Exceptional sequences of maximal length on some surfaces isogenous to a higher product
aa r X i v : . [ m a t h . AG ] M a y EXCEPTIONAL SEQUENCES OF MAXIMAL LENGTH ON SOMESURFACES ISOGENOUS TO A HIGHER PRODUCT
KYOUNG-SEOG LEE
Abstract.
Let S = ( C × D ) /G be a surface isogenous to a higher product of unmixed typewith p g = q = 0 , G = ( Z / or ( Z / . We construct exceptional sequences of linebundles of maximal length and quasiphantom categories on S . Introduction
Derived category of an algebraic variety is an interesting invariant containing much infor-mation about the variety. Algebraic varieties having equivalent derived categories share manygeometric properties [23].One of the most powerful tools to study derived categories is the notion of semiorthogonaldecomposition. A semiorthogonal decomposition divides a derived category into simpler sub-categories and we can study the derived category via these simpler subcategories. One way toget a semiorthogonal decomposition is to construct an exceptional sequence. When we have anexceptional sequence then we get an admissible triangulated subcategory generated by the ex-ceptional sequence and its orthogonal complement which give a semiorthogonal decomposition.There are lots of studies about semiorthogonal decompositions of derived categories of smoothprojective varieties, especially for rational or Fano varieties. Many rational varieties haveexceptional sequences which generate the derived categories of them. Especially every smoothprojective rational surface has an exceptional sequence which generates its derived category[28],[35]. It is known that every toric variety also has an exceptional sequence which generatesits derived category [25]. For a Fano variety, the structure sheaf is an exceptional object andthere exist at least one semiorthogonal decomposition [30].It is expected that the behaviours of derived categories of varieties with nonnegative Kodairadimensions will be very different from those of rational or Fano varieties. For example it isknown that there is no nontrivial semiorthogonal decomposition for curves with genus greaterthan or equal to 1 [34] or varieties having trivial canonical bundles. In particular they do nothave any exceptional object.
Key words and phrases.
Derived category, exceptional sequence, quasiphantom category, surface isogenousto a higher product.
For a surface of general type with p g = q = 0 , the structure sheaf is an exceptionalobject and there is already a semiorthogonal decomposition. If there is another exceptionalobject in the orthogonal complement of the structure sheaf then we can divide the orthogonalcomplement into two smaller pieces. Then it is an interesting question how much we canextend the exceptional sequence in the derived category. It is easy to show that the length ofthe exceptional sequence is bounded by the rank of the Grothendieck group. When the lengthof exceptional sequence is the rank of Grothendieck group we call the exceptional sequence isof maximal length.Recently there are some constructions of exceptional sequences of maximal lengths on sur-faces of general type with p g = q = 0 . In [1], [9], [10], [12], [18], [19], [20], [33] the authorsconstructed exceptional sequences of maximal lengths consisting of line bundles on surfacesof general type with p g = q = 0 . The triangulated subcategories generated by exceptionalsequences are not full in these cases. The categories of orthogonal complements of these excep-tional sequences have vanishing Hochschild homologies and finite Grothendieck groups. Theyare called the quasiphantom categories. It seems that these semiorthogonal decompositionscontain much information about the geometry of surfaces of general type and may provide lotsof unexpected feature of the derived categories of algebraic varieties. For example in [10] theauthors constructed the first counterexample to the nonvanishing conjecture, and in [11] the au-thors gave the first counterexample to geometric Jordan-H¨older property using the constructionof [10] (see also [29]).Let S = ( C × D ) /G be a surface isogenous to a higher product of unmixed type with p g = q = 0 . When G is abelian, Bauer and Catanese proved there are 4 possible groups,( Z / , ( Z / , ( Z / , ( Z / , and described their moduli spaces in [3]. Galkin and Shinderconstructed exceptional sequences of maximal length on the Beauville surface, the ( Z / casein [20]. Motivated by their work we constructed exceptional sequences of maximal length onthe surfaces isogenous to a higher product of unmixed type with p g = q = 0 and G = ( Z / in [33]. Therefore it is a natural question whether the other surfaces isogenous to a higherproduct of unmixed type with p g = q = 0 admit exceptional sequences of maximal lengthwhen G is ( Z / , or ( Z / .In this paper we construct exceptional sequences of line bundles of maximal length on surfacesisogenous to a higher product of unmixed type with p g = q = 0 , G = ( Z / or G = ( Z / . Theorem 0.1.
Let S = ( C × D ) /G be a surface isogenous to a higher product of unmixedtype with p g = q = 0 , G = ( Z / or G = ( Z / . There are exceptional sequences of linebundles of maximal length on S . The orthogonal complements of the admissible subcategories XCEPTIONAL SEQUENCES ON SOME SURFACES ISOGENOUS TO A HIGHER PRODUCT 3 generated by these exceptional sequences in the derived category of S are quasiphantomcategories.The idea of the constructions are as follows. Let S = ( C × D ) /G be a surface isogenous toa higher product of unmixed type with p g = q = 0 . In the previous paper [33], we proved thatone cannot construct any exceptional sequence of maximal length consisting of line bundles on S using G -equivariant line bundles on C and D when G = ( Z / or G = ( Z / . Inthis paper we use G -invariant line bundles instead of G -equivariant line bundles on C and D to construct exceptional sequences of maximal length on S . To be more precise, we showthat there are G -invariant theta characteristics and G -invariant torsion line bundles on C and D whose box products become G -equivariant line bundles on C × D . To do this weexplicitly compute the Schur multipliers of these invariant line bundles in the cocyle level. Weshow that one can produce exceptional sequences of maximal length on S by this way. Weexpect that one can construct exceptional sequences of maximal lengths on surfaces isogenousto a higher product with nonabelian quotient groups in a similar way.We also compute the Hochschild cohomologies of the quasiphantom categories and prove thatfor some exceptional sequences we made the DG algebras of endomorphisms are deformationinvariant. Acknowledgement.
I am grateful to my advisor Young-Hoon Kiem for his invaluable adviceand many suggestions for the first draft of this paper. Without his support and encourage-ment, this work could not have been accomplished. I thank Fabrizio Catanese, Igor Dolgachevfor answering my questions, and helpful conversations. I would like to thank Seoul NationalUniversity for its support during the preparation of this paper.
Notations.
We will work over C . A curve will mean a smooth projective curve. A surface willmean a smooth projective surface. Derived category of a variety will mean the bounded derivedcategory of coherent sheaves on the variety. G denotes a finite group and b G = Hom ( G, C ∗ )denotes the character group of G . Here ∼ denotes linear equivalence of divisors.1. Surfaces isogenous to a higher product
In this section we recall the definition and some basic facts about surfaces isogenous to ahigher product. For details, see [3].
Definition 1.1.
A surface S is called isogenous to a higher product if S = ( C × D ) /G where C , D are curves with genus at least 2 and G is a finite group acting freely on C × D .When G acts via a product action, S is called of unmixed type. KYOUNG-SEOG LEE
Remark 1.2. [3] Let S be a surface isogenous to a higher product of unmixed type. Then S is a surface of general type. When p g = q = 0 , one can prove that K S = 8 , C/G ∼ = D/G ∼ = P and | G | = ( g C − g D −
1) where g C and g D denote the genus of C and D ,respectively.Bauer and Catanese proved that there are four families of surfaces isogenous to a higherproduct of unmixed type with p g = q = 0 , G is abelian. Moreover they computed thedimensions of the families they form in [3]. Theorem 1.3. [3] Let S be a surface isogenous to a higher product ( C × D ) /G of unmixedtype with p g = q = 0 . If G is abelian, then G is one of the following groups :(1) ( Z / , and these surfaces form an irreducible connected component of dimension 5 intheir moduli space;(2) ( Z / , and these surfaces form an irreducible connected component of dimension 4 intheir moduli space;(3) ( Z / , and these surfaces form an irreducible connected component of dimension 2 intheir moduli space;(4) ( Z / , and S is the Beauville surface.Recently Shabalin [38] and Bauer, Catanese and Frapporti [4], [5] have computed the firsthomology groups of these surfaces. Theorem 1.4. [4], [5], [38] Let S be a surface isogenous to a higher product ( C × D ) /G of unmixed type with p g = q = 0 , G is an abelian group. Then we have the followingisomorphisms :(1) H ( S, Z ) ∼ = ( Z / ⊕ ( Z / for G = ( Z / ;(2) H ( S, Z ) ∼ = ( Z / for G = ( Z / ;(3) H ( S, Z ) ∼ = ( Z / for G = ( Z / ;(4) H ( S, Z ) ∼ = ( Z / for G = ( Z / . Remark 1.5.
Let S be a surface with p g = q = 0 and K S = 8 . From the exponentialsequence 0 → Z → O → O ∗ → P ic ( S ) ∼ = H ( S, Z ) . Because q = 0 we get b = 0 . Noether’s formula χ ( O X ) = 1 = 112 (8 + 2 b − b + b ) = 112 ( K S + χ top ( S )) XCEPTIONAL SEQUENCES ON SOME SURFACES ISOGENOUS TO A HIGHER PRODUCT 5 implies that these surfaces have b = 2 .Now we compute the Grothendieck groups of these surfaces. Lemma 1.6. [20, Lemma 2.7], [33, Lemma 2.6] Let S be a surface with p g = q = 0 isogenousto a higher product ( C × D ) /G of unmixed type and let G be abelian. Then K ( S ) ∼ = Z ⊕ P ic ( S ) . Derived categories of surfaces isogenous to a higher product of unmixedtype with p g = q = 0 , G = ( Z / or G = ( Z / We recall some basic notions to describe derived categories of algebraic varieties.
Definition 2.1. (1) An object E in a triangulated category D is called exceptional if Hom ( E, E [ i ]) = ( C if i = 0,0 otherwise.(2) A sequence E , · · · , E n of exceptional objects is called an exceptional sequence if Hom ( E i , E j [ k ]) = 0 , ∀ i > j, ∀ k. When S is a surface with p g = q = 0 , every line bundle on S is an exceptional object in D b ( S ) . Now we define the notion of semi-orthogonal decomposition. Definition 2.2. [23] Let D be a triangulated category.(1) A full triangulated subcategory D ′ ⊂ D is called admissible if the inclusion has a rightadjoint.(2) A sequence of full admissible triangulated subcategories D , · · · , D n ⊂ D is semi-orthogonalif for all i > j D j ⊂ D ⊥ i . (3) Such a sequence is called a semi-orthogonal decomposition if D , · · · , D n generate D .Next we define the notion of quasiphantom category. Definition 2.3. [21, Definition 1.8] Let X be a smooth projective variety. Let A be anadmissible triangulated subcategory of D b ( X ) . Then A is called a quasiphantom categoryif the Hochchild homology of A vanishes, and the Grothendieck group of A is finite. If theGrothendieck group of A also vanishes, then A is called a phantom category. KYOUNG-SEOG LEE
Now we discuss about previous construction of exceptional sequences of maximal length onthe surfaces isogenous to a higher product of unmixed type with p g = q = 0 , G = ( Z / or G = ( Z / . When G = ( Z / , the surface isogenous to a higher product is called theBeauville surface. Galkin and Shinder constructed exceptional sequences of maximal length onthe Beauville surface in [20]. Motivated by their work, we constructed exceptional sequencesof maximal length on the surfaces isogenous to a higher product when G = ( Z / in [33].Recently Coughlan has constructed exceptional sequences of maximal length on these surfacesvia different approach in [12]. Theorem 2.4. [12], [20], [33] Let S = ( C × D ) /G be a surface isogenous to a higher productof unmixed type with p g = q = 0 , G = ( Z / or G = ( Z / . There are exceptionalsequences of line bundles of maximal length on S . The orthogonal complements of the ad-missible subcategories generated by these exceptional sequences in the derived category of S are quasiphantom categories.In this paper we will show that similar statements are true for surfaces isogenous to a higherproduct of unmixed type with p g = q = 0 , G = ( Z / or G = ( Z / .3. Theta characteristics
In this section we collect some facts about Z / C be a curve with involution σ , B be the quotient curve C/σ , π : C → B be the quotient map, and R ⊂ C be the set of ramification points. The doublecovering corresponds to a line bundle ρ on B such that ρ = O B ( π ∗ R ) , see [2], [7], [8].Beauville classifies all σ -invariant line bundles on C in [8]. Lemma 3.1. [8, Lemma 1] Consider the map φ : Z R → P ic ( C ) which maps r ∈ R to O C ( r ) . Its image lies in the subgroup P ic ( C ) σ of σ -invariant line bundles. When R = ∅ , φ induces a short exact sequence0 → Z / → ( Z / R → P ic ( C ) σ /π ∗ P ic ( B ) → , and the kernel is generated by (1 , · · · , . Beauville also showed how to compute the cohomologies of these invariant line bundles.
Proposition 3.2. [8, Proposition 1] Let M be a σ -invariant line bundle on C . Then(1) M ∼ = π ∗ L ( E ) for some L ∈ P ic ( B ) and E ⊂ R . Any pair ( L ′ , E ′ ) satisfying M ∼ = π ∗ L ′ ( E ′ ) is equal to ( L, E ) or ( L ⊗ ρ − ( π ∗ E ) , R − E ) .(2) There is a natural isomorphism H ( C, M ) ∼ = H ( B, L ) ⊕ H ( B, L ⊗ ρ − ( π ∗ E )) . XCEPTIONAL SEQUENCES ON SOME SURFACES ISOGENOUS TO A HIGHER PRODUCT 7
When the invariant line bundle is a theta characteristic, the above proposition becomes asfollows.
Proposition 3.3. [8, Proposition 2] Let κ be a σ -invariant theta characteristic on C .Then(1) κ ∼ = π ∗ L ( E ) for some L ∈ P ic ( B ) and E ⊂ R with L ∼ = K B ⊗ ρ ( − π ∗ E ) . If another pair( L ′ , E ′ ) satisfies κ ∼ = π ∗ L ′ ( E ′ ) , we have ( L ′ , E ′ ) = ( L, E ) or ( L ′ , E ′ ) = ( K B ⊗ L − , R − E ) .(2) h ( κ ) = h ( L ) + h ( L ) .The above theorems are our main tools to compute the cohomologies of line bundles whichwe construct in this paper. 4. α -sheaves In this section we collect some facts in order to compute the 2-cocycles corresponding to the G -invariant line bundles on curves. In this paper G is an abelian group and G acts on C ∗ trivially. From this assumption the definition and computation of group cohomology becomemuch simpler than usual.4.1. generalities on group cohomology. In 4.1 and 4.2 we recall some definitions and prop-erties about group cohomology following [24].
Definition 4.1. [24] Let G be an abelian group and G acts on C ∗ trivially.(1) A function α : G × · · · × G | {z } n → C ∗ is called an n-cochain.(2) An n-cochain α is called normalized if α ( g , · · · , g n ) = 1 whenever any of the g i = e ∈ G. (3) A 2-cochain α is called 2-cocycle if α ( x, y ) α ( xy, z ) = α ( y, z ) α ( x, yz ) for all x, y, z ∈ G ,and we denote the abelian group of 2-cocycles by Z ( G, C ∗ ) . (4) A 2-cocycle α is called 2-coboundary if there exist a 1-cochain t : G → C ∗ such that α ( x, y ) = t ( x ) t ( y ) t ( xy ) − , and we denote the abelian group of 2-coboundaries by B ( G, C ∗ ) .(5) H ( G, C ∗ ) := Z ( G, C ∗ ) /B ( G, C ∗ ) is called the Schur multiplier of G .4.2. Schur multiplier.
Now we focus on H ( G, C ∗ ) . We only consider groups of the form G ∼ = ( Z / r for some r ∈ N . Consider a decomposition of G = N × T ∼ = ( Z / a × ( Z / b for some a, b ∈ N . Definition 4.2.
A cocycle α ∈ Z ( G, C ∗ ) is normal if α ( n, t ) = 1 , for all n ∈ N, t ∈ T. When α is a normal cycle then the following lemmas will enable us to compute α . KYOUNG-SEOG LEE
Lemma 4.3. [24] (1) Each α ∈ Z ( G, C ∗ ) is cohomologous to a normal cocycle β such that α | T × T = β | T × T .(2) If α is a normal cocycle, then α ( nt, n ′ t ′ ) = α ( t, t ′ ) α ( t, n ′ ) α ( n, n ′ ) , n.n ′ ∈ N, t, t ′ ∈ T. In particular, a normal cocycle α is uniquely determined by α | N × N , α | T × T , α | T × N . Lemma 4.4. [24] α | N × N , α | T × T , α | T × N determine a normal cocycle α of Z ( G, C ∗ ) if andonly if the following four conditions hold:(1) α | N × N ∈ Z ( N, C ∗ ) .(2) α | T × T ∈ Z ( T, C ∗ ) .(3) α ( tt ′ , n ) = α ( t, n ) α ( t ′ , n ) , n ∈ N, t, t ′ ∈ T .(4) α ( t, nn ′ ) = α ( t, n ) α ( t, n ′ ) , n, n ′ ∈ N, t ∈ T .4.3. Generalities on α -sheaves. Let X be an algebraic variety over C , let G be a finitegroup acting on X , and let α be a 2-cocyle of G with coefficients in C ∗ . Elagin introducedthe notion of α -sheaves, and proved some properties of them in [16], [17]. Definition 4.5. [17] An α -sheaf on X is a coherent sheaf F together with isomorphisms θ g : g ∗ F → F for all g ∈ G such that θ gh = α ( g, h ) θ h ◦ h ∗ θ g for any pair g, h ∈ G . Remark 4.6.
We can set α to be normalized canonically in our geometric case. Proposition 4.7. [17, Proposition 1.2] The α -sheaves on X form abelian category. Let α , β be 2-cocycles of G . let F and G be α - and β - sheaves on X . Then F ⊗ G is an αβ sheaf on X . Proof. θ gh ⊗ θ gh = α ( g, h ) β ( g, h )( θ h ◦ h ∗ θ g ) ⊗ ( θ h ◦ h ∗ θ g ) = α ( g, h ) β ( g, h )( θ h ⊗ θ h ) ◦ h ∗ ( θ g ⊗ θ g )for any pair g, h ∈ G. (cid:3) Basic example : O (1) bundle on P with ( Z / -action. In this subsection weconsider P with G = ( Z / -action and the G -invariant line bundle O (1) . Let G = ( Z / e ⊕ ( Z / e , and G acts on P as e · [ x : y ] = [ − x : y ] and e · [ x : y ] =[ y : x ] . Let U = Spec ( C [ xy ]) and V = Spec ( C [ yx ]) be an affine open covering of P and let O (1) ∼ = O ([1 : 0]) be a G -invariant line bundle. In this case we can compute [ α ] ∈ H ( G, C ∗ )by explicit computations as follows.Fix four isomorphisms XCEPTIONAL SEQUENCES ON SOME SURFACES ISOGENOUS TO A HIGHER PRODUCT 9 θ ( U ) : 0 ∗ O ([1 : 0])( U ) ∼ = C [ xy ] ⊗ C [ xy ] C [ xy ] → C [ xy ] ∼ = O ([1 : 0])( U ) , ⊗ ,θ ( V ) : 0 ∗ O ([1 : 0])( V ) ∼ = xy C [ yx ] ⊗ C [ yx ] C [ yx ] → xy C [ yx ] ∼ = O ([1 : 0])( V ) , xy ⊗ xy ,θ e ( U ) : e ∗ O ([1 : 0])( U ) ∼ = C [ xy ] ⊗ C [ xy ] C [ xy ] → C [ xy ] ∼ = O ([1 : 0])( U ) , ⊗ ,θ e ( V ) : e ∗ O ([1 : 0])( V ) ∼ = xy C [ yx ] ⊗ C [ yx ] C [ yx ] → xy C [ yx ] ∼ = O ([1 : 0])( V ) , xy ⊗
7→ − xy ,θ e ( U ) : e ∗ O ([1 : 0])( U ) ∼ = xy C [ yx ] ⊗ C [ yx ] C [ xy ] → C [ xy ] ∼ = O ([1 : 0])( U ) , xy ⊗ ,θ e ( V ) : e ∗ O ([1 : 0])( V ) ∼ = C [ xy ] ⊗ C [ xy ] C [ yx ] → xy C [ yx ] ∼ = O ([1 : 0])( V ) , ⊗ xy ,θ e + e ( U ) : ( e + e ) ∗ O ([1 : 0])( U ) ∼ = xy C [ yx ] ⊗ C [ yx ] C [ xy ] → C [ xy ] ∼ = O ([1 : 0])( U ) , xy ⊗
7→ − ,θ e + e ( V ) : ( e + e ) ∗ O ([1 : 0])( V ) ∼ = C [ xy ] ⊗ C [ xy ] C [ yx ] → xy C [ yx ] ∼ = O ([1 : 0])( V ) , ⊗ xy , To compute α ( e , e ) , α ( e , e ) from θ e + e = α ( e , e ) θ e ◦ e ∗ θ e ,θ e + e = α ( e , e ) θ e ◦ e ∗ θ e , consider the following isomorphisms. e ∗ θ e ( U ) : e ∗ e ∗ O ([1 : 0])( U ) ∼ = xy C [ yx ] ⊗ C [ yx ] C [ yx ] ⊗ C [ yx ] C [ xy ] → xy C [ yx ] ⊗ C [ yx ] C [ xy ] ∼ = e ∗ O ([1 : 0])( U ) ,xy ⊗ ⊗
7→ − xy ⊗ ,e ∗ θ e ( V ) : e ∗ e ∗ O ([1 : 0])( V ) ∼ = C [ xy ] ⊗ C [ xy ] C [ xy ] ⊗ C [ xy ] C [ yx ] → C [ xy ] ⊗ C [ xy ] C [ yx ] ∼ = e ∗ O ([1 : 0])( V ) , ⊗ ⊗ ⊗ , e ∗ θ e ( U ) : e ∗ e ∗ O ([1 : 0])( U ) ∼ = xy C [ yx ] ⊗ C [ yx ] C [ xy ] ⊗ C [ xy ] C [ xy ] → C [ xy ] ⊗ C [ xy ] C [ xy ] ∼ = e ∗ O ([1 : 0])( U ) ,xy ⊗ ⊗ ⊗ ,e ∗ θ e ( V ) : e ∗ e ∗ O ([1 : 0])( V ) ∼ = C [ xy ] ⊗ C [ xy ] C [ yx ] ⊗ C [ yx ] C [ yx ] → xy C [ yx ] ⊗ C [ yx ] C [ yx ] ∼ = e ∗ O ([1 : 0])( V ) , ⊗ ⊗ xy ⊗ . Then we get α ( e , e ) = 1 , α ( e , e ) = − α ( e , e ) , α ( e , e ) from θ = α ( e , e ) θ e ◦ e ∗ θ e ,θ = α ( e , e ) θ e ◦ e ∗ θ e , we consider the following isomorphisms. e ∗ θ e ( U ) : e ∗ e ∗ O ([1 : 0])( U ) ∼ = C [ xy ] ⊗ C [ xy ] C [ xy ] ⊗ C [ xy ] C [ xy ] → C [ xy ] ⊗ C [ xy ] C [ xy ] ∼ = e ∗ O ([1 : 0])( U ) , ⊗ ⊗ ⊗ ,e ∗ θ e ( V ) : e ∗ e ∗ O ([1 : 0])( V ) ∼ = xy C [ yx ] ⊗ C [ yx ] C [ yx ] ⊗ C [ yx ] C [ yx ] → xy C [ yx ] ⊗ C [ yx ] C [ yx ] ∼ = e ∗ O ([1 : 0])( V ) ,xy ⊗ ⊗
7→ − xy ⊗ ,e ∗ θ e ( U ) : e ∗ e ∗ O ([1 : 0])( U ) ∼ = C [ xy ] ⊗ C [ xy ] C [ yx ] ⊗ C [ yx ] C [ xy ] → xy C [ yx ] ⊗ C [ yx ] C [ xy ] ∼ = e ∗ O ([1 : 0])( U ) , ⊗ ⊗ xy ⊗ ,e ∗ θ e ( V ) : e ∗ e ∗ O ([1 : 0])( V ) ∼ = xy C [ yx ] ⊗ C [ yx ] C [ xy ] ⊗ C [ xy ] C [ yx ] → C [ xy ] ⊗ C [ xy ] C [ yx ] ∼ = e ∗ O ([1 : 0])( V ) ,xy ⊗ ⊗ ⊗ . Then we get α ( e , e ) = 1 , α ( e , e ) = 1 .Let N = ( Z / e and let T = ( Z / e be a decomposition of G = N × T . The aboveisomorphisms are selected so that our α to be normalized. Finally we can compute α as XCEPTIONAL SEQUENCES ON SOME SURFACES ISOGENOUS TO A HIGHER PRODUCT 11 below using the general properties of the normal cycles stated in the previous subsection. α e e e + e e e e + e Pullback of invariant line bundle.
Let σ be an involution of C and suppose that T = h σ i extends to a group of automorphisms G = N ⊕ T of C . Let B = C/ h σ i be thequotient curve and π : C → B be the quotient map. Let L be an N -invariant line bundleson B whose Schur multiplier is β ∈ H ( N, C ∗ ) . In this case we can define θ g : g ∗ π ∗ L → π ∗ L and compute the Schur multiplier of α ∈ H ( G, C ) of π ∗ L using N -invariant structure of L . Lemma 4.8. (1) π ∗ L is a G -invariant line bundle on C .(2) α ( n, n ′ ) = β ( n, n ′ ) , ∀ n, n ′ ∈ N. (3) α ( σ, σ ) = 1 ,(4) α ( σ, n ) = α ( n, σ ) = 1 , ∀ n ∈ N . Proof.
We have the following three commutative diagram. C π (cid:31) (cid:31) ❅❅❅❅❅❅❅❅ σ / / C π (cid:127) (cid:127) ⑦⑦⑦⑦⑦⑦⑦⑦ BC π (cid:15) (cid:15) n / / C π (cid:15) (cid:15) B n / / BC π (cid:15) (cid:15) nσ = σn / / C π (cid:15) (cid:15) B n / / B Then (1) is obvious. From the above diagrams we have isomorphisms n ∗ π ∗ L ∼ = π ∗ n ∗ L , σ ∗ π ∗ L ∼ = π ∗ σ ∗ L , ( nσ ) ∗ π ∗ L ∼ = π ∗ n ∗ L . These isomorphisms enable us to define θ n : n ∗ π ∗ L → π ∗ L , θ σ : σ ∗ π ∗ L → π ∗ L , θ nσ : ( nσ ) ∗ π ∗ L → π ∗ L on C via the pullback of θ n : n ∗ L → L on B . It suffices to check that these isomorphisms satisfy (2), (3), (4) in a fixed affine chart.We leave this to readers. (cid:3) Remark 4.9.
Let G = N ⊕ T be the decomposition of G as above. Because α is a normalcocyle we can compute α ( g, g ′ ) for all g, g ′ ∈ G from the general properties of normal cycles.5. G = ( Z / case Let S = ( C × D ) /G be a surface isogenous to a higher product with p g = q = 0 and G = ( Z / . In this case the curves are a hyperelliptic curve of genus 3 and an elliptic-hyperelliptic curve of genus 5. Let C be the hyperelliptic curve of genus 3 and D be theelliptic-hyperelliptic curve of genus 5. Let π C : C → P , π D : D → P be the quotientmaps. Then π C has 5 branch points and π D has 6 branch points. We may assume thatthe stabilizer elements of C are ( e , e , e , e , e + e ) , and the stabilizer elements of D are( e + e , e + e , e + e + e , e + e , e + e , e + e + e ) . Let E , E , E , E , E be thecorresponding set-theoretic fibers on C and let F , F , F , F , F , F be the correspondingset-theoretic fibers on D . For detail, see [3].First we construct G -invariant theta characteristic κ C on C . Lemma 5.1.
There exist G -invariant theta characteristic κ C on C such that h ( κ C ) = h ( κ C ) = 0 . Proof. C is a hyperelliptic curve of genus 3. Let π C : C → P be the quotient by ( Z / e action. We set κ C = π C ∗ O ( − ⊗ O C ( E ) be a line bundle. From the Proposition of [8]we see that κ C is a theta characteristic with h ( κ C ) = h ( κ C ) = 0 . Since O C ( E ) is G -equivariant line bundle, it suffice to show that π C ∗ O ( −
1) is a G -invariant line bundle.The action of G induces ( Z / e ⊕ ( Z / e -action on P , and O ( −
1) is an invariant linebundle on P . Therefore π C ∗ O ( −
1) is a G -invariant line bundle on C . (cid:3) Next we construct G -invariant 2 -torsion line bundle η D on D having the same Schurmultiplier as that of κ C . Let x, y, z ∈ G be the stabilizer elements of the G -action on D and let F x be a set-theoretic fiber which is a half of the ramification points of the corresponding( Z / x -quotient of D . Then G -action induces an ( Z / y ⊕ ( Z / z -action on F x whichis free. Then we can decompose F x = F x ⊔ F x such that one ( Z / y -action preserves each F x , F x and ( Z / z -action exchanges them. Let us denote η y,zD = O ( F x − F x ) . Note that η y,z is not trivial by Lemma 3.1. Lemma 5.2. η y,zD is a G -invariant 2 -torsion line bundle on D . XCEPTIONAL SEQUENCES ON SOME SURFACES ISOGENOUS TO A HIGHER PRODUCT 13
Proof.
Consider the ( Z / x ⊕ ( Z / y -action on D , and let π x,yD : D → P be the quotientmorphism. 2 F x ∼ F x follows since 2 F x ∼ F x ∼ ( π x,yD ) ∗ pt . Therefore η y,zD is a 2 -torsionline bundle. Since ( Z / x ⊕ ( Z / y -action fixes η y,zD and z ∗ ( η y,zD ) ∼ = ( η y,zD ) ⊗− ∼ = η y,zD , η y,zD is a G -invariant line bundle on D . (cid:3) Therefore we get three 2 -torsion line bundles η e + e ,e + e + e D , η e + e + e ,e + e D , η e + e ,e + e D on D , and let η D = η e + e ,e + e + e D ⊗ η e + e + e ,e + e D ⊗ η e + e ,e + e D . Proposition 5.3.
Let κ C be the G -invariant theta characteristics constructed above, andlet η C be the G -invariant torsion line bundles constructed above. Then α ( κ C ) α ( η D ) = 0 ∈ H ( G, C ∗ ) . Proof.
Using the above basic example we can compute the cocycles of α ( κ C ) , α ( η D ) ∈ H ( G, C ∗ ) . Since O C ( E ) is a G -equivariant line bundle, it suffice to compute α (( π C ) ∗ O ( − . Fromthe calculation of basic example we get the following table for α (( π C ) ∗ O ( − .α e e e + e e e + e e + e e + e + e e e e + e e e + e e + e e + e + e α ( η e + e ,e + e + e D ) as follows. α e e e + e e e + e e + e e + e + e e e e + e e e + e e + e e + e + e α ( η e + e + e ,e + e D ) as follows. α e e e + e e e + e e + e e + e + e e e e + e e e + e e + e e + e + e α ( η e + e ,e + e D ) as follows. α e e e + e e e + e e + e e + e + e e e e + e e e + e e + e e + e + e α ( κ C ) α ( η D ) becomes as follows. α e e e + e e e + e e + e e + e + e e e e + e e e + e e + e e + e + e t (0) = 1 , t ( e ) = √− , t ( e ) = √− , t ( e ) = √− , t ( e + e ) = 1 , t ( e + e ) = 1 , t ( e + e ) = − , t ( e + e + e ) = −√− , and check that α ( x, y ) = t ( x ) t ( y ) t ( xy ) − , ∀ x, y ∈ G . (cid:3) XCEPTIONAL SEQUENCES ON SOME SURFACES ISOGENOUS TO A HIGHER PRODUCT 15
Remark 5.4.
It follows that κ C ⊠ η D is a G -equivariant line bundle on C × D . Lemma 5.5.
The 6 effective G invariant divisors of degree 4 divide into 3 divisor classes F ∼ F , F ∼ F , F ∼ F which are not linearly equivalent to each other. Proof.
Consider ( Z / e + e ) ⊕ ( Z / e + e + e ) -action on D and let π e + e ,e + e + e : D → P be the quotient map. F ∼ F follows since they are pullbacks of the point of P via π e + e ,e + e + e . Similarly we get F ∼ F , F ∼ F . Consider ( Z / e + e ) -action on D and let π e + e : D → E be the quotient map. Then F is a pullback of a line bundle on E , and F is half of the ramification point of π e + e . From the Lemma 3.1 we get F ≁ F .Similarly we see that F ≁ F . (cid:3) From the above two lemmas, we find that every G -invariant effective divisor on E ofdegree 4 is linearly equivalent to F or F or F . With the same notation, we have thefollowing lemma. Lemma 5.6. h ( D, O D ( F + F − F )) = 0 ,h ( D, O D ( F + F − F )) = 0 ,h ( D, O D ( − F − F + F )) = 0 ,h ( D, O D ( − F − F + F )) = 8 . Proof.
From the Riemann-Roch formula we find that h ( D, O D ( F + F − F )) − h ( D, O D ( F + F − F )) = 1 + 4 − . Therefore it suffices to show that h ( D, O D ( F + F − F )) = 0 . We know that F , F , F are G -invariant divisors on D and hence there is a G -action on H ( D, O D ( F + F − F )) . If h ( D, O D ( F + F − F )) = 0 , then there is a G -eigensection f ∈ H ( D, O D ( F + F − F )) ,and F + F − F + ( f ) should be a G -invariant effective divisor of degree 4. Every G -invariant effective divisor of degree 4 on D is linearly equivalent to F or F or F by theabove lemma. It follows that F + F − F ∼ F or F + F − F ∼ F or F + F − F ∼ F .Then F − F ∼ F − F ∼ F + F ∼ F ∼ F which contradicts the assumptionthat F , F and F are not linearly equivalent to each other.From the Riemann-Roch theorem we get h ( D, O D ( − F − F + F )) − h ( D, O D ( − F − F + F )) = 1 − − − . and h ( D, O D ( − F − F + F )) = 0because the degree of O D ( − F − F + F ) is negative. (cid:3) Remark 5.7.
Because the G -action on C × D is free we have D b ( S ) ≃ D bG ( C × D ) andevery G -equivariant line bundle on C × D corresponds to a line bundle on S . Thereforewe regards a G -equivariant line bundle on C × D as a line bundle on S Theorem 5.8.
Let S = ( C × D ) /G be a surface isogenous to a higher product with p g = q = 0 , G = ( Z / . For any choice of four characters χ , χ , χ , χ , the following sequence O C × D ( χ ) , q ∗ O D ( − F − F + F )( χ ) , κ − C ⊠ η D ( χ ) , κ − C ⊠ ( η D ⊗ O D ( − F − F + F ))( χ )is a exceptional sequence of line bundles of maximal length in D b ( S ) . Proof.
Since p g = q = 0 , every line bundle on S is exceptional. From the K¨unneth formulawe find that h j ( C × D, q ∗ O D ( F + F − F )) = 0 , ∀ j,h j ( C × D, κ C ⊠ η D ) = 0 , ∀ j,h j ( C × D, κ C ⊠ ( η D ⊗ O D ( F + F − F ))) = 0 , ∀ j,h j ( C × D, κ C ⊠ ( η D ⊗ O D ( − F − F + F ))) = 0 , ∀ j. Therefore the G -invariant parts are also trivial. Hence, we find that O C × D ( χ ) , q ∗ O D ( − F − F + F )( χ ) , κ − C ⊠ η D ( χ ) , κ − C ⊠ ( η D ⊗ O D ( − F − F + F ))( χ ) form an exceptional sequence.Since the rank of K ( S ) is 4, the maximal length of exceptional sequences on S is 4. (cid:3) Proposition 5.9.
Let A be the orthogonal complement of an exceptional sequence O C × D ( χ ) , q ∗ O D ( − F − F + F )( χ ) , κ − C ⊠ η D ( χ ) , κ − C ⊠ ( η D ⊗ O D ( − F − F + F ))( χ ) . Then A is a quasiphantom category whose Grothendieck group is isomorphic to ( Z / ⊕ ( Z / . Proof.
Since the Betti number of S is 4, we see that the orthogonal complement of an excep-tional sequence is a quasiphantom category from Kuznetsov’s theorem [31]. (cid:3) Then we can compute the Hochschild cohomologies of the quasiphantom categories. See [32]for the definitions and more details.
Proposition 5.10.
The pseudoheight of the exceptional sequence O C × D ( χ ) , q ∗ O D ( − F − F + F )( χ ) , κ − C ⊠ η D ( χ ) , κ − C ⊠ ( η D ⊗ O D ( − F − F + F ))( χ ) is 4 and the height is 4. XCEPTIONAL SEQUENCES ON SOME SURFACES ISOGENOUS TO A HIGHER PRODUCT 17
Proof.
From the K¨unneth formula and degree computation we find that O C × D ( χ ) , q ∗ O D ( − F − F + F )( χ ) , κ − C ⊠ η D ( χ ) , κ − C ⊠ ( η D ⊗ O D ( − F − F + F ))( χ ) , O C × D ( χ ) ⊗ ω − S , q ∗ O D ( − F − F + F )( χ ) ⊗ ω − S , κ − C ⊠ η D ( χ ) ⊗ ω − S , κ − C ⊠ ( η D ⊗O D ( − F − F + F ))( χ ) ⊗ ω − S is Hom-free. This sequence cannot be cyclically Ext -connected by Serre duality and Kodairavanishing theorem. (cid:3) Therefore we get the following consequence about the Hochschild cohomologies of our quasiphan-tom categories.
Corollary 5.11.
Let A be the orthogonal complement of the exceptional collection O C × D ( χ ) , q ∗ O D ( − F − F + F )( χ ) , κ − C ⊠ η D ( χ ) , κ − C ⊠ ( η D ⊗ O D ( − F − F + F ))( χ ) . Then wehave HH i ( S ) = HH i ( A ) , for i = 0 , , HH ( S ) ⊂ HH ( A ) .6. G = ( Z / case Let S = ( C × D ) /G be a surface isogenous to a higher product with p g = q = 0 and G = ( Z / . In this case the curves are elliptic-hypereilliptic curves of genus 5. Let C and D be the elliptic-hypereilliptic curves of genus 5. Let π C : C → P , π D : D → P be thequotient maps. Then π C and π D have 5 branch points. We may assume that the stabilizerelements of C are ( e , e , e , e , e := e + e + e + e ) , and the stabilizer elements of D are( e + e , e + e , e + e , e + e , e + e ) . Let E , E , E , E , E be the corresponding set-theoreticfibers on C and let F , F , F , F , F be the corresponding set-theoretic fibers on D . Fordetail, see [3].Now we construct G -invariant theta charactristics on C and D . Let x, y, z, w, x + y + z + w ∈ G be the stabilizer elements. Consider ( Z / x ⊕ ( Z / y -action on C andlet π x,y : C → P be the ( Z / x ⊕ ( Z / y -quotient of C . Then G -action induces an( Z / z ⊕ ( Z / w -action on P . Note that the set theoretic fiber of π C with stabilizer element z is the union of pullbacks of two points p, q on P by π x,y . Take one such point p on P and denote E x,y,z,w = ( π x,y ) ∗ p . In this way we can define three G -invariant line bundles O C ( E e ,e ,e ,e ) , O C ( E e ,e ,e,e ) , O C ( E e ,e,e ,e ) on C . Let us denote κ C = O C ( E e ,e ,e ,e ) ,and η C = O C ( E e ,e ,e,e − E e ,e,e ,e ) . Lemma 6.1. (1) E x,y,z,w is a G -invariant theta characteristic on C .(2) η C is a G -invariant torsion line bundle on C .(3) κ C ⊗ η C is a G -invariant theta characteristic with h ( C, κ C ⊗ η C ) = h ( C, κ C ⊗ η C ) = 2 . Proof. (1) follows immediately since ( π x,y ) ∗ p + ( π x,y ) ∗ q is a canonical divisor on C and( π x,y ) ∗ p ∼ ( π x,y ) ∗ q . (2) is obvious. Consider e -action on C . Then κ C ⊗ η C is a pullbackof degree 2 line bundles on C/ h e i . From Proposition 3.3 and Riemann-Roch formula we have(3). (cid:3) Similarly we can define three theta characteristics O D ( F e + e ,e + e ,e + e ,e + e ) , O D ( F e + e ,e + e ,e + e ,e + e ) , O D ( F e + e ,e + e ,e + e ,e + e ) on D in a similar way. Let us denote κ D = O D ( F e + e ,e + e ,e + e ,e + e ) , η D = O D ( F e + e ,e + e ,e + e ,e + e − F e + e ,e + e ,e + e ,e + e ) Lemma 6.2. (1) F x,y,z,w is a G -invariant theta characteristic on D .(2) η D is a G -invariant torsion line bundle on D .(3) κ D ⊗ η D is a G -invariant theta characteristic with h ( D, κ D ⊗ η D ) = h ( D, κ D ⊗ η D ) = 0 . Proof.
The only nontrivial part is (3). Consider e + e -action on D , and let π e + e : D → E be the quotient map. Then κ D ⊗ η D ∼ = ( π e + e ) ∗ ( L ) ⊗ O D ( F e + e ,e + e ,e + e ,e + e ) for some L .From the Proposition 3.3 we get h ( D, κ D ⊗ η D ) = h ( L ) + h ( L ) . Consider e + e -actionof E . Then we can prove that L is a noneffective theta characteristic on E by the sameargument. Therefore we get h ( D, κ D ⊗ η D ) = h ( D, κ D ⊗ η D ) = 0 . (cid:3) Proposition 6.3.
Let κ C , κ D be the G -invariant theta characteristics constructed above,and let η C , η D be the G -invariant 2 -torsion line bundles constructed above. Then α ( κ C ) α ( η D ) = α ( η C ) α ( κ D ) = 0 ∈ H ( G, C ∗ ) . Proof.
As the ( Z / -case we can compute α ( κ C ) , α ( η C ) , α ( κ D ) , α ( η D ) (see the table inthe last part of the paper). Finally we can show that α ( κ C ) α ( η D ) is a coboundary by giving t (0) = 1 , t ( e ) = 1 , t ( e ) = 1 , t ( e + e ) = − , t ( e ) = √− , t ( e + e ) = √− , t ( e + e ) = −√− , t ( e + e + e ) = √− , t ( e ) = 1 , t ( e + e ) = − , t ( e + e ) = 1 , , t ( e + e + e ) =1 , t ( e + e ) = −√− , t ( e + e + e ) = √− , t ( e + e + e ) = √− , t ( e + e + e + e ) = √− . and check that α ( x, y ) = t ( x ) t ( y ) t ( xy ) − , ∀ x, y ∈ G .Similarly we can show that α ( η C ) α ( κ D ) is a coboundary by giving t (0) = 1 , t ( e ) = √− , t ( e ) =1 , t ( e + e ) = √− , t ( e ) = √− , t ( e + e ) = − , t ( e + e ) = √− , t ( e + e + e ) = − , t ( e ) = √− , t ( e + e ) = − , t ( e + e ) = √− , t ( e + e + e ) = − , t ( e + e ) =1 , t ( e + e + e ) = −√− , t ( e + e + e ) = 1 , t ( e + e + e + e ) = −√− . and checkthat α ( x, y ) = t ( x ) t ( y ) t ( xy ) − , ∀ x, y ∈ G .Therefore we have α ( κ C ) α ( η D ) = α ( η C ) α ( κ D ) = 0 ∈ H ( G, C ∗ ) (cid:3) Remark 6.4. (1) Because the G -action on C × D is free we have D b ( S ) ≃ D bG ( C × D ) andevery G -equivariant line bundle on C × D corresponds to a line bundle on S . Therefore XCEPTIONAL SEQUENCES ON SOME SURFACES ISOGENOUS TO A HIGHER PRODUCT 19 we regards a G -equivariant line bundle on C × D as a line bundle on S .(2) The above proposition implies that we can regard κ C ⊠ η D , η C ⊠ κ D as line bundles on S . Theorem 6.5.
Let S = ( C × D ) /G be a surface isogenous to a higher product with p g = q = 0 , G = ( Z / . Then there exist two characters χ , χ ∈ b G such that the followingsequence O , κ − C ⊠ η D ( χ ) , η C ⊠ κ − D ( χ ) , ( κ − C ⊗ η C ) ⊠ ( κ − D ⊗ η D )( χ + χ )is a exceptional sequence of maximal length on D b ( S ) . Proof.
Since p g = q = 0 , every line bundle on S is exceptional. For simplicity let O , L − , L − , ( L ⊗ L ) − be the line bundles on S corresponding to O , κ − C ⊠ η D ( χ ) , η C ⊠ κ − D ( χ ) , ( κ − C ⊗ η C ) ⊠ ( κ − D ⊗ η D )( χ + χ ) on C × D . We have to show that H i ( S, L ) = 0 , H i ( S, L ) = 0 , H i ( S, L − ⊗ L ) = 0 , H i ( S, L ⊗ L ) = 0 for all i . Note that Riemann-Rochformula implies that χ ( S, L ) = 0 , χ ( S, L ) = 0 , χ ( S, L − ⊗ L ) = 0 , χ ( S, L ⊗ L ) = 0 .From Proposition 3.3 we see that h ( C, κ C ) = h ( C, κ C ) = 2 , h ( D, κ D ) = h ( D, κ D ) = 2 ,and from Riemann-Roch we get h ( C, η C ) = 0 , h ( C, η C ) = 4 , h ( D, η D ) = 0 , h ( D, η D ) =4 . From this we get h ( C × D, κ C ⊠ η D ) = 8 and h ( C × D, η C ⊠ κ D ) = 8 . Therefore thereexist χ , χ ∈ b G such that H ( C × D, κ C ⊠ η D ( χ )) G = 0 , H ( C × D, η C ⊠ κ D ( χ )) G = 0 .Let L − = κ − C ⊠ η D ( χ ) , L − = η C ⊠ κ − D ( χ ) . Then we have H ( S, L ) = H ( S, L ) = 0and H ( S, L ) = 0 follows since χ ( L ) = 0 . Similarly we get H i ( S, L ) = 0 , for all i .It remains to show that H i ( S, L ⊗ L ) = H i ( S, L − ⊗ L ) = 0 , for all i . Recall that L − ⊗ L corresponds to ( κ − C ⊗ η C ) ⊠ ( κ D ⊗ η D )( χ + χ ) on C × D . From the K¨unnethformula we find that H ( C × D, ( κ − C ⊗ η C ) ⊠ ( κ D ⊗ η D )) = 0 for degree reason. Finally H ( C × D, ( κ − C ⊗ η C ) ⊠ ( κ D ⊗ η D )) = H ( C, κ − C ⊗ η C ) ⊗ H ( D, κ D ⊗ η D ) ⊕ H ( C, κ − C ⊗ η C ) ⊗ H ( D, κ D ⊗ η D ) = 0 , H ( C × D, ( κ − C ⊗ η C ) ⊠ ( κ D ⊗ η D )) = H ( C, κ − C ⊗ η C ) ⊗ H ( D, κ D ⊗ η D ) = 0since κ D ⊗ η D is a noneffective theta characteristic on D . We can prove H i ( S, L ⊗ L ) = 0 ,for all i similarly. Therefore we get the desired result. Since the rank of K ( S ) is 4, themaximal length of exceptional sequences on S is 4. (cid:3) Proposition 6.6.
Let A be the orthogonal complement of an exceptional sequence O , κ − C ⊠ η D ( χ ) , η C ⊠ κ − D ( χ ) , ( κ − C ⊗ η C ) ⊠ ( κ − D ⊗ η D )( χ + χ ) . Then A is a quasiphantom categorywhose Grothendieck group is isomorphic to ( Z / . Proof.
Since the Betti number of S is 4, we see that the orthogonal complement of an excep-tional sequence is a quasiphantom category from Kuznetsov’s theorem [31]. (cid:3) We also prove that the DG algebra of endomorphisms of the exceptional sequences con-structed above are deformation invariant.
Proposition 6.7.
The DG algebra of endomorphisms of T = O⊕ κ − C ⊠ η D ( χ ) ⊕ η C ⊠ κ − D ( χ ) ⊕ ( κ − C ⊗ η C ) ⊠ ( κ − D ⊗ η D )( χ + χ ) does not change under small deformations of the complexstructure of S . Proof.
From the K¨unneth formula we get the followings. H ( C × D, κ − C ⊠ η D ) = 0 ,H ( C × D, η C ⊠ κ − D ) = 0 ,H ( C × D, ( κ − C ⊗ η C ) ⊠ ( κ − D ⊗ η D )) = 0 .H ( C × D, ( κ C ⊗ η C ) ⊠ ( κ − D ⊗ η D )) = C . Consider the minimal model of the DG algebra of endomorphism of T = O⊕ κ − C ⊠ η D ( χ ) ⊕ η C ⊠ κ − D ( χ ) ⊕ ( κ − C ⊗ η C ) ⊠ ( κ − D ⊗ η D )( χ + χ ) . We see that the minimal model of the DG algebra RHom ∗ ( T, T ) is formal by degree reason. We also see that m is also 0 since there is no Ext or Ext between objects. By semicontinuity we see that the dimension of H ∗ ( RHom ∗ ( T, T ))is constant and the algebra structure of the minimal algebra H ∗ ( RHom ∗ ( T, T )) does notchange by small deformations of the complex structure of S . (cid:3) Proposition 6.8.
The pseudoheight of the exceptional collection O , κ − C ⊠ η D ( χ ) , η C ⊠ κ − D ( χ ) , ( κ − C ⊗ η C ) ⊠ ( κ − D ⊗ η D )( χ + χ ) is 4 and the height is 4. Proof.
From the K¨unneth formula and degree computation we find that O , κ − C ⊠ η D ( χ ) , η C ⊠ κ − D ( χ ) , ( κ − C ⊗ η C ) ⊠ ( κ − D ⊗ η D )( χ + χ ) , O ⊗ ω − S , κ − C ⊠ η D ( χ ) ⊗ ω − S , η C ⊠ κ − D ( χ ) ⊗ ω − S , ( κ − C ⊗ η C ) ⊠ ( κ − D ⊗ η D )( χ + χ ) ⊗ ω − S is Hom-free. This sequence cannot be cyclically Ext -connectedby Serre duality and Kodaira vanishing theorem. (cid:3) Therefore we get the following consequence about the Hochschild cohomologies of the or-thogonal complements of our exceptional sequences.
Corollary 6.9.
Let A be the orthogonal complement of the exceptional collection O , κ − C ⊠ η D ( χ ) , η C ⊠ κ − D ( χ ) , ( κ − C ⊗ η C ) ⊠ ( κ − D ⊗ η D )( χ + χ ) . Then we have HH i ( S ) = HH i ( A ) ,for i = 0 , , HH ( S ) ⊂ HH ( A ) . XCEPTIONAL SEQUENCES ON SOME SURFACES ISOGENOUS TO A HIGHER PRODUCT 21
References [1]
V. Alexeev and D. Orlov , Derived categories of Burniat surfaces and exceptional collections. Math.Ann. 357 (2013), no. 2, 743-759.[2]
W. P. Barth, K. Hulek, C. A. M. Peters and A. Van de Ven , Compact complex surfaces, second ed. .Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics[Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] 4(Springer-Verlag, Berlin, 2004).[3]
I. Bauer and F. Catanese , Some new surfaces with p g = q = 0 . The Fano Conference
Univ. TorinoTurin (2004), 123-142.[4]
I. Bauer and F. Catanese , private communication.[5]
I. Bauer, F. Catanese and D. Frapporti , The fundamental group and torsion group of Beauvillesurfaces. 2014, Preprint, arXiv:1402.2109.[6]
I. Bauer, F. Catanese and R. Pignatelli , Surfaces of general type with geometric genus zero: a survey.
Complex and differential geometry
Springer Proc. Math. 8 (Springer, Heidelberg, 2011) 1-48.[7]
A. Beauville , Complex algebraic surfaces . Translated from the 1978 French original by R. Barlow, withassistance from N. I. Shepherd-Barron and M. Reid. Second edition. London Mathematical Society StudentTexts 34 (Cambridge University Press, Cambridge, 1996).[8]
A. Beauville , Vanishing thetanulls on curves with involutions. Rend. Circ. Mat. Palermo (2) 62 (2013),no. 1, 61-66.[9]
Ch. B¨ohning, H-Ch. Graf von Bothmer L. Katzarkov and P. Sosna , Determinantal Barlow sur-faces and phantom categories. 2012, Preprint, arXiv:1210.0343.[10]
Ch. B¨ohning, H-Ch. Graf von Bothmer and P. Sosna , On the derived category of the classicalGodeaux surface. Adv. Math. 243 (2013), 203-231.[11]
Ch. B¨ohning, H-Ch. Graf von Bothmer and P. Sosna , On the Jordan-H¨older property for geometricderived categories. 2012, Preprint, arXiv:1211.1129.[12]
S. Coughlan , Enumerating exceptional collections on some surfaces of general type with p g = 0 . 2014,Preprint, arXiv:1402.1540.[13] I. Dolgachev , Algebraic surfaces with q = p g = 0 . Algebraic surfaces , C.I.M.E. Summer Sch. 76(Springer, Heidelberg, 2010) 97-215.[14]
I. Dolgachev , Invariant stable bundles over modular curves X(p). Recent progress in algebra (Tae-jon/Seoul, 1997), 65-99, Contemp. Math., 224, Amer. Math. Soc., Providence, RI, 1999.[15]
I. Dolgachev , Lectures on invariant theory . London Mathematical Society Lecture Note Series 296 (Cam-bridge University Press, Cambridge, 2003).[16]
A. Elagin , On an equivariant derived category of bundles of projective spaces. (Russian) Tr. Mat. Inst.Steklova 264 (2009), Mnogomernaya Algebraicheskaya Geometriya, 63–68; translation in Proc. Steklov Inst.Math. 264 (2009), no. 1, 56-61 ISBN: 5-7846-0109-1; 978-5-7846-0109-4.[17]
A. Elagin , Semi-orthogonal decompositions for derived categories of equivariant coherent sheaves. (Rus-sian) Izv. Ross. Akad. Nauk Ser. Mat. 73 (2009), no. 5, 37–66; translation in Izv. Math. 73 (2009), no. 5,893-920 [18]
N. Fakhruddin , Exceptional collections on 2-adically uniformised fake projective planes, 2013,arXiv:1310.3020.[19]
S. Galkin and L. Katzarkov and A. Mellit and E. Shinder , Minifolds and phantoms. Preprint,2013, arXiv:1305.4549.[20]
S. Galkin and E. Shinder , Exceptional collections of line bundles on the Beauville surface. Adv. Math.244 (2013), 1033-1050.[21]
S. Gorchinskiy and D. Orlov , Geometric phantom categories. Publ. Math. Inst. Hautes Etudes Sci.117 (2013), 329-349.[22]
R. Hartshorne , Algebraic geometry . Graduate Texts in Mathematics No. 52. (Springer-Verlag, NewYork-Heidelberg, 1977).[23]
Huybrechts , Fourier-Mukai transforms in algebraic geometry . Oxford Mathematical Monographs. TheClarendon Press, Oxford University Press, Oxford, 2006. viii+307 pp. ISBN: 978-0-19-929686-6; 0-19-929686-3.[24]
G. Karpilovsky , The Schur Multiplier . London Mathematical Society Monographs. New Series, 2. TheClarendon Press, Oxford University Press, New York, 1987. x+302 pp. ISBN: 0-19-853554-6.[25]
Y. Kawamata , Derived categories of toric varieties. Michigan Math. J. 54 (2006), no. 3, 517-535.[26]
B. Keller , Introduction to A -infinity algebras and modules.
Homology Homotopy Appl.
S.-I. Kimura , Chow groups are finite dimensional, in some sense.
Math. Ann.
331 (2005), no. 1 173-201.[28]
A. King , Tilting bundles on some rational surfaces. Unpublished manuscript, 1997.[29]
A. Kuznetsov , A simple counterexample to the Jordan-H¨older property for derived categories. 2013,arXiv:1304.0903.[30]
A. Kuznetsov , Derived categories of Fano threefolds. (Russian) Tr. Mat. Inst. Steklova 264 (2009), Mno-gomernaya Algebraicheskaya Geometriya, 116–128; translation in Proc. Steklov Inst. Math. 264 (2009), no.1, 110-122 ISBN: 5-7846-0109-1; 978-5-7846-0109-4.[31]
A. Kuznetsov , Hochschild homology and semiorthogonal decompositions. Preprint, 2009,arXiv:0904.4330.[32]
A. Kuznetsov , Height of exceptional collections and Hochschild cohomology of quasiphantom categories.Preprint, 2012, arXiv:1211.4693.[33]
K-S. Lee , Derived categories of surfaces isogenous to a higher product. Preprint, 2013, arXiv:1303.0541.[34]
S. Okawa
Semi-orthogonal decomposability of the derived category of a curve. Adv. Math. 228 (2011),no. 5, 2869-2873.[35]
D. Orlov
Projective bundles, monoidal transformations, and derived categories of coherent sheaves. (Rus-sian) Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 4, 852–862; translation in Russian Acad. Sci. Izv. Math.41 (1993), no. 1, 133-141[36]
J.-P. Serre , Linear representations of finite groups . Translated from the second French edition by LeonardL. Scott. Graduate Texts in Mathematics Vol. 42 (Springer-Verlag, New York-Heidelberg, 1977).[37]
P. Seidel , Fukaya categories and Picard-Lefschetz theory . Zurich Lectures in Advanced Mathematics.(European Mathematical Society , Zrich, 2008)
XCEPTIONAL SEQUENCES ON SOME SURFACES ISOGENOUS TO A HIGHER PRODUCT 23 [38]
T. Shabalin , Homology of some surfaces with p g = q = 0 isogenous to a product. 2013, Preprint,arXiv:1311.4048. Department of Mathematics, Seoul National University, Seoul 151-747, Korea
E-mail address : [email protected] Table 1. α ( O C ( E e ,e ,e ,e )) α e e e + e e e + e e + e e + e + e e e + e e + e e + e + e e + e e + e + e e + e + e e + e + e + e e e e + e e e + e e + e e + e + e e e + e e + e e + e + e e + e e + e + e e + e + e e + e + e + e4
E-mail address : [email protected] Table 1. α ( O C ( E e ,e ,e ,e )) α e e e + e e e + e e + e e + e + e e e + e e + e e + e + e e + e e + e + e e + e + e e + e + e + e e e e + e e e + e e + e e + e + e e e + e e + e e + e + e e + e e + e + e e + e + e e + e + e + e4 XCEPTIONAL SEQUENCES ON SOME SURFACES ISOGENOUS TO A HIGHER PRODUCT 25
Table 2. α ( O C ( E e ,e ,e,e )) α e e e + e e e + e e + e e + e + e e e + e e + e e + e + e e + e e + e + e e + e + e e + e + e + e e e e + e e e + e e + e e + e + e e e + e e + e e + e + e e + e e + e + e e + e + e e + e + e + e4
Table 2. α ( O C ( E e ,e ,e,e )) α e e e + e e e + e e + e e + e + e e e + e e + e e + e + e e + e e + e + e e + e + e e + e + e + e e e e + e e e + e e + e e + e + e e e + e e + e e + e + e e + e e + e + e e + e + e e + e + e + e4 Table 3. α ( O C ( E e ,e,e ,e )) α e e e + e e e + e e + e e + e + e e e + e e + e e + e + e e + e e + e + e e + e + e e + e + e + e e e e + e e e + e e + e e + e + e e e + e e + e e + e + e e + e e + e + e e + e + e e + e + e + e4
Table 2. α ( O C ( E e ,e ,e,e )) α e e e + e e e + e e + e e + e + e e e + e e + e e + e + e e + e e + e + e e + e + e e + e + e + e e e e + e e e + e e + e e + e + e e e + e e + e e + e + e e + e e + e + e e + e + e e + e + e + e4 Table 3. α ( O C ( E e ,e,e ,e )) α e e e + e e e + e e + e e + e + e e e + e e + e e + e + e e + e e + e + e e + e + e e + e + e + e e e e + e e e + e e + e e + e + e e e + e e + e e + e + e e + e e + e + e e + e + e e + e + e + e4 XCEPTIONAL SEQUENCES ON SOME SURFACES ISOGENOUS TO A HIGHER PRODUCT 27
Table 4. α ( O D ( F e + e ,e + e ,e + e ,e + e )) α e e e + e e e + e e + e e + e + e e e + e e + e e + e + e e + e e + e + e e + e + e e + e + e + e e e e + e e e + e e + e e + e + e e e + e e + e e + e + e e + e e + e + e e + e + e e + e + e + e Table 5. α ( O D ( F e + e ,e + e ,e + e ,e + e )) α e e e + e e e + e e + e e + e + e e e + e e + e e + e + e e + e e + e + e e + e + e e + e + e + e e e e + e e e + e e + e e + e + e e e + e e + e e + e + e e + e e + e + e e + e + e e + e + e + e XCEPTIONAL SEQUENCES ON SOME SURFACES ISOGENOUS TO A HIGHER PRODUCT 29
Table 6. α ( O D ( F e + e ,e + e ,e + e ,e + e )) α e e e + e e e + e e + e e + e + e e e + e e + e e + e + e e + e e + e + e e + e + e e + e + e + e e e e + e e e + e e + e e + e + e e e + e e + e e + e + e e + e e + e + e e + e + e e + e + e + e Table 7. α ( κ C ⊠ η D ) α e e e + e e e + e e + e e + e + e e e + e e + e e + e + e e + e e + e + e e + e + e e + e + e + e e e e + e e e + e e + e e + e + e e e + e e + e e + e + e e + e e + e + e e + e + e e + e + e + e XCEPTIONAL SEQUENCES ON SOME SURFACES ISOGENOUS TO A HIGHER PRODUCT 31
Table 8. α ( η C ⊠ κ D ) α e e e + e e e + e e + e e + e + e e e + e e + e e + e + e e + e e + e + e e + e + e e + e + e + e e e e + e e e + e e + e e + e + e e e + e e + e e + e + e e + e e + e + e e + e + e e + e + e + e4