DERIVED EQUIVALENCE OF K3 SURFACES

YIRUI XIONG

1. Introduction

K3 surface S is a smooth projective surface with trivial canonical bundle (Calabi-Yau 1 2-fold) and H (S, OS) = 0. The most important property is that the lattice and Hodge structures on its second cohomology group determines completely the geometry:

Theorem 1.1 (Global Torelli theorem, see [9] chapter 7.5). Two K3 surfaces S1 and S2 are 2 ∼ 2 isomorphic if and only if there is a Hodge isometry H (S1, Z) = H (S2, Z) (for definition of Hodge isometry, see definition 3.1)

The result is originally due to Pijateckii-Sapiro and Safarevich in the algebraic case [15], and to Burns and Rapoport in the analytic case [5]. More recently, after the works of Mukai [12] and Orlov[13], the bounded derived categories of coherent sheaves of K3 surfaces have been extensively studied. In particular, there is a homological version of global Torelli theorem:

Theorem 1.2. Let X and Y be K3 surfaces, then the following statements are equivalent:

(1) X and Y are derived equivalent: there exists a triangle equivalence Φ:Db(X) → Db(Y ); 2 2 2 (2) there is a Hodge isometry φ : He (X, Z) → He (Y, Z), where He (X, Z) is the extended Hodge lattice introduced by Mukai; (3) Y is isomorphic to a fine and 2-dimensional moduli space of stable sheaves on X.

The main goal of this note is to explain the above theorem. The basic tool is Fourier- Mukai transform, which is extremely useful for studying derived equivalence between smooth projective varieties. We will review the basic definitions and properties of Fourier-Mukai functors in section 2. Then after reviewing basic facts on moduli space of stable sheaves on K3 surface, we will complete the proof. Throughout this note, we are working on smooth projective varieties X. And we denote the bounded derived category of coherent sheaves on X by Db(X) := Db(CohX). We denote b the Grothendieck group of D (X) by K0(X), which is a free abelian group generated by isomorphism classes in Db(X), and subject to relations [C] = [A] + [B] if A, B, C fit into distinguished triangle:

A / B / C / A[1] 1 2 YIRUI XIONG

2. Fourier-Mukai transforms

2.1. Orlov’s result. Let X and Y be smooth projective varieties, we denote the product of X and Y over C by X × Y , and the maps: X × Y

πX πY

{ XY# The Fourier-Mukai transforms give a canonical way to construct derived functors between Db(X) and Db(Y ). And Orlov’s result tells us all derived functors can be constructed in that way.

Definition 2.1 (Fourier-Mukai transform). Let P ∈ Db(X × Y ), then the associated Fourer- Mukai transform is the functor: X→Y b b ΦP :D (X) −→ D (Y ) • ∗ • F 7−→ (πY )∗(πX F ⊗ P). P is called Fourier-Mukai kernel.

Example 2.2. Let O∆ be the structure sheaf of ∆ ⊂ X × X, then we have b b ΦO∆ = id :D (X) → D (X)

Example 2.3. Let f : X → Y be a projective morphism. Then we denote Γf ⊂ X × Y the ∼ X→Y b b ∗ ∼ Y →X b b graph of f, we have Rf∗ = ΦO :D (X) → D (Y ) and Lf = ΦO :D (Y ) → D (X). Γf Γf We verify the first equality and the second is similar:

Note that if we denote map i : X → X × Y , x 7→ (x, f(x)), then OΓf = i∗OX , and we have commutative diagrams: i X / X × Y

πX id #  X and i X / X × Y

πY f  # Y Thus we have X→Y • ∗ • ΦO (F ) = (πY )∗(πX F ⊗ OΓf ) Γf ∗ • = (πY )∗(πX F ⊗ i∗OX ) ∗ ∗ • (projection formula) = (πY )∗i∗(i πX F ⊗ OX ) • = (πy ◦ i)∗(F ) • = Rf∗(F ).

One of the important features of Fourier-Mukai transforms is it contains natural left and right adjoints. DERIVED EQUIVALENCE OF K3 SURFACES 3

Definition 2.4. For any object P ∈ Db(X × Y ), let

∨ ∗ ∨ ∗ PL := P ⊗ πY ωY [dimY ], PR := P ⊗ πX ωX [dimX]

∨ where P := RHom(P, OX×Y ).

X→Y b b Proposition 2.5 ([11]). We keep notations as above, then let F := ΦP :D (X) → D (Y ), then

G := ΦY →X :Db(Y ) −→ Db(X) PL H := ΦY →X :Db(Y ) −→ Db(X) PR are left, resp., right adjoints to F . In particular, when F is an equivalence, then F −1 =∼ G =∼ H.

Proposition 2.6 (Composition, [11]). Let P ∈ Db(X × Y ), Q ∈ Db(Y × Z), then

X→Z ∼ Y →Z X→Y ΦR = ΦQ ◦ ΦP ,

∗ ∗ where R := (πXZ )∗(πXY P ⊗ πYZ Q).

The following theorem enables us to study derived equivalences between projective varieties using Fourier-Mukai transforms:

Theorem 2.7 ([13][14]). Let X and Y be two smooth projective varieties, and let

F :Db(X) −→ Db(Y ) be equivalence, then there exists P ∈ Db(X × Y ) unique up to isomorphism, such that

∼ X→Y F = ΦP .

Remark 2.8. In a slightly more general form, we only need to assume to be fully faithful with left and right adjoints. However, such theorem is false without fully faithful assumptions. Recently, [16] found an explicit example where the above theorem failed.

Proposition 2.9. Let X and Y be derived equivalence, then dim X = dim Y and ωX , ωY ⊗k ∼ have same orders. Here the order is the smallest positive integer k such that ωX = OX .

b b X→Y b Proof. By Orlov’s result, we know F :D (X) → D (Y ) must be in form ΦP :D (X) → Db(Y ) and its inverse is given by ΦY →X and ΦY →X (proposition 2.5). Again, by using the PL PR uniqueness of Fourier-Mukai kernel we have ∼ PL = PR.

And the above equality is equivalent to the following:

∨ ∨ ∗ ∗ P 'P ⊗ πX ωX ⊗ πY ωY [dim(X) − dim(Y )] in Db(X × Y ). Since we work in the bounded derived category, the above shift must be zero, otherwise repeat the isomorphism, we will get P∨ =∼ 0 in Db(X × Y ), which is absurd. For the second result, we need the following general lemma: 4 YIRUI XIONG

0 Lemma 2.10 ([1]). Let T and T be C-linear categories of finite-type with Serre functors ST 0 and ST 0 , then any linear equivalence F : T → T commutes with Serre functors, we have ∼ F ◦ ST = ST 0 ◦ F.

k ∼ k ∼ Now we continue the proof, suppose ωX = OX , then SX [−kn] = id and

−1 k ∼ k ∼ F ◦ SY [−kn] ◦ F = SX [−kn] = id.

k ∼ k ∼ Thus SY [−kn] = id, therefore ωY = OY . 

2.2. Cohomological Fourier-Mukai transforms. In this subsection we pass Fourier-Mukai transforms between derived categories to cohomology groups. Firstly there exists a natural map

b [−]:D (X) −→ K0(X) X X F • 7−→ [F •] = (−1)i[F i] = (−1)i[Hi(F •)].

Given projective morphism f : X → Y , then we have a natural map

∗ f : K0(Y ) −→ K0(X) X hF i 7−→ (−1)i[Lf i(F )].

And we introduce

f! : K0(X) −→ K0(Y ) X i i hMi 7−→ (−1) [R f∗(M)].

Proposition 2.11. Given projective morphism f : X → Y , then we have the following commutative diagrams:

f ∗ Db(Y ) / Db(X)

[−] [−]   K0(Y ) / K0(X) f ∗ and

f∗ Db(X) / Db(Y )

[−] [−]   K0(X) / K0(Y ). f!

Now let e ∈ K0(X × Y ), then the K-theoretic Fourier-Mukai transform is defined to be

K ∗ Φe : K0(X) → K0(Y ), [M] 7→ [(πY )!(e ⊗ πX (M)]

Due to above compatibilities, we have DERIVED EQUIVALENCE OF K3 SURFACES 5

Proposition 2.12. Keep the notations as above, we have the following commutative diagram

ΦP Db(X) / Db(Y )

[−] [−]   K0(X) / K0(Y ) Φ[P]

• Now we turn to rational cohomology groups H (X, Q), given morphism f : X → Y , we have natural map ∗ • • f :H (Y, Q) −→ H (X, Q) and by using Poincare duality, we have map

i i+2 dim(Y )−2 dim(X) f∗ :H (X, Q) −→ H (Y, Q).

• Definition 2.13. For any α ∈ H (X × Y, Q), the cohomological Fourier-Mukai transform is defined to be

H • • Φα :H (X, Q) −→ H (Y,Q) ∗ β 7−→ (πY )∗ (α ∪ πX (β)) .

The natural way to pass from K-groups to cohomology groups is using Chern character • ch(−): K0(X) → H (X, Q). However, in general we do not have the following commutative diagram: K ΦP K0(X) / K0(Y )

ch(−) ch(−)   H•(X, ) / H•(Y, ) Q H Q Φch(P) To make the diagram commutative, we need to use Grothendieck-Riemann-Roch theorem:

Theorem 2.14 ([7]). Let f : X → Y be projective morphisms between smooth projective varieties, then we have commutative diagram:

f! K0(X) / K0(Y )

ch(−).tdX ch(−).tdY

•  •  H (X, Q) / H (Y, Q) f∗ where tdX is the Todd class of tangent sheaf on X. By taking Y = {pt}, we will obtain Hirzebruch-Riemann-Roch theorem.

This enables us to give

• b Definition 2.15 (Mukai vector). Let e ∈ K0(X) or F ∈ D (X), then the Mukai vector is defined as following:

p • • v(e) := ch(e). tdX , resp. v(F ) := v([F ]).

Now we have 6 YIRUI XIONG

Lemma 2.16. Keep the same notations, we have commutative diagram:

K Φe K0(X) / K0(Y )

v(−) v(−)   H•(X, ) / H•(Y, ) Q H Q Φv(e)

We can get similar composition formula as in proposition 2.6 (this is left to reader). Then we have

Proposition 2.17. Suppose P ∈ Db(X × Y ) defines an equivalence

b b ΦP :D (X) −→ D (Y )

H L i L j then Φv(P) : H (X, Q) → H (Y, Q) is a bijection.

H However, we remark that Φv(P) does not preserve grading in general (even if ΦP is an equivalence). • Now we consider the Hodge structure on H (X, Q). The Hodge decomposition

n M p,q H (X, C) = H (X) p+q=n where Hp,q(X) =∼ Hq(X, Ωp). By Lefschtez (1, 1)-theorem, all characteristic classes are of type (p, p) in Hp,p(X), thus

M p,p 2p • v(−): K0(X) −→ H (X) ∩ H (X, Q) ⊂ H (X, Q).

We thus have

b b H • Proposition 2.18. If ΦP :D (X) → D (Y ) defines an equivalence, then Φv(P) :H (X, Q) → • H (Y, Q) yields isomorphism M M Hp,q(X) =∼ Hp,q(Y ). (2.1) p−q=i p−q=i Corollary 2.19. Let X be a K3 surface, Y is a smooth projective variety which is derived equivalent to X, then Y is also K3 surface.

∼ Proof. By proposition 3.1 we know dim Y = 2 and ωY = OY . Thus Y is K3 or abelian 1 surface. Now we prove H (Y, OY ) = 0. Set i = −1, then by proposition 2.18 we have

(h0,1 + h1,2)(X) = (h0,1 + h1,2)(Y ), by Hodge theory and Serre duality, we know h0,1(X) = h1,2(X), h0,1(Y ) = h1,2(Y ), thus 0,1 0,1 h (X) = h (Y ) = 0. We have shown that Y is K3 surface. 

3. Derived equivalence of K3 surfaces

In this section we fix X K3 surface. DERIVED EQUIVALENCE OF K3 SURFACES 7

• 3.1. Extended Hodge lattice. Recall the Mukai pairing defined on H (X, Z):

h(α0, α1, α2), (β0, β1, β2)i := α1.β1 − α0.β1 − α2.β0 ∈ Z 2i where αi, βi ∈ H (X, Z). Now we have Hirzebruch-Riemann-Roch theorem in form X (E,F ) = −hv(E), v(F )i, for E,F ∈ Coh(X).

• Definition 3.1 (Mukai). The extended Hodge lattice on H (X, Z) is denoted by H(e X, Z), where

1,1 0 1,1 4 He (X) := H (X, Z) ⊕ H (X) ⊕ H (X, Z), 2,0 He (X) := H2,0(X), 0,2 He (X) := H0,2(X). 1,1 Under this definition, the Mukai vector v(E) ∈ He (X), for any coherent sheaf E. We say Φ : H(e X, Z) → H(e Y, Z) are Hodge isometry if Φ preserves Mukai pairing and Hodge decomposition. And since such decomposition is orthogonal with respect to the paring, thus 2,0 2,0 Hodge isometry is equivalent to say Φ preserves Mukai pairing and map He (X) to He (Y ).

b b Proposition 3.2. Suppose ΦP :D (X) → D (Y ) is derived equivalence of K3 surfaces for b H • • some P ∈ D (X × Y ), then Φv(P) :H (X, Z) → H (Y, Z) preserves Mukai pairing.

Proof. This follows from general result for arbitrary projective variety, see [6]. 

Now we need to show Φv(P) preserves Hodge decomposition. We need the following im- portant observation, which is by Mukai.

Lemma 3.3 ([12]). Let X, Y be K3 surfaces, then for any E ∈ Db(X × Y ), we have v(E) ∈ • H (X × Y, Z).

Proof. By definition we have

v(E) = ch(E).ptd(X × Y ) ∗ p ∗ p = ch(E).πX tdX .πY tdY ∗ ∗ = ch(E).πX (1, 0, 1).πY (1, 0, 1)

1 2

where we denote ch(E) = rk(E)E, c1(E), 2 c1(E) − 2c2(E) , ch3(E), ch4(E) . Clearly p td(X × Y ), rk(E), c1(E) are integral. And the pairing on K3 surface is even ([]), thus ∗ ∗ for c1(E) = πX α + πY β, we have 2 ∗ 2 ∗ ∗ ∗ 2 c1(E) = πX α + 2πX α.πY β + πY β is divided by 2. Now we need to check ch3(E) and ch4(E) are integral. Note that by Kunneth formula,

6 2 4 M 4 2 ch3(E) ∈ H (X × Y, Q) = H (X) ⊗ H (Y ) H (X) ⊗ H (Y ) 8 4 4 ch4(E) ∈ H (X × Y, Q) = H (X) ⊗ H (Y ). 8 YIRUI XIONG

We have Grothendieck-Riemann-Roch theorem for πX :

∗ (πX )∗(ch(E).πY tdY ) = ch ((πX )!E) , (3.1)

•,4 • where the right side is integral since it lives on K3 surface. We look at e ∈ H (X, Q) ⊗ 4 ∗ H (Y, Q) in ch(E).πY tdY , then by definition of (πX )∗, we have •,4 • • (πX )∗(e ) = eX ∈ H (X, Q), • •,4 • 4 then by equality (3.1), we know eX is integral, thus e = eX ⊗ ω, where ω ∈ H (Y ) is the fundamental cocycle. Thus e•,4 is integral. Interchanging X and Y we can get e4,• ∈ 4 • 4,2 2,4 4,4 H (X) ⊗ H (Y ) is also integral. Thus ch3(E) = e + e , ch4(E) = e are both integral form. 

b b Corollary 3.4. Suppose ΦP :D (X) → D (Y ) are derived equivalence, then Φv(P) : H(e X, Z) → H(e Y, Z) is Hodge isometry.

3.2. Moduli space of stable sheaves on K3 surface, and finish of proof. We first recall the definition of moduli space of semi-stable sheaves on X.

1,1 Definition 3.5. Fix a polarization H on X, and Mukai vector v ∈ He (X), then moduli functor is defined to be

H MX (v): Schemes −→ Sets S 7−→ {flat families E → X × S

such that E|X⊗{s} is H-semi-stable with Mukai vector v / ' H ∼ H
where ' is S- equivalence [10]. If MX (v) = Hom −,MX (v) as functors for some scheme H H MX (v). Then MX (v) is called fine moduli space. And the sheaf correspond to id ∈ H H
Hom MX (v),MX (v) in above equivalence is called universal sheaf.

H We list some general results for MX (v) on K3 surface: Proposition 3.6 ([12]). (1) Suppose that v = (r, l, s) satisfies gcd(r, hl.Hi, s) = 1, then M H (v) is a fine moduli space for moduli functor and M H (v) only parametrizes stable vector bundles. 0 1,1 0 (2) Suppose there exists a vector v ∈ He (X, Z) suh that hv, v i = 1, then there exists an ample line bundle H such that gcd(r, hl.Hi, s) = 1.

In particular, when v is primitive, that is, there isn’t an integer λ 6= ±1, such that v = λv0, then there exists a vector v0 such that hv, v0i = 1 (since Mukai paring is positive-definite). Thus we summarize the result as following:

Corollary 3.7. Suppose v is primitive, then there exists H ample line bundle such that M H (v) is a fine moduli space, in particular, M H (v) only parametrizes stable bundles.

From now on we always assume Mukai vector v is primitive. By general theory of moduli space, we know the tangent space at a point [E] ∈ M H (v) is H ∼ 1 T[E]M (v) = Ext (E,E). DERIVED EQUIVALENCE OF K3 SURFACES 9

The dimension could be computed via Riemann-Roch theorem: since

X (E,E) = −hv, vi, and dim Hom(E,E) = 1 since stable objects are simple, and dim Hom(E,E) = dim Ext2(E,E) by Serre duality. Thus we have

dim Ext1(E,E) = 2 + hv, vi

The dimension of tangent space is independent of the points in M H (v), thus we have

Proposition 3.8. M H (v) is smooth projective variety of dimension 2 + hv, vi.

We need the following criterion for equivalence of Fourier-Mukai transforms.

Lemma 3.9 ([2][4]). Let X,Y be smooth projective varieties of dimension n, and P ∈ b b b D (X × Y ). Denote the corresponding Fourier-Mukai transform: ΦP :D (X) → D (Y ). Then Φ is an equivalence if and only if it satisfies following conditions: for any closed point x ∈ X

(1) we have

HomDb(Y ) (Φ(Ox), Φ(Ox)) = C

where COx denotes the skyscrapper sheaf at x; (2) we have ∼ Φ(Ox) ⊗ ωY = Φ(Ox),

(3) for two different points x1 6= x2, we have

i HomDb(Y ) (Φ(Ox1 ), Φ(Ox2 )) = 0, for 0 ≤ i ≤ n.

Now we return back to K3 surface case, and we assume v is primitive and hv, vi = 0. Then M H (v) is 2 dimensional, and we denote the universal sheaf by P.

Corollary 3.10. Let X be K3 surface, we have

b H b ΦP :D (MX (v)) → D (X) as equivalence.

H Proof. Now for each point y ∈ M (v), Py is stable sheaf and Py ⊗ ωM H (v) = Py. We verify the conditions as in above lemma:

(1) For any closed point y ∈ M H (v), we have (a)

HomDb(X) (Φ(Oy), Φ(Oy))

= HomDb(X)(Py, Py) = C, (b) the second condition is obvious by K3 property, 10 YIRUI XIONG

(c) for y1 6= y2, then Py1 6= Py2 , and they have same Hilbert polynomial (fol-

lows from flatness), thus HomDb(X) (Py1 , Py2 ) = 0. By Serre duality we have 2 1 Ext (Py1 , Py2 ) = 0. For Ext (Py1 , Py2 ), we use again Riemann-Roch theorem

X (Py1 , Py2 ) = −hv, vi = 0, 1 thus Ext (Py1 , Py2 ) = 0.



By proposition 2.9, since K3 is preserved under derived equivalence, thus we have

H Corollary 3.11. MX (v) is again K3 surface.

Finish of proof. 1) ⇒ 2) is contained in corollary 3.4. We now show (2) ⇒ (3). Given Hodge isometry φ : H(e Y, Z) → H(e X, Z), we put v = φ(0, 0, 1), then hv, vi = φh(0, 0, 1), (0, 0, 1)i = 0. By proper automorphism of H(e X, Z), we may assume v is primitive. Now we consider M H (v) on X, then we have ψ :Db(M H (v)) −→ Db(X) as derived equivalence by corollary 3.10. And again by proof of (a) ⇒ (b) we have Hodge isometry

0 H φ : H(e M (v), Z) −→ H(e X, Z) (0, 0, 1) 7−→ v. Now consider the composition of Hodge isometries,

φ φ0−1 H H(e Y, Z) / H(e X, Z) / H(e M (v), Z) which sends (0, 0, 1) to (0, 0, 1). This can be restricted to Hodge isometry between ordinary Hodge decompositions: 2 2 H H (Y, Z) → H (M (v), Z). By global Torelli theorem 1.1, we know Y =∼ M H (v). The proof for (c) ⇒ (a) follows from corollary 3.10. 

3.3. Example: derived equivalence for Elliptic K3 surface [3].

Definition 3.12. An elliptic K3 surface S is a K3 surface together with a flat morphism π : X → P1 such that for general fibres π−1(t) is a smooth projective. curve of genus 1.

−1 Denote algebraic equivalent class of π (t) by D0. Let

λX = min{λ |D divisor such that D.D0 = λ}

Now consider moduli space Y = MX (0, rD0, d) where r, d ∈ Z such that (1) d is coprime to r;

(2) d is coprime to λX .

Then we can have v = (0, rD0, d) primitive, thus Y is a fine moduli space. Thus there exists a universal sheaf P ∈ Db(X × Y ). We have DERIVED EQUIVALENCE OF K3 SURFACES 11

b Proposition 3.13 ([3] ). Keep the notations as above, we have ΦP :D (MX (0, rD0, d)) → Db(X) as an equivalence.

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