1 Introduction
In the pioneering work [Reference Dimitrov, Haiden, Katzarkov and Kontsevich11], Dimitrov–Haiden–Katzarkov–Kontsevich introduced various categorical analogies of classical theory from dynamical systems. In particular, they defined the entropy of an endofunctor on a triangulated category with a split generator. One of their motivations comes from the connection between the theory of stability conditions on triangulated categories and the Teichmüller theory of surfaces [Reference Bridgeland and Smith10, Reference Gaiotto, Moore and Neitzke.12]. In this connection, a stability condition on a triangulated category corresponds to a measured foliation (a quadratic differential) on a surface, and the mass of stable objects corresponds to the length of geodesics. Thus the mass of objects plays the role of “volume” in some sense. In [Reference Dimitrov, Haiden, Katzarkov and Kontsevich11], they also suggested that there is a connection between the growth rate of mass of objects under the mapping by an endofunctor and the entropy of the endofunctor. In this paper, we establish fundamental properties of the mass growth and clarify the relationship between it and the entropy. We also show that they coincide under a certain condition. The result in this paper is motivated by the well-known classical work “Volume growth and entropy” by Yomdin [Reference Yomdin22] on classical dynamical systems.
1.1 Fundamental properties of mass growth
First, we introduce the mass growth with respect to endofunctors. Let
${\mathcal{D}}$
be a triangulated category and
$K({\mathcal{D}})$
be its Grothendieck group. A stability condition
$\unicode[STIX]{x1D70E}=(Z,{\mathcal{P}})$
on
${\mathcal{D}}$
[Reference Bridgeland8] is a pair of a linear map
$Z:K({\mathcal{D}})\rightarrow \mathbb{C}$
and a family of full subcategories
${\mathcal{P}}(\unicode[STIX]{x1D719})\subset {\mathcal{D}}$
for
$\unicode[STIX]{x1D719}\in \mathbb{R}$
satisfying some axioms (see Definition 2.8). A nonzero object in
${\mathcal{P}}(\unicode[STIX]{x1D719})$
is called a semistable object of phase
$\unicode[STIX]{x1D719}$
. One of the axioms implies that any nonzero object
$E\in {\mathcal{D}}$
can be decomposed into semistable objects with decreasing phases, that is, there is a sequence of exact triangles, called a Harder–Narasimhan filtration,
with
$A_{i}\in {\mathcal{P}}(\unicode[STIX]{x1D719}_{i})$
and
$\unicode[STIX]{x1D719}_{1}>\unicode[STIX]{x1D719}_{2}>\cdots >\unicode[STIX]{x1D719}_{m}$
. Through the Harder–Narasimhan filtration, the mass of
$E$
with a parameter
$t\in \mathbb{R}$
(see Definition 3.1) is defined by
$$\begin{eqnarray}m_{\unicode[STIX]{x1D70E},t}(E):=\mathop{\sum }_{i=1}^{m}|Z(A_{i})|e^{\unicode[STIX]{x1D719}_{i}t}.\end{eqnarray}$$
Thus, a given stability condition defines the “volume” of objects in some sense. Actually in the connection between spaces of stability conditions and Teichmüller spaces, the mass of stable objects gives the length of corresponding geodesics [Reference Bridgeland and Smith10, Reference Gaiotto, Moore and Neitzke.12, Reference Haiden, Katzarkov and Kontsevich14, Reference Ikeda15]. For an endofunctor
$F:{\mathcal{D}}\rightarrow {\mathcal{D}}$
, we want to consider the growth rate of mass of objects under the mapping by
$F$
. Therefore, we introduce the following quantity. The mass growth with respect to
$F$
is the function
$h_{\unicode[STIX]{x1D70E},t}(F):\mathbb{R}\rightarrow [-\infty ,\infty ]$
defined by
$$\begin{eqnarray}h_{\unicode[STIX]{x1D70E},t}(F):=\sup _{E\in {\mathcal{D}}}\bigg\{\limsup _{n\rightarrow \infty }\frac{1}{n}\log (m_{\unicode[STIX]{x1D70E},t}(F^{n}E))\bigg\}.\end{eqnarray}$$
(As conventions, set
$m_{\unicode[STIX]{x1D70E},t}(0)=0$
and
$\log 0=-\infty$
.) Fundamental properties of
$h_{\unicode[STIX]{x1D70E},t}(F)$
are stated as the main result of this paper. We also recall the space of stability conditions to consider the behavior of
$h_{\unicode[STIX]{x1D70E},t}(F)$
under the deformation of
$\unicode[STIX]{x1D70E}$
. In [Reference Bridgeland8], it was shown that the set of stability conditions
$\text{Stab}({\mathcal{D}})$
has a natural topology and in addition,
$\text{Stab}({\mathcal{D}})$
becomes a complex manifold.
Next, we recall the entropy of endofunctors from [Reference Dimitrov, Haiden, Katzarkov and Kontsevich11]. Let
${\mathcal{D}}$
be a triangulated category with a split generator and
$F:{\mathcal{D}}\rightarrow {\mathcal{D}}$
be an endofunctor. In [Reference Dimitrov, Haiden, Katzarkov and Kontsevich11], they introduced the function
$h_{t}(F):\mathbb{R}\rightarrow [-\infty ,\infty )$
, called the entropy of
$F$
(see Definition 2.4), and showed various fundamental properties of
$h_{t}(F)$
. In addition, they investigated the relationship between the entropy
$h_{t}(F)$
and the mass growth
$h_{\unicode[STIX]{x1D70E},t}(F)$
(see [Reference Dimitrov, Haiden, Katzarkov and Kontsevich11, Section 4.5]). Our result is the following.
Theorem 1.1. (Theorem 3.5 and Proposition 3.10)
Let
${\mathcal{D}}$
be a triangulated category,
$F:{\mathcal{D}}\rightarrow {\mathcal{D}}$
be an endofunctor, and
$\unicode[STIX]{x1D70E}$
be a stability condition on
${\mathcal{D}}$
. Assume that
${\mathcal{D}}$
has a split generator
$G$
. Then the mass growth
$h_{\unicode[STIX]{x1D70E},t}(F)$
satisfies the following.
-
(1) If a stability condition
$\unicode[STIX]{x1D70F}$
lies in the same connected component as
$\unicode[STIX]{x1D70E}$
in the space of stability conditions
$\text{Stab}({\mathcal{D}})$
, then
$$\begin{eqnarray}h_{\unicode[STIX]{x1D70E},t}(F)=h_{\unicode[STIX]{x1D70F},t}(F).\end{eqnarray}$$
-
(2) The mass growth of the generator
$G$
determines
$h_{\unicode[STIX]{x1D70E},t}(F)$
, that is,
$$\begin{eqnarray}h_{\unicode[STIX]{x1D70E},t}(F)=\limsup _{n\rightarrow \infty }\frac{1}{n}\log (m_{\unicode[STIX]{x1D70E},t}(F^{n}G)).\end{eqnarray}$$
-
(3) An inequality
holds.
$$\begin{eqnarray}h_{\unicode[STIX]{x1D70E},t}(F)\leqslant h_{t}(F)<\infty\end{eqnarray}$$
In the case
$t=0$
, this result was stated in [Reference Dimitrov, Haiden, Katzarkov and Kontsevich11, Section 4.5] by using the triangle inequality for mass (see Proposition 3.3). However, there are no literature to prove the triangle inequality for mass even if
$t=0$
. Therefore, we give a detailed proof of it with a parameter
$t$
in Section 3.2, which is the most technical part of this paper.
1.2 Lower bound by the spectral radius
We consider the lower bound of the mass growth when
$t=0$
. Since
$F:{\mathcal{D}}\rightarrow {\mathcal{D}}$
preserves exact triangles,
$F$
induces a linear transformation
$[F]:K({\mathcal{D}})\rightarrow K({\mathcal{D}})$
. We extend
$[F]$
on
$K({\mathcal{D}})\otimes \mathbb{C}$
naturally. The spectral radius of
$[F]$
is defined by
$$\begin{eqnarray}\unicode[STIX]{x1D70C}([F]):=\max \{|\unicode[STIX]{x1D706}|\mid \unicode[STIX]{x1D706}\text{ is an eigenvalue of }[F]\text{ on }K({\mathcal{D}})\otimes \mathbb{C}\}.\end{eqnarray}$$
Theorem 1.2. (Proposition 3.11)
In the case
$t=0$
, we have an inequality
$$\begin{eqnarray}\log \unicode[STIX]{x1D70C}([F])\leqslant h_{\unicode[STIX]{x1D70E},0}(F)\leqslant h_{0}(F)\end{eqnarray}$$
for any stability condition
$\unicode[STIX]{x1D70E}\in \text{Stab}({\mathcal{D}})$
.
As known results, if
${\mathcal{D}}$
is saturated, then it was shown in [Reference Dimitrov, Haiden, Katzarkov and Kontsevich11, Theorem 2.9] that for a linear map
$\text{HH}_{\ast }(F):\text{HH}_{\ast }({\mathcal{D}})\rightarrow \text{HH}_{\ast }({\mathcal{D}})$
induced on the Hochschild homology of
${\mathcal{D}}$
, the inequality
$\log \unicode[STIX]{x1D70C}(\text{HH}_{\ast }(F))\leqslant h_{0}(F)$
holds under some condition for eigenvalues of
$\text{HH}_{\ast }(F)$
. They also conjectured that the inequality holds without that condition. Our result, Theorem 1.2, holds without any conditions for
$[F]$
. However, we use the existence of stability conditions on
${\mathcal{D}}$
. For many examples in [Reference Dimitrov, Haiden, Katzarkov and Kontsevich11, Reference Kikuta17, Reference Kikuta and Takahashi18], it was shown that the equality
$\log \unicode[STIX]{x1D70C}([F])=h_{0}(F)$
holds. Kikuta–Takahashi gave a certain conjecture on the equality in [Reference Kikuta and Takahashi18, Conjecture 5.3].
1.3 Equality between mass growth and entropy
The remaining important question is to ask when the equality
$h_{\unicode[STIX]{x1D70E},t}(F)=h_{t}(F)$
holds. In the following, we give a sufficient condition for the equality. For a stability condition
$\unicode[STIX]{x1D70E}=(Z,{\mathcal{P}})$
, we can associate an abelian category, called the heart of
${\mathcal{P}}$
, as the extension-closed subcategory generated by objects in
${\mathcal{P}}(\unicode[STIX]{x1D719})$
for
$\unicode[STIX]{x1D719}\in (0,1]$
. Denote it by
${\mathcal{P}}((0,1])$
. A stability condition
$\unicode[STIX]{x1D70E}=(Z,{\mathcal{P}})$
is called algebraic if the heart
${\mathcal{P}}((0,1])$
is a finite length abelian category with finitely many simple objects (see Definitions 2.7 and 2.11).
Theorem 1.3. (Theorem 3.14)
Let
$F:{\mathcal{D}}\rightarrow {\mathcal{D}}$
be an endofunctor. If a connected component
$\text{Stab}^{\circ }({\mathcal{D}})\subset \text{Stab}(D)$
contains an algebraic stability condition, then
${\mathcal{D}}$
has a split generator
$G$
and for any
$\unicode[STIX]{x1D70E}\in \text{Stab}^{\circ }({\mathcal{D}})$
we have
$$\begin{eqnarray}h_{t}(F)=h_{\unicode[STIX]{x1D70E},t}(F)=\lim _{n\rightarrow \infty }\frac{1}{n}\log (m_{\unicode[STIX]{x1D70E},t}(F^{n}G)).\end{eqnarray}$$
Note that in the above theorem, the stability condition
$\unicode[STIX]{x1D70E}$
is not necessarily an algebraic stability condition.
We see a typical example, which satisfies the condition in Theorem 1.3 from Section 4.1. Let
$A=\bigoplus _{k}A^{k}$
be a dg-algebra such that
$H^{0}(A)$
is a finite-dimensional algebra and
$H^{k}(A)=0$
for
$k>0$
. Denote by
${\mathcal{D}}_{fd}(A)$
the derived category of dg-modules over
$A$
with finite-dimensional total cohomology, that is,
$\sum _{k}\dim H^{k}(M)<\infty$
. Then there is a bounded t-structure whose heart is isomorphic to the abelian category of finite-dimensional modules over
$H^{0}(A)$
. As a result, we can construct algebraic stability conditions on
${\mathcal{D}}_{fd}(A)$
. Thus, in the context of representation theory, Theorem 1.3 works well. As an application, we compute the entropy of spherical twists in Section 4.2.
On the other hand, for derived categories coming from algebraic geometry, we cannot find algebraic hearts in general. Only in special cases, for example in the case that the derived category has a full strong exceptional collection, the work by Bondal [Reference Bondal7] enables us to find algebraic hearts. It is an important problem to answer whether the equality
$h_{\unicode[STIX]{x1D70E},t}(F)=h_{t}(F)$
holds without the existence of algebraic stability conditions.
1.4 Categorical theory versus classical theory
We compare our result with the well-known classical result “Volume growth and entropy” by Yomdin [Reference Yomdin22]. Let
$M$
be a compact smooth manifold and
$f:M\rightarrow M$
be a smooth map. The map
$f$
induces a linear map
$f_{\ast }:H_{\ast }(M;\mathbb{R})\rightarrow H_{\ast }(M;\mathbb{R})$
on the homology group
$H_{\ast }(M;\mathbb{R})$
. For the map
$f$
, we can define the topological entropy
$h_{\text{top}}(f)$
[Reference Adler, Konheim and McAndrew1] and the inequality
$\log \unicode[STIX]{x1D70C}(f_{\ast })\leqslant h_{\text{top}}(f)$
was conjectured in [Reference Shub21]. In [Reference Yomdin22], Yomdin introduced the volume growth
$v(f)$
by using a Riemannian metric on
$M$
and showed that
$$\begin{eqnarray}\log \unicode[STIX]{x1D70C}(f_{\ast })\leqslant v(f)\leqslant h_{\text{top}}(f).\end{eqnarray}$$
Our result Theorem 1.2 looks like the categorical analogue of this classical result. On the other hand, the difference between categorical theory and classical theory is that the categorical entropy
$h_{t}(F)$
and the mass growth
$h_{\unicode[STIX]{x1D70E},t}(F)$
have the parameter
$t$
, which measures the growth rate of degree shifts in a triangulated category. This point is an essentially new feature of categorical theory.
Notations. We work over a field
$\mathbb{K}$
. All triangulated categories in this paper are
$\mathbb{K}$
-linear and their Grothendieck groups are free of finite rank, that is,
$K({\mathcal{D}})\cong \mathbb{Z}^{n}$
for some
$n$
. An endofunctor
$F:{\mathcal{D}}\rightarrow {\mathcal{D}}$
refers to an exact endofunctor, that is,
$F$
preserves all exact triangles and commutes with degree shifts. The natural logarithm is extended to
$\log :[0,\infty )\rightarrow [-\infty ,\infty )$
by setting
$\log 0:=-\infty$
.
2 Preliminaries
In this section, we prepare basic terminologies mainly from [Reference Bridgeland8, Reference Dimitrov, Haiden, Katzarkov and Kontsevich11].
2.1 Complexity and entropy
First, we recall the notion of complexity and entropy from [Reference Dimitrov, Haiden, Katzarkov and Kontsevich11, Section 2].
Let
${\mathcal{D}}$
be a triangulated category. A triangulated subcategory is called thick if it is closed under taking direct summands. For an object
$E\in {\mathcal{D}}$
, we denote by
$\langle E\rangle \subset {\mathcal{D}}$
the smallest thick triangulated subcategory containing
$E$
. An object
$G\in {\mathcal{D}}$
is called a split generator if
$\langle G\rangle ={\mathcal{D}}$
. This implies that for any object
$E\in {\mathcal{D}}$
, there is some object
$E^{\prime }\in {\mathcal{D}}$
such that we have a sequence of exact triangles
with
$n_{i}\in \mathbb{Z}$
. We note that the object
$E^{\prime }$
and the above sequence are not unique.
Definition 2.1. [Reference Dimitrov, Haiden, Katzarkov and Kontsevich11, Definition 2.1]
Let
$E_{1}$
and
$E_{2}$
be objects in
${\mathcal{D}}$
. The complexity of
$E_{2}$
relative to
$E_{1}$
is the function
$\unicode[STIX]{x1D6FF}_{t}(E_{1},E_{2}):\mathbb{R}\rightarrow [0,\infty ]$
defined by
By definition, we have an inequality
$0<\unicode[STIX]{x1D6FF}_{t}(G,E)<\infty$
for a split generator
$G\in {\mathcal{D}}$
and a nonzero object
$E\in {\mathcal{D}}$
. We recall fundamental inequalities for complexity.
Proposition 2.2. [Reference Dimitrov, Haiden, Katzarkov and Kontsevich11, Proposition 2.3]
For
$E_{1},E_{2},E_{3}\in {\mathcal{D}}$
,
-
(1)
$\unicode[STIX]{x1D6FF}_{t}(E_{1},E_{3})\leqslant \unicode[STIX]{x1D6FF}_{t}(E_{1},E_{2})\unicode[STIX]{x1D6FF}_{t}(E_{2},E_{3})$
, -
(2)
$\unicode[STIX]{x1D6FF}_{t}(E_{1},E_{2}\oplus E_{3})\leqslant \unicode[STIX]{x1D6FF}_{t}(E_{1},E_{2})+\unicode[STIX]{x1D6FF}_{t}(E_{1},E_{3})$
, -
(3)
$\unicode[STIX]{x1D6FF}_{t}(F(E_{1}),F(E_{2}))\leqslant \unicode[STIX]{x1D6FF}_{t}(E_{1},E_{2})$
for an endofunctor
$F:{\mathcal{D}}\rightarrow {\mathcal{D}}$
.
Similar to [Reference Dimitrov, Haiden, Katzarkov and Kontsevich11, Proposition 2.3], it is easy to check the following. This generalizes Proposition 2.2(2) to the nonsplit case.
Lemma 2.3. For objects
$D,E_{1},E_{2},E_{3}\in {\mathcal{D}}$
, if there is an exact triangle
$E_{1}\rightarrow E_{2}\rightarrow E_{3}\rightarrow E_{1}[1]$
, then
$$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{t}(D,E_{2})\leqslant \unicode[STIX]{x1D6FF}_{t}(D,E_{1})+\unicode[STIX]{x1D6FF}_{t}(D,E_{3}).\end{eqnarray}$$
Now, we introduce the notion of the entropy of endofunctors. The entropy of an endofunctor
$F$
measures the growth rate of complexity
$\unicode[STIX]{x1D6FF}_{t}(G,F^{n}G)$
as
$n\rightarrow \infty$
.
Definition 2.4. [Reference Dimitrov, Haiden, Katzarkov and Kontsevich11, Definition 2.5]
Let
${\mathcal{D}}$
be a triangulated category with a split generator
$G$
and let
$F:{\mathcal{D}}\rightarrow {\mathcal{D}}$
be an endofunctor. The entropy of
$F$
is the function
$h_{t}(F):\mathbb{R}\rightarrow [-\infty ,\infty )$
defined by
$$\begin{eqnarray}h_{t}(F):=\lim _{n\rightarrow \infty }\frac{1}{n}\log \unicode[STIX]{x1D6FF}_{t}(G,F^{n}G).\end{eqnarray}$$
By [Reference Dimitrov, Haiden, Katzarkov and Kontsevich11, Lemma 2.6], it follows that
$h_{t}(F)$
is well defined and
$h_{t}(F)<\infty$
.
2.2 Bounded t-structures and the associated cohomology
Definition 2.5. [Reference Beĭlinson, Bernstein and Deligne6]
A t-structure on
${\mathcal{D}}$
is a full subcategory
${\mathcal{F}}\subset {\mathcal{D}}$
satisfying the following conditions:
-
(a)
${\mathcal{F}}[1]\subset {\mathcal{F}}$
; -
(b) define
${\mathcal{F}}^{\bot }:=\{F\in {\mathcal{D}}|\operatorname{Hom}(D,F)=0$
for all
$D\in {\mathcal{F}}\}$
, then for every object
$E\in {\mathcal{D}}$
, there is an exact triangle
$D\rightarrow E\rightarrow F\rightarrow D[1]$
in
${\mathcal{D}}$
with
$D\in {\mathcal{F}}$
and
$F\in {\mathcal{F}}^{\bot }$
.
In addition, the t-structure
${\mathcal{F}}\subset {\mathcal{D}}$
is said to be bounded if
${\mathcal{F}}$
satisfies the condition
$$\begin{eqnarray}{\mathcal{D}}=\mathop{\bigcup }_{i,j\in \mathbb{Z}}{\mathcal{F}}^{\bot }[i]\cap {\mathcal{F}}[j].\end{eqnarray}$$
For a t-structure
${\mathcal{F}}\subset {\mathcal{D}}$
, we define the heart
${\mathcal{H}}\subset {\mathcal{D}}$
by
$$\begin{eqnarray}{\mathcal{H}}:={\mathcal{F}}^{\bot }[1]\cap {\mathcal{F}}.\end{eqnarray}$$
It was proved in [Reference Beĭlinson, Bernstein and Deligne6] that
${\mathcal{H}}$
becomes an abelian category. Bridgeland gave the characterization of the heart of a bounded t-structure as follows.
Lemma 2.6. [Reference Bridgeland8, Lemma 3.2]
Let
${\mathcal{H}}\subset {\mathcal{D}}$
be a full additive subcategory. Then
${\mathcal{H}}$
is the heart of a bounded t-structure if and only if the following conditions hold:
-
(a) if
$k_{1}>k_{2}\in \mathbb{Z}$
and
$A_{i}\in {\mathcal{H}}[k_{i}]\;(i=1,2)$
, then
$\operatorname{Hom}_{{\mathcal{D}}}(A_{1},A_{2})=0$
; -
(b) for
$0\neq E\in {\mathcal{D}}$
, there is a finite sequence of integers and a sequence of exact triangles
$$\begin{eqnarray}k_{1}>k_{2}>\cdots >k_{m}\end{eqnarray}$$
with
$A_{i}\in {\mathcal{H}}[k_{i}]$
for all
$i$
.
The above filtration in the condition (b) defines the
$k$
th cohomology
$H^{k}(E)\in {\mathcal{H}}$
of the object
$E$
by
$$\begin{eqnarray}H^{k}(E):=\left\{\begin{array}{@{}ll@{}}A_{i}[-k_{i}]\quad & \text{if }k=-k_{i}\\ 0\quad & \text{otherwise}.\end{array}\right.\end{eqnarray}$$
This cohomology becomes a cohomological functor from
${\mathcal{D}}$
to
${\mathcal{H}}$
, that is, if there is an exact triangle
$D\rightarrow E\rightarrow F\rightarrow E[1]$
, then we can obtain a long exact sequence
$$\begin{eqnarray}\cdots \rightarrow H^{k-1}(F)\rightarrow H^{k}(D)\rightarrow H^{k}(E)\rightarrow H^{k}(F)\rightarrow H^{k+1}(D)\rightarrow \cdots\end{eqnarray}$$
in the abelian category
${\mathcal{H}}$
. In the last of this section, we introduce a special class of bounded t-structures.
Definition 2.7. We say that the heart of a bounded t-structure is algebraic if it is a finite length abelian category with finitely many isomorphism classes of.
If
${\mathcal{D}}$
has an algebraic heart
${\mathcal{H}}$
with simple objects
$S_{1},\ldots ,S_{n}$
, then it is easy to see that the direct sum
$G:=\bigoplus _{i=1}^{n}S_{i}$
becomes a split generator of
${\mathcal{D}}$
.
2.3 Bridgeland stability conditions
In [Reference Bridgeland8], Bridgeland introduced the notion of a stability condition on a triangulated category as follows.
Definition 2.8. Let
${\mathcal{D}}$
be a triangulated category and
$K({\mathcal{D}})$
be its Grothendieck group. A stability condition
$\unicode[STIX]{x1D70E}=(Z,{\mathcal{P}})$
on
${\mathcal{D}}$
consists of a group homomorphism
$Z:K({\mathcal{D}})\rightarrow \mathbb{C}$
, called a central charge, and a family of full additive subcategories
${\mathcal{P}}(\unicode[STIX]{x1D719})\subset {\mathcal{D}}$
for
$\unicode[STIX]{x1D719}\in \mathbb{R}$
satisfying the following conditions:
-
(a) if
$0\neq E\in {\mathcal{P}}(\unicode[STIX]{x1D719})$
, then
$Z(E)=m(E)\exp (i\unicode[STIX]{x1D70B}\unicode[STIX]{x1D719})$
for some
$m(E)\in \mathbb{R}_{{>}0}$
; -
(b) for all
$\unicode[STIX]{x1D719}\in \mathbb{R}$
,
${\mathcal{P}}(\unicode[STIX]{x1D719}+1)={\mathcal{P}}(\unicode[STIX]{x1D719})[1]$
; -
(c) if
$\unicode[STIX]{x1D719}_{1}>\unicode[STIX]{x1D719}_{2}$
and
$A_{i}\in {\mathcal{P}}(\unicode[STIX]{x1D719}_{i})\;(i=1,2)$
, then
$\operatorname{Hom}_{{\mathcal{D}}}(A_{1},A_{2})=0$
; -
(d) for
$0\neq E\in {\mathcal{D}}$
, there is a finite sequence of real numbers and a sequence of exact triangles
$$\begin{eqnarray}\unicode[STIX]{x1D719}_{1}>\unicode[STIX]{x1D719}_{2}>\cdots >\unicode[STIX]{x1D719}_{m}\end{eqnarray}$$
with
$A_{i}\in {\mathcal{P}}(\unicode[STIX]{x1D719}_{i})$
for all
$i$
.
We write
$\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70E}}^{+}(E):=\unicode[STIX]{x1D719}_{1}$
and
$\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70E}}^{-}(E):=\unicode[STIX]{x1D719}_{m}$
. Nonzero objects in
${\mathcal{P}}(\unicode[STIX]{x1D719})$
are called
$\unicode[STIX]{x1D70E}$
-semistable of phase
$\unicode[STIX]{x1D719}$
in
$\unicode[STIX]{x1D70E}$
. The sequence of exact triangles in (d) is called a Harder–Narasimhan filtration of
$E$
with semistable factors
$A_{1},\ldots ,A_{m}$
of phases
$\unicode[STIX]{x1D719}_{1}>\cdots >\unicode[STIX]{x1D719}_{m}$
.
In addition to the above axioms, we always assume that our stability conditions have the support property in [Reference Kontsevich and Soibelman19]. Let
$\Vert \,\cdot \,\Vert$
be some norm on
$K({\mathcal{D}})\otimes \mathbb{R}$
. A stability condition
$\unicode[STIX]{x1D70E}=(Z,{\mathcal{P}})$
satisfies the support property if there is some constant
$C>0$
such that
$$\begin{eqnarray}\frac{|Z(E)|}{\Vert [E]\Vert }>C\end{eqnarray}$$
for all
$\unicode[STIX]{x1D70E}$
-semistable objects
$E\in {\mathcal{D}}$
.
For an interval
$I\subset \mathbb{R}$
, we denote by
${\mathcal{P}}(I)$
the extension-closed subcategory generated by objects in
${\mathcal{P}}(\unicode[STIX]{x1D719})$
for
$\unicode[STIX]{x1D719}\in I$
, namely
$$\begin{eqnarray}{\mathcal{P}}(I):=\{E\in {\mathcal{D}}\mid \unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70E}}^{\pm }(E)\in I\}\cup \{0\}.\end{eqnarray}$$
From a stability condition
$(Z,{\mathcal{P}})$
, we can construct a bounded t-structure
${\mathcal{F}}:={\mathcal{P}}((0,\infty ))$
and its heart is given by
${\mathcal{H}}={\mathcal{P}}((0,1])$
.
2.4 Algebraic stability conditions
In [Reference Bridgeland8], Bridgeland gave the alternative description of a stability condition on
${\mathcal{D}}$
as a pair of a bounded t-structure and a central charge on its heart. By using this description, we construct algebraic stability conditions.
Definition 2.9. Let
${\mathcal{H}}$
be an abelian category and let
$K({\mathcal{H}})$
be its Grothendieck group. A central charge on
${\mathcal{H}}$
is a group homomorphism
$Z:K({\mathcal{H}})\rightarrow \mathbb{C}$
such that for any nonzero object
$0\neq E\in {\mathcal{H}}$
, the complex number
$Z(E)$
lies in the semiclosed upper half-plane
$\mathbb{H}:=\{re^{i\unicode[STIX]{x1D70B}\unicode[STIX]{x1D719}}~\in \mathbb{C}\mid r\in \mathbb{R}_{{>}0},\unicode[STIX]{x1D719}\in (0,1]\}$
.
For any nonzero object
$E\in {\mathcal{H}}$
, define the phase of
$E$
by
$$\begin{eqnarray}\unicode[STIX]{x1D719}(E):=\frac{1}{\unicode[STIX]{x1D70B}}\arg Z(E)\in (0,1].\end{eqnarray}$$
An object
$0\neq E\in {\mathcal{H}}$
is called
$Z$
-semistable if every subobject
$0\neq A\subset E$
satisfies
$\unicode[STIX]{x1D719}(A)\leqslant \unicode[STIX]{x1D719}(E)$
. A Harder–Narasimhan filtration of
$0\neq E\in {\mathcal{H}}$
is the filtration
$$\begin{eqnarray}0=E_{0}\subset E_{1}\subset \cdots \subset E_{m-1}\subset E_{m}=E,\end{eqnarray}$$
whose extension factors
$F_{i}:=E_{i}/E_{i-1}$
are
$Z$
-semistable with decreasing phases
$$\begin{eqnarray}\unicode[STIX]{x1D719}(F_{1})>\cdots >\unicode[STIX]{x1D719}(F_{m}).\end{eqnarray}$$
A central charge
$Z$
is said to have the Harder–Narasimhan property if any nonzero object of
${\mathcal{H}}$
has a Harder–Narasimhan filtration. The following gives another definition of a stability condition.
Proposition 2.10. [Reference Bridgeland8, Proposition 5.3]
Giving a stability condition on
${\mathcal{D}}$
is equivalent to giving a heart
${\mathcal{H}}$
of a bounded structure on
${\mathcal{D}}$
and a central charge on
${\mathcal{H}}$
with the Harder–Narasimhan property.
In Proposition 2.10, the pair
$(Z,{\mathcal{H}})$
is constructed from a stability condition
$(Z,{\mathcal{P}})$
by setting
${\mathcal{H}}:={\mathcal{P}}((0,1])$
.
Definition 2.11. A stability condition
$(Z,{\mathcal{P}})$
is called algebraic if the corresponding heart
${\mathcal{H}}={\mathcal{P}}((0,1])$
is algebraic (for the definition of algebraic hearts, see Definition 2.7).
Algebraic stability conditions are constructed from algebraic hearts as follows. Let
${\mathcal{H}}\subset {\mathcal{D}}$
be an algebraic heart with simple objects
$S_{1},\ldots ,S_{n}$
. Then the Grothendieck group is given by
$K({\mathcal{H}})\cong \bigoplus _{i=1}^{n}\mathbb{Z}[S_{i}]$
. Take
$(z_{1},\ldots ,z_{n})\in \mathbb{H}^{n}$
and define the central charge
$Z:K({\mathcal{H}})\rightarrow \mathbb{C}$
by the linear extension of
$Z(S_{i}):=z_{i}$
. Then
$Z$
has the Harder–Narasimhan property by [Reference Bridgeland8, Proposition 2.4]. Thus
$(Z,{\mathcal{H}})$
becomes an algebraic stability condition on
${\mathcal{D}}$
.
2.5 Harder–Narasimhan polygons
In this section, we discuss the Harder–Narasimhan polygon following [Reference Bayer5]. This plays a key role in showing the triangle inequality for mass in Section 3.2. The following is based on [Reference Bayer5, Section 3].
Definition 2.12. Let
${\mathcal{H}}$
be an abelian category and
$Z$
be a central charge on it. For an object
$E\in {\mathcal{H}}$
, the Harder–Narasimhan polygon
$\operatorname{HN}^{Z}(E)$
of
$E$
is the convex hull of the subset
$\{Z(A)\in \mathbb{C}\mid A\subset E\}\subset \mathbb{C}$
in the complex plane.
It is clear from the definition that if
$F\subset E$
, then
$\operatorname{HN}^{Z}(F)\subset \operatorname{HN}^{Z}(E)$
. The Harder–Narasimhan polygon
$\operatorname{HN}^{Z}(E)$
is called polyhedral on the left if it has finitely many extremal points
$0=z_{0},z_{1},\ldots ,z_{k}=Z(E)$
such that
$\operatorname{HN}^{Z}(E)$
lies to the right of the path
$z_{0}z_{1}\ldots z_{k}$
. This implies that the intersection of
$\operatorname{HN}^{Z}(E)$
and the closed half-plane to the left of the line through
$0$
and
$Z(E)$
becomes a polygon with vertices
$z_{0},z_{1},\ldots ,z_{k}$
(see Figure 1).
Figure 1. Harder–Narasimhan polygon.
Proposition 2.13. [Reference Bayer5, Proposition 3.3]
The object
$E$
has a Harder–Narasimhan filtration if and only if
$\operatorname{HN}^{Z}(E)$
is polyhedral on the left. In particular, if the Harder–Narasimhan filtration of
$E$
is given by
$$\begin{eqnarray}0=E_{0}\subset E_{1}\subset E_{2}\subset \cdots E_{k}=E,\end{eqnarray}$$
then extremal points of
$\operatorname{HN}^{Z}(E)$
are given by
$z_{i}=Z(E_{i})$
for
$i=0,1,\ldots ,k$
.
2.6 Topology on the space of stability conditions
In [Reference Bridgeland8], Bridgeland introduced a natural topology on the space of stability conditions and showed that this space becomes a complex manifold. In the following, we recall his construction. Let
$\text{Stab}({\mathcal{D}})$
be the set of stability conditions on a triangulated category
${\mathcal{D}}$
with the support property. For stability conditions
$\unicode[STIX]{x1D70E}=(Z,{\mathcal{P}})$
and
$\unicode[STIX]{x1D70F}=(W,{\mathcal{Q}})$
in
$\text{Stab}(D)$
, set
$$\begin{eqnarray}\displaystyle d({\mathcal{P}},{\mathcal{Q}}):=\sup _{0\neq E\in {\mathcal{D}}}\{|\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70E}}^{-}(E)-\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70F}}^{-}(E)|,|\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70E}}^{+}(E)-\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70F}}^{+}(E)|\}\in [0,\infty ] & & \displaystyle \nonumber\end{eqnarray}$$
and
$$\begin{eqnarray}\Vert Z-W\Vert _{\unicode[STIX]{x1D70E}}:=\sup \bigg\{\frac{|Z(E)-W(E)|}{|Z(E)|}\biggm\vert\,\!E\;\text{is}\;\unicode[STIX]{x1D70E}\text{-}\text{semistable}\bigg\}\in [0,\infty ].\end{eqnarray}$$
Define a subset
$B_{\unicode[STIX]{x1D716}}(\unicode[STIX]{x1D70E})\subset \text{Stab}({\mathcal{D}})$
by
$$\begin{eqnarray}B_{\unicode[STIX]{x1D716}}(\unicode[STIX]{x1D70E}):=\{\unicode[STIX]{x1D70F}=(W,{\mathcal{Q}})\in \text{Stab}({\mathcal{D}})\mid d({\mathcal{P}},{\mathcal{Q}})<\unicode[STIX]{x1D716},\Vert Z-W\Vert _{\unicode[STIX]{x1D70E}}<\sin (\unicode[STIX]{x1D70B}\unicode[STIX]{x1D716})\}\end{eqnarray}$$
for
$0<\unicode[STIX]{x1D716}<\frac{1}{4}$
.
In [Reference Bridgeland8, Section 6], it was shown that the family of subsets
$$\begin{eqnarray}\{B_{\unicode[STIX]{x1D716}}(\unicode[STIX]{x1D70E})\subset \text{Stab}({\mathcal{D}})\mid \unicode[STIX]{x1D70E}\in \text{Stab}({\mathcal{D}}),0<\unicode[STIX]{x1D716}<{\textstyle \frac{1}{4}}\}\end{eqnarray}$$
becomes an open basis of a topology on
$\text{Stab}({\mathcal{D}})$
. In [Reference Bridgeland8], Bridgeland showed a crucial theorem.
Theorem 2.14. [Reference Bridgeland8, Theorem 1.2]
The projection map of central charges
$$\begin{eqnarray}\unicode[STIX]{x1D70B}:\text{Stab}({\mathcal{D}})\longrightarrow \operatorname{Hom}_{\mathbb{Z}}(K({\mathcal{D}}),\mathbb{C}),\quad (Z,{\mathcal{P}})\mapsto Z\end{eqnarray}$$
is a local isomorphism of topological spaces. In particular,
$\unicode[STIX]{x1D70B}$
induces a complex structure on
$\text{Stab}({\mathcal{D}})$
.
3 Mass growth of objects and categorical entropy
3.1 Mass with a parameter and complexity
In this section, we introduce the mass growth of objects and show fundamental properties of it.
Definition 3.1. [Reference Dimitrov, Haiden, Katzarkov and Kontsevich11, Section 4.5]
Take a stability condition
$\unicode[STIX]{x1D70E}=(Z,{\mathcal{P}})$
on
${\mathcal{D}}$
. Let
$E\in {\mathcal{D}}$
be a nonzero object with semistable factors
$A_{1},\ldots ,A_{m}$
of phases
$\unicode[STIX]{x1D719}_{1}>\cdots >\unicode[STIX]{x1D719}_{m}$
. The mass of
$E$
with a parameter
$t\in \mathbb{R}$
is the function
$m_{\unicode[STIX]{x1D70E},t}(E):\mathbb{R}\rightarrow \mathbb{R}_{{>}0}$
defined by
$$\begin{eqnarray}m_{\unicode[STIX]{x1D70E},t}(E):=\mathop{\sum }_{i=1}^{m}|Z(A_{i})|e^{\unicode[STIX]{x1D719}_{i}t}.\end{eqnarray}$$
When
$t=0$
,
$m_{\unicode[STIX]{x1D70E},0}(E)$
is called the mass of
$E$
and simply written as
$m_{\unicode[STIX]{x1D70E}}(E):=m_{\unicode[STIX]{x1D70E},0}(E)$
. As a convention, set
$m_{\unicode[STIX]{x1D70E},t}(E):=0$
if
$E\cong 0$
.
In the following, if
$\unicode[STIX]{x1D70E}$
is clear in the context, we often drop it from the notation and write
$m_{t}(E)$
. Similar to the growth rate of complexity of a generator with respect to endofunctors, we consider the growth rate of mass of objects.
Definition 3.2. [Reference Dimitrov, Haiden, Katzarkov and Kontsevich11, Section 4.5]
Let
$\unicode[STIX]{x1D70E}$
be a stability condition on
${\mathcal{D}}$
and
$F:{\mathcal{D}}\rightarrow {\mathcal{D}}$
be an endofunctor. The mass growth with respect to
$F$
is the function
$h_{\unicode[STIX]{x1D70E},t}(F):\mathbb{R}\rightarrow [-\infty ,\infty ]$
defined by
$$\begin{eqnarray}h_{\unicode[STIX]{x1D70E},t}(F):=\sup _{E\in {\mathcal{D}}}\bigg\{\limsup _{n\rightarrow \infty }\frac{1}{n}\log (m_{\unicode[STIX]{x1D70E},t}(F^{n}E))\bigg\}.\end{eqnarray}$$
In the rest of this section, we study fundamental properties of
$h_{\unicode[STIX]{x1D70E},t}(F)$
. The triangle inequality for
$m_{\unicode[STIX]{x1D70E},t}$
plays an important role.
Proposition 3.3. For objects
$D,E,F\in {\mathcal{D}}$
, if there is an exact triangle
$D\rightarrow E\rightarrow F\rightarrow D[1]$
, then
$$\begin{eqnarray}m_{\unicode[STIX]{x1D70E},t}(E)\leqslant m_{\unicode[STIX]{x1D70E},t}(D)+m_{\unicode[STIX]{x1D70E},t}(F).\end{eqnarray}$$
The proof of Proposition 3.3 is given in Section 3.2.
Proposition 3.4. Let
$\unicode[STIX]{x1D70E}$
be a stability condition on
${\mathcal{D}}$
. Then
$$\begin{eqnarray}m_{\unicode[STIX]{x1D70E},t}(E)\leqslant m_{\unicode[STIX]{x1D70E},t}(D)\unicode[STIX]{x1D6FF}_{t}(D,E)\end{eqnarray}$$
for any objects
$D,E\in {\mathcal{D}}$
.
Proof. It is sufficient to show the case
$E\in \langle D\rangle$
. Then by the definition of complexity
$\unicode[STIX]{x1D6FF}_{t}(D,E)$
, for any
$\unicode[STIX]{x1D716}>0$
, there is a sequence of exact triangles
such that
$$\begin{eqnarray}\mathop{\sum }_{i=1}^{k}e^{n_{i}t}<\unicode[STIX]{x1D6FF}_{t}(D,E)+\unicode[STIX]{x1D716}.\end{eqnarray}$$
Note that
$m_{\unicode[STIX]{x1D70E},t}$
satisfies
$m_{\unicode[STIX]{x1D70E},t}(D[n])=m_{\unicode[STIX]{x1D70E},t}(D)\cdot e^{nt}$
for
$D\in {\mathcal{D}}$
and
$n\in \mathbb{Z}$
. By using the inequality in Proposition 3.3 repeatedly, we have
$$\begin{eqnarray}\displaystyle m_{\unicode[STIX]{x1D70E},t}(E)\leqslant m_{\unicode[STIX]{x1D70E},t}(E\oplus E^{\prime }) & {\leqslant} & \displaystyle \mathop{\sum }_{i=1}^{k}m_{\unicode[STIX]{x1D70E},t}(D[n_{i}])\leqslant m_{\unicode[STIX]{x1D70E},t}(D)\cdot \biggl(\mathop{\sum }_{i=1}^{k}e^{n_{i}t}\biggr)\nonumber\\ \displaystyle & {\leqslant} & \displaystyle m_{\unicode[STIX]{x1D70E},t}(D)\unicode[STIX]{x1D6FF}_{t}(D,E)+\unicode[STIX]{x1D716}\cdot m_{\unicode[STIX]{x1D70E},t}(D)\nonumber\end{eqnarray}$$
for any
$\unicode[STIX]{x1D716}>0$
. This implies the result.◻
Now, we show fundamental properties of the mass growth.
Theorem 3.5. Let
$F:{\mathcal{D}}\rightarrow {\mathcal{D}}$
be an endofunctor and
$\unicode[STIX]{x1D70E}$
be a stability condition on
${\mathcal{D}}$
. Assume that
${\mathcal{D}}$
has a split generator
$G\in {\mathcal{D}}$
. Then the mass growth
$h_{\unicode[STIX]{x1D70E},t}(F)$
satisfies the followings.
-
(1)
$h_{\unicode[STIX]{x1D70E},t}(F)$
is given by
$$\begin{eqnarray}h_{\unicode[STIX]{x1D70E},t}(F)=\limsup _{n\rightarrow \infty }\frac{1}{n}\log (m_{\unicode[STIX]{x1D70E},t}(F^{n}G)).\end{eqnarray}$$
-
(2) We have an inequality
where
$$\begin{eqnarray}h_{\unicode[STIX]{x1D70E},t}(F)\leqslant h_{t}(F)<\infty ,\end{eqnarray}$$
$h_{t}(F)$
is the entropy of
$F$
(see Definition 2.4).
Proof. By Proposition 2.2(3) and Proposition 3.4, we have
$$\begin{eqnarray}m_{t}(F^{n}E)\leqslant m_{t}(F^{n}G)\unicode[STIX]{x1D6FF}_{t}(F^{n}G,F^{n}E)\leqslant m_{t}(F^{n}G)\unicode[STIX]{x1D6FF}_{t}(G,E)\end{eqnarray}$$
for any object
$E\in {\mathcal{D}}$
. Hence
$$\begin{eqnarray}\limsup _{n\rightarrow \infty }\frac{1}{n}\log m_{t}(F^{n}E)\leqslant \limsup _{n\rightarrow \infty }\frac{1}{n}\log m_{t}(F^{n}G)\end{eqnarray}$$
and this inequality implies
$(1)$
. Again by Proposition 3.4, we have
$$\begin{eqnarray}m_{t}(F^{n}G)\leqslant m_{t}(G)\unicode[STIX]{x1D6FF}_{t}(G,F^{n}G).\end{eqnarray}$$
Hence
$$\begin{eqnarray}\limsup _{n\rightarrow \infty }\frac{1}{n}\log m_{t}(F^{n}G)\leqslant \lim _{n\rightarrow \infty }\frac{1}{n}\log \unicode[STIX]{x1D6FF}_{t}(G,F^{n}G)\end{eqnarray}$$
and this inequality implies
$(2)$
.◻
3.2 Triangle inequality for mass with a parameter
We prove Proposition 3.3. Recall the notation
$\mathbb{H}=\{re^{i\unicode[STIX]{x1D70B}\unicode[STIX]{x1D719}}\mid r>0,\unicode[STIX]{x1D719}\in (0,1]\}$
. For a complex number
$z\in \mathbb{H}$
, define a function of
$t\in \mathbb{R}$
by
$$\begin{eqnarray}g_{t}(z):=|z|e^{\unicode[STIX]{x1D719}(z)t},\end{eqnarray}$$
where
$\unicode[STIX]{x1D719}(z)$
is the phase of
$z$
given by
$\unicode[STIX]{x1D719}(z):=(1/\unicode[STIX]{x1D70B})\arg z\in (0,1]$
. We start by showing the triangle inequality for
$g_{t}(z)$
.
Lemma 3.6. For
$z_{1},z_{2}\in \mathbb{H}$
, an inequality
$$\begin{eqnarray}g_{t}(z_{1}+z_{2})\leqslant g_{t}(z_{1})+g_{t}(z_{2})\end{eqnarray}$$
holds.
Figure 2. Triangle consisting of vertices
$0,z_{1},z_{1}+z_{2}$
.
Proof. Set
$\unicode[STIX]{x1D719}_{1}:=\unicode[STIX]{x1D719}(z_{1}),\unicode[STIX]{x1D719}_{2}:=\unicode[STIX]{x1D719}(z_{2})$
and
$\unicode[STIX]{x1D719}_{3}:=\unicode[STIX]{x1D719}(z_{1}+z_{2})$
. If
$\unicode[STIX]{x1D719}_{1}=\unicode[STIX]{x1D719}_{2}$
, the result is trivial. Without loss of generality, we reduce to the case
$\unicode[STIX]{x1D719}_{1}>\unicode[STIX]{x1D719}_{2}$
. By applying the law of sine for the triangle consisting of vertices
$0,z_{1},z_{1}+z_{2}$
(see Figure 2), we obtain
$$\begin{eqnarray}|z_{1}+z_{2}|=d\sin (\unicode[STIX]{x1D70B}a+\unicode[STIX]{x1D70B}b),\qquad |z_{1}|=d\sin \unicode[STIX]{x1D70B}b,\qquad |z_{2}|=d\sin \unicode[STIX]{x1D70B}a,\end{eqnarray}$$
where
$a=\unicode[STIX]{x1D719}_{1}-\unicode[STIX]{x1D719}_{3}$
,
$b=\unicode[STIX]{x1D719}_{3}-\unicode[STIX]{x1D719}_{2}$
and
$d$
is the diameter of the circumcircle of the triangle. By inputting these parameters, the inequality
$g_{t}(z_{1}+z_{2})\leqslant g_{t}(z_{1})+g_{t}(z_{2})$
becomes
$$\begin{eqnarray}\sin (\unicode[STIX]{x1D70B}a+\unicode[STIX]{x1D70B}b)\leqslant e^{at}\sin \unicode[STIX]{x1D70B}b+e^{-bt}\sin \unicode[STIX]{x1D70B}a,\end{eqnarray}$$
where
$0<a,b<1$
. Dividing by
$\sin \unicode[STIX]{x1D70B}a\sin \unicode[STIX]{x1D70B}b$
and applying the addition formula, we have
$$\begin{eqnarray}\frac{e^{at}-\cos \unicode[STIX]{x1D70B}a}{\sin \unicode[STIX]{x1D70B}a}+\frac{e^{-bt}-\cos \unicode[STIX]{x1D70B}b}{\sin \unicode[STIX]{x1D70B}b}\geqslant 0.\end{eqnarray}$$
After setting
$c=-b$
, the above inequality is equivalent to
$f(a)\geqslant f(c)$
for
$-1<c<0<a<1$
, where
$$\begin{eqnarray}f(x)=\frac{e^{xt}-\cos \unicode[STIX]{x1D70B}x}{\sin \unicode[STIX]{x1D70B}x}.\end{eqnarray}$$
It is easy to check that
$f(x)$
is increasing in the intervals
$(-1,0)$
and
$(0,1)$
, and the limit of
$f(x)$
at the zero is given by
$\lim _{x\rightarrow \pm 0}f(x)=t/\unicode[STIX]{x1D70B}$
.◻
The triangle inequality for
$g_{t}(z)$
implies the following.
Lemma 3.7. Let
$z_{1},\ldots ,z_{k}$
and
$w_{1},\ldots ,w_{l}$
be the complex numbers in
$\mathbb{H}$
with
$z_{k}=w_{l}$
and set
$z_{0}=w_{0}=0$
. If they satisfy the following conditions (see the left of Figure 3):
-
(a)
$\unicode[STIX]{x1D719}(z_{i}-z_{i-1})>\unicode[STIX]{x1D719}(z_{i+1}-z_{i})$
and
$\unicode[STIX]{x1D719}(w_{j}-w_{j-1})>\unicode[STIX]{x1D719}(w_{j+1}-w_{j})$
for
$i=1,\ldots ,k$
and
$j=1,\ldots ,l$
; -
(b) the polygon
$w_{0}w_{1}w_{2}\ldots w_{l}w_{0}$
contains the polygon
$z_{0}z_{1}z_{2}\ldots z_{k}z_{0}$
,
then
$$\begin{eqnarray}\mathop{\sum }_{i=1}^{k}g_{t}(z_{i}-z_{i-1})\leqslant \mathop{\sum }_{j=1}^{l}g_{t}(w_{j}-w_{j-1}).\end{eqnarray}$$
Figure 3. Polygons and a triangulation of the encircled domain.
Proof. By the condition
$(b)$
, there is a unique domain encircled by two paths
$z_{0}z_{1}z_{2}\ldots z_{k}$
and
$w_{0}w_{1}w_{2}\ldots w_{l}$
. By the convexity condition
$(a)$
, we can triangulate this domain as follows. First, we extend lines
$z_{i}z_{i+1}$
for
$i=0,\ldots ,k-2$
to intersect
$w_{0}w_{1}w_{2}\ldots w_{l}$
. Then we obtain polygons which contain exactly one
$z_{i}z_{i+1}$
for some
$i$
. Next, we triangulate the polygon containing
$z_{i}z_{i+1}$
by drawing lines from
$z_{i}$
to
$w_{j}$
in the polygon. Then we can obtain the triangulated domain as in the right of Figure 3. Applying the triangle inequality for
$g_{t}(z)$
(Lemma 3.2) repeatedly, we obtain the result.◻
Lemma 3.8. Let
$\unicode[STIX]{x1D70E}=(Z,{\mathcal{P}})$
be a stability condition and
${\mathcal{H}}={\mathcal{P}}((0,1])$
be the associated heart. If there is a short exact sequence
$$\begin{eqnarray}0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0\end{eqnarray}$$
in
${\mathcal{H}}$
and
$C\in {\mathcal{P}}(1)$
, then
$$\begin{eqnarray}m_{t}(A)\leqslant m_{t}(B)+e^{-t}m_{t}(C).\end{eqnarray}$$
Proof. Let
$$\begin{eqnarray}\displaystyle & 0=A_{0}\subset A_{1}\subset A_{2}\subset \cdots \subset A_{k}=A & \displaystyle \nonumber\\ \displaystyle & 0=B_{0}\subset B_{1}\subset B_{2}\subset \cdots \subset B_{l-1}=B & \displaystyle \nonumber\end{eqnarray}$$
be Harder–Narasimhan filtrations of
$A$
and
$B$
. Set
$z_{i}:=Z(A_{i})$
for
$i=0,1,\ldots ,k$
,
$w_{j}:=Z(B_{j})$
for
$j=0,1,\ldots ,l-1$
and
$w_{l}:=Z(B)-Z(C)=Z(A)$
. Then by definition of the Harder–Narasimhan filtration, these complex numbers satisfy the condition
$(a)$
in Lemma 3.7. Consider the Harder–Narasimhan polygons
$\operatorname{HN}^{Z}(A)$
and
$\operatorname{HN}^{Z}(B)$
(see Definition 2.12). By Proposition 2.13, complex numbers
$z_{0},z_{1},\ldots ,z_{k}$
and
$w_{0},w_{1},\ldots ,w_{l-1}$
are extremal points of
$\operatorname{HN}^{Z}(A)$
and
$\operatorname{HN}^{Z}(B)$
, respectively. Thus, the intersection of
$\operatorname{HN}^{Z}(A)$
and the left of the line through
$0$
and
$z_{k}=Z(A)$
is the polygon
$z_{0}z_{1}z_{2}\ldots z_{k}z_{0}$
and the intersection of
$\operatorname{HN}^{Z}(B)$
and the left of the line through
$0$
and
$w_{l}=Z(A)$
is the polygon
$w_{0}w_{1}w_{2}\ldots w_{l}w_{0}$
. Since
$\operatorname{HN}^{Z}(A)\subset \operatorname{HN}^{Z}(B)$
, the polygon
$w_{0}w_{1}w_{2}\ldots w_{l}w_{0}$
contains the polygon
$z_{0}z_{1}z_{2}\ldots z_{k}z_{0}$
and this implies the condition
$(b)$
in Lemma 3.7. Since
$$\begin{eqnarray}\displaystyle m_{t}(A)=\mathop{\sum }_{i=1}^{k}g_{t}(z_{i}-z_{i-1}),\qquad m_{t}(B)=\mathop{\sum }_{j=1}^{l-1}g_{t}(w_{j}-w_{j-1}),\qquad e^{-t}m_{t}(C)=g_{t}(w_{l}-w_{l-1}), & & \displaystyle \nonumber\end{eqnarray}$$
applying Lemma 3.7, we obtain the result. ◻
Proof of Proposition 3.3.
Assume that there is an exact triangle
$D\rightarrow E\rightarrow F\rightarrow D[1]$
. From a Harder–Narasimhan filtration of
$D$
, we can construct the dual of it:
with
$A_{i}\in {\mathcal{P}}(\unicode[STIX]{x1D719}_{i})$
and
$\unicode[STIX]{x1D719}_{m}>\unicode[STIX]{x1D719}_{m-1}>\cdots \unicode[STIX]{x1D719}_{1}$
. Applying the octahedra axiom for the above sequence together with the exact triangle
$D\rightarrow E\rightarrow F\rightarrow D[1]$
, we can construct a sequence of exact triangles
Since
$m_{t}(D)=\sum _{i=1}^{m}|Z(A_{i})|e^{t\unicode[STIX]{x1D719}_{i}}$
and
$A_{i}$
is semistable, the problem is reduced to the case that
$D$
is semistable. Without loss of generality, we can assume
$D\in {\mathcal{P}}(1)$
. By taking the cohomology associated with the heart
${\mathcal{H}}={\mathcal{P}}((0,1])$
(see Section 2.2), we have a long exact sequence
$$\begin{eqnarray}0\rightarrow H^{-1}(E)\rightarrow H^{-1}(F)\rightarrow H^{0}(D)\rightarrow H^{0}(E)\rightarrow H^{0}(F)\rightarrow 0\end{eqnarray}$$
and isomorphisms
$H^{i}(E)\cong H^{i}(F)$
for
$i\neq -1,0$
in
${\mathcal{H}}$
. If
$1>\unicode[STIX]{x1D719}^{+}(H^{0}(E))$
, then the map
$f:H^{0}(D)\rightarrow H^{0}(E)$
is zero. Hence, the long exact sequence splits into
$0\rightarrow H^{-1}(E)\rightarrow H^{-1}(F)\rightarrow H^{0}(D)\rightarrow 0$
and
$H^{0}(E)\cong H^{0}(F)$
. From Lemma 3.8, we have
$$\begin{eqnarray}m_{t}(H^{-1}(E))e^{t}\leqslant m_{t}(H^{-1}(F))e^{t}+m_{t}(D).\end{eqnarray}$$
Thus, we obtain the result. If the map
$f:H^{0}(D)\rightarrow H^{0}(E)$
is not zero, then the long exact sequence splits into two short exact sequences
$$\begin{eqnarray}\displaystyle & 0\rightarrow H^{-1}(E)\rightarrow H^{-1}(F)\rightarrow \operatorname{Ker}f\rightarrow 0 & \displaystyle \nonumber\\ \displaystyle & 0\rightarrow \text{Im}\,f\rightarrow H^{0}(E)\rightarrow H^{0}(F)\rightarrow 0. & \displaystyle \nonumber\end{eqnarray}$$
Let
$E_{+}\in {\mathcal{P}}(1)$
be the semistable factor of
$E$
with phase one. Note that
$m_{t}(D)=m_{t}(\operatorname{Ker}f)+m_{t}(\text{Im}\,f)$
since
$\operatorname{Ker}f\subset D\in {\mathcal{P}}(1)$
and
$\text{Im}\,f\subset E_{+}\in {\mathcal{P}}(1)$
. Again by Lemma 3.8, we have
$$\begin{eqnarray}m_{t}(H^{-1}(E))e^{t}\leqslant m_{t}(H^{-1}(F))e^{t}+m_{t}(\operatorname{Ker}f)\end{eqnarray}$$
and it is easy to check that
$m_{t}(H^{0}(E))=m_{t}(\text{Im}\,f)+m_{t}(H^{0}(F))$
.◻
3.3 Mass growth and deformation of stability conditions
The aim of this section is to show that for a stability condition
$\unicode[STIX]{x1D70E}$
and an endofunctor
$F$
, the mass growth
$h_{\unicode[STIX]{x1D70E},t}(F)$
is stable under the continuous deformation of
$\unicode[STIX]{x1D70E}$
. The following inequality was shown in [Reference Bridgeland8, Proposition 8.1] when
$t=0$
.
Proposition 3.9. Let
$\unicode[STIX]{x1D70E}=(Z,{\mathcal{P}})\in \text{Stab}({\mathcal{D}})$
be a stability condition on
${\mathcal{D}}$
. If
$\unicode[STIX]{x1D70F}=(W,{\mathcal{Q}})\in B_{\unicode[STIX]{x1D716}}(\unicode[STIX]{x1D70E})$
with small enough
$\unicode[STIX]{x1D716}>0$
, then there are functions
$C_{1},C_{2}:\mathbb{R}\rightarrow \mathbb{R}_{{>}0}$
such that
$$\begin{eqnarray}C_{1}(t)m_{\unicode[STIX]{x1D70F},t}(E)<m_{\unicode[STIX]{x1D70E},t}(E)<C_{2}(t)m_{\unicode[STIX]{x1D70F},t}(E)\end{eqnarray}$$
for all
$0\neq E\in {\mathcal{D}}$
.
Proof. We use an argument similar to the proof of [Reference Bridgeland8, Proposition 8.1]. It is sufficient to show that for
$\unicode[STIX]{x1D70F}=(W,{\mathcal{Q}})\in B_{\unicode[STIX]{x1D716}}(\unicode[STIX]{x1D70E})$
with small enough
$\unicode[STIX]{x1D716}>0$
, there are some constants
$C>1$
and
$r>0$
such that
$$\begin{eqnarray}m_{\unicode[STIX]{x1D70F},t}(E)<Ce^{r|t|}m_{\unicode[STIX]{x1D70E},t}(E)\end{eqnarray}$$
for any nonzero object
$E\in {\mathcal{D}}$
. We first consider the case
$\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70E}}^{+}(E)-\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70E}}^{-}(E)<\unicode[STIX]{x1D702}$
for
$0<\unicode[STIX]{x1D702}<\frac{1}{4}$
. In this case, it was shown in the proof of [Reference Bridgeland8, Proposition 8.1] that there is a constant
$C(\unicode[STIX]{x1D716},\unicode[STIX]{x1D702})>1$
such that
$$\begin{eqnarray}m_{\unicode[STIX]{x1D70F}}(E)\leqslant C(\unicode[STIX]{x1D716},\unicode[STIX]{x1D702})m_{\unicode[STIX]{x1D70E}}(E)\end{eqnarray}$$
and
$C(\unicode[STIX]{x1D716},\unicode[STIX]{x1D702})\rightarrow 1$
as
$\max \{\unicode[STIX]{x1D716},\unicode[STIX]{x1D702}\}\rightarrow 0$
. Note that
$\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70E}}^{+}(E)-\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70E}}^{-}(E)<\unicode[STIX]{x1D702}$
implies
$\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70E}}^{\pm }(E)\in (\unicode[STIX]{x1D713},\unicode[STIX]{x1D713}+\unicode[STIX]{x1D702})$
for some
$\unicode[STIX]{x1D713}\in \mathbb{R}$
. Since
$d({\mathcal{P}},{\mathcal{Q}})<\unicode[STIX]{x1D716}$
, we have
$\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70F}}^{\pm }(E)\in (\unicode[STIX]{x1D713}-\unicode[STIX]{x1D716},\unicode[STIX]{x1D713}+\unicode[STIX]{x1D716}+\unicode[STIX]{x1D702})$
. By definition of
$m_{\unicode[STIX]{x1D70E},t}(E)$
and
$m_{\unicode[STIX]{x1D70F},t}(E)$
, it follows that
$$\begin{eqnarray}m_{\unicode[STIX]{x1D70F},t}(E)\leqslant m_{\unicode[STIX]{x1D70F}}(E)\exp (\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70F}}^{+}(E)|t|),\qquad m_{\unicode[STIX]{x1D70E}}(E)\exp (\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70E}}^{-}(E)|t|)\leqslant m_{\unicode[STIX]{x1D70E},t}(E).\end{eqnarray}$$
Since
$\unicode[STIX]{x1D713}<\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70E}}^{-}(E)$
and
$\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70F}}^{+}(E)<\unicode[STIX]{x1D713}+\unicode[STIX]{x1D716}+\unicode[STIX]{x1D702}$
, we have an inequality
$$\begin{eqnarray}m_{\unicode[STIX]{x1D70F},t}(E)\leqslant C(\unicode[STIX]{x1D716},\unicode[STIX]{x1D702})e^{(\unicode[STIX]{x1D716}+\unicode[STIX]{x1D702})|t|}m_{\unicode[STIX]{x1D70E},t}(E).\end{eqnarray}$$
Next, we consider a general nonzero object
$E$
. Take a real number
$\unicode[STIX]{x1D719}$
and a positive integer
$n$
. For
$k\in \mathbb{Z}$
, define intervals
$$\begin{eqnarray}I_{k}:=[\unicode[STIX]{x1D719}+kn\unicode[STIX]{x1D716},\unicode[STIX]{x1D719}+(k+1)n\unicode[STIX]{x1D716}),\qquad J_{k}:=[\unicode[STIX]{x1D719}+kn\unicode[STIX]{x1D716}-\unicode[STIX]{x1D716},\unicode[STIX]{x1D719}+(k+1)n\unicode[STIX]{x1D716}+\unicode[STIX]{x1D716})\end{eqnarray}$$
and let
$\unicode[STIX]{x1D6FC}_{k}$
and
$\unicode[STIX]{x1D6FD}_{k}$
be the truncation functors projecting into the subcategories
${\mathcal{Q}}(I_{k})$
and
${\mathcal{P}}(J_{k})$
, respectively. Then, as in the proof of [Reference Bridgeland8, Proposition 8.1], we have
$\unicode[STIX]{x1D6FC}_{k}\circ \unicode[STIX]{x1D6FD}_{k}=\unicode[STIX]{x1D6FC}_{k}$
and therefore
$m_{\unicode[STIX]{x1D70F},t}(\unicode[STIX]{x1D6FC}_{k}(E))\leqslant m_{\unicode[STIX]{x1D70F},t}(\unicode[STIX]{x1D6FD}_{k}(E))$
. As a result, for small enough
$n\unicode[STIX]{x1D716}$
, we have
$$\begin{eqnarray}m_{\unicode[STIX]{x1D70F},t}(E)=\mathop{\sum }_{k}m_{\unicode[STIX]{x1D70F},t}(\unicode[STIX]{x1D6FC}_{k}(E))\leqslant \mathop{\sum }_{k}m_{\unicode[STIX]{x1D70F},t}(\unicode[STIX]{x1D6FD}_{k}(E))<C(\unicode[STIX]{x1D716},(n+2)\unicode[STIX]{x1D716})e^{(n+3)\unicode[STIX]{x1D716}|t|}\mathop{\sum }_{k}m_{\unicode[STIX]{x1D70E},t}(\unicode[STIX]{x1D6FD}_{k}(E)).\end{eqnarray}$$
On the other hand, we can choose
$\unicode[STIX]{x1D719}$
so that
$$\begin{eqnarray}\mathop{\sum }_{k}m_{\unicode[STIX]{x1D70E},t}(\unicode[STIX]{x1D6FD}_{k}(E))\leqslant \frac{n+2}{n}m_{\unicode[STIX]{x1D70E},t}(E).\end{eqnarray}$$
By taking the limits
$\unicode[STIX]{x1D716}\rightarrow 0$
and
$n\rightarrow \infty$
in keeping with
$n\unicode[STIX]{x1D716}\rightarrow 0$
, the result follows.◻
From Proposition 3.9, we immediately have the following.
Proposition 3.10. Let
$F:{\mathcal{D}}\rightarrow {\mathcal{D}}$
be an endofunctor, and
$\unicode[STIX]{x1D70E}$
and
$\unicode[STIX]{x1D70F}$
be stability conditions on
${\mathcal{D}}$
. If
$\unicode[STIX]{x1D70E}$
and
$\unicode[STIX]{x1D70F}$
lie in the same connected component in
$\text{Stab}({\mathcal{D}})$
, then
$$\begin{eqnarray}h_{\unicode[STIX]{x1D70E},t}(F)=h_{\unicode[STIX]{x1D70F},t}(F).\end{eqnarray}$$
Proof. Let
$\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}\in \text{Stab}({\mathcal{D}})$
be stability conditions such that
$\unicode[STIX]{x1D70F}\in B_{\unicode[STIX]{x1D716}}(\unicode[STIX]{x1D70E})$
for small enough
$\unicode[STIX]{x1D716}>0$
. Then Proposition 3.9 implies
$h_{\unicode[STIX]{x1D70E},t}(F)=h_{\unicode[STIX]{x1D70F},t}(F)$
. Thus,
$h_{\unicode[STIX]{x1D70E},t}(F)$
is locally constant on the topological space
$\text{Stab}({\mathcal{D}})$
.◻
3.4 Lower bound of the mass growth by the spectral radius
Let
$F:{\mathcal{D}}\rightarrow {\mathcal{D}}$
be an endofunctor. Since
$F$
preserves exact triangles in
${\mathcal{D}}$
,
$F$
induces a linear transformation
$[F]:K({\mathcal{D}})\rightarrow K({\mathcal{D}})$
. We extend
$[F]$
on
$K({\mathcal{D}})\otimes \mathbb{C}$
naturally. The spectral radius of
$[F]$
is defined by
$$\begin{eqnarray}\unicode[STIX]{x1D70C}([F]):=\max \{|\unicode[STIX]{x1D706}|\mid \unicode[STIX]{x1D706}\text{ is an eigenvalue of }[F]\text{ on }K({\mathcal{D}})\otimes \mathbb{C}\}.\end{eqnarray}$$
Proposition 3.11. For any stability condition
$\unicode[STIX]{x1D70E}\in \text{Stab}({\mathcal{D}})$
, we have an inequality
$$\begin{eqnarray}\log \unicode[STIX]{x1D70C}([F])\leqslant h_{\unicode[STIX]{x1D70E},0}(F).\end{eqnarray}$$
Proof. We recall the assumption
$K({\mathcal{D}})\cong \mathbb{Z}^{\oplus n}$
. Set
$K({\mathcal{D}})_{\mathbb{C}}:=K({\mathcal{D}})\otimes \mathbb{C}$
. Let
$A_{1},\ldots ,A_{n}\in {\mathcal{D}}$
be objects whose classes
$[A_{1}],\ldots ,[A_{n}]$
form a basis of
$K({\mathcal{D}})_{\mathbb{C}}$
. Take an eigenvector
$$\begin{eqnarray}v=\mathop{\sum }_{i=1}^{n}a_{i}[A_{i}]\in K({\mathcal{D}})_{\mathbb{C}}\quad (a_{i}\in \mathbb{C})\end{eqnarray}$$
for the eigenvalue
$\unicode[STIX]{x1D706}\in \mathbb{C}$
of
$[F]$
satisfying
$|\unicode[STIX]{x1D706}|=\unicode[STIX]{x1D70C}([F])$
. First, we consider the case that a stability condition
$\unicode[STIX]{x1D70E}=(Z,{\mathcal{P}})$
satisfies
$Z(v)\neq 0$
. Note that the mass satisfies
$|Z(E)|\leqslant m_{\unicode[STIX]{x1D70E}}(E)$
and
$m_{\unicode[STIX]{x1D70E}}(E\oplus E^{\prime })=m_{\unicode[STIX]{x1D70E}}(E)+m_{\unicode[STIX]{x1D70E}}(E^{\prime })$
for any objects
$E,E^{\prime }\in {\mathcal{D}}$
. Then
$$\begin{eqnarray}\displaystyle |\unicode[STIX]{x1D706}|^{k}|Z(v)|=|Z(\unicode[STIX]{x1D706}^{k}v)|=|Z([F]^{k}v)| & {\leqslant} & \displaystyle \mathop{\sum }_{i=1}^{n}|a_{i}|\cdot |Z(F^{k}A_{i})|\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \mathop{\sum }_{i=1}^{n}l_{i}m_{\unicode[STIX]{x1D70E}}(F^{k}A_{i})=m_{\unicode[STIX]{x1D70E}}\biggl(F^{k}\biggl(\bigoplus _{i=1}^{n}A_{i}^{\oplus l_{i}}\biggr)\biggr),\nonumber\end{eqnarray}$$
where
$l_{1},\ldots ,l_{n}$
are positive integers satisfying
$|a_{i}|\leqslant l_{i}$
. Since
$|Z(v)|>0$
, we have
$$\begin{eqnarray}\log \unicode[STIX]{x1D70C}([F])=\lim _{k\rightarrow \infty }\frac{1}{k}\log (|\unicode[STIX]{x1D706}|^{k}|Z(v)|)\leqslant \limsup _{k\rightarrow \infty }\frac{1}{k}\log (m_{\unicode[STIX]{x1D70E}}(F^{k}E))\leqslant h_{\unicode[STIX]{x1D70E},0}(F),\end{eqnarray}$$
where
$E=\bigoplus _{i=1}^{n}A_{i}^{\oplus l_{i}}$
. Next, consider the case
$Z(v)=0$
. Then by Theorem 2.14, we can deform
$\unicode[STIX]{x1D70E}=(Z,{\mathcal{P}})$
to
$\unicode[STIX]{x1D70E}^{\prime }=(Z^{\prime },{\mathcal{P}}^{\prime })$
so that
$Z^{\prime }(v)\neq 0$
. Again we have
$\log \unicode[STIX]{x1D70C}([F])\leqslant h_{\unicode[STIX]{x1D70E}^{\prime },0}(F)$
and Proposition 3.10 implies
$h_{\unicode[STIX]{x1D70E},0}(F)=h_{\unicode[STIX]{x1D70E}^{\prime },0}(F)$
.◻
3.5 Mass growth via algebraic stability conditions
If a triangulated category has an algebraic stability condition, then we can show that the mass growth coincides with the entropy. Let
${\mathcal{H}}\subset {\mathcal{D}}$
be an algebraic heart with simple objects
$S_{1},\ldots ,S_{n}$
. Then the Grothendieck group is given by
$$\begin{eqnarray}K({\mathcal{D}})\cong \bigoplus _{i=1}^{n}\mathbb{Z}[S_{i}].\end{eqnarray}$$
The class of an object
$E\in {\mathcal{H}}$
is written as
$[E]=\sum _{i=1}^{n}d_{i}[S_{i}]$
with
$d_{i}\in \mathbb{Z}_{{\geqslant}0}$
. We define the dimension of
$E$
by
$\dim E:=\sum _{i=1}^{n}d_{i}\in \mathbb{Z}_{{\geqslant}0}$
. Then the dimension gives the upper bound of the complexity for objects in
${\mathcal{H}}$
.
Lemma 3.12. Let
${\mathcal{H}}\subset {\mathcal{D}}$
be an algebraic heart with simple objects
$S_{1},\ldots ,S_{n}$
. Then for the split generator
$G:=\bigoplus _{i=1}^{n}S_{i}$
, we have an inequality
$$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{t}(G,E)\leqslant \dim E.\end{eqnarray}$$
Proof. Since
${\mathcal{H}}$
is a finite length abelian category, for any object
$E\in {\mathcal{H}}$
there is a Jordan–Hölder filtration
$$\begin{eqnarray}0=E_{0}\subset E_{1}\subset E_{2}\subset \cdots \subset E_{l}=E\end{eqnarray}$$
of length
$l=\dim E$
with
$E_{i}/E_{i-1}\in \{S_{1},\ldots ,S_{n}\}$
. As a result, we can construct a filtration
$$\begin{eqnarray}0=E_{0}^{\prime }\subset E_{1}^{\prime }\subset E_{2}^{\prime }\subset \cdots \subset E_{l}^{\prime }=E\oplus E^{\prime }\end{eqnarray}$$
of length
$l=\dim E$
with
$E_{i}^{\prime }/E_{i-1}^{\prime }=G$
and this implies the result.◻
Following Section 2.4, we construct the special algebraic stability condition. For an algebraic heart
${\mathcal{H}}\subset {\mathcal{D}}$
with simple objects
$S_{1},\ldots ,S_{n}$
, define the central charge
$$\begin{eqnarray}Z_{0}:K({\mathcal{D}})\cong \bigoplus _{i=1}^{n}\mathbb{Z}[S_{i}]\rightarrow \mathbb{C}\end{eqnarray}$$
by
$Z_{0}(S_{i}):=i$
. Then the pair
$\unicode[STIX]{x1D70E}_{0}:=(Z_{0},{\mathcal{H}})$
becomes an algebraic stability condition. By definition, the mass of an object
$E\in {\mathcal{H}}$
is given by
$$\begin{eqnarray}m_{\unicode[STIX]{x1D70E}_{0},t}(E)=\dim E\cdot e^{(1/2)t}.\end{eqnarray}$$
Together with Lemma 3.12, we obtain the following inequality.
Proposition 3.13. For any
$E\in {\mathcal{D}}$
, the generator
$G=\bigoplus _{i=1}^{n}S_{i}$
and the algebraic stability condition
$\unicode[STIX]{x1D70E}_{0}=(Z_{0},{\mathcal{H}})$
, we have an inequality
$$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{t}(G,E)\leqslant e^{-(1/2)t}m_{\unicode[STIX]{x1D70E}_{0},t}(E).\end{eqnarray}$$
Proof. For an object
$E\in {\mathcal{D}}$
, we denote by
$H^{k}(E)\in {\mathcal{H}}$
the cohomology associated with the heart
${\mathcal{H}}$
(see Section 2.2). By using Lemmas 2.3 and 3.12, we have
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}_{t}(G,E)\leqslant \mathop{\sum }_{k}\unicode[STIX]{x1D6FF}_{t}(G,H^{k}(E))e^{-kt}\leqslant \mathop{\sum }_{k}\dim H^{k}(E)e^{-kt}. & & \displaystyle \nonumber\end{eqnarray}$$
On the other hand, the definition of
$m_{\unicode[STIX]{x1D70E}_{0},t}(E)$
implies
$$\begin{eqnarray}m_{\unicode[STIX]{x1D70E}_{0},t}(E)=\mathop{\sum }_{k}m_{\unicode[STIX]{x1D70E}_{0},t}(H^{k}(E))e^{-kt}=\mathop{\sum }_{k}\dim H^{k}(E)e^{(1/2)t}e^{-kt}.\end{eqnarray}$$
Thus, we obtain the result. ◻
We show the main result of this section.
Theorem 3.14. Let
$F:{\mathcal{D}}\rightarrow {\mathcal{D}}$
be an endofunctor. If a connected component
$\text{Stab}^{\circ }({\mathcal{D}})\subset \text{Stab}({\mathcal{D}})$
contains an algebraic stability condition, then
${\mathcal{D}}$
has a split generator
$G$
and for any
$\unicode[STIX]{x1D70E}\in \text{Stab}^{\circ }({\mathcal{D}})$
, we have
$$\begin{eqnarray}h_{t}(F)=h_{\unicode[STIX]{x1D70E},t}(F)=\lim _{n\rightarrow \infty }\frac{1}{n}\log (m_{\unicode[STIX]{x1D70E},t}(F^{n}G)).\end{eqnarray}$$
Proof. Let
${\mathcal{H}}$
be an algebraic heart with simple objects
$S_{1},\ldots ,S_{n}$
and set
$G=\bigoplus _{i=1}^{n}S_{i}$
. Then
$G$
is a split generator of
${\mathcal{D}}$
. Consider the special algebraic stability condition
$\unicode[STIX]{x1D70E}_{0}=(Z_{0},{\mathcal{H}})$
which is constructed in this section. By Proposition 3.10, it is sufficient to show that
$$\begin{eqnarray}h_{\unicode[STIX]{x1D70E}_{0},t}(F)=h_{t}(F).\end{eqnarray}$$
By [Reference Dimitrov, Haiden, Katzarkov and Kontsevich11, Lemma 2.6], the limit
$$\begin{eqnarray}h_{t}(F)=\lim _{n\rightarrow \infty }\frac{1}{n}\log \unicode[STIX]{x1D6FF}_{t}(G,F^{n}G)\end{eqnarray}$$
converges. On the other hand, by Proposition 3.4 and Proposition 3.13, we have
$$\begin{eqnarray}e^{(1/2)t}\unicode[STIX]{x1D6FF}_{t}(G,F^{n}G)\leqslant m_{\unicode[STIX]{x1D70E}_{0},t}(F^{n}G)\leqslant m_{\unicode[STIX]{x1D70E}_{0},t}(G)\unicode[STIX]{x1D6FF}_{t}(G,F^{n}G).\end{eqnarray}$$
Hence the limit
$$\begin{eqnarray}\lim _{n\rightarrow \infty }\frac{1}{n}\log (m_{\unicode[STIX]{x1D70E}_{0},t}(F^{n}G))\end{eqnarray}$$
converges and coincides with
$h_{t}(F)$
.◻
4 Applications
4.1 Entropy on the derived categories of nonpositive dg-algebras
In this section, we discuss the entropy of endofunctors on the derived categories of nonpositive dg-algebras. In this case, we can describe the entropy as the growth rate of the Hilbert–Poincaré polynomial of a generator.
Let
$A=\bigoplus _{k\in \mathbb{Z}}A^{k}$
be a dg-algebra over
$\mathbb{K}$
satisfying the following conditions:
-
(a)
$H^{k}(A)=0$
for
$i>0$
; -
(b)
$H^{0}(A)$
is a finite-dimensional algebra over
$\mathbb{K}$
.
Let
${\mathcal{D}}(A)$
be the derived category of dg-modules over
$A$
and
${\mathcal{D}}_{fd}(A)$
be the full subcategory of
${\mathcal{D}}(A)$
consisting of dg-modules with finite-dimensional total cohomology, that is,
$$\begin{eqnarray}{\mathcal{D}}_{fd}(A):=\bigg\{M\in {\mathcal{D}}(A)\biggm\vert\mathop{\sum }_{k}\dim H^{k}(M)<\infty \bigg\}.\end{eqnarray}$$
Define the full subcategory
${\mathcal{F}}\subset {\mathcal{D}}_{fd}(A)$
by
$$\begin{eqnarray}{\mathcal{F}}:=\{M\in {\mathcal{D}}_{fd}(A)\mid H^{k}(M)=0\text{ for }k>0\}.\end{eqnarray}$$
Then
${\mathcal{F}}$
becomes a bounded t-structure on
${\mathcal{D}}_{fd}(A)$
. The heart
${\mathcal{H}}_{s}$
of
${\mathcal{F}}$
is called the standard heart. It is known that the
$0$
th cohomology functor
$H^{0}:{\mathcal{D}}_{fd}(A)\rightarrow \operatorname{mod -}H^{0}(A)$
induces an equivalence of abelian categories:
$$\begin{eqnarray}H^{0}:{\mathcal{H}}_{s}\xrightarrow[{}]{{\sim}}\operatorname{mod -}H^{0}(A),\end{eqnarray}$$
where
$\operatorname{mod -}H^{0}(A)$
is the abelian category of finite-dimensional
$H^{0}(A)$
-modules. (For details, see [Reference Amiot2, Section 2].) Since
$H^{0}(A)$
is a finite-dimensional algebra,
${\mathcal{H}}_{s}$
is an algebraic heart. Thus, we can construct an algebraic stability condition on
${\mathcal{D}}_{fd}(A)$
. Applying Theorem 3.14, we obtain the following.
Proposition 4.1. Let
$\text{Stab}^{\circ }({\mathcal{D}}_{fd}(A))$
be the connected component which contains stability conditions with the standard heart
${\mathcal{H}}_{s}$
. Then for any stability condition
$\unicode[STIX]{x1D70E}\in \text{Stab}^{\circ }({\mathcal{D}}_{fd}(A))$
and an endofunctor
$F:{\mathcal{D}}_{fd}(A)\rightarrow {\mathcal{D}}_{fd}(A)$
, we have
$$\begin{eqnarray}h_{t}(F)=h_{\unicode[STIX]{x1D70E},t}(F).\end{eqnarray}$$
Next, we describe
$h_{t}(F)$
by using the Hilbert–Poincaré polynomial.
Definition 4.2. For a dg-module
$M\in {\mathcal{D}}_{fd}(A)$
, define the Hilbert–Poincaré polynomial of
$M$
by
$$\begin{eqnarray}P_{t}(M):=\mathop{\sum }_{k\in \mathbb{Z}}\dim H^{k}(M)e^{-kt}\in \mathbb{Z}[e^{t},e^{-t}].\end{eqnarray}$$
As in Section 3.5, we construct the special stability condition
$\unicode[STIX]{x1D70E}_{0}=(Z_{0},{\mathcal{H}}_{s})$
by using the standard heart
${\mathcal{H}}_{s}$
. Then by definition of
$\unicode[STIX]{x1D70E}_{0}$
, we have
$$\begin{eqnarray}m_{\unicode[STIX]{x1D70E}_{0},t}(M)=e^{(1/2)t}P_{t}(M)\end{eqnarray}$$
for any dg-module
$M\in {\mathcal{D}}_{fd}(A)$
. As a result, the entropy is described as follows.
Proposition 4.3. Let
$F:{\mathcal{D}}_{fd}(A)\rightarrow {\mathcal{D}}_{fd}(A)$
be an endofunctor and
$G\in {\mathcal{D}}_{fd}(A)$
be a split generator. Then the entropy of
$F$
is given by
$$\begin{eqnarray}h_{t}(F)=\lim _{n\rightarrow \infty }\frac{1}{n}\log P_{t}(F^{n}G).\end{eqnarray}$$
4.2 Entropy of spherical twists
In this section, we compute the entropy of Seidel–Thomas spherical twists on the derived categories of Calabi–Yau algebras associated with acyclic quivers. Let
$Q$
be an acyclic quiver with vertices
$\{1,\ldots ,n\}$
and
$\unicode[STIX]{x1D6E4}_{N}Q$
be the Ginzburg
$N$
-Calabi–Yau dg-algebra associated with
$Q$
for
$N\geqslant 2$
. (For the definition of
$\unicode[STIX]{x1D6E4}_{N}Q$
, see [Reference Ginzburg13, Section 4.2] or [Reference Keller16, Section 6.2].) Set
${\mathcal{D}}_{Q}^{N}:={\mathcal{D}}_{fd}(\unicode[STIX]{x1D6E4}_{N}Q)$
. By [Reference Keller16, Theorem 6.3], the category
${\mathcal{D}}_{Q}^{N}$
is an
$N$
-Calabi–Yau category, that is, there is a natural isomorphism
$$\begin{eqnarray}\operatorname{Hom}(E,F)\xrightarrow[{}]{{\sim}}\operatorname{Hom}(F,E[N])^{\ast }\end{eqnarray}$$
for
$E,F\in {\mathcal{D}}_{Q}^{N}$
. (Here
$V^{\ast }$
denotes the dual space of a
$\mathbb{K}$
-linear space
$V$
.) In the Calabi–Yau category, we can consider a certain class of objects, called spherical objects. An object
$S\in {\mathcal{D}}_{Q}^{N}$
is called
$N$
-spherical if
$$\begin{eqnarray}\displaystyle \operatorname{Hom}(S,S[i])=\left\{\begin{array}{@{}ll@{}}\mathbb{K}\quad & \text{if }i=0,N\\ 0\quad & \text{otherwise}.\end{array}\right. & & \displaystyle \nonumber\end{eqnarray}$$
For a spherical object
$S\in {\mathcal{D}}_{Q}^{N}$
, Seidel–Thomas [Reference Seidel and Thomas20] defined an exact autoequivalence
$\unicode[STIX]{x1D6F7}_{S}\in \operatorname{Aut}({\mathcal{D}}_{Q}^{N})$
, called a spherical twist, by the exact triangle
$$\begin{eqnarray}\operatorname{Hom}^{\bullet }(S,E)\otimes S\longrightarrow E\longrightarrow \unicode[STIX]{x1D6F7}_{S}(E)\end{eqnarray}$$
for any object
$E\in {\mathcal{D}}_{Q}^{N}$
. The inverse functor
$\unicode[STIX]{x1D6F7}_{S}^{-1}\in \operatorname{Aut}({\mathcal{D}}_{Q}^{N})$
is given by
$$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{S}^{-1}(E)\longrightarrow E\longrightarrow S\otimes \operatorname{Hom}^{\bullet }(E,S)^{\ast }.\end{eqnarray}$$
The Ginzburg dg-algebra
$\unicode[STIX]{x1D6E4}_{N}Q$
satisfies the conditions in Section 4.1 when
$N\geqslant 2$
. (In the case
$N=2$
, we need some modification.) Hence, the category
${\mathcal{D}}_{Q}^{N}$
has the standard algebraic heart
${\mathcal{H}}_{s}\subset {\mathcal{D}}_{Q}^{N}$
generated by simple
$\unicode[STIX]{x1D6E4}_{N}Q$
-modules
$S_{1},\ldots ,S_{n}$
corresponding to vertices
$\{1,\ldots ,n\}$
of
$Q$
. In addition, these objects
$S_{1},\ldots ,S_{n}$
are
$N$
-spherical by [Reference Keller16, Lemma 4.4]. Thus, we can define spherical twists
$\unicode[STIX]{x1D6F7}_{S_{1}},\ldots ,\unicode[STIX]{x1D6F7}_{S_{n}}\in \operatorname{Aut}({\mathcal{D}}_{Q}^{N})$
. In the following, we compute the entropy of spherical twists
$\unicode[STIX]{x1D6F7}_{S_{1}},\ldots ,\unicode[STIX]{x1D6F7}_{S_{n}}$
by using Proposition 4.3. For simplicity, write
$\unicode[STIX]{x1D6F7}_{i}:=\unicode[STIX]{x1D6F7}_{S_{i}}$
.
Lemma 4.4. For a spherical twist
$\unicode[STIX]{x1D6F7}_{i}\in \operatorname{Aut}({\mathcal{D}}_{Q}^{N})$
and a spherical object
$S_{j}\in {\mathcal{D}}_{Q}^{N}$
, the Hilbert–Poincaré polynomial of
$\unicode[STIX]{x1D6F7}_{i}^{k}S_{j}\;(k\geqslant 0)$
is given by
$$\begin{eqnarray}P_{t}(\unicode[STIX]{x1D6F7}_{i}^{k}S_{j})=\left\{\begin{array}{@{}ll@{}}e^{k(1-N)t}\quad & \text{if }i=j\\ 1+q_{ij}\mathop{\sum }_{l=0}^{k-1}e^{l(1-N)t}\quad & \text{if }q_{ij}>0\\ 1+q_{ji}e^{(2-N)t}\mathop{\sum }_{l=0}^{k-1}e^{l(1-N)t}\quad & \text{if }q_{ji}>0\\ 1\quad & \text{otherwise},\end{array}\right.\end{eqnarray}$$
where
$q_{ij}$
is the number of arrows from
$i$
to
$j$
in
$Q$
.
Proof. First, we note that
$$\begin{eqnarray}\dim \operatorname{Hom}(S_{i},S_{j}[m])=\left\{\begin{array}{@{}ll@{}}1\quad & \text{if }i=j\text{ and }m=0,N\\ q_{ij}\quad & \text{if }q_{ij}>0\text{ and }m=1\\ q_{ji}\quad & \text{if }q_{ji}>0\text{ and }m=N-1\\ 0\quad & \text{otherwise}.\end{array}\right.\end{eqnarray}$$
This follows by the definition of
$S_{1},\ldots ,S_{n}$
for
$m=0$
, by [Reference Assem, Simson and Skowronski4, Lemma 2.12] for
$m=1$
, and the Serre duality for
$m=N-1,N$
. By the definition of spherical twists, it is easy to see that
$\unicode[STIX]{x1D6F7}_{i}^{k}S_{i}=S_{i}[k(1-N)]$
and hence
$P_{t}(\unicode[STIX]{x1D6F7}_{i}^{k}S_{i})=e^{k(1-N)t}$
. If
$i\neq j$
and
$q_{ij}=q_{ji}=0$
, then
$\unicode[STIX]{x1D6F7}_{i}^{k}S_{j}=S_{j}$
and hence
$P_{t}(\unicode[STIX]{x1D6F7}_{i}^{k}S_{j})=1$
. Consider the case
$q_{ij}>0$
. Since
$$\begin{eqnarray}\operatorname{Hom}^{\bullet }(S_{i},S_{j})\otimes S_{i}=\bigoplus _{m\in \mathbb{Z}}\operatorname{Hom}(S_{i}[m],S_{j})\otimes S_{i}[m]\cong S_{i}^{\oplus q_{ij}}[-1],\end{eqnarray}$$
we have an exact triangle
$$\begin{eqnarray}S_{j}\rightarrow \unicode[STIX]{x1D6F7}_{i}S_{j}\rightarrow S_{i}^{\oplus q_{ij}}\rightarrow S_{j}[1].\end{eqnarray}$$
Applying the spherical twist
$\unicode[STIX]{x1D6F7}_{i}$
for the above exact triangle repeatedly, we obtain a sequence of exact triangles
This implies the result in the case
$q_{ij}>0$
and a similar argument gives the result in the case
$q_{ji}>0$
.◻
Proposition 4.5. Let
$Q$
be a connected acyclic quiver and assume that
$Q$
is not a quiver with one vertex and no arrows. Then the entropy of spherical twists
$\unicode[STIX]{x1D6F7}_{1},\ldots ,\unicode[STIX]{x1D6F7}_{n}$
is given by
$$\begin{eqnarray}h_{t}(\unicode[STIX]{x1D6F7}_{i})=\left\{\begin{array}{@{}ll@{}}0\quad & \text{if }t\geqslant 0\\ (1-N)t\quad & \text{if }t<0.\end{array}\right.\end{eqnarray}$$
Proof. We use the generator
$G=\bigoplus _{j=1}^{n}S_{j}$
. Then
$P_{t}(\unicode[STIX]{x1D6F7}_{i}^{k}G)=\sum _{j=1}^{n}P_{t}(\unicode[STIX]{x1D6F7}_{i}^{k}S_{j})$
. Recall from Proposition 4.4 that
$$\begin{eqnarray}P_{t}(\unicode[STIX]{x1D6F7}_{i}^{k}S_{j})=1+q_{ij}\mathop{\sum }_{l=0}^{k-1}e^{l(1-N)t}=1+q_{ij}\frac{1-e^{k(1-N)t}}{1-e^{(1-N)t}}\end{eqnarray}$$
in the case
$q_{ij}>0$
and
$$\begin{eqnarray}P_{t}(\unicode[STIX]{x1D6F7}_{i}^{k}S_{j})=1+q_{ji}e^{(2-N)t}\mathop{\sum }_{l=0}^{k-1}e^{l(1-N)t}=1+q_{ji}e^{(2-N)t}\frac{1-e^{k(1-N)t}}{1-e^{(1-N)t}}\end{eqnarray}$$
in the case
$q_{ji}>0$
. First, we consider the case
$t>0$
. Then the above two terms converge to some positive real numbers as
$k\rightarrow \infty$
since
$(1-N)t<0$
. By the assumption on
$Q$
, the sum
$\sum _{j=1}^{n}P_{t}(\unicode[STIX]{x1D6F7}_{i}^{k}S_{j})$
contains at least one of the above two. As a result,
$\sum _{j=1}^{n}P_{t}(\unicode[STIX]{x1D6F7}_{i}^{k}S_{j})$
also converges to some positive real number as
$k\rightarrow \infty$
. Thus
$$\begin{eqnarray}h_{t}(\unicode[STIX]{x1D6F7}_{i})=\lim _{k\rightarrow \infty }\frac{1}{k}\log P_{t}(\unicode[STIX]{x1D6F7}_{i}^{k}G)=0\end{eqnarray}$$
when
$t>0$
. Next, consider the case
$t<0$
. Similarly, we can show that
$e^{-k(1-N)t}\sum _{j=1}^{n}P_{t}(\unicode[STIX]{x1D6F7}_{i}^{k}S_{j})$
converges to some positive real number as
$k\rightarrow \infty$
since
$-(1-N)t<0$
. Thus
$$\begin{eqnarray}\displaystyle h_{t}(\unicode[STIX]{x1D6F7}_{i}) & = & \displaystyle \lim _{k\rightarrow \infty }\frac{1}{k}\log P_{t}(\unicode[STIX]{x1D6F7}_{i}^{k}G)=\lim _{k\rightarrow \infty }\frac{1}{k}\log e^{k(1-N)t}e^{-k(1-N)t}P_{t}(\unicode[STIX]{x1D6F7}_{i}^{k}G)\nonumber\\ \displaystyle & = & \displaystyle (1-N)t+\lim _{k\rightarrow \infty }\frac{1}{k}\log e^{-k(1-N)t}P_{t}(\unicode[STIX]{x1D6F7}_{i}^{k}G)=(1-N)t\nonumber\end{eqnarray}$$
when
$t<0$
. Finally, we can easily check that
$h_{t}(\unicode[STIX]{x1D6F7}_{i})=0$
when
$t=0$
.◻
Remark 4.6. The subgroup of autoequivalences generated by spherical twists
$$\begin{eqnarray}\text{Sph}({\mathcal{D}}_{Q}^{N}):=\langle \unicode[STIX]{x1D6F7}_{1},\ldots ,\unicode[STIX]{x1D6F7}_{n}\rangle \subset \operatorname{Aut}({\mathcal{D}}_{Q}^{N})\end{eqnarray}$$
is called the Seidel–Thomas braid group. Here we only computed the entropy of generators
$\unicode[STIX]{x1D6F7}_{1},\ldots ,\unicode[STIX]{x1D6F7}_{n}$
. It is an important problem to compute the entropy
$h_{t}(\unicode[STIX]{x1D6F7})$
for a general element
$\unicode[STIX]{x1D6F7}\in \text{Sph}({\mathcal{D}}_{Q}^{N})$
.
4.3 Lower bound of the entropy on the derived categories of surfaces
Let
$X$
be a smooth projective variety over
$\mathbb{C}$
and denote by
$\text{D}^{b}(X)$
the bounded derived category of coherent sheaves on
$X$
. Define the Euler form
$\unicode[STIX]{x1D712}:K(\text{D}^{b}(X))\times K(\text{D}^{b}(X))\rightarrow \mathbb{Z}$
by
$$\begin{eqnarray}\unicode[STIX]{x1D712}(E,F):=\mathop{\sum }_{i\in \mathbb{Z}}(-1)^{i}\dim _{\mathbb{ C}}\operatorname{Hom}_{\text{D}^{b}(X)}(E,F[i]).\end{eqnarray}$$
The numerical Grothendieck group
$N(X)$
is the quotient of
$K(\text{D}^{b}(X))$
by the radical of the Euler form
$\unicode[STIX]{x1D712}$
. Let
$\operatorname{End}^{FM}(\text{D}^{b}(X))$
be the semigroup consisting of Fourier–Mukai type endofunctors. Since these endofunctors preserve the radical of
$\unicode[STIX]{x1D712}$
, they induce linear maps on
$N(X)$
, that is, the semigroup homomorphism
$$\begin{eqnarray}\operatorname{End}^{FM}(\text{D}^{b}(X))\rightarrow \operatorname{End}(N(X)),\quad F\mapsto [F]\end{eqnarray}$$
is well defined (see [Reference Kikuta and Takahashi18, Section 5.1]). A stability condition
$\unicode[STIX]{x1D70E}=(Z,{\mathcal{P}})$
is called numerical if
$Z:K(\text{D}^{b}(X))\rightarrow \mathbb{C}$
factors through the numerical Grothendieck group
$N(X)$
.
In [Reference Arcara and Bertram3, Reference Bridgeland9], a numerical stability condition on
$\text{D}^{b}(X)$
was constructed when
$\dim _{\mathbb{C}}X=2$
. Applying Theorem 3.5 and Proposition 3.11, we obtain the following lower bound of the entropy.
Proposition 4.7. Let
$X$
be a smooth projective surface over
$\mathbb{C}$
and
$F:\text{D}^{b}(X)\rightarrow \text{D}^{b}(X)$
be a Fourier–Mukai type endofunctor. Then
$$\begin{eqnarray}\log \unicode[STIX]{x1D70C}([F])\leqslant h_{0}(F),\end{eqnarray}$$
where
$\unicode[STIX]{x1D70C}([F])$
is the spectral radius of the induced linear map
$[F]:N(X)\rightarrow N(X)$
and
$h_{0}(F)$
is the entropy of
$F$
at
$t=0$
.
Remark 4.8. Let
$X$
be a smooth projective variety over
$\mathbb{C}$
. In [Reference Kikuta and Takahashi18], they conjectured that the equality
$\log \unicode[STIX]{x1D70C}([F])=h_{0}(F)$
holds for any autoequivalence
$F\in \operatorname{Aut}(\text{D}^{b}(X))$
. This conjecture was shown for a curve in [Reference Kikuta17] and for a variety with ample canonical bundle or ample anticanonical bundle in [Reference Kikuta and Takahashi18].
Acknowledgments
The author would like to thank Kohei Kikuta, Genki Ouchi, and Atsushi Takahashi for valuable discussions and comments.
