LECTURE 6 References 1. the Grothendieck Group Let a Be An

DERIVED CATEGORIES: LECTURE 6

EVGENY SHINDER

References [AO] V. Alexeev, D. Orlov, Derived categories of Burniat surfaces and exceptional collections, arXiv:1208.4348v2 [BBS] Christian B¨ohning,Hans-Christian Graf von Bothmer, Pawel Sosna, On the derived category of the classical Godeaux surface, arXiv:1206.1830v1 [BBKS] Christian B¨ohning,Hans-Christian Graf von Bothmer, Ludmil Katzarkov, Pawel Sosna: Deter- minantal Barlow surfaces and phantom categories, arXiv:1210.0343 [GO] Sergey Gorchinskiy, Dmitri Orlov: Geometric Phantom Categories, arXiv:1209.6183 [GS] S.Galkin, E.Shinder, Exceptional collections of line bundles on the Beauville surface, arXiv:1210.3339 [math.AG]

1. The Grothendieck group

Let A be an abelian category. The Grothendieck group K0(A) is deﬁned as an abelian group with generators [A] for each isomorphism class A ∈ A and relations of the form [A] = [A0] + [A00] for each short exact sequence 0 → A0 → A → A00 → 0 in A. Analogously, if C is a triangulated category, then the Grothendieck group K0(C) is deﬁned as an abelian group with generators [A] for each isomorphism class A ∈ C and relations of the form [A] = [A0] + [A00] for each distinguished triangle A0 → A → A00 → A0[1]. Note that it follows that [A[1]] = −[A]. Lemma 1.1. For an abelian category A the natural morphism b K0(A) → K0(D (A)) is an isomorphism.

b Proof. Let φ : K0(A) → K0(D (A)) be the morphism which assigns [A] to [A[0]]. We need to construct the inverse to φ. b Let ψ : Iso(D (A)) → K0(A) be the morphism deﬁned as X ψ(A•) := (−1)p[Hp(A•)]. p For a distinguished triangle A• → B• → C• → A•[1] 1 we get a long exact sequence · · · → Hp(A•) → Hp(B•) → Hp(C•) → Hp+1(A•) → ..., thus • • • ψ(B ) = ψ(A ) + ψ(C ) ∈ K0(A). This shows that ψ descends to a well-deﬁned homomorphism b ψ : K0(D (A)) → K0(A). By construction we have ψ(φ([A])) = [A] for A ∈ A, so that φ is injective. Filtering an object by its terms we also see that φ is surjective. Hence φ is an isomorphism and φ−1 = ψ.

Deﬁnition 1.2. Let X be a variety. The Grothendieck group K0(X) is deﬁned as b K0(X) := K0(Coh(X)) ' K0(D (X)). Proposition 1.3. If C = hA, Bi is a semi-orthogonal decomposition of triangulated categories, then there is a direct sum decomposition

K0(C) = K0(A) ⊕ K0(B). Proof. A triangulated functor between triangulated categories induces a morphism on Grothendieck groups. Thus the embeddings A ⊂ C, B ⊂ C give a canonical morphism K0(A) ⊕ K0(B) → K0(C) and its inverse is given by the sum of the morphisms induced by the two projections C → A, C → B.

K0(X) is endowed with the Euler bilinear form X (1.1) χ(F, G) = Extp(F, G). p One uses the pairing (1.1) to deﬁne a numerically exceptional collection as a sequence b of objects E1,...,Er ∈ D (X) such that χ([Ej], [Ei]) = 0 for j > i. Note that if a sequence is exceptional, then it is also numerically exceptional, but not vice-versa.

2. Phantoms and quasi-phantoms b Let A ⊂ D (X) be an admissible subcategory. It has been thought that if HH∗(A) = 0 or K0(A) = 0, then A = 0. Recent constructions [BBS], [AO], [GS], [GO], [BBKS] showed that this is not the case. Following [GO] we give the deﬁnitions:

Deﬁnition 2.1. If A= 6 0, K0(A) = 0, A is called a phantom. If A= 6 0, HH∗(A) = 0 and K0(A) is ﬁnite torsion, then A is called a quasi-phantom.

Lemma 2.2. Let X be a smooth projective variety with an exceptional collection E1,...,Er. b P Write D (X) = hE1,...,Er, Ai. Assume that the sum of all Betti numbers bk(X) is equal to r. Then HH∗(A) = 0. 2 Proof. We have r ∗ r r C = H (X, C) = HH∗(X) = HH∗(pt) ⊕ HH∗(A) = C ⊕ HH∗(A), therefore HH∗(A) = 0. Remark 2.3. The conditions of Lemma 2.2 can be only be satisﬁed if all cohomology classes on X are of type (p, p). Indeed as shown in the proof of the Lemma we have ∗ r H (X, C) = HH∗(pt) , however HH∗(pt) = HH0(pt) = C only can contribute to (p, p) classes. In particular if X is a surface, then pg(X) = q(X) = 0 is required.

3. The Beauville surface and its properties In what follows G is an abelian group 2 G = (Z/5) = Z/5 · e1 ⊕ Z/5 · e2 2 acting on a three dimensional vector space V with induced action on P = P(V ) given by e1 · (X : Y : Z) = (ζ5X : Y : Z)

e2 · (X : Y : Z) = (X : ζ5Y : Z), where ζ5 is the 5-th root of unity. Let C be the plane G-invariant Fermat quintic curve X5 + Y 5 + Z5 = 0.

1 We consider the scheme-theoretic quotient C/G which is isomorphic to P and the quotient map 1 π : C → P 1 of degree 25. Explicitly we may pick coordinates on P such that π is given by the formula π(X : Y : Z) = (X5 : Y 5). 1 One easily checks that there are three ramiﬁcation points on P corresponding to the orbits where G acts non-freely:

j D1 = {(0 : −ζ5 : 1), j = 0 ... 4} j (3.1) D2 = {(−ζ5 : 0 : 1), j = 0 ... 4} j j D3 = {(ζ5 : −ζ5 : 0), j = 0 ... 4}

Stabilizers of the points in Di, i = 1, 2, 3 are equal to

G1 = Z/5 · e1 (3.2) G2 = Z/5 · e2 G3 = Z/5 · (e1 + e2) respectively. 3 Lemma 3.1. The equivariant Picard group P icG(C) splits as a direct sum

G P ic (C) = Gb ⊕ Z ·O(1).

The canonical class is uniquely divisible by 2, and if we write KC (1) for the resulting line bundle, KC (1) and OC (1) differ by torsion, more precisely, we have KC (1) = OC (1)[3, 3]. We introduce the curve C0 which is defined by the same equation X5 + Y 5 + Z5 = 0 as C but has a different G-action. We pick the G-action on C0 to be defined as 2 4 e1 · (X : Y : Z) = (ζ5 X : ζ5 Y : Z) 3 e2 · (X : Y : Z) = (ζ5X : ζ5 Y : Z)

For this action points in divisors Di, i = 1, 2, 3 defined as in (3.1) have stabilizers 0 G1 = Z/5 · (e1 + 2e2) 0 (3.3) G2 = Z/5 · (e1 + 3e2) 0 G3 = Z/5 · (e1 + 4e2) respectively. We let T = C × C0 with the diagonal G-action. Since the stabilizers in (3.2) and (3.3) are distinct, the G-action on T is free. One can check that the corresponding 2 smooth quotient Beauville surface S = T/G is of general type with pg = q = 0,K = 8 (Chapter X, Exercise 4 in [?]). The Noether formula gives b2 = 2. Since pg = q = 0, the exponential exact sequence gives an identification 2 P ic(S) = H (S, Z). Modulo torsion P ic(S) is an indefinite unimodular lattice of rank 2, that is a hyperbolic plane. We introduce G-linearized line bundles O(i, j) and K(i, j) for i, j ∈ Z as follows: ∗ ∗ O(i, j) = p1(O(i)) ⊗ p2(O(j)) ∗ ∗ K(i, j) = p1(K(i)) ⊗ p2(K(j)) = O(i, j)[3i + 3j, 3i + 2j]. Proposition 3.2. 1. The Picard group of S splits as

G P ic(S)(= P ic (T )) = Gb · [O] ⊕ Z · [O(1, 0)] ⊕ Z · [O(0, 1)]. 4 2 2. The Grothendieck group has a decomposition K0(S) = Z ⊕ (Z/5) . 3. The canonical class ωS is equal to K(2, 2) = O(2, 2)[2, 0]. 4. The intersection pairing is given by

(O(i1, j1)(χ1) ·O(i2, j2)(χ2)) = i1j2 + j1i2. 5. The Euler characteristic of a line bundle L = O(i, j)(χ) is equal to (i − 1)(j − 1).

Proof. See [GS], Proposition 2.4 and Lemma 2.7. 4 4. Exceptional and numerically exceptional collections on the Beauville surface Lemma 4.1. A sequence O,L1,L2,L3 of line bundles on S is numerically exceptional if and only if it belongs to one of the following four numerical types (that is each object is allowed to be twisted by an arbitrary character): (Ic) O, O(−1, 0), O(c − 1, −1), O(c − 2, −1), c ∈ Z (IIc) O, O(0, −1), O(−1, c − 1), O(−1, c − 2), c ∈ Z (IIIc) O, O(−1, c), O(−1, c − 1), O(−2, −1), c ∈ Z (IVc) O, O(c, −1), O(c − 1, −1), O(−1, −2), c ∈ Z. Proof. It easily follows from 3.2 (5) that all listed sequences are numerically exceptional. For the reverse implication see [GS], Lemma 3.1. We now investigate which of the numerically exceptional collections of Lemma 4.1 can be lifted to exceptional collections. Here by a lift we mean a lift with respect to the morphism 2 2 Z ⊕ Gb = P ic(S) → P ic(S)/tors = Z , that is a choice of a character χ ∈ Gb. We will need a detailed study of the characters that may appear in the cohomology groups of sheaves on T . For a G-linearized line bundle on T we deﬁne the acyclic set of L as ∗ ∗ A(L) := {χ ∈ Hom(G, C ): χ∈ / [H (T,L)]} By deﬁnition L(χ) is acyclic if and only if −χ ∈ A(L). Since by Proposition 3.2(1), any line bundle on S is isomorphic to some K(i, j)(χ), we see from the next lemma that there are 39 isomorphism classes of acyclic line bundles on S. Lemma 4.2. The only nonempty acyclic sets of line bundles K(i, j) on S are: A(K(1, −2)) = {[0, 0]} A(K(1, −1)) = {[0, 3], [2, 0], [3, 2]} A(K(1, 0)) = {[0, 0], [0, 1], [0, 2], [1, 4], [2, 3], [3, 0], [4, 0]} A(K(1, 1)) = {[0, 0], [1, 2], [2, 1], [2, 2], [3, 3], [3, 4], [4, 3]} A(K(1, 2)) = {[0, 0], [0, 3], [0, 4], [1, 0], [2, 0], [3, 2], [4, 1]} A(K(1, 3)) = {[0, 2], [2, 3], [3, 0]} A(K(1, 4)) = {[0, 0]} A(K(−1, 1)) = {[0, 0]} A(K(0, 1)) = {[0, 0], [3, 3], [3, 4], [4, 3]} A(K(2, 1)) = {[0, 0], [1, 2], [2, 1], [2, 2]} A(K(3, 1)) = {[0, 0]}.

Proof. See [GS], Lemma 3.3. 5 Theorem 4.3. The following list contains all exceptional collections of length 4 consist- ing of line bundles on S (up to a common twist by a line bundle):

(I1) O, K(−1, 0), K(0, −1), K(−1, −1)

(IV1) O, K(1, −1), K(0, −1), K(−1, −2)

(I−1) O, K(−1, 0), K(−2, −1), K(−3, −1) (4.1) (IV−1) O, K(−1, −1), K(−2, −1), K(−1, −2)

(II0 = IV0) O, K(0, −1), K(−1, −1), K(−1, −2)

(I0) O, K(−1, 0), K(−1, −1), K(−2, −1).

Proof. See [GS], Theorem 3.5. Corollary 4.4. The Beauville surface S admits quasi-phantom subcategories A with 2 K0(A) = (Z/5) . Proof. Taking orthogonals to collections in Theorem 4.3 and using Lemma 2.2 we see that HH∗(A) = 0. An argument using additivity of K0 and Proposition 3.2(2) shows that 2 K0(A) = (Z/5) .