A trichotomy for the autoequivalence groups on smooth projective surfaces

TL;DR

This paper classifies autoequivalence groups of derived categories on smooth projective surfaces into three types based on the maximal dimension of Fourier--Mukai kernels, providing a new trichotomy and conjectures for the case of dimension 2.

Contribution

It introduces a novel trichotomy classification for autoequivalence groups based on Fourier--Mukai kernel dimensions and proposes a conjecture for the dimension 2 case, with partial proofs.

Findings

Maximal Fourier--Mukai kernel dimension is 2, 3, or 4.

Conjecture formulated for the dimension 2 case.

Partial proofs provided for special cases.

Abstract

We study autoequivalence groups of the derived categories on smooth projective surfaces, and show a trichotomy of types according to the maximal dimension of Fourier--Mukai kernels for autoequivalences. This number is $2$, $3$ or $4$, and we also pose a conjecture on the description of autoequivalence groups if it is $2$, and prove it in some special cases.

Equations

$$\mathrm{Comp}({\Phi }^{\mathcal{P}})$$

$$N_X:$$

$$\in \{\dim (X),\dim (X)+1,\dots ,2\dim (X)\}$$

$${\mathrm{Br}}_Z(S)=⟨T_{\alpha }\mid \alpha \in D_Z(S)\text{ spherical object}⟩(\subset \mathrm{Auteq}D(S)).$$

$$\mathrm{Auteq}D(S)=⟨{\mathrm{Br}}_Z(S),\mathrm{Pic}(S)⟩⋊\mathrm{Aut}(S)\times \mathbb{Z}[1].$$

$$\mathrm{Auteq}D(X)=\mathrm{Pic}(S)⋊\mathrm{Aut}(S)\times \mathbb{Z}[1].$$

$$R(X,D):=\underset{m\ge 0}{⨁}{H}^0(X,{\mathcal{O}}_X(mD)).$$

$$\kappa (D):=\{\begin{array}{ll}\text{ the transcendence degree of }R(X,D)-1& \text{ if }R(X,D)\ne \mathbb{C}\\ -\infty & \text{ otherwise.}\end{array}$$

$$\nu (D):=\max \{k\in \mathbb{Z}\mid D^k\cdot H^{\dim (X)-k}\ne 0\},$$

$$\nu (D)\ge \kappa (D)$$

$${\Phi }^{\mathcal{P}}:=\mathbb{R}{p}_{Y\ast }(\mathcal{P}\stackrel{\mathbb{L}}{\otimes }{p}_X^{\ast }(-)):D(X)\to D(Y).$$

$$A(X):=\mathrm{Pic}(X)⋊\mathrm{Aut}(X)\times \mathbb{Z}[1],$$

$$\Phi ={\phi }_{\ast }\circ ((-)\otimes \mathcal{L})\circ [i]$$

$$({\mathcal{O}}_{{\Gamma }_{\phi }}\otimes p_1^{\ast }\mathcal{L})[i],$$

$${\Phi }^{\mathcal{P},H}:H^{\ast }(X,\mathbb{Q})\to H^{\ast }(Y,\mathbb{Q}),$$

$${\Phi }^{\mathcal{P},H}\circ v(-)=v(-)\circ {\Phi }^{\mathcal{P}}$$

$$\alpha \otimes {\omega }_X\cong \alpha .$$

$$\mathrm{ch}(\mathcal{F})=\mathrm{ch}(\mathcal{F})\cdot \mathrm{ch}({\omega }_X)=\mathrm{ch}(\mathcal{F})\cdot (1+c_1({\omega }_X)+\frac{1}{2}{c}_1(\omega {)}^2+\cdots )$$

$$0=\mathrm{ch}(\mathcal{F})\cdot (c_1({\omega }_X)+\frac{1}{2}{c}_1(\omega {)}^2+\cdots ).$$

$$K_X\cdot C=0.$$

$${\mathrm{Hom}}_{D(X)}^k(\alpha ,\alpha )\cong \{\begin{array}{ll}0& (k\ne 0,\dim (X))\\ \mathbb{C}& (k=0,\dim (X)).\end{array}$$

$$\mathcal{C}:=\mathrm{Cone}(p_1^{\ast }{\alpha }^{\vee }\stackrel{\mathbb{L}}{\otimes }{p}_2^{\ast }\alpha \to {\mathcal{O}}_{{\Delta }_X})\in D(X_1\times X_2)$$

$${\mathrm{RHom}}_{D(X)}(\alpha ,\beta ){\stackrel{\mathbb{L}}{\otimes }}_{\mathbb{C}}\alpha \to \beta \to T_{\alpha }(\beta )$$

$$d(E)=c_1(E)\cdot F,$$

$${\mathrm{Br}}_Z(S)=⟨T_{\alpha }\mid \alpha \in D_Z(S)\text{ spherical object}⟩(\subset \mathrm{Auteq}D(S)),$$

$$\Gamma_{0}(m):=\Bigl{\{}\begin{pmatrix}c&a\\ d&b\end{pmatrix}\in\operatorname{SL}(2,\mathbb{Z})\bigm{|}d\in m\mathbb{Z}\Bigr{\}}$$

$$1\to ⟨{\mathrm{Br}}_Z(S),\otimes {\mathcal{O}}_S(D)\mid D\cdot F=0,\text{ F is a fiber}⟩⋊\mathrm{Aut}(S)\times \mathbb{Z}[2]\to \mathrm{Auteq}D(S)$$

$$\displaystyle\stackrel{{\scriptstyle\Theta}}{{\to}}\Big{\{}\begin{pmatrix}c&a\\ d&b\end{pmatrix}\in\Gamma_{0}(\lambda_{S})\bigm{|}J_{S}(b)\cong S\Big{\}}$$

$$\Phi ={\Phi }_{X\to Y}^{\mathcal{P}}:D(X)\to D(Y)$$

$$\mathrm{Supp}(\mathcal{P})=\mathrm{Supp}(\mathcal{Q}).$$

$$\mathcal{P}{|}_{x\times X}\cong \Phi ({\mathcal{O}}_x)\text{and}\mathrm{Supp}(\mathcal{P}{|}_{x\times X})={\Gamma }_x$$

Full text

1 Introduction

The study of derived categories $D(X)=D^{b}(\operatorname{Coh}(X))$ of coherent sheaves on a smooth projective varieties $X$ has become an important topic in algebraic geometry over the last decades. It is an interesting and basic problem to describe the group $\operatorname{Auteq}D(X)$ of autoequivalences of $D(X)$. In this article, we consider the autoequivalence group of smooth projective surfaces.

Let us introduce an integer $N_{X}$, which plays the key role of a trichotomy of types of the autoequivalence group on smooth projective surfaces. First recall that an Orlov’s deep result states that every autoequivalence on a smooth projective variety $X$ is given by a Fourier–Mukai transform $\Phi^{\mathcal{P}}$ with unique kernel $\mathcal{P}\in D(X\times X)$ (see §2.1). Let us define

[TABLE]

to be the set of irreducible components $W_{0}$ of $\operatorname{Supp}(\mathcal{P})$ which dominates $X$ by the first projection $p_{1}\colon X\times X\to X$, which turns out to be non-empty by 3.1 (i)). Define

[TABLE]

and call it the Fourier–Mukai support dimension of $X$.

For smooth projective surfaces $S$, (conjectural) descriptions of the group $\operatorname{Auteq}D(S)$ and the geometry of surfaces are quite different, depending on the value $N_{S}$. The following is the first main result in this article.

Theorem 1.1** (=Theorem 5.3).**

We have the following.

1. (i)

$N_{S}=4$* if and only if $K_{S}\equiv 0$.* 2. (ii)

$N_{S}=3$* if and only if $S$ has a minimal elliptic fibration and $K_{S}\not\equiv 0$.* 3. (iii)

$N_{S}=2$* if and only if $S$ has no minimal elliptic fibration and $K_{S}\not\equiv 0$.*

In the case $N_{S}=4$, Theorem 1.1 implies that $S$ is one of K3, abelian, bielliptic or Enriques surfaces. Bayer and Bridgeland describe the autoequivalence group of K3 surfaces with the Picard number $1$ [BB17]. Orlov finds a description of the autoequivalence group of abelian varieties (not necessarily surfaces) [Or02]. Recently, Potter finds a description of the autoequivalence group of bielliptic surfaces [Po17].

Let us consider the case $N_{S}=3$. In this case, Theorem 1.1 implies that $S$ has a minimal elliptic fibration $\pi\colon S\to C$ and $K_{S}\not\equiv 0$. Suppose furthermore that each reducible fiber of $\pi$ is non-multiple, and forms a cycle of $(-2)$-curves, i.e. it is of type $\mathrm{I}_{n}$ for some $n>1$. Then, the autoequivalence group $\operatorname{Auteq}D(S)$ is described in [Ue16]. See also Conjecture 2.1.

Finally, let us consider the case $N_{S}=2$. Let us set $Z$ the union of all $(-2)$-curves on $S$, and define

[TABLE]

Here, a functor $T_{\alpha}$ is a special kind of an autoequivalence, called a twist functor (see §2.2). Then, we pose the following conjecture:

Conjecture 1.2** (cf. Conjecture 6.7).**

If $N_{S}=2$, then we have

[TABLE]

The classical Bondal–Orlov Theorem states that if $\pm K_{X}$ is ample for a smooth projective variety $X$, we have

[TABLE]

Because there are no $(-2)$-curves on a smooth projective surface $S$ with ample $\pm K_{S}$ we can regard Conjecture 1.2 as a variant of their result. The following is the second main result of this article.

Theorem 1.3** (cf. Theorem 6.8).**

Let $S$ be a smooth projective surface with $N_{S}=2$. Then Conjecture 1.2 holds true, if $Z$ is a disjoint union of configurations of $(-2)$-curves of type $A$.

Theorem 1.3 is a generalization of [IU05, Theorem 1.5] and [BP14, Theorem 1]. We show Theorem 6.8, which is slightly stronger than Theorem 1.3.

Notation and conventions.

We follow the notation and terminology of [Ha77] unless otherwise stated. All varieties will be defined over the complex number field $\mathbb{C}$ in this article. A point on a variety will always mean a closed point.

By a minimal elliptic surface, we will always mean a smooth projective surface $S$ together with a smooth projective curve $C$ and a relatively minimal morphism $\pi\colon S\to C$ whose general fiber is an elliptic curve. Here a relatively minimal morphism means a morphism whose fibers contains no $(-1)$-curves. Such a morphism $\pi$ is called an minimal elliptic fibration.

We denote by $D(X)$ the bounded derived category of coherent sheaves on an algebraic variety $X$ For any subset $Z(\subset X)$, we denote the full triangulated subcategory of $D(X)$ consisting of objects supported on $Z$ by $D_{Z}(X)$. Here, the support of an object $\alpha\in D(X)$ is, by definition, the union of the set-theoretic supports of its cohomology sheaves $\mathcal{H}^{i}(\alpha)$. Note that the support is always closed subset because $\alpha$ is a bounded complex of coherent sheaves. We denote the dimension of the support of $\alpha$ by $\dim(\alpha)$.

An object $\alpha$ in $D(X)$ is said to be rigid if $\mathop{\mathrm{Hom}}\nolimits^{1}_{D(X)}(\alpha,\alpha)=0$.

Given a closed embedding of schemes $i\colon Z\hookrightarrow X$, we denote the derived pullback $\mathbb{L}i^{*}\alpha$ simply by $\alpha|_{Z}$.

For algebraic varieties $X,Y$, we denote the diagonal in $X\times X$ by $\Delta_{X}$, and denote the projections by $p_{X}\colon X\times Y\to X$ and $p_{Y}\colon X\times Y\to Y$, or $p_{1}\colon X\times Y\to X$ and $p_{2}\colon X\times Y\to Y$.

For an abelian variety $X$, we denote the dual variety $\mathop{\mathrm{Pic}}\nolimits^{0}X$ by $\widehat{X}$.

$\operatorname{Auteq}\mathcal{T}$ denotes the group of isomorphism classes of $\mathbb{C}$-linear exact autoequivalences of a $\mathbb{C}$-linear triangulated category $\mathcal{T}$.

For a Cartier divisor $D$ on a normal projective variety $X$, we define a graded $\mathbb{C}$-algebra by

[TABLE]

Recall that the Iitaka dimension $\kappa(X,D)=\kappa(D)(\in\{-\infty,0,1,\ldots,\dim(X)\})$ of $D$ is

[TABLE]

We call $\kappa(K_{X})$ the Kodaira dimension of $X$, and simply denote it by $\kappa(X)$. Assume furthermore that $D$ is a nef divisor. Then, recall the numerical Iitaka dimension $\nu(X,D)=\nu(D)(\in\{0,1,\ldots,\dim(X)\})$ of $D$ by

[TABLE]

where $H$ is an ample divisor on $X$. In general, it is known that the inequality

[TABLE]

holds.

Let $X$ be a minimal model, that is, $X$ is a normal projective variety with $\mathbb{Q}$-factorial terminal singularities and $K_{X}$ is nef. We call $\nu(K_{X})$ the numerical Kodaira dimension of $X$, and simply denote it by $\nu(X)$. The abundance conjecture states that if $X$ is a minimal model, then the equality $\kappa(X)=\nu(X)$ holds. It is known to be true for surfaces and $3$-folds. See [KMM] for these terminology and results.

2 Fourier–Mukai transforms

2.1 Fourier–Mukai transforms

Let $X$ and $Y$ be smooth projective varieties. For an object $\mathcal{P}\in D(X\times Y)$, we define an exact functor $\Phi^{\mathcal{P}}$, called the integral functor with kernel $\mathcal{P}$, by

[TABLE]

We also sometimes write $\Phi^{\mathcal{P}}$ as $\Phi^{\mathcal{P}}_{X\to Y}$ to emphasize that it is a functor from $D(X)$ to $D(Y)$.

By the result of Orlov (see [Hu06, Theorem 5.14]), for a fully faithful functor $\Phi\colon D(X)\to D(Y)$, there is an object $\mathcal{P}\in D(X\times Y)$, unique up to isomorphism, such that $\Phi\cong\Phi^{\mathcal{P}}.$ If an integral functor $\Phi^{\mathcal{P}}$ is an equivalence, it is called a Fourier–Mukai transform.

Note that every autoequivalence is given as an integral functor by the Orlov’s result, and hence let us consider standard autoequivalences as examples of Fourier–Mukai transforms. The autoequivalence group $\operatorname{Auteq}D(X)$ always contains the group

[TABLE]

generated by standard autoequivalences, namely the functors of tensoring with line bundles, push forward along automorphisms, and the shift functor $[1]$. Any standard equivalence $\Phi$ are given by the following form;

[TABLE]

for an automorphism $\varphi$, an integer $i$ and a line bundle $\mathcal{L}$. Then, $\Phi$ is the Fourier–Mukai transform with the kernel

[TABLE]

whose support is $\Gamma_{\varphi}$, where $\Gamma_{\varphi}$ is the graph of $\varphi$.

Every Fourier–Mukai transform $\Phi^{\mathcal{P}}$ induces a cohomological Fourier–Mukai transform

[TABLE]

which is an isomorphism of the total cohomologies, and the commutativity

[TABLE]

holds (see [Hu06, §5.2]). Here we put $v(-):=\operatorname{ch}(-)\sqrt{\text{td}(X)}$.

If there exists a Fourier–Mukai transform between $D(X)$ and $D(Y)$, then we call $X$ a Fourier–Mukai partner of $Y$.

2.2 Calabi–Yau objects and spherical objects

Let $X$ be a smooth projective variety. An object $\alpha\in D(X)$ is called a Calabi–Yau object if it satisfies

[TABLE]

For example, a [math]-dimensional sheaf on a smooth projective variety and a line bundle $\mathcal{L}$ on a $(-2)$-curve $C$ on a smooth projective surface are Calabi–Yau objects.

Take a Calabi–Yau object $\alpha$, an autoequivalence $\Phi\in\operatorname{Auteq}D(X)$ and a closed subscheme $D$ of $\operatorname{Supp}(\mathcal{P})$. Then all cohomology sheaves $\mathcal{H}^{i}(\alpha)$ and $\alpha|_{D}$ are Calabi–Yau objects. It is known that the Serre functor $(-)\otimes\omega_{X}[\dim(X)]$ commutes with the equivalence $\Phi$ (cf. [Hu06, Lemma 1.30]), and thus, $\Phi(\alpha)$ is also a Calabi–Yau object.

Next, let us consider a sheaf $\mathcal{F}\in\operatorname{Coh}(X)$ which is a Calabi–Yau object (we call it a Calabi–Yau sheaf). Then, we have

[TABLE]

and hence,

[TABLE]

For a Calabi–Yau object $\alpha\in D(X)$ and an irreducible curve $C$ contained in $\operatorname{Supp}(\alpha)$, every cohomology sheaf $\mathcal{H}^{i}(\alpha|_{C})$ is a Calabi–Yau sheaf. Hence, equality (3) yields

[TABLE]

If there exists a Calabi–Yau object $\alpha$ in $D(X)$ with $\operatorname{Supp}(\alpha)=X$, we can find $i\in\mathbb{Z}$ such that $\operatorname{rank}\mathcal{H}^{i}(\alpha)>0$. Since $\mathcal{H}^{i}(\alpha)$ is also a Calabi–Yau sheaf, equality (3) implies that $c_{1}(\omega_{X})$ is torsion.

Next we introduce an important class of examples of autoequivalences. We say that an object $\alpha\in D(X)$ is spherical if $\alpha$ is a Calabi–Yau object and it satisfies

[TABLE]

For example, a line bundle on a K3 surface $X$ and a line bundle $\mathcal{L}$ on a $(-2)$-curve on a smooth projective surface $X$ are spherical objects in $D(X)$ (see [Ue16, §2.2]).

Put $X=X_{1}=X_{2}$. For a spherical object $\alpha\in D(X)$, we consider the mapping cone

[TABLE]

of the natural evaluation $p_{1}^{*}\alpha^{\vee}\stackrel{{\scriptstyle\mathbb{L}}}{{\otimes}}p_{2}^{*}\alpha\to\mathcal{O}_{\Delta_{X}}$. Then the integral functor $T_{\alpha}:=\Phi^{\mathcal{C}}_{X_{1}\to X_{2}}$ defines an autoequivalence of $D(X)$, called the twist functor along the spherical object $\alpha$ (cf. [Hu06, Proposition 8.6]). By (4), there is a exact triangle

[TABLE]

for $\beta\in D(X)$.

2.3 Fourier–Mukai transforms on elliptic surfaces

Refer [Br98] to the results in this subsection. Let $\pi\colon S\to C$ be a minimal elliptic surface. For an object $E$ of $D(S)$, we define the fiber degree of $E$ as

[TABLE]

where $F$ is a general fiber of $\pi$. Let us denote by $\lambda_{S}$ the highest common factor of the fiber degrees of objects of $D(S)$. Equivalently, $\lambda_{S}$ is the smallest number $d$ such that there exists a holomorphic $d$-section of $\pi$. Consider an integer $b$ coprime to $\lambda_{S}$. There exists a smooth, $2$-dimensional component $J_{S/C}(b)=J_{S}(b)$ of the moduli space of pure one-dimensional stable sheaves on $S$, the general point of which represents a rank $1$, degree $b$ stable vector bundle supported on a smooth fiber of $\pi$. There is a natural morphism $J_{S}(b)\to C$, taking a point representing a sheaf supported on the fiber $\pi^{-1}(x)$ of $S$ to the point $x$. This morphism is a minimal elliptic fibration. Obviously, $J_{S}(0)\cong J(S)$, the Jacobian surface associated to $S$, and $J_{S}(1)\cong S$.

There exists a universal sheaf $\mathcal{U}$ on $J_{S}(b)\times_{C}S$ such that the resulting functor $\Phi^{\mathcal{U}}_{J_{S}(b)\to S}$ is an equivalence.

Let us set $Z$ the union of all $(-2)$-curves on $S$. Define

[TABLE]

and denote the congruence subgroup of $\operatorname{SL}(2,\mathbb{Z})$ by

[TABLE]

for $m\in\mathbb{Z}$. Then, we pose the following conjecture.

Conjecture 2.1**.**

Suppose that a smooth projective surface $S$ has a minimal elliptic fibration $\pi\colon S\to C$ and $K_{S}\not\equiv 0$. Then, we have a short exact sequence

[TABLE]

Here $\Theta$ is induced by the action of $\operatorname{Auteq}D(S)$ on the even degree part $H^{0}(F,\mathbb{Z})\oplus H^{2}(F,\mathbb{Z})\cong\mathbb{Z}^{2}$ of the integral cohomology group of a smooth fiber $F$.

Suppose that each reducible fiber of $\pi$ is non-multiple, and forms a cycle of $(-2)$-curves, i.e. it is of type $\mathrm{I}_{n}$ for some $n>1$. Then Conjecture 2.1 is shown to be true in [Ue16]. See also [Ue17].

3 Support of the kernel of Fourier–Mukai transforms

In this section, we consider the support of the kernel of Fourier–Mukai transforms. Many results and ideas are due to Kawamata [Ka02], but for easy reference, we often refer Huybrecht’s book [Hu06].

Let $X$ and $Y$ be smooth projective varieties, and suppose that

[TABLE]

is a Fourier–Mukai transform. In this case, we have $\dim(X)=\dim(Y)$ (cf. [Hu06, Corollary 5.21]), and the quasi-inverse of $\Phi$ is given by $\Phi^{\mathcal{Q}}$, where $\mathcal{Q}=\mathcal{P}^{\vee}\otimes p_{X}^{*}\omega_{X}$. It is known (cf. [Hu06, Lemma 3.32]) that

[TABLE]

Let us denote by $\Gamma$ the support of $\mathcal{P}$. For $x\in X$, $\Gamma_{x}$ denotes the fiber over $x$ by $p_{X}|_{\Gamma}$. Notice that

[TABLE]

(see [Hu06, Lemma 3.29]), which implies that $\Gamma_{x}=\operatorname{Supp}(\Phi(\mathcal{O}_{x}))$ as sets. Furthermore, we note the following lemma.

Lemma 3.1**.**

1. (i)

There exists an irreducible component of $\Gamma$ which dominates $X$ by $p_{X}$, and a similar statement holds for $p_{Y}$. 2. (ii)

$\operatorname{Supp}(\Phi(\mathcal{O}_{x}))$* is connected for any $x\in X$.* 3. (iii)

If $\dim(\Phi(\mathcal{O}_{x}))=\dim(X)$ holds, then $K_{X}\equiv 0$ and $K_{Y}\equiv 0$. 4. (iv)

Let $W$ be an irreducible component of $\Gamma$, and $\nu\colon\tilde{W}\to W$ be the normalization. Then $\nu^{*}p_{X}^{*}\omega_{X}^{\otimes m}\cong\nu^{*}p_{Y}^{*}\omega_{Y}^{\otimes m}$ for some $m>0$.

Proof.

(i) See [Hu06, Lemma 6.4] for the first statement. Apply the first statement for the quasi-inverse $\Phi^{\mathcal{Q}}_{Y\to X}$ and use (6). Then, we can show the second.

(ii) See [Hu06, Lemma 6.11].

(iii) If the equality holds, $\operatorname{Supp}(\Phi(\mathcal{O}_{x}))=Y$. Since $\Phi(\mathcal{O}_{x})$ is a Calabi–Yau object, $c_{1}(Y)$ is torsion as in §2.2. Then the result follows by Remark 3.4 (ii).

(iv) See [Hu06, Lemma 6.9].∎

Let us define

[TABLE]

the set of irreducible components $W_{0}$ of $\Gamma=\operatorname{Supp}(\mathcal{P})$ which dominates $X$ by $p_{X}$. Note that $\operatorname{Comp}(\Phi^{\mathcal{P}}_{X\to Y})\neq\emptyset$ by Lemma 3.1 (i).

Lemma 3.2**.**

Take an irreducible component $W$ of $\Gamma$.

1. (i)

We see $\dim(W)\leq\dim(p_{X}(W))+\dim(p_{Y}(W))$. If furthermore $\dim(W)=\dim(p_{X}(W))+\dim(Y)$ holds, then $K_{X}\equiv 0$. 2. (ii)

If $\dim(W)=\dim(X)$ and $W\in\operatorname{Comp}(\Phi^{\mathcal{P}}_{X\to Y})$, then $W$ is the unique irreducible component dominating $X$ by $p_{X}$. Furthermore, it also dominates $Y$ by $p_{Y}$. 3. (iii)

If $\dim(W)=2\dim(X)$, then $W=X\times Y$.

Proof.

(i) The first result follows from the fact $W\subset p_{X}(W)\times p_{Y}(W)$. For the second, denote by $W_{x}$ the fiber of $p_{X}|_{W}\colon W\to p_{X}(W)$ over a point $x\in p_{X}(W)$. Then, $\dim(Y)\leq\dim(W_{x})\leq\dim(\Phi(\mathcal{O}_{x}))$, and hence $\operatorname{Supp}(\Phi(\mathcal{O}_{x}))=Y$. Then Lemma 3.1 (iii) completes the proof.

(ii) Note that if $\dim(\Phi(\mathcal{O}_{x}))=0$ for $x\in X$, there is a point $y\in Y$ and an integer $n$ such that $\Phi(\mathcal{O}_{x})\cong\mathcal{O}_{y}[n]$ by [Hu06, Lemma 4.5]. We also notice that there are no other components dominating $X$, since $\operatorname{Supp}(\Phi(\mathcal{O}_{x}))$ is connected by Lemma 3.1 (ii). Hence, for general points $x\neq x^{\prime}\in X$, we have $\dim(\Phi(\mathcal{O}_{x}))=\dim(\Phi(\mathcal{O}_{x^{\prime}}))=0$, and $\mathop{\mathrm{Hom}}\nolimits^{i}(\Phi(\mathcal{O}_{x}),\Phi(\mathcal{O}_{x^{\prime}}))=0$ for all $i$. Then, $\operatorname{Supp}(\Phi(\mathcal{O}_{x}))\cap\operatorname{Supp}(\Phi(\mathcal{O}_{x^{\prime}}))=\emptyset$. In particular, $W$ also dominates $Y$ by $p_{Y}$.

(iii) This is obvious, since $\dim(X)=\dim(Y)$ and $W\subset X\times Y$ by definition. ∎

The equation (6) implies that Lemma 3.2 (i) and (ii) still hold after replacing $p_{X}$ with $p_{Y}$.

For an irreducible closed subvariety $V$ of $X$, we set

[TABLE]

Lemma 3.3**.**

Let $V$ be an irreducible closed subvariety of $X$, and take $W_{0}\in\operatorname{Comp}(\Phi^{\mathcal{P}}_{X\to Y})$.

1. (i)

Suppose that

[TABLE]

holds. Then, we have

[TABLE] 2. (ii)

Suppose that $K_{X}|_{V}$ is big. Then, the inequality (9) holds. Assume furthermore that $K_{X}|_{V}$ is nef and the equality in (9) holds. Then, $K_{X}$ is nef. 3. (iii)

Suppose that $-K_{X}|_{V}$ is big. Then, the inequality (9) holds. Assume furthermore that $-K_{X}|_{V}$ is nef and the equality in (9) holds. Then, $-K_{X}$ is nef.

Proof.

(i) Set $W_{0V}:=p_{X}^{-1}(V)\cap W_{0}(\subset X\times Y)$. Then, we have

[TABLE]

If the projection $p_{Y}$ contracts a curve $C^{\prime}$ on $W_{0V}$, then $p_{X}(C^{\prime})$ is a curve on $V$. Denote the normalization $\widetilde{W}_{0}\to W_{0}$ by $\nu_{0}$, and take an irreducible curve $\widetilde{C^{\prime}}$ on $\widetilde{W}_{0}$ with $\nu_{0}(\widetilde{C^{\prime}})=C^{\prime}$. Then

[TABLE]

holds by Lemma 3.1 (iv). Hence, condition (8) implies that $p_{Y}|_{W_{0V}}$ is generically finite on the image, and hence $\dim(W_{0V})\leq\dim(Y)$. The result follows from the equality $\dim(X)=\dim(Y)$ and inequality (10).

(ii) Take the normalization $\mu\colon\widetilde{V}\to V$. Since $\mu^{*}(K_{X}|_{V})$ is also big, Kodaira’s lemma yields that $\mu^{*}(K_{X}|_{V})$ is $\mathbb{Q}$-linearly equivalent to $A+B$, where $A$ is an ample $\mathbb{Q}$-divisor and $B$ is an effective $\mathbb{Q}$-divisor on $\widetilde{V}$. Then, we have

[TABLE]

and hence, $p_{Y}|_{W_{0V}}$ is generically finite on the image as in the proof of (i) and inequality (9) holds by (i). If equality in (9) holds, then inequality (10) implies $\dim(Y)\leq\dim(W_{0V})$, and thus $\dim(Y)=\dim(W_{0V})$. Hence, it turns out that $p_{Y}|_{W_{0V}}$ is surjective. Since the linear equivalence

[TABLE]

holds for some $m>0$ by Lemma 3.1 (iv), $K_{Y}$ is nef by the assumption that $K_{X}|_{V}$ is nef . Hence, $K_{X}$ is nef (see Remark 3.4(ii)). The statement (iii) can be proved in a similar way. ∎

Remark 3.4**.**

1. (i)

If $K_{X}$ is big, i.e. $X$ is of general type, then Lemma 3.3 (ii) for $V=X$ yields $\dim(W_{0})=\dim(X)$. In a similar way, if $-K_{X}$ is big, then $\dim(W_{0})=\dim(X)$ holds by Lemma 3.3 (iii). These are actually shown by Kawamata in the proof of [Ka02, Theorem 2.3 (2)]. 2. (ii)

If $K_{X}$ is nef and $Y$ is a Fourier–Mukai partner of $X$, then $K_{Y}$ is nef and $\nu(X)=\nu(Y)$ holds. A similar statement is true for anticanonical divisors $-K_{X}$ and $-K_{Y}$. See [Ka02, Theorem 2.3] and [Hu06, Propositions 6.15, 6.18] for the proof. 3. (iii)

Let $\{\varphi_{i}\}$ be the set of all extremal contractions on $X$. Define $V$ to be a fiber of maximal dimension among all fibers of all $\varphi_{i}$. Then $-K_{X}|_{V}$ is ample, and hence, Lemma 3.3 (iii) implies that inequality (9) holds.

Lemma 3.5**.**

Let $D$ be a nef Cartier divisor and $H$ be a very ample divisor on a normal projective variety $X$. Set $V=\bigcap_{i=1}^{\dim(X)-\nu(D)}H_{i}$ for general members $H_{i}\in|H|$. Then

[TABLE]

Proof.

It follows from the definition on the numerical Iitaka-Kodaira dimension that $D|_{V}$ is a nef and big divisor on $V$. Then by Kodaira’s lemma, $D|_{V}$ is $\mathbb{Q}$-linearly equivalent to $A+B$, where $A$ is an ample $\mathbb{Q}$-divisor and $B$ is an effective $\mathbb{Q}$-divisor on $V$. Hence, $\overline{\bigcup_{D\cdot C=0,C\subset V}C}\subset B$, and then the result follows. ∎

Proposition 3.6**.**

Fix $W_{0}\in\operatorname{Comp}(\Phi^{\mathcal{P}}_{X\to Y})$.

1. (i)

Assume that $K_{X}$ is nef. Then, we have

[TABLE] 2. (ii)

Assume that $-K_{X}$ is nef. Then, we have

[TABLE] 3. (iii)

Assume that $\kappa(X)\geq 0$. Suppose that the minimal model conjecture and the abundance conjecture hold. Then, we have

[TABLE]

Proof.

(i) and (ii) are direct consequences of Lemmas 3.3 (i) and 3.5.

(iii) We may assume $\kappa(X)>0$, since otherwise the statement is obvious. Run the minimal model program for $X$. Then, we obtain a birational map $\phi\colon X\dashrightarrow X_{m}$, where $X_{m}$ is a minimal model. Take a common resolution $f\colon X^{\prime}\to X$ and $g\colon X^{\prime}\to X_{m}$. Then [Ka02, Lemma 4.4] states that there is an integer $n>0$ such that $D:=n(f^{*}K_{X}-g^{*}K_{X_{m}})$ is effective.

Let $H$ be a very ample divisor on $X_{m}$. Set $V_{m}:=\bigcap_{i=1}^{\dim(X)-\kappa(X)}H_{i}$ for general members $H_{i}\in|H|$. Then, take its strict transform on $X$ and denote it by $V$. We see that $V$ satisfies the condition (8). Indeed, assume

[TABLE]

for a contradiction. Let us set

[TABLE]

where $U$ is the open subset of $X$ on which $\phi$ is an isomorphism. Then we know

[TABLE]

by (12) and the choice of $V_{m}$. Take irreducible curves $C\in\widetilde{\mathcal{C}_{V}}$ and $C^{\prime}$ on $X^{\prime}$ satisfying $f(C^{\prime})=C$. Then, we have

[TABLE]

On the other hand, we have $g^{*}K_{X_{m}}\cdot C^{\prime}\geq 0$ and $D\cdot C^{\prime}\geq 0$, since $K_{X_{m}}$ is nef, and $f(\operatorname{Supp}(D))\cap U=\emptyset$. Therefore, we have $g^{*}K_{X_{m}}\cdot C^{\prime}=D\cdot C^{\prime}=0$, and thus

[TABLE]

In particular, we know that

[TABLE]

and hence

[TABLE]

Here, the first inclusion follows from (13). This produces a contradiction to

[TABLE]

obtained by Lemma 3.5 and $\nu(X_{m})=\kappa(X_{m})=\kappa(X)$. Therefore, the result follows from Lemma 3.3 (i). ∎

Corollary 3.7**.**

Take $W_{0}\in\operatorname{Comp}(\Phi^{\mathcal{P}}_{X\to Y})$.

1. (i)

If $\dim(W_{0})=2\dim(X)$, then we have $K_{X}\equiv 0$. 2. (ii)

If $\dim(W_{0})=2\dim(X)-1$ and $K_{X}\not\equiv 0$, then either $K_{X}$ is nef and $\nu(X)=1$ or else $-K_{X}$ is nef and $\nu(-K_{X})=1$.

Proof.

(i) We can see $\operatorname{Supp}(\Phi^{\mathcal{P}}_{X\to Y}(\mathcal{O}_{x}))=Y$ by Lemma 3.2 (iii). Then Lemma 3.1 (iii) implies the statement.

(ii) Lemma 3.3 (ii) and (iii) yield that $K_{X}$ or $-K_{X}$ is nef. Thus, Proposition 3.6 (i) and (ii) imply the result. ∎

Remark 3.8**.**

Suppose that $\kappa(X)=0$ or $1$. Then Corollary 3.7 implies that equality in (11) cannot be attained unless $X$ is a minimal model. On the other hand, in the case $\kappa(X)\geq 2$, equality in (11) may possibly hold for a non-minimal model $X$ as follows.

Set $X:=S\times\widehat{E}$ and $Y:=S\times E$ for an elliptic curve $E$ and a smooth projective surface $S$ of general type. Then it satisfies $\kappa(X)=2$. Consider a Poincaré bundle $\mathcal{P}_{E}$ on $\widehat{E}\times E$. If $h\colon\Delta_{S}\times\widehat{E}\times E\to\widehat{E}\times E$ denotes the projection, then $h^{*}\mathcal{P}_{E}\in D(X\times Y)$ gives rise to a Fourier–Mukai transform between $D(X)$ and $D(Y)$ (see [Hu06, Exercise 5.20]). Assume furthermore that $S$ is not minimal. Then $X$ is not minimal, but equality in (11) holds.

4 Fourier–Mukai support dimensions

Let $X$ and $Y$ be smooth projective varieties, and consider a Fourier–Mukai transform

[TABLE]

We give names to some special types of Fourier–Mukai transforms:

Definition 4.1**.**

A Fourier–Mukai transform $\Phi^{\mathcal{P}}_{X\to Y}$ is said to be $K$-equivalent type, if there is an element $W_{0}\in\operatorname{Comp}(\Phi^{\mathcal{P}}_{X\to Y})$ such that $\dim(W_{0})=\dim(X)$. Similarly, it is said to be Calabi–Yau type, if there is an element $W_{0}\in\operatorname{Comp}(\Phi^{\mathcal{P}}_{X\to Y})$ such that $\dim(W_{0})=2\dim(X)$.

Note that in both cases, it turns out that the set $\operatorname{Comp}(\Phi^{\mathcal{P}}_{X\to Y})$ consists of the unique element by Lemma 3.2 (ii) and (iii).

Example 4.2**.**

1. (i)

Any standard autoequivalences are $K$-equivalent type by the description of the kernel object in (1). 2. (ii)

Let $\alpha$ be a spherical object in $D(X)$. Note that $\mathop{\mathbb{R}\mathrm{Hom}}\nolimits_{D(X)}(\alpha,\mathcal{O}_{x})=0$ if and only if $x\not\in\operatorname{Supp}(\alpha)$ by [BM02, Lemma 4.2]. Then, by the triangle (5) for $\beta=\mathcal{O}_{x}$, we see that

[TABLE]

for $x\in\operatorname{Supp}(\alpha)$, and

[TABLE]

for $x\not\in\operatorname{Supp}(\alpha)$. Consequently, equations (7) imply

[TABLE]

where $\mathcal{C}\in D(X\times X)$ is the kernel object of $T_{\alpha}$, given in (4).

Let $C$ be a $(-2)$-curve on a smooth projective surface $X$. Then, the twist functor $T_{\mathcal{O}_{C}}$ is $K$-equivalent type. On the other hand, the twist functor $T_{\mathcal{O}_{X}}$ along the structure sheaf $\mathcal{O}_{X}$ on a K3 surface $X$ is Calabi–Yau type.

Remark 4.3**.**

1. (i)

Let $\Phi^{\mathcal{P}}_{X\to Y}$ be a Fourier–Mukai transform of $K$-equivalent type, and take the unique element $W_{0}\in\operatorname{Comp}(\Phi^{\mathcal{P}}_{X\to Y})$. Then, Kawamata shows in the proof of [Ka02, Theorem 2.3 (2)] that $p_{X}|_{W_{0}}$ and $p_{Y}|_{W_{0}}$ are birational morphisms, and that $W_{0}$ is the graph of the birational map $(p_{Y}|_{W_{0}})\circ(p_{X}|_{W_{0}})^{-1}$ between $X$ and $Y$. Moreover, if we take a resolution of singularities $f\colon Z\to W_{0}$, then the linear equivalence $f^{*}(p_{X}|_{W_{0}})^{*}(K_{X})\sim f^{*}(p_{Y}|_{W_{0}})^{*}(K_{Y})$ holds (use Lemma 3.1 (iv), and we can take $m=1$ on the resolution $Z$. See [Hu06, Proposition 6.19]). In other words, varieties $X$ and $Y$ are K-equivalent. 2. (ii)

For a given Fourier–Mukai transform $\Phi^{\mathcal{P}}_{X\to Y}$, if $\dim(\Phi^{\mathcal{P}}_{X\to Y}(\mathcal{O}_{x}))=0$ for a point $x\in X$, then $\Phi^{\mathcal{P}}_{X\to Y}(\mathcal{O}_{x})=\mathcal{O}_{y}[i]$ for some point $y\in Y$ and $i\in\mathbb{Z}$ by [Hu06, Lemma 4.5]. Moreover, a Fourier–Mukai transform $\Phi^{\mathcal{P}}_{X\to Y}$ is a $K$-equivalent type if and only if $\dim(\Phi^{\mathcal{P}}_{X\to Y}(\mathcal{O}_{x}))=0$ holds for a general point $x\in X$. Consequently, we can see that a composition of Fourier–Mukai transforms of $K$-equivalent type is again $K$-equivalent type.

Next let us consider the case $X=Y$, i.e. $\Phi=\Phi^{\mathcal{P}}\in\operatorname{Auteq}D(X)$. Then we define

[TABLE]

and call it by the Fourier–Mukai support dimension of $X$. Obviously, we have

[TABLE]

Let us consider two extreme cases below; the case $N_{X}=\dim(X)$ and the case $N_{X}=2\dim(X)$.

4.1 $K$-equivalent type

Define

[TABLE]

Then, Remark 4.3(ii) tells us that $\operatorname{Auteq}_{\}}D(X)$ is a subgroup of $\operatorname{Auteq}D(X)$. It is easy to see by definition and Remark 4.3(i) that the following conditions are equivalent;

• •

$\operatorname{Auteq}_{\operatorname{K-equiv}}D(X)=\operatorname{Auteq}D(X)$.

• •

$N_{X}=\dim(X)$.

• •

For any $\Phi^{\mathcal{P}}\in\operatorname{Auteq}D(X)$, the set $\operatorname{Comp}(\Phi^{\mathcal{P}}{})$ consists of the unique element $W_{0}(\subset X\times X)$, which is the graph of a birational automorphism of $X$.

If one, and hence, all of these conditions are satisfied, the autoequivalence group $\operatorname{Auteq}D(X)$ (or, simply $X$) is said to be $K$-equivalent type.

Proposition 4.4** (Kawamata).**

Let $X$ a smooth projective variety with $\pm K_{X}$ big. Then $\operatorname{Auteq}D(X)$ is $K$-equivalent type.

Proof.

It follows from Remark 3.4 (i). ∎

4.2 Calabi–Yau type

By definition, the following conditions are equivalent;

• •

There is an autoequivalence $\Phi$ of $D(X)$ such that $\Phi$ is Calabi–Yau type.

• •

$N_{X}=2\dim(X)$.

If one, and hence, two of these conditions are satisfied, the autoequivalence group $\operatorname{Auteq}D(X)$ (or, simply $X$) is said to be Calabi–Yau type. Note that Corollary 3.7 yields $K_{X}\equiv 0$ in this case. It is natural to ask whether the converse is true or not.

Problem 4.5**.**

Suppose that $K_{X}\equiv 0$. Then, is $\operatorname{Auteq}D(X)$ Calabi–Yau type?

We give an affirmative answer to Problem 4.5 for abelian varieties in Proposition 4.6, for curves $X$ in Theorem 4.7, and for surfaces $X$ in Theorem 5.3.

Proposition 4.6**.**

Let $X$ be an abelian variety. Then, $\operatorname{Auteq}D(X)$ is Calabi–Yau type.

Proof.

Let us put $d:=\dim(X)$, and consider the normalizes Poincaré bundle $\mathcal{P}$ on $X\times\widehat{X}$. Then, the integral functor $\Phi^{\mathcal{P}}_{X\to\widehat{X}}$ is an equivalence (cf. [Hu06, Proposition 9.19]), and the cohomological Fourier–Mukai transform $\Phi^{\mathcal{P},H}_{X\to\widehat{X}}$ induces an isomorphism between the total cohomologies $H^{*}(X,\mathbb{Q})$ and $H^{*}(\widehat{X},\mathbb{Q})$, which restricts an isomorphism $H^{n}(X,\mathbb{Q})$ and $H^{2d-n}(\widehat{X},\mathbb{Q})$ for any $n$. The last isomorphism coincides with

[TABLE]

where $\text{PD}_{n}$ is Poincaré duality (see [Hu06, Lemma 9.23]). Here note that there is a natural isomorphism between $H^{2d-n}(\widehat{X},\mathbb{Q})$ and $H^{2d-n}(X,\mathbb{Q})^{*}$ by the construction of the dual abelian variety $\widehat{X}$.

For an ample line bundle $\mathcal{L}$ on $\widehat{X}$, consider the Fourier–Mukai transform

[TABLE]

with a kernel object $\mathcal{Q}\in D(X\times X)$. The cohomological Fourier–Mukai transform induced by the autoequivalence $(-)\otimes\mathcal{L}$ of $D(\widehat{X})$ is just multiplying by $\operatorname{ch}(\mathcal{L})$. For a point $x\in X$, we have $\operatorname{ch}(\mathcal{O}_{x})=(0,\ldots,0,1)\in H^{2*}(X,\mathbb{Q})$, and hence (14) yields

[TABLE]

Therefore, the [math]-th cohomology component of $\Phi^{\mathcal{Q},H}(\operatorname{ch}(\mathcal{O}_{x}))$ is $\operatorname{ch}_{d}(\mathcal{L})=\frac{1}{d!}c_{1}(\mathcal{L})^{d}$, which is not [math]. This means $\operatorname{Supp}(\Phi^{\mathcal{Q}}(\mathcal{O}_{x}))=X$, and hence $\operatorname{Supp}\mathcal{Q}=X\times X$ (see the equations (7)). In particular, we obtain $N_{X}=2\dim(X)$. ∎

Now, we can show a dichotomy of the autoequivalence groups of smooth projective curves.

Theorem 4.7** (Dichotomy).**

Let $C$ be a smooth projective curve with the genus $g(C)$, and $N_{C}\in\{1,2\}$ be its Fourier–Mukai support dimension.

1. (i)

$N_{C}=2$* (Calabi–Yau type) if and only if $g(C)=1$, namely $C$ is an elliptic curve.* 2. (ii)

$N_{C}=1$* ($K$-equivalent type) if and only if $g(C)\neq 1$, namely $C$ is a projective line or a curve of general type.*

Proof.

If $C$ is not an elliptic curve, then $\pm K_{C}$ is ample. Hence, Proposition 4.4 tells us that $N_{C}=1$. Since elliptic curves are $1$-dimensional abelian varieties, Proposition 4.6 completes the proof. ∎

Theorem 4.7 shows that Fourier–Mukai support dimensions of smooth projective curves reflect their geometry. We obtain a similar result for smooth projective surfaces in Theorem 5.3.

5 Trichotomy of autoequivalence groups on smooth projective surfaces

In this section we show a similar result in the $2$-dimensional case to Theorem 4.7.

Lemma 5.1**.**

Let $S$ be smooth projective surface, and take $\Phi^{\mathcal{P}}\in\operatorname{Auteq}D(S)$ and $W_{0}\in\operatorname{Comp}(\mathcal{P})$. Then we have $\dim(\mathcal{P})=\dim(W_{0})$. In particular,

[TABLE]

holds.

Proof.

Set $\Gamma:=\operatorname{Supp}(\mathcal{P})$. Note that $2\leq\dim(W_{0})\leq\dim(\Gamma)\leq 4$. Obviously, $\dim(\Gamma)=4$ is equivalent to $\dim(W_{0})=4$.

Suppose that $\dim(W_{0})=2$. Then Lemma 3.2 (ii) implies that $W_{0}$ is the unique irreducible component of $\Gamma$ dominating $S$ by $p_{1}$, and is also the unique irreducible component dominating $S$ by $p_{2}$. Hence, Lemma 3.2 (i) forces that there are no $3$-dimensional irreducible components. In particular, $\dim(\Gamma)=2$. To the contrary, if $\dim(\Gamma)=2$, then $\dim(W_{0})=2$ follows. This completes the proof. ∎

Remark 5.2**.**

Let $X$ be a Calabi–Yau $3$-fold, i.e. it satisfies $\omega_{X}\cong\mathcal{O}_{X}$ and $H^{1}(X,\mathcal{O}_{X})=0$, and suppose that $X$ contains $E\cong\mathbb{P}^{2}$. Note that the normal bundle $\mathcal{N}_{E/X}$ is isomorphic to $\mathcal{O}_{\mathbb{P}^{2}}(-3)$. Then, we can see that $\mathcal{O}_{E}$ is a spherical object of $D(X)$ (We leave the proof of this fact to readers. Use [Hu06, Proposition 11.8] and the Local-to-Global Ext spectral sequence). The kernel of the twist functor $T_{\mathcal{O}_{E}}$ has two irreducible components. One is supported on the diagonal $\Delta_{X}$ in $X\times X$ and the other one on $E\times E$ (see Example 4.2 (ii)). Hence, Lemma 5.1 is false for higher dimensional varieties.

Now, we are in a position to show a trichotomy of autoequivalence groups on smooth projective surfaces.

Theorem 5.3** (Trichotomy).**

Let $S$ be a smooth projective surface and $N_{S}\in\{2,3,4\}$ be the Fourier–Mukai support dimension of $S$.

1. (i)

$N_{S}=4$* (Calabi–Yau type) if and only if $K_{S}\equiv 0$.* 2. (ii)

$N_{S}=3$* if and only if $S$ has a minimal elliptic fibration and $K_{S}\not\equiv 0$.* 3. (iii)

$N_{S}=2$* ($K$-equivalent type) if and only if $S$ has no minimal elliptic fibration and $K_{S}\not\equiv 0$.*

Proof.

(i) For each surface $S$ with $K_{S}\equiv 0$, let us give an example of autoequivalence whose kernel object has $4$-dimensional support.

First, take a K3 surface $S$ and let $\mathcal{P}$ be the ideal sheaf $I_{\Delta_{S}}$ of the diagonal $\Delta_{S}$ in $S\times S$. For $x\in S$, the integral functor $\Phi^{\mathcal{P}}$ satisfies $\Phi^{\mathcal{P}}(\mathcal{O}_{x})=I_{x}$, the ideal sheaf of the point $x$, and then [BM01, Corollary 2.8] implies that $\Phi^{\mathcal{P}}$ is an autoequivalence.

For an abelian surface $S$, we have already shown $N_{S}=4$ in Proposition 4.6.

Take an Enriques surface $T$. Then there is a K3 surface $S$ with an involution $\iota$ on $S$ such that $T$ is the quotient of $S$ by $\left<\iota\right>\cong\mathbb{Z}/2\mathbb{Z}$. Then it turns out that the autoequivalence $\Phi^{\mathcal{P}}$ given above for a K3 surface $S$ descends to an autoequivalence of $D(T)$, and its kernel has a $4$-dimensional support. See [BM98, Example 5.2] for details.

For a bielliptic surface $S$, there are elliptic curves $E_{1},E_{2}$ and a finite group $G$ acting diagonally on $E_{1}\times E_{2}$ such that $S=(E_{1}\times E_{2})/G$. Therefore, $S$ has two minimal elliptic fibrations $\pi_{i}\colon S\to E_{i}/G$. Take a universal sheaf $\mathcal{U}_{i}$ on $J_{S/(E_{i}/G)}(1)\times_{E_{i}/G}S$ ($i=1,2$) given in §2.3. Fix an isomorphism between $J_{S/(E_{i}/G)}(1)$ and $S$, and regard $\Phi_{i}:=\Phi_{J_{S/(E_{i}/G)}(1)\to S}^{\mathcal{U}_{i}}$ as an autoequivalence of $D(S)$. Then the kernel of the composition $\Phi_{1}\circ\Phi_{2}$ is $4$-dimensional.

Conversely, it follows from Corollary 3.7 (i) that the equality $N_{S}=4$ implies the equality $K_{S}\equiv 0$.

(ii) First note that the equality $N_{S}=3$ implies that either $K_{S}$ is nef and $\nu(S)=1$, or that $-K_{S}$ is nef and $\nu(-K_{S})=1$ by Corollary 3.7 and Lemma 5.1. Moreover, if $K_{S}$ is nef, it is known that $S$ has a minimal elliptic fibration (cf. [Be96, Proposition IX.2]). Therefore, we consider the only case $-K_{S}$ is nef and $\nu(-K_{S})=1$. Note that in this case, there is a smooth rational curve $C$ on $S$ with $K_{S}\cdot C<0$.

Take an autoequivalence $\Phi=\Phi^{\mathcal{P}}$ of $D(S)$ with $\dim(\mathcal{P})=3$. Then Lemma 5.1 implies that there is a $3$-dimensional irreducible component $W_{0}$ of $\operatorname{Supp}(\mathcal{P})$ dominating $S$ by $p_{1}$. Let us denote by $W_{0x}(\subset\{x\}\times S)$ the fiber of the morphism $p_{1}|_{W_{0}}\colon W_{0}\to S$ over a point $x\in S$, and regard it as a divisor on $S$ by the isomorphism $\{x\}\times S\cong S$. If $\dim W_{0x}=2$ for some $x$, Lemma 3.1 (iii) supplies a contradiction to $K_{S}\not\equiv 0$. Hence, every fiber of $p_{1}|_{W_{0}}$ is $1$-dimensional, and therefore $p_{1}|_{W_{0x}}$ is flat (cf. [Ha77, Exercise III.10.9]). Take points $x,y\in C$. Since $C$ is isomorphic to $\mathbb{P}^{1}$, the points $x$ and $y$ are rationally equivalent [math]-cycles on $S$. Hence, the divisors $W_{0x}$ and $W_{0y}$ on $S$ are linearly equivalent (see [Fu98, Theorems 1.1.4, 1.1.7]).

If $\bigcap_{x\in C}W_{0x}\neq\emptyset$, then we see $C\subset\operatorname{Supp}(\Phi^{-1}(\mathcal{O}_{z}))$ for a point $z\in\bigcap_{x\in C}W_{0x}$. This contradicts $K_{S}\cdot C\neq 0$, since $\Phi^{-1}(\mathcal{O}_{z})$ is a Calabi–Yau object. Hence, we conclude $\bigcap_{x\in C}W_{0x}=\emptyset$, and therefore the complete linear system $\mathfrak{d}:=|W_{0x}|$ is base point free. Moreover, note that $K_{S}\cdot W_{0x}=0$ for each $x\in C$, since $W_{0x}$ is contained in $\operatorname{Supp}(\Phi(\mathcal{O}_{x}))$. Furthermore, the Hodge index theorem implies $W_{0x}\cdot W_{0x}=0$. Then we can see that $\mathfrak{d}$ defines a minimal elliptic fibration, after taking the Stein factorization if necessary.

Conversely, if $S$ has a minimal elliptic fibration, take a universal sheaf $\mathcal{U}$ on $J_{S}(1)\times S$. Then $\Phi^{\mathcal{U}}_{J_{S}(1)\to S}$ is a Fourier–Mukai transform. Since $J_{S}(1)\cong S$ and $\dim(\mathcal{U})=3$, we obtain $N_{S}=3$.

(iii) The result follows from (i) and (ii). ∎

For a smooth projective curve $C$, every Fourier–Mukai partner of $C$ is isomorphic to $C$. Therefore, it is obvious that Fourier–Mukai support dimension is a derived invariant for smooth projective curves. In the surface case, a similar result holds.

Corollary 5.4**.**

Fourier–Mukai support dimension is a derived invariant for smooth projective surfaces, i.e. if smooth projective surfaces $S$ and $T$ are Fourier–Mukai partners, then $N_{S}=N_{T}$.

Proof.

Let $T$ be a Fourier–Mukai partner of a smooth projective surface $S$. Suppose that $T$ is not isomorphic to $S$. Then $S$ is either a K3 surface, an abelian surface or a minimal elliptic surface, and moreover, $T$ is also a surface of the same type as $S$ ([BM01, Ka02]). Therefore, Theorem 5.3 implies the conclusion. ∎

Conjecture 5.5**.**

1. (i)

Fourier–Mukai support dimension is a derived invariant for smooth projective varieties, i.e. if smooth projective varieties $X$ and $Y$ are Fourier–Mukai partners, then $N_{X}=N_{Y}$. 2. (ii)

Let $X$ and $Y$ be a smooth projective varieties. Assume that $X$ is of $K$-equivalent type and that there is a Fourier–Mukai transform $\Phi^{\mathcal{P}}_{X\to Y}$. Then, $\Phi^{\mathcal{P}}_{X\to Y}$ is $K$-equivalent type.

Remark 5.6**.**

1. (i)

Conjecture 5.5 (ii) implies Conjecture 5.5 (i) for smooth projective varieties of $K$-equivalent type. In fact, suppose that Conjecture 5.5 (ii) is true. Then $Y$ in Conjecture 5.5 (ii) is also $K$-equivalent type, since a composition of Fourier–Mukai transforms of $K$-equivalent type is $K$-equivalent type. In particular, Conjecture 5.5 (i) is true for $X$ of $K$-equivalent type. 2. (ii)

Kawamata predicts in [Ka02, Conjecture 1.2] that birationally equivalent, derived equivalent smooth projective varieties are $K$-equivalent, but a counterexample to his conjecture is discovered by the author in [Ue04]. Conjecture 5.5 (ii) is a special version of Kawamata’s conjecture, since $X$ and $Y$ are $K$-equivalent by Remark 4.3 (i) when a Fourier–Mukai transform $\Phi^{\mathcal{P}}_{X\to Y}$ is $K$-equivalent type.

6 Autoequivalence groups of $K$-equivalent type

Let $S$ be smooth projective surface and take $\Phi=\Phi^{\mathcal{P}}\in\operatorname{Auteq}_{\operatorname{K-equiv}}D(S)$. Set

[TABLE]

Then $\dim\Gamma=2$ by Lemma 5.1, and Lemma 3.2 (ii) yields that there is the unique component $W_{0}$ of $\Gamma$ dominating $S$ by both of $p_{1}$ and $p_{2}$. Note that $\Gamma_{x}$ is at most $1$-dimensional for $x\in S$.

Let us denote by $Z$ the union of $(-2)$-curves on $S$. The set $Z$ has finitely many connected components, since the Picard number $\rho(S)$ is finite. But it can possibly have infinitely many irreducible components. If a K3 surface $S$ contains $(-2)$-curves, and $S$ admits the infinite automorphism group, then the set $Z$ on $S$ is an example of such.

We first show Proposition 6.4 below. We need several claims to prove it. Take a $(-2)$-curve $C$ on $S$ and $\mathcal{L}\in\mathop{\mathrm{Pic}}\nolimits(C)$. We regard $\mathcal{L}$ as an object of $D(S)$ in a natural way.

Claim 6.1**.**

We have $\dim(\Phi(\mathcal{L}))=1$. Moreover, every cohomology sheaf $\mathcal{H}^{i}(\Phi(\mathcal{L}))$ is rigid and pure $1$-dimensional.

Proof.

Note that $\operatorname{Supp}(\Phi(\mathcal{L}))$ is contained in $p_{2}(p_{1}^{-1}(C)\cap\Gamma)$. Then, we see $\dim(\Phi(\mathcal{L}))\leq 1$, since $W_{0}$ is the unique component dominating $S$ by $p_{2}$. Since $\mathcal{L}$ is rigid on $S$, but $\mathcal{O}_{x}$ is not rigid for any $x\in S$, we have $\dim(\Phi(\mathcal{L}))>0$ by [Hu06, Lemma 4.5]. Moreover, [IU05, Proposition 3.5] implies that $\mathcal{H}^{i}(\Phi(\mathcal{L}))$ is rigid, and it is pure $1$-dimensional by [IU05, Lemma 3.9]. ∎

Claim 6.2**.**

We have $\operatorname{Supp}(\Phi(\mathcal{L}))\subset Z$.

Proof.

Take an irreducible component $D$ of $\operatorname{Supp}(\Phi(\mathcal{L}))$. Then, we see $D\cdot K_{S}=0$, since $\Phi(\mathcal{L})$ is a Calabi–Yau object.

Take an integer $i$ such that $\operatorname{Supp}(\mathcal{H}^{i}(\Phi(\mathcal{L})))$ contains the irreducible curve $D$. Let us set

[TABLE]

and

[TABLE]

where the closed subset $E$ does not contain $D$. Then consider the short exact sequence

[TABLE]

in $\operatorname{Coh}(S)$, where $\mathcal{H}^{0}_{E}(\mathcal{M})$ is the subsheaf with supports in $E$ (cf. [Ha77, Exercise II.1.20]).

Note that $\operatorname{Supp}(\mathcal{K})=D$ and hence $\dim(\mathcal{H}^{0}_{E}(\mathcal{M}))\cap D\leq 0$. Assume for a contradiction that $\mathcal{K}$ is not pure $1$-dimensional. Then, there is a local section $s$ of $\mathcal{K}$ such that $s(x)\neq 0$ for some point $x\in S$, but $s(y)=0$ for all point $y\in S$ except $x$. Let $t$ be a local section of $\mathcal{M}$ which is a lift of $s$. If $x\not\in E$, then $\phi$ is an isomorphism around the point $x$, and hence $t$ generates a [math]-dimensional subsheaf of $\mathcal{M}$, which contradicts Claim 6.1. Suppose that $x\in E$. Then, $t$ gives a local section of $\mathcal{H}^{0}_{E}(\mathcal{M})$, and hence $s=\phi(t)$ should be [math], which also gives a contradiction. Therefore we can conclude that $\mathcal{K}$ is pure $1$-dimensional. Thus, Serre duality yields

[TABLE]

and then, [KO95, Lemma 2.2 (2)] implies that $\mathcal{K}$ is rigid. Therefore, we have

[TABLE]

Consequently, we have $D^{2}<0$ and hence, $D$ is a $(-2)$-curve. ∎

Claim 6.3**.**

$\operatorname{Supp}(\Phi(\mathcal{O}_{x}))$* is contained in $Z$ for any point $x\in Z$.*

Proof.

Take a $(-2)$-curve $C$ containing $x$. Then, there is an exact triangle

[TABLE]

which implies

[TABLE]

This completes the proof by Claim 6.2. ∎

Proposition 6.4**.**

Let $S$ be a smooth projective surface. Then, there is a group homomorphism

[TABLE]

Proof.

Claim 6.3 and [Ue16, Lemma 2.4] complete the proof. ∎

We define the group $\operatorname{Br}_{Z}(S)$ generated by twist functors along spherical objects supported in $Z$;

[TABLE]

The following is crucial to show Theorems 6.6 and 6.8.

Lemma 6.5**.**

Let $S$ be a smooth projective surface.

1. (i)

For a point $x\in S$ and $\Phi\in\operatorname{Auteq}D(S)$, suppose that $\operatorname{Supp}(\Phi(\mathcal{O}_{x}))$ is at most $1$-dimensional, and contains no $(-2)$-curves. Then, $\Phi(\mathcal{O}_{x})$ is a shift of a sheaf. 2. (ii)

Suppose that $\Phi\in\operatorname{Auteq}D_{Z}(S)$ preserves the cohomology class $\operatorname{ch}(\mathcal{O}_{x})\in H^{4}(S,\mathbb{Q})$ for all points $x\in Z$. Then, there is an autoequivalence $\Psi\in\operatorname{Br}_{Z}(S)$, an integer $i$ and a point $y\in Z$ satisfying $\Psi\circ\Phi(\mathcal{O}_{x})\cong\mathcal{O}_{y}[i]$.

Proof.

(i) Recall that every $1$-dimensional component $C$ of $\operatorname{Supp}(\mathcal{H}^{i}(\Phi(\mathcal{O}_{x})))$ satisfies $K_{S}\cdot C=0$ and is not a $(-2)$-curve, and then we obtain $C^{2}\geq 0$. Thus the Riemann–Roch theorem yields

[TABLE]

(see [Ue16, §2.2]). In particular,

[TABLE]

holds. Now the statement follows from [BM01, Lemma 2.9].

(ii) Notice that the proof of [IU05, Key proposition] works in our situation, and that the assumption $\Phi^{H}$ preserves the class $\operatorname{ch}(\mathcal{O}_{x})$ is needed in the proof of [IU05, Condition 7.5] (see also [Ue16, footnote in pp. 572]). Then the result follows. ∎

Let us define

[TABLE]

Theorem 6.6**.**

Let $S$ be a smooth projective surface. Then, there is a short exact sequence

[TABLE]

Proof.

Take $\Phi=\Phi^{\mathcal{P}}\in\operatorname{Auteq}_{\operatorname{K-equiv}}D(S)$ and a point $x\in S\backslash Z$. Then, we know by Claim 6.3 that $\operatorname{Supp}(\Phi(\mathcal{O}_{x}))\cap Z=\emptyset$ as in [Ue16, Corollary 3.5]. Moreover Lemma 6.5 (i) implies that $\Phi(\mathcal{O}_{x})$ is a shift of a sheaf. Hence, [Br99, Lemma 4.3] implies that $\mathcal{P}|_{p_{1}^{-1}(S\backslash Z)}$ is a shift of a sheaf, flat over $S\backslash Z$ by $p_{1}$. Consequently, we see $\dim(\Phi(\mathcal{O}_{x}))=0$. Furthermore, assume $\Phi\in\ker\iota_{Z}$. Then, we can say $\dim(\Phi(\mathcal{O}_{x}))=0$ for all point $x\in S$, and thus it follows from [Ue16, Lemma 2.2] that

[TABLE]

for $\mathcal{L}\in\mathop{\mathrm{Pic}}\nolimits_{Z}(S)$, $\phi\in\operatorname{Aut}_{Z}(S)$. Therefore, we obtain the result. ∎

Conjecture 6.7**.**

Let $S$ be a smooth projective surface. Then

[TABLE]

Consequently,

[TABLE]

Theorem 6.8**.**

Let $S$ be a smooth projective surface. Then Conjecture 6.7 holds true, if $Z$ is a disjoint union of configurations of $(-2)$-curves of type $A$.

Proof.

Note that $\Phi\in\operatorname{Auteq}^{\dagger}_{\operatorname{K-equiv}}D_{Z}(S)$ preserves the class $\operatorname{ch}(\mathcal{O}_{x})$ for $x\in Z$. Then for any point $x\in Z$, Lemma 6.5 (ii) assures that there is an autoequivalence $\Psi\in\operatorname{Br}_{Z}(S)$, an integer $i$ and a point $y\in Z$ satisfying $\Psi\circ\Phi(\mathcal{O}_{x})\cong\mathcal{O}_{y}[i]$. Now the result follows from [Hu06, Corollary 5.23]. ∎

Recall that if $\operatorname{Auteq}D(S)$ is $K$-equivalent type, then $\operatorname{Auteq}D(S)=\operatorname{Auteq}_{\operatorname{K-equiv}}D(S)$. Therefore, Theorem 6.8 implies Theorem 1.3.