Table Of ContentCATEGORICAL ENTROPY FOR FOURIER-MUKAI TRANSFORMS ON GENERIC
ABELIAN SURFACES.
KO¯TAYOSHIOKA
Abstract. Inthisnote,weshallcomputethecategoricalentropyofanautoequivalenceonagenericabelian
surface.
7
1
0
2
n 0. Introduction
a
J In [1], Dimitrov, Haiden, Katzarkov, and Kontsevich introduced a categorical entropy ht(Φ) (t R) for
∈
5 anendofunctorΦofatriangulatedcategorywithasplitgenerator. Forendofunctorsofthederivedcategory
1 D(X) of coherent sheaves on a smooth projective variety X, Kikuta and Takahashi [3], [4] studied the
entropy. In particular they proved that h0 coincides with the topological entropy for an endofunctor Lf∗
]
G induced by a surjective endomorphism f of smooth projective variety [4, Thm. 5.4] and an autoequivalence
ofD(X)ifdimX =1[3]or K isample[4,Thm. 5.6]. TheyalsoconjecturedthataGromov-Yomdintype
A ± X
resultholds,thatis,h (Φ)=logρ(Φ)([4,Conjecture5.3])whereρ(Φ)isthespectralradiusoftheactionofΦ
0
.
h onthe algebraiccohomologygroupH∗(X,Q) . Itseems that there areonly afew example ofcomputation
alg
t ofcategoricalentropy,anditmaybe interestingto addmoreexamples. Inthis note,weshallgiveanalmost
a
m trivial example of the computation. Thus we shall compute the entropy for special autoequivalences on
abelian surfaces. For an abelian variety, Orlov [9] proved that the kernel of an equivalence is a sheaf up to
[
shift. So we can expect that the entropy which measures the complexity of an equivalence is simple. For a
1 special equivalence on an abelian surface, we shall check that our expectation is true.
v
To be more precise, let X be an abelian surface and H an ample divisor. We set L := Z ZH Z̺ ,
9 ⊕ ⊕ X
where̺ is thefundamentalclassofX. LetΦ:D(X) D(X)be aFourier-Mukaifunctorwhichpreserves
0 X
→
0 L. In Theorem 2.2, we shall compute ht(Φ). Combining a recent paper of Ikeda [2], we also check that the
4 conjecture of Kikuta and Takahashi holds for equivalences on abelian surfaces (Proposition 2.10).
0 We also remark that there is a symplectic manifold of generalized Kummer type which has an automor-
.
1 phism of infinite order. This is an analogue of a recent result of Ouchi [11]. By the absence of spherical
0 objects, our example is almost trivial.
7
1
:
v
i
X 1. Fourier-Mukai transforms on an abelian surface
r 1.1. Notation. We denote the category of coherent sheaves on X by Coh(X) and the bounded derived
a
category of Coh(X) by D(X). A Mukai lattice of X consists of H2∗(X,Z) := 2 H2i(X,Z) and an
i=0
integral bilinear form , on H2∗(X,Z):
h i L
x +x +x ̺ ,y +y +y ̺ :=(x ,y ) x y x y Z,
0 1 2 X 0 1 2 X 1 1 0 2 2 0
h i − − ∈
where x ,y H2(X,Z), x ,x ,y ,y Z and ̺ H4(X,Z) is the fundamental class of X. We also
1 1 0 2 0 2 X
∈ ∈ ∈
introducethealgebraicMukailatticeasthepairofH∗(X,Z) :=Z NS(X) Zand , onH∗(X,Z) .
alg alg
⊕ ⊕ h i
For x=x +x +x ̺ with x ,x Z and x H2(X,Z), we also write x=(x ,x ,x ). For E D(X),
0 1 2 X 0 2 1 0 1 2
∈ ∈ ∈
v(E):=ch(E) denotes the Mukai vector of E.
For E D(X Y), we set
∈ ×
ΦE (x):=Rp (E p∗ (x)), x D(X),
X→Y Y∗ ⊗ X ∈
2010 Mathematics Subject Classification. Primary14D20.
Keywords and phrases. Categoricalentropy, abeliansurfaces.
Theauthor issupportedbytheGrant-in-aidforScientificResearch(No.26287007, 24224001), JSPS.
1
where p ,p are projections from X Y to X and Y respectively. Let Eq(D(X),D(Y)) be the set of
X Y
×
equivalences between D(X) and D(Y). We set
Eq (D(Y),D(Z))
0
:= ΦE[2k] Eq(D(Y),D(Z)) E Coh(Y Z), k Z ,
Y→Z ∈ ∈ × ∈
n (cid:12) o
(Z):= Eq (D(Y),D(Z)(cid:12)),
E 0 (cid:12)
Y
[
:= (Z)= Eq (D(Y),D(Z)).
E E 0
Z Y,Z
[ [
Note that is a groupoid with respect to the composition of the equivalences.
E
For an object E D(X) with rkE =0, we set µ(E):=c (E)/rkE.
1
∈ 6
1.2. Semi-homogeneoussheaves. Wecollectsomepropertiesofsemi-homogeneoussheavesonanabelian
surface [6].
Proposition 1.1. For a coherent sheaf E on X, the following conditions are equivalent.
(i) E is a semi-homogeneous sheaf.
(ii) E is a semi-stable sheaf with v(E)2 =0 with respect to an ample divisor H.
h i
Proposition 1.2 (cf. [9]). Let Φ : D(X) D(Y) be an equivalence. For a semi-homogeneous sheaf E on
→
X, there is an integer n such that Φ(E)[n] is a semi-homogeneous sheaf.
Proposition 1.3 ([14, Prop. 4]). Let E and F be semi-homogeneous sheaves.
(i) Assume that E and F are locally free sheaves.
(a) If v(E),v(F) >0, then Hom(E,F)=Ext2(E,F)=0.
h i
(b) If v(E),v(F) <0, then (µ(E),H)=(µ(F),H), Ext1(E,F)=0 and
h i 6
Hom(E,F)=0, (µ(E),H)>(µ(F),H)
(1.1)
(Ext2(E,F)=0, (µ(F),H)>(µ(E),H).
(ii) Assume that E is locally free and F is a torsion sheaf.
(a) If v(E),v(F) >0, then Hom(E,F)=Ext2(E,F)=0.
h i
(b) If v(E),v(F) <0, then Ext1(E,F)=Ext2(E,F)=0.
h i
(iii) Assume that E and F are torsion sheaves. Then v(E),v(F) 0. If v(E),v(F) > 0, then
h i ≥ h i
Hom(E,F)=Ext2(E,F)=0.
Remark 1.4. If (D2) > 0, then D is ample if and only if (D,H ) > 0 for an ample divisor H . Since
0 0
v(E),v(F) =rkErkF((µ(E) µ(F))2)/2, v(E),v(F) <0impliesµ(E) µ(F)isampleorµ(F) µ(E)
−h i − h i − −
is ample.
1.3. Cohomological Fourier-Mukai transforms. We collect some results on the Fourier-Mukai trans-
forms on abelian surfaces X with rkNS(X) = 1. Let H be the ample generator of NS(X). We shall
X
describe the action of Fourier-Mukai transforms on the cohomology lattices in [12]. For Y FM(X), we
∈
have (H2)=(H2). We set D :=(H2)/2. In [12, sect. 6.4], we constructed an isomorphism of lattices
Y X X
ι : (H∗(X,Z) , , ) ∼ (Sym (Z,D),B),
X alg h i −−→ 2
r d√D
(r,dH ,a) ,
X 7→ d√D a
(cid:18) (cid:19)
where Sym (Z,D) is given by
2
x y√D
Sym (Z,D):= x,y,z Z ,
2 ((cid:18)y√D z (cid:19) (cid:12)(cid:12) ∈ )
(cid:12)
and the bilinear form B on Sym2(Z,D) is given by (cid:12)
(cid:12)
B(X ,X ):=2Dy y (x z +z x )
1 2 1 2 1 2 1 2
−
x y √D
for X = i i Sym (Z,D) (i=1,2).
i y √D z ∈ 2
(cid:18) i i (cid:19)
Each Φ Eq (D(X),D(Y)) gives an isometry
X→Y ∈ 0
(1.2) ι ΦH ι−1 O(Sym (Z,D)),
Y ◦ X→Y ◦ X ∈ 2
where O(Sym (Z,D)) is the isometry group of the lattice (Sym (Z,D),B). Thus we have a map
2 2
η : O(Sym (Z,D))
E → 2
2
which preserves the structures of multiplications.
Definition 1.5. We set
a√r b√s a,b,c,d,r,s Z, r,s>0
G:= ∈ ,
c√s d√r rs=D, adr bcs= 1
((cid:18) (cid:19)(cid:12)(cid:12) − ± )
Gb :=G SL(2,R). (cid:12)(cid:12)
∩ (cid:12)
We have an action of G on the lattice (Sym (Z,D),B):
· b 2
b r d√D r d√D
(1.3) g :=g tg, g G.
· d√D a d√D a ∈
(cid:18) (cid:19) (cid:18) (cid:19)
Thus we have a homomorphism: b
α:G/ 1 O(Sym (Z,D)).
{± }→ 2
Theorem 1.6 ([12, Thm. 6.16, Prop. 6.19]). Let Φ Eq (D(Y),D(X)) be an equivalence.
b ∈ 0
(1) v := v(Φ( )) and v := Φ(̺ ) are positive isotropic Mukai vectors with v ,v = 1 and we
1 Y 2 Y 1 2
O h i −
can write
v =(p2r ,p q H ,q2r ), v =(p2r ,p q H ,q2r ),
1 1 1 1 1 X 1 2 2 2 2 2 2 X 2 1
p ,q ,p ,q ,r ,r Z, p ,r ,r >0,
1 1 2 2 1 2 1 1 2
∈
r r =D, p q r p q r =1.
1 2 1 2 1 2 1 2
−
(2) We set
p √r p √r
tθ(Φ):= 1 1 2 2 G/ 1 .
± q1√r2 q2√r1 ∈ {± }
(cid:18) (cid:19)
Then tθ(Φ) is uniquely determined by Φ and we have a map
tθ : G/ 1 .
E → {± }
(3) The action of tθ(Φ) on Sym (Z,D) is the action of Φ on the algebraic Mukai lattice:
2
ι Φ(v)=tθ(Φ) ι (v).
X Y
◦ ·
Thus we have the following commutative diagram:
(1.4) tθE(cid:15)(cid:15) ❖❖❖❖❖❖❖❖η❖❖❖❖❖❖''
G/ 1 // O(Sym (Z,D))
{± } α 2
For Φ:Eq (D(X),D(X)) preservingthe sublattice L:=Z ZH Z̺ , it is easy to see that the same
0 b ⊕ X⊕ X
proof of Theorem 1.6 works. Thus replacing H∗(X,Z) by L, we have a similar claims to Theorem 1.6.
alg
From now on, we identify the Mukai lattice L with Sym (Z,D) via ι . Then for g G and v L, g v
2 X ∈ ∈ ·
means ι (g v)=g ι (v). We also set H :=H . By [15, Lem. 2.5], we have the following.
X X X
· ·
b
Lemma 1.7. For
a b√D
(1.5) A= , a,b,c,d Z,ad bcD=1,
c√D d ∈ −
(cid:18) (cid:19)
there is a Fourier-Mukai transform ΦEA :D(X) D(X) such that ΦEA (L)=L with tθ(ΦEA )=A.
X→X → X→X X→X
Replacing A by A if necessary, we assume that trA 0.
− ≥
E
Definition 1.8. For a matrix A in (1.5) such that trA 0, let Φ A : D(X) D(X) such that
E Coh(X X) and ΦEA (L)=L with tθ(ΦEA )=A.≥ X→X →
A ∈ × X→X X→X
We note that
(1.6) v((E ) )=(b2D,bdH,d2), v((E∨) )=(b2D, baH,a2).
A |{x}×X A |X×{x} −
Remark 1.9. IfE′ alsoinducesthe samecohomologicaltransformΦ,then thereis anisomorphismf :X
A →
X and a line bundle P Pic0(X) such that E′ =(1 f)∗(E ) p∗(P).
∈ A X × A ⊗ 2
Let T be a subgroupof Eq (D(X),D(X)) induced by Aut (X) Pic0(X), where Aut (X) is the group
0 E H × H
of isomorphisms preserving H. Then T preserves L and Φ A mod T is determined by A.
X→X
Remark 1.10. If X is an abelian surface with End(X)=Z, then NS(X)=ZH with (H2)=2D and for all
∼
autoequivalences Φ, tθ(Φ) is of the form (1.5).
3
Remark 1.11. For an equivalence ΦE :D(X) D(X) suchthat E Coh(X X) and ΦE preserves
X→X → ∈ × X→X
L. We set
(1.7) B :=tθ(ΦE )= p1√r1 p2√r2
X→X q1√r2 q2√r1
(cid:18) (cid:19)
as in Theorem 1.6 (2). Then B2 is of the form (1.5). If (p +q )p >0 or p =0, then (ΦEB )2 ΦEB2
mod T. If (p +q )p < 0 or p +q = 0, then (ΦEB )21 Φ2EB22[−2] mod2T. Hence forXth→eXcom≡putXat→ioXn
E 1 2 2 1 2 X→EX ≡ X→X
of h (Φ B ) is reduced to the computation of h (Φ A ). In particular, if NS(X) = ZH, then we can
t X→X t X→X
compute h for all equivalences.
t
Since the eigen equation of A in (1.5) is x2 (a+d)x+1 = 0, A has two real eigen value α > β = 1/α
−
unless trA=0,1,2. If trA=0,1, then A4 =E,A6 =E. If trA=2, then (A E)2 =0, and hence
−
(1.8) An =E+n(A E).
−
Assume that trA>2. Then
(1.9) α,β Q,
6∈
since Z is integrally closed and 0<β <1. We set
1 1
(1.10) P := (A βE), Q:= (A αE).
α β − β α −
− −
Then
(1.11) An =αnP +βnQ.
Let E be a semi-homogeneous sheaf on X with v(E)=(p2,pqH,q2D), p2,pq,q2D Z. Then
∈
p
(1.12) ι (v(L))=utu, u= .
X q√D
(cid:18) (cid:19)
We set
p
(1.13) n =Anu=αnPu+βnQu.
q √D
n
(cid:18) (cid:19)
Then
p
(1.14) ι (v(Φn(E)))= n p q √D .
X q √D n n
n
(cid:18) (cid:19)
(cid:0) (cid:1)
Let
1
(1.15) u =
α s√D
(cid:18) (cid:19)
be an eigen vector with respect to α. Thus s= α−a. By (1.9), (A βE)u=0, and hence
bD − 6
q
n
(1.16) lim =s.
n→∞pn
The following is obvious.
Lemma 1.12. Assume that x2 trAx+detA=(x α)(x β), α,β C. Then the eigen equation of the
− − − ∈
representation matrix of the action of A on Sym (R,D) is
2
(1.17) (x α2)(x αβ)(x β2)=(x2 ((trA)2 2detA)x+(detA)2)(x detA).
− − − − − −
In particular, if α β , then α2 is the spectral radius of this representation.
| |≥| | | |
2. Computation of entropy
For an endofunctor Φ : D(X) D(X), let h (Φ) : R R be the entropy defined in [1, Defn.
t
→ → {−∞}∪ E
2.5]. In this section, we shall compute the entropy for the equivalence Φ := Φ A by using the following
X→X
result.
Proposition 2.1 ([1, Thm. 2.7] and [4, Prop. 3.8]). Let G,G′ be split generators of D(X) and F an
endofunctor of D(X) of Fourier-Mukai type such that FnG,FnG′ =0 for all n>0. Then
6∼
1
(2.1) h (F)= lim logδ′(G,FnG′)
t n→∞n t
where
(2.2) δ′(M,N):= dimHom(M,N[m])e−mt, M,N D(X).
t ∈
m∈Z
X
4
E
Theorem 2.2. Let Φ:=Φ A be an equivalence associated to A (see Definition 1.8).
X→X
(1) Assume b=0. Then h (Φ)=logρ(Φ)=0.
t
(2) Assume that b>0. Then
logρ(Φ), trA 2
≥
(2.3) h (Φ)= logρ(Φ) 2t, trA=1
t − 3
logρ(Φ) t, trA=0.
−
(3) Assume that b<0. Then
logρ(Φ) 2t, trA 2
− ≥
(2.4) h (Φ)= logρ(Φ) 4t, trA=1
t − 3
logρ(Φ) t, trA=0.
−
Remark 2.3. If trA 2, then ρ(Φ)=1.
≤
Proof. Let N be a line bundle with c (N)=mH, m>0. We take
1
(2.5) G= N⊗i, G′ = N⊗i
−3≤i≤−1 1≤i≤3
M M
as split generators of D(X) ([10]).
E
(1) Assume that b = 0. Then Φ A is an equivalence defined by Aut(X) Pic(X). Then it is easy to
X→X ×
see that
E E
(2.6) h (Φ A )=0=logρ(Φ A ).
t X→X X→X
We next prove (2) and (3). So we assume that b=0.
6
(I). We first treat the case where trA>2. We set
p 1
(2.7) i,n =An .
q √D im√D
i,n
(cid:18) (cid:19) (cid:18) (cid:19)
Then v(Φn(N⊗i)) = (p2 ,p q H,q2 D). For a sufficiently large n, q /p is sufficiently close to s by
i,n i,n i,n i,n i,n i,n
(1.16). Hence we can take an integer m such that
q
i,n
(2.8) > m
p −
i,n
for all sufficiently large n (depending on m). We note that µ((E∨) )= a H and
A |X×{x} −bD
a α a a α
(2.9) s+ = − + = .
bD bD bD bD
Let E be a semi-homogeneous sheaf with µ(E)=xH. We set µ(Φn(E))=x H. Then
n
c+dx
n
(2.10) x = .
n+1
a+bDx
n
Assume that b>0. Then (2.9) implies s> a . If s x < α−1, then
−bD | − | |b|D
c+dx x s
(2.11) s | − |.
a+bDx − ≤ α
(cid:12) (cid:12)
ΦHnen(Ece)xnCaolsho(Xsa)tifsofiresnthe0sabmyePcroopndosititioionn(cid:12)(cid:12)(cid:12). 1If.3E. also sat(cid:12)(cid:12)(cid:12)isfies x>−baD, then xn >−baD for all n≥0. Hence
We s∈et Φn(N⊗i)=E≥i[φ (n)], Ei Coh(X) (cf. Proposition 1.2). For E :=Ei , µ(E)= pi,n0H. Hence
n i n ∈ n0 qi,n0
by (1.16), we get Φn′(Ei ) Coh(X) for n′ 0 and n 0. Therefore φ (n) 2Z is constant for a
n0 ∈ ≥ 0 ≫ i ∈
sufficiently large n. Hence there are l such that Hom(G,Φn(N⊗i)[k])=0 for k =l and
i i
6
δ′(G,Φn(G′))= χ(N⊗i,Φn(N⊗j))e−ljt
t
−3≤i≤−11≤j≤3
(2.12) X X
= (αna +βnb )2e−ljt
i,j i,j
−3≤i≤−11≤j≤3
X X
where
1 1
αna +βnb =det ,(αnP +βnQ)
i,j i,j im√D jm√D
(2.13) (cid:18)(cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
1 1
=det ,An
im√D jm√D
(cid:18)(cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
(see (1.11)). By (2.8), a =0 for all sufficiently large n. Hence logδ′(G,Φn(G′)) 2nlog α and
i,j 6 t ∼ | |
5
1
(2.14) h (Φ)= lim logδ′(G,Φn(G′))=2log α =logρ(Φ).
t n→∞n t | |
Assume that b<0. Then (2.9) implies s< a . If s x < α−1, then
−bD | − | |b|D
c+dx x s
(2.15) s | − |.
a+bDx − ≤ α
(cid:12) (cid:12)
Hence xn also satisfies the same condition(cid:12)(cid:12)(cid:12). If E also sa(cid:12)(cid:12)(cid:12)tisfies x < −baD, then xn < −baD. Hence Φ(E)[2] ∈
Coh(X). We set (Φ[2])n(N⊗i) = Ei[ψ (n)], Ei Coh(X). In this case, ψ (n) is constant for a sufficiently
n i n ∈ i
large n. Hence there are l such that Hom(G,Φn(N⊗i)[k]) = 0 for k = l +2n and logχ(G,Φn(G′))
i i
6 ∼
2nlog α. Hence
| |
1
(2.16) h (Φ)= lim logδ′(G,Φn(G′))=2log α 2t=logρ(Φ) 2t.
t n→∞n t | |− −
(II). Assume that trA= 2. Then An = E+n(A E). We set s := 1−a. Let E be a semi-homogeneous
− bD
sheaf with µ(E)=xH and set µ(Φn(E))=x H. Since
n
nc+(n(d 1)+1)x
x = − ,
n
1+n(a 1)+nbDx
(2.17) −
bDx+(a 1)
x s= − ,
n
− (1+n(a 1+bDx))bD
−
we have lim x =s. In the same way as in the case trA>2, we see that
n→∞ n
logρ(Φ), b>0
(2.18) h (Φ)=
t
(logρ(Φ) 2t, b<0.
−
(III). Assume that trA=1. Then A3 = E. It is easy to see that
−
(2.19) (ΦEXA→X)2 ≡(ΦΦEXEXAA→→22XX[−m2]odmTod T bb><00
and
(2.20) (ΦEA )3 [−2] mod T b>0
X→X ≡([ 4] mod T b<0
−
(see Remark 1.9). Indeed ΦEA ((E ) ) M (u∨) for b > 0 and ΦEA ((E ) )[2] M (u∨)
X→X A |{x}×X ∈ H X→X A |{x}×X ∈ H
for b<0, where u:=v((E ) )=(b2D,baH,a2). Hence
A |X×{x}×X
(2.21) h (ΦEA )= −32t b>0
t X→X ( 4t b<0.
−3
Assume that trA=0. Then (ΦEA )2 [ 2] mod T implies
X→X ≡ −
E
(2.22) h (Φ A )= t.
t X→X −
(cid:3)
Remark 2.4. OurassumptionofΦ(L)=Lisverystrongingeneral. IndeedH isnotpreservedbytheaction
of Aut(X) in general, and there is an abelian surface with an automorphism of positive entropy.
Remark 2.5. In [2], Ikeda introduced a mass growth h of a stability condition σ and studied several
σ,t
properties. In our example,
(2.23) h (Φ)=h (Φ),
σ,t t
where σ be a stability condition such that Z (E):= ezH,v(E) (z H) and φ (k )=1:
σ σ x
h i ∈
WeonlyexplainthecasewheretrA>2. WefirstnotethatΦpreservesσ-stabilityforsemi-homogeneous
sheaves. We set
1 1
(2.24) αna (z)+βnb (z)=det ,(αnP +βnQ) .
j j z√D jm√D
(cid:18)(cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
Then by (1.11), we get Z (Φn(N⊗j))=(αna (z)+βnb (z))2. Assume that b<0. Then φ ((Φ[2])n(N⊗j))
σ j j σ
is bounded. Hence by using [2, Thm. 1.1], we get h (Φ)=log α2 2t=h (Φ).
σ,t t
| | −
6
2.1. Gromov-Yomdin type conjecture by Kikuta and Takahashi. Let X be an arbitrary abelian
surface. We shall compute h (Φ) and check the Kikuta and Takahashi’s conjecture [4, Conjecture 5.3] for
0
some endofunctors of D(X).
Lemma 2.6. Let D be a divisor with (D2) > 0 and η NS(X). There is no isotropic vector v = 0 in
∈ 6
(Qeη+Qeη+D+Qeη+2D)⊥.
Proof. Replacing v by ve−η, we may assume that η =0. We set v =(r,ξ,a)=0. Then (ξ2)=2ra. By the
6
conditions, we get
(D2)
(2.25) a=0, (D,ξ) r =0, 2(D,ξ) 2r(D2)=0.
− 2 −
Hence (D,ξ) = r(D2) = 0. Since (D2) > 0, we get r = a = (ξ2) = 0. Since D⊥ is negative definite, we get
ξ =0. Therefore our claim holds. (cid:3)
Lemma 2.7. For a semi-homogeneous sheaf E and a line bundle L,
(2.26) dimHom(L,E[k]) max 4χ(L(pH),E) p=0, 1, 2 ,
≤ { | || ± ± }
k∈Z
X
where H is an ample divisor on X.
Proof. We first note that E is a semi-stable sheaf with respect to H by Proposition 1.1. Then E is S-
equivalent to E such that E are stable with v(E )=v(E ). Hence it is sufficient to prove the claim for
i i i i j
⊕
a stable semi-homogeneous sheaf E. We first assume that χ(L,E) = 0. Then E is not 0-dimensional. We
can easily show the following claim.
(i) If (c (E L∨),H)>0, then dimHom(L,E)=dimHom(L,E[1]) and Hom(L,E[2])=0.
1
⊗
(ii) If (c (E L∨),H)<0, then dimHom(L,E)=0 and dimHom(L,E[1])=dimHom(L,E[2]).
1
⊗
(iii) If (c (E L∨),H) = 0, then dimHom(L,E) = dimHom(L,E[2]) 1, dimHom(L,E[1]) 2 and
1
⊗ ≤ ≤
v(L)=v(E).
For the case of (iii), obviously the claim holds by Lemma 2.6. So we shall treat the case of (i) and (ii). In
these cases, E is not 0-dimensional. Replacing H by its translate, we can take an injective homomorphism
E E(H). Then we get
→
dimHom(L,E) dimHom(L(pH),E)
≤
for p 0. We also see that
≤
dimHom(E,L) dimHom(E,L(pH))
≤
for p 0. By Lemma 2.6, χ(L(pH),E)= 0 for an integer p 0,1,2 and χ(L(pH),E) =0 for an integer
≥ 6 ∈{ } 6
p 0, 1, 2 . Hence by using Proposition 1.3, we get
∈{ − − }
dimHom(L,E) max 0,χ(L(pH),E) p= 1, 2
(2.27) ≤ { | − − }
dimExt2(L,E) max 0,χ(L(pH),E) p=1,2 .
≤ { | }
Thus
(2.28) dimHom(L,E[k]) max 2 χ(L(pH),E) p=0, 1, 2 .
≤ { | || ± ± }
k∈Z
X
By Proposition 1.3, the same claim also holds if χ(L,E)=0. (cid:3)
6
Proposition 2.8. Let Φ : D(X) D(X) be an endofunctor which is a composite of equivalences, f∗ and
→
f , where f :X X is a finite morphism. Then h (Φ) logρ(Φ).
∗ 0
→ ≤
Proof. We use split generators G,G′ in (2.5). For a finite morphism f : X X, f∗ and f send semi-
∗
→
homogeneous sheaves to semi-homogeneous sheaves. By Proposition 1.2, autoequivaences also send semi-
homogeneoussheavestosemi-homogeneoussheavesuptoshift. HenceΦn(N⊗j)isasemi-homogeneoussheaf
up to shift. By Lemma 2.7, we have
(2.29) dimHom(N⊗i,Φn(N⊗j)[k]) max 4χ((N(pH))⊗i,Φn(N⊗j)) p=0, 1, 2 .
≤ { | || ± ± }
k∈Z
X
For any real number λ>ρ(Φ), we have
1
(2.30) lim Φn(N⊗j)=0,
n→∞λn
and hence
1
(2.31) lim χ((N(pH))⊗i,Φn(N⊗j) =0.
n→∞λn| |
7
Then we have χ((N(pH))⊗i,Φn(N⊗j) λn for sufficiently large n. Combining (2.29) with this estimate,
| | ≤
we see that
1
(2.32) h (Φ)= lim logδ′(G,Φn(G′)) logλ.
0 n→∞n 0 ≤
Since λ is arbitrary, we get h (Φ) logρ(Φ). (cid:3)
0
≤
Remark 2.9. Let X be a simple abelian variety of dimX = d, that is, there is no subabelian variety Y
of X with 0 < dimY < d. Then for any line bundle L, K(L) := x X T∗(L) = L is a finite set,
{ ∈ | x ∼ }
unless L Pic0(X). Hence if c (L)=0, then (Ld)=0. Then for a semi-homogeneous sheaf E, we see that
1
∈ 6 6
χ(E)=0 or ch(E) Z ch( ). Hence we also get h (Φ) logρ(Φ).
>0 X 0
6 ∈ O ≤
By [2, Thm. 1.2], h (Φ) logρ(Φ). Therefore we get the following result which support a Gromov-
0
≥
Yomdin type conjecture in [4, Conjecture 5.3].
Proposition 2.10. Let X be an abelian surface and Φ:D(X) D(X) an endofunctor in Proposition 2.8.
→
Then h (Φ)=logρ(Φ).
0
3. An example of automorphism on the moduli of stable sheaves
Let M (v) be the moduli space of stable sheaves E on X with v(E) = v and K (v) be a fiber of the
H H
albanesemapa:M (v) Alb(M (v))=X Pic0(X). K (v)isanirreduciblesymplectic manifoldwhich
H H H
→ ×
is derormation equivalent to a generalized Kummer manifold [13]. In [15, Prop. 3.50], we constructed an
example of moduli space M (v) which have an automorphism of infinite order. Thus there is a Fourier-
H
Mukai transformΦ which induces an isomorphismg :M (v) M (v) such that g is infinite order. In this
H H
→
example, it is easy to see that a similar claim to [11] holds. Thus by using [8], we get
dimK (v)
(3.1) H h (Φ)=h(g′)
0
2
whereg′ :K (v) K (v)isanautomorphisminducedg byaFourier-MukaitransformΦ:D(X) D(X)
H H
→ →
in [15, Prop. 3.50]:
Let be a quasi-universal family on M (v) X. By T∗( ) P (x X,P Pic0(X)), we have an
E H × x E ⊗ y ∈ y ∈
isomorphismψ :MH(v)→MH(v) suchthat(ψ×1X)∗(E)⊗L∼=Tx∗(E)⊗Py⊗L′,whereL,L′ arepull-backs
of locally free sheaves on M (v). Then
H
c (p (ch((ψ 1 )∗( ))α∨))=c (p (ch(T∗( ) P )α∨))
1 MH(v)! × X E 1 MH(v)! x E ⊗ y
(3.2) =c (p (ch( )T∗ (α∨)))
1 MH(v)! E −x
=c (p (ch( )α∨))
1 MH(v)! E
for α v⊥. Thus ψ∗(θ (α)) = θ (α). Hence for an isomorphism g : M (v) M (v) induced by Φ, we
v v H H
∈ →
have an isomorphism g′ :K (v) K (v) which induces the isomorphism Φ:v⊥ v⊥.
H H
→ →
4. Appendix
Let (X,H) be a principally polarized abelian variety of dimX = d. In [5], cohomological action of the
P
group G of Fourier-Mukai transforms generated by Φ and (H) is described. In particular the
X→X ⊗OX
action on the cohomology group generated by H is the action of SL(2,Z) on the d-th symmetric product
of Q2. Then we can also compute h (ΦE ) of ΦE G, where E is a coherent sheaf on X X. In
particular h (ΦE )=log αd dt tif trXA→<X 2. X→X ∈ ×
t X→X | | − −
Remark 4.1. For a semi-homogeneous sheaf E with rkE > 0, if c (E) is ample, then Hi(E) = 0 for i = 0
1
6
(cf. [6, Prop. 7.3]).
Acknowledgement. I would like to thank Atsushi Takahashi for useful comments.
References
[1] Dimitrov,G.,Haiden,F.,Katzarkov, L.,Kontsevich,M.,Dynamical systemsand categories,arXiv:1307.8418
[2] Ikeda, A.,Mass growth of objectsand categorical entropy,arXiv:1612.00995
[3] Kikuta,K.,On entropy for autoequivalences of the derived category of curves,arXiv:1601.06682
[4] Kikuta,K.,Takahashi, A.,Onthe categorical entropy and the topological entropy,arXiv:1602.03463
[5] Maciocia, A., Piyaratne, D., Fourier-Mukai Transforms and Bridgeland Stability Conditions on Abelian Threefolds II,
arXiv:1310.0299
[6] Mukai,S.,Semi-homogeneous vectorbundles on an abelian variety,J.Math.KyotoUniv.18(1978), no.2,239–272.
[7] Mukai,S.,Duality between D(X) and D(Xb) with itsapplication to Picard sheaves,NagoyaMath.J.81(1981), 153–175.
[8] Oguiso,O.,A remark on dynamical degrees of automorphisms of hyperK¨ahler manifolds, ManuscriptaMath.130(2009)
101-111.
8
[9] Orlov, D., Derived categories of coherent sheaves on abelian varieties and equivalences between them,
arXiv:alg-geom/9712017,Izv.RAN,Ser.Mat.,v.66,N3(2002).
[10] Orlov,D.,Remarksongeneratorsanddimensionsoftriangulatedcategories,arXiv:0804.1163,MoscowMath.J.9(2009),
no.1,153–159.
[11] Ouchi,G.,Automorphisms ofpositiveentropyonsomehyperKahlermanifoldsviaderivedautomorphisms ofK3surfaces,
arXiv:1608.05627
[12] Yanagida,S.,Yoshioka,K.,Semi-homogeneoussheaves,Fourier-MukaitransformsandmoduliofstablesheavesonAbelian
surfaces,J.reineangew.Math.684(2013), 31–86.
[13] Yoshioka,K.,Moduli spaces of stable sheaves on abelian surfaces,Math.Ann.321(2001), 817–884, math.AG/0009001
[14] Yoshioka,K.,Fourier-Mukai transform on abelian surfaces,Math.Ann.345(2009), 493–524
[15] Yoshioka, K., Bridgeland’s stability and the positive cone of the moduli spaces of stable objects on an abelian surface,
arXiv:1206.4838,Adv.Stud.PureMath.69(2016), 473–537.
DepartmentofMathematics,Facultyof Science,Kobe University, Kobe,657,Japan
E-mail address: [email protected]
9
