\gamma} \subseteq \mathcal{M}_{\gamma}$ which is smooth and open.

The canonical projection $\pi :{\mathcal{M}}^{\mathsf{f}}_{\gamma} \to \mathcal{M}_{\gamma}$ sends a pair $(E,s)$ to the graded object associated with the Jordan–Hölder filtration of $E$ in $\mathcal{A}^\mu$ . This map is projective, and its fibre over any $E\in {\mathcal{M}}^{\mathsf{s}}_{\gamma}$ is isomorphic to the projective space $P(\Phi (E))\simeq \mathbb{P}^{\phi (\gamma )-1}$ .

Following (49), we define the quiver $Q_{\boldsymbol{\gamma }}$ with vertices $1,\dots ,n$ and the number of arrows from $i$ to $j$ equal to

(56) \begin{align} a_{ij}=\dim {\rm Ext}^1(E_i,E_j)=\delta _{ij}-\chi (\gamma _i,\gamma _j). \end{align}

We define $Q'_{\boldsymbol{\gamma }}$ to be the quiver obtained from $Q_{\boldsymbol{\gamma }}$ by adding one vertex $*$ and adding

(57) \begin{align} {\bf w}_i=\dim \Phi (E_i)=\phi (\gamma _i) \end{align}

arrows $*\to i$ for all vertices $i$ in $Q_{\boldsymbol{\gamma }}$ .

The Euler form of $Q_{\boldsymbol{\gamma }}$ is given by

\begin{align*}\chi _{Q_{\boldsymbol{\gamma }}}({\bf m},{\bf m}')=\sum _{i}m_im'_i-\sum _{i,j}a_{ij}m_im'_j =\sum _{i,j}(\delta _{ij}-a_{ij})m_im'_j ,\quad {\bf m},{\bf m}'\in \mathbb{Z}^n.\end{align*}

If $\gamma =\sum _i m_i\gamma _i$ and $\gamma '=\sum _i m'_i\gamma _i$ , then

\begin{align*}\chi (\gamma ,\gamma ')=\sum _{i,j}\chi (\gamma _i,\gamma _j)m_im'_j =\chi _{Q_{\boldsymbol{\gamma }}}({\bf m},{\bf m}').\end{align*}

This means that the map $\mathbb{Z}^n\to \Gamma$ , ${\bf m}\mapsto \sum _i m_i\gamma _i$ preserves the Euler forms.

4.3 Virtual smallness

Let us now assume that the Euler form is symmetric on $\Lambda$ .

Proof. Let $S_{\boldsymbol{\gamma },{\bf m}}\subseteq \mathcal{M}_{\gamma}$ be the stratum corresponding to $\boldsymbol{\gamma }=(\gamma _1,\dots ,\gamma _n)\in \Gamma ^n$ and ${\bf m}=(m_1,\dots ,m_n)\in \mathbb{N}^n$ such that $\sum _i m_i\gamma _i=\gamma$ . Let $Q=Q_{\boldsymbol{\gamma }}$ and $Q'=Q'_{\boldsymbol{\gamma }}$ be as in (56). By our assumptions, the quiver $Q$ is symmetric (cf. Remark 4.9). By Theorem 4.10, for $E=\bigoplus _{i=1}^nE_i^{m_i}\in S_{\boldsymbol{\gamma },{\bf m}}$ , we can identify the fibre $\pi ^{-1}(E)$ with the moduli space $\mathcal{M}^{\mathsf{f},{\rm nil}}_{{\bf m}}$ of stable nilpotent representations of $Q'$ . Let $R_{{\bf m}}=\bigoplus _{a:i\to j}{\rm Hom}(\mathbb{C}^{m_i},\mathbb{C}^{m_j})$ be the space of all representations of $Q$ having dimension vector ${\bf m}$ , and let $R^{{\rm nil}}_{{\bf m}}\subset R_{{\bf m}}$ be the subscheme of nilpotent representations. These schemes are equipped with the action of the group $G_{{\bf m}}=\prod _{i\in Q_0}{\bf GL}_{m_i}$ . As $Q$ is a symmetric quiver, we have (see [Reference Meinhardt and ReinekeMR19])

\begin{align*}{\tfrac 12}\chi _Q({\bf m},{\bf m})+\dim R^{{\rm nil}}_{{\bf m}}-\dim G_{{\bf m}}\leqslant \sum _{i=1}^nm_i\left ({\tfrac 12}\chi _Q(e_i,e_i)-1\right ),\end{align*}

where $e_i$ is the standard basis vector of $\mathbb{Z}^{n}$ . We have

\begin{align*}\dim \mathcal{M}^{\mathsf{f},{\rm nil}}_{{\bf m}} =\dim R^{{\rm nil}}_{{\bf m}}(Q)-\dim G_{{\bf m}}+{\bf w}\cdot {\bf m} \end{align*}

as $G_{{\bf m}}$ acts freely on stable framed representations. Note that ${\bf w}\cdot {\bf m}=\phi (\gamma )$ . Therefore,

\begin{align*}\dim \mathcal{M}_{{\bf m}}^{\mathsf{f},{\rm nil}}\leqslant \sum _{i=1}^nm_i\left ({\tfrac 12}\chi (\gamma _i,\gamma _i)-1\right ) -{\tfrac 12}\chi (\gamma ,\gamma )+\phi (\gamma ). \end{align*}

The dimension of the stable locus ${\mathcal{M}}^{\mathsf{s}}_{\gamma} \subseteq \mathcal{M}_{\gamma}$ is $1-\chi (\gamma ,\gamma )$ , and hence $\dim S_{\boldsymbol{\gamma },{\bf m}}=\sum _{i=1}^n(1-\chi (\gamma _i,\gamma _i))$ (if $S_{\boldsymbol{\gamma },{\bf m}}$ is non-empty). The defect of $\pi$ over $S_{\boldsymbol{\gamma },{\bf m}}$ is equal to (see (18))

(58) \begin{align} \delta (\pi ,S_{\boldsymbol{\gamma },{\bf m}}) &=2\dim \mathcal{M}_{{\bf m}}^{\mathsf{f},{\rm nil}}+\dim S_{\boldsymbol{\gamma },{\bf m}}-\dim {\mathcal{M}}^{\mathsf{f}}_{\gamma} \nonumber \\[5pt] &=2\dim \mathcal{M}_{{\bf m}}^{\mathsf{f},{\rm nil}} +\sum _{i=1}^n(1-\chi (\gamma _i,\gamma _i)) +\left (\chi (\gamma ,\gamma )-\phi (\gamma )\right )\nonumber \\[5pt] & \le \sum _{i=1}^nm_i\left (\chi (\gamma _i,\gamma _i)-2\right ) +\sum _{i=1}^n(1-\chi (\gamma _i,\gamma _i))+\phi (\gamma )\nonumber \\[5pt] & =\sum _{i=1}^n(m_i-1)\left (\chi (\gamma _i,\gamma _i)-1\right ) -\sum _{i=1}^nm_i+\phi (\gamma ) \leqslant -\sum _{i=1}^nm_i+\phi (\gamma )\leqslant \phi (\gamma )-1, \end{align}

where we used the fact that $m_i\geqslant 1$ and $\chi (\gamma _i,\gamma _i)\le 1$ . The last inequality in (58) can be an equality only for $n=1$ and $m_1=1$ . In this case the stratum $S_{\boldsymbol{\gamma },{\bf m}}$ is the stable locus ${\mathcal{M}}^{\mathsf{s}}_{\gamma}$ . The fibre over $E\in {\mathcal{M}}^{\mathsf{s}}_{\gamma}$ can be identified with the projective space $P(\Phi (E))\simeq \mathbb{P}^{\phi (\gamma )-1}$ . Therefore, the defect of $\pi$ over ${\mathcal{M}}^{\mathsf{s}}_{\gamma}$ is

\begin{align*}\delta (\pi ,{\mathcal{M}}^{\mathsf{s}}_{\gamma} ) =2(\phi (\gamma )-1)+(1-\chi (\gamma ,\gamma ))+(\chi (\gamma ,\gamma )-\phi (\gamma )) =\phi (\gamma )-1. \end{align*}

5.1 Moduli spaces

We follow the conventions from Section 4.2. Let $C$ be a smooth projective curve of genus $g$ over $\mathbb{C}$ . Let $\mathcal{A}={\rm Coh} C$ and $\mathcal{A}^\mu ={\rm Coh}^\mu C\subseteq {\rm Coh} C$ be the category of semistable vector bundles having a fixed slope $\mu \in \mathbb{Q}$ . In a similar way to (52), we define

\begin{align*}\Lambda _+=\left \{(r,d)\in \mathbb{Z}_{\gt 0}\times \mathbb{Z}\,\middle |\,d/r=\mu \right \},\quad \Lambda =\Lambda _+\sqcup \left \{0\right \}.\end{align*}

Note that $\Lambda =\mathbb{N}\gamma _0\simeq \mathbb{N}$ for a unique element $\gamma _0=(r_0,d_0)\in \Lambda _+$ .

For $\gamma =(r,d)\in \Lambda$ , let $\mathcal{M}_{\gamma}$ (respectively ${\mathcal{M}}^{\mathsf{s}}_{\gamma}$ ) be the moduli space of semistable (respectively stable) vector bundles over $C$ having class $\gamma$ (rank $r$ and degree $d$ ). Similarly, let $\mathfrak{M}_{\gamma}$ be the moduli stack of semistable vector bundles over $C$ having class $\gamma$ .