Exceptional collections on 2-adically uniformised fake projective planes
aa r X i v : . [ m a t h . AG ] M a y EXCEPTIONAL COLLECTIONS ON -ADICALLY UNIFORMISED FAKEPROJECTIVE PLANES NAJMUDDIN FAKHRUDDIN
Abstract.
We show that there exist exceptional collections of length 3 consisting of line bundles on thethree fake projective planes that have a 2-adic uniformisation with torsion free covering group. We alsocompute the Hochschild cohomology of the right orthogonal of the subcategory of the bounded derivedcategory of coherent sheaves generated by these exceptional collections. Introduction
Mumford showed in [12] that discrete cocompact torsion-free subgroups Γ ⊂ PGL ( Q ) which act transi-tively on the vertices of the Bruhat–Tits building of PGL , Q give rise to so called fake projective planes—surfaces of general type with the same Betti numbers as P —and he gave one example of such a subgroup.All such subgroups were classified by Cartwright, Mantero, Steger and Zappa [4]; there are two others andthey give rise to two other fake projective planes [8].More recently, all fake projective planes over C were classified by Prasad and Yeung [14] and Cartwrightand Steger [5]. However, from the algebro-geometric point of view these surfaces are still not well understood;for example, it is still not known whether Bloch’s conjecture on zero cycles on surfaces with p g = 0 holdsfor any of these surfaces. Another question about surfaces of general type with p g = 0 that has arisenvery recently is the existence of exceptional collections of maximal possible length in their bounded derivedcategories of coherent sheaves. The first such example was found by B¨ohning, von Bothmer and Sosna [3]and subsequently several other examples have been found. In the article [7] of Galkin, Katzarkov, Mellit andShinder, the authors conjecture that such exceptional collections exist on all fake projective planes admittinga cube root of the canonical bundle. The following is a consequence of the main result of this paper, Theorem4.1. Theorem.
Let M be a fake projective plane having a -adic uniformisation with a torsion free coveringgroup. There is an exceptional collection of length in D b ( M ) consisting of line bundles. We note that 3 is the smallest possible length of an exceptional collection in D b ( X ), for X any smoothprojective variety, so that the right orthogonal A to the exceptional collection is a quasiphantom subcate-gory of D b ( X ), i.e., such that the Hochschild homology HH • ( A ) = 0. Using a spectral sequence recentlyconstructed by Kuznetsov [11] which has a very simple form in our setting, we are also able to compute theHochschild cohomology of A . While we do not prove Bloch’s conjecture for these surfaces, we formulate ageneral conjecture, Conjecture 4.2, on the K of quasiphantom subcategories which we hope might lead toa proof.Our method of proof depends crucially on the fact that the fake projective planes with 2-adic uniformisa-tions have natural regular proper models over Spec( Z ). The special fibre in all cases is an explicit irreduciblerational surface whose normalisation is the blowup of P F along its rational points. The main technical resultis the computation of all the cohomology groups of a natural class of line bundles on these fake projectiveplanes, Proposition 3.1. The particular case of this relevant to the construction of exceptional collectionsis proved by using a Galois theoretic argument and specialisation, eventually reducing this to an explicitcomputation on P F .After the first version of this paper was put on arXiv, we learned from L. Katzarkov that the authors of[7] had proved their conjecture for 6 fake projective planes over C , distinct from the ones we have considered,and by different methods. This is included in v2 of [7]. Line bundles on fake projective planes X the formal scheme over Spec( Z ) corresponding to PGL ( Q ) constructed by Mustafin[13] and Kurihara [9]; the reader may also consult [12] for an exposition in the case we use. The irreduciblecomponents of the special fibre of X are in bijection with the vertices of the Bruhat–Tits builiding ofPGL , Q and each of these components is isomorphic to the surface B obtained by blowing up P F at allits F -rational points. There is a faithful action of PGL ( Q ) on X which is transitive on the irreduciblecomponents of the special fibre and the stabilizer of each component is isomorphic to PGL ( Z ). This actionrestricts to a faithful action on b Ω , the two dimensional Drinfeld upper half space over Q , which is thegeneric fibre (as a rigid analytic space over Q ) of X . Moreover, b Ω is an admissible open subset (in thesense of rigid analytic geometry) of P Q and the PGL ( Q ) action on it is compatible with this inclusion andthe natural action of PGL ( Q ) on P Q .If Γ is a discrete torsion-free cocompact subgroup of PGL ( Q ), then one may form the quotient formalscheme X / Γ. The dualising sheaf ω X descends to a line bundle on X / Γ which is ample on the special fibre,hence by Grothendieck’s existence theorem, X / Γ is the formal completion of a unique regular projectivescheme over Spec( Z ). If Γ acts transitively on the irreducible components of X , then Mumford shows thatthe generic fibre M of M , the projective scheme over Spec( Z ) corresponding to X / Γ, is a fake projectiveplane. The special fibre M of M is an irreducible surface over F whose normalisation is isomorphic to B . Lemma 2.1.
Let F be any finite unramified extension of Q . For all line bundles L on M F we have | c ( L ) . In particular, ω M does not have a cube root defined over F .Proof. Let A F be the ring of integers of F . Since M is regular and F is unramified, it follows that M F := M ⊗ Z A F is also regular. Since M is geometrically irreducible, it follows that the restrictionmap Pic( M F ) → Pic( M F ) is an isomorphism. In particular, any line bundle L on M F extends uniquely toa line bundle L on M F . Since M is a fake projective plane, Pic( M F ) modulo its (finite) torsion subgroupis isomorphic to Z . Since c ( ω M ) = 9 = 0, it follows that there exists a positive integers m, n so that L ⊗ m is isomorphic to ω ⊗ n M .From the computations of [12, p. 238], it follows that the degree of ω M , which is the restriction of ω M to M , on the image of any exceptional divisor in B is 1. Since the degree of L on the curve must also bean integer, it follows from L ⊗ m ∼ = ω ⊗ n M that m | n , so 9 = c ( ω M ) | c ( L ) . (cid:3) ω M does have cube roots defined over certain cubic extensions of Q . The existenceof cube roots over some extension also follows from the classification results of Prasad and Yeung [14], andthe basic principle of our proof is the same. However, the argument below is elementary and is essentiallyimmediate from the construction of the groups Γ. More importantly, it also gives precise information aboutthe field of definition of the cube roots.Let Q be an algebraic closure of Q . There is a natural surjection q : SL ( Q ) → PGL ( Q ); we let G denote the group q − (PGL ( Q )) so that there is a short exact sequence1 → µ ( Q ) → G → PGL ( Q ) → . Suppose Γ ′ is a subgroup of G mapping isomorphically onto Γ by q and so that all elements of Γ ′ are definedover a finite extension k of Q . Then Γ ′ acts on b Ω ⊗ Q k via q and the base change of the action of Γ on b Ω .We denote by O ( −
1) the inverse of the standard generator of Pic( P Q ) as well as its restriction to b Ω and b Ω ⊗ Q k . Since the line bundle O ( −
1) on P k has a natural SL ,k linearisation, the inclusion of Γ ′ in SL ( k )gives rise to a linearisation of O ( −
1) on b Ω ⊗ Q k , so it descends to a line bundle L on M k = ( b Ω ⊗ Q k ) / Γ.Since ω M is the line bundle on M corresponding to O ( −
3) with its induced linearisation, it follows that L ⊗ ∼ = ω M k .2.3. We now check that subgroups Γ ′ as above exist for all the three fake projective planes and alsodetermine the extensions k corresponding to these subgroups. This requires explicit knowledge of the groupsΓ, so we have to consider Mumford’s example and the CMSZ examples separately. However, in both casesΓ is contained in a larger lattice Γ which can be lifted to a lattice Γ ′ in G in a very simple way. XCEPTIONAL COLLECTIONS ON FAKE PROJECTIVE PLANES 3
The Mumford lattice.
Mumford’s lattice Γ is a sublattice of index 21 of the subgroup Γ of PGL ( Q )generated by the images of the matrices σ = λ −
10 1 − , τ = λ λ and ρ = λ − λ /
20 0 λ / where λ is a certain element of Q of the form 2 u with u a unit [12, § σ, τ ∈ SL ( Q ) while det( ρ ) = λ / µ be a cube root of det( ρ ), k = Q ( µ ) and ρ ′ = µ − ρ . Clearly ρ ′ ∈ SL ( k ) and the image in PGL ( k ) of the subgroup Γ ′ of SL ( k ) generated by σ , τ and ρ ′ is equal to Γ . Moreover, since k does not contain a primitive cube root of 1, the only scalar matrixin Γ is the identity. It follows that Γ ′ maps isomorphically onto Γ . We then let Γ ′ be the inverse image ofΓ in Γ ′ . The three choices for µ give rise to 3 such subgroups Γ ′ . The three subgroups Γ ′ which they giverise to are also distinct since, by Lemma 2.1, none of the Γ ′ can be subgroups of SL ( Q ).2.3.2. The CMSZ lattices.
The lattices constructed by Cartwright, Mantero, Steger and Zappa [4, p. 181]are both sublattices of Γ , the image of the subgroup of GL ( Q ) generated by the elements a = − ( S − /
41 0 10 1 ( S − / and s = − − ( S − / − − ( S − /
40 0 1 where S ∈ Z is the square root of −
15 which is congruent to 1 modulo 4 [4, p. 182]. Clearly, s ∈ SL ( Q )while det( a ) = ( S − /
4. Since ( S − S + 1) = −
16, it follows that in fact the valuation of S − S − / k be the extension of Q obtained by adjoing a cube root asabove and then modifying a , it follows as in the previous case that we get three distinct lifts of Γ (for bothchoices of Γ).2.3.3. In each of the cases discussed above, it can be seen that Hom(Γ , µ ) has order three, so the threelifts that we have constructed are in fact all.3. Cohomology of line bundles
Henceforth, M denotes any one of the fake projective planes considered earlier. We let K be the Galoisclosure of any of the cubic extensions k of Q of the previous section, so it is a Galois extension of Q with Galois group S , containing the unramified quadratic extension F = Q ( ζ ), with ζ + ζ + 1 = 0; theextension K/F is totally ramified. We will compute the dimensions of the cohomology groups of all linebundles on M K contained in the subgroup P of Pic( M K ) generated by the cube roots of ω M K constructedabove and the line bundles of order two coming from characters of Γ in Q × . Using the explicit description ofthe groups Γ given in [12] and [4] or the figures at the end of [8], one can see that this group is isomorphic to Z × Z / × ( Z / for the Mumford surface and one of the CMSZ surfaces and to Z × Z / L , deg( L ), to be the positive square root of c ( L ) if L is ample and its negativeotherwise. Since ω M is ample, in order to compute the cohomology of all line bundles it suffices, by Serreduality, to consider only line bundles L which are ample. Proposition 3.1.
For any L ∈ P let h i ( L ) denote the dimension of H i ( M K , L ) . Let L ∈ P be ample andlet d = deg( L ) .(1) If d ∈ { , } , then h i ( L ) = 0 for all i .(2) If d = 3 , then h i ( L ) = 0 for i = 0 , and h ( L ) = 1 if L ∼ = ω M else h ( L ) = 1 and h i ( L ) = 0 for i = 1 , .(3) If d > , then h ( L ) = ( d − d − / and h i ( L ) = 0 for i = 1 , .Furthermore, h ( L ) = 0 for any L ∈ P . It seems reasonable to expect that a similar result holds for all line bundles on all fake projective planes.
NAJMUDDIN FAKHRUDDIN
Proof.
Since c ( ω M ) = 9, we have c ( L ) · c ( ω M ) = 3 d . Moreover, χ ( O M ) = 1, so by the Riemann–Rochtheorem for surfaces we have χ ( L ) = ( d − d − / d > L ⊗ ω − is ample so the result in this case follows from the Kodaira vanishing theorem.We now consider the case d = 2 and assume that h ( L ) >
0. By the definition of P , L is isomorphic to L ⊗ ⊗ T , where L is one of the cube roots of ω M K constructed in § T is either a trivial line bundle orhas order 2. The action of Gal( K/ Q ) on P preserves all the bundles of order 2 and permutes the three cuberoots of ω M K , so it follows that L has two other Galois conjugates, say L ′ and L ′′ , and L ⊗ L ′ ⊗ L ′′ ∼ = ω ⊗ M K ⊗ T .Let k/ Q be the cubic extension over which L is defined and let D be any divisor (defined over k ) inthe linear system corresponding to L . It follows that D has two Galois conjugates, D ′ and D ′′ , such that O ( D ′ ) ∼ = L ′ and O ( D ′′ ) ∼ = L ′′ . Then C K := D + D ′ + D ′′ is a Galois invariant divisor in the linear systemcorresponding to ω ⊗ M K ⊗ T , so it is the base change of a divisor C on M . Let C be the specialisation of C in M . Since T corresponds to a character of Γ of order at most 2, the specialisation of T is trivial, hence C is a Cartier divisor in the linear system corresponding to ω ⊗ M .We claim that the Weil divisor associated to C is divisible by 3 in the group of Weil divisors on M . Itsuffices to prove this over M , F , and since F/ Q is unramified (so specialisation commutes with base change)it is enough to consider the specialisation of C F as a divisor on M F ⊂ M R , where R is the ring of integersin F .Observe that no prime divisor in the support of D can be preserved by Gal( K/F ) ∼ = Z /
3, since anysuch divisor would descend to a divisor of degree 1 or 2 defined over F which is not possible by Lemma 2.1(since F/ Q is unramified). It follows that each prime divisor Z in the support of C F splits into a sum ofthree prime divisors over K . We show that the specialisation of any such Z has multiplicity 3 along eachcomponent of its support.Let Z be the Zariski closure of Z in M R and f Z its normalisation. Since Z splits into 3 components over K , the function field of Z must contain K . Thus, since f Z is normal, the morphism f Z → Spec( R ) factorsthough Spec( S ), where S is the ring of integers of K . The specialisation of f Z is given by the valuations of auniformizer of R with respect to the discrete valuation corresponding to the generic point of each irreduciblecomponent of the closed fibre. Since K/F is ramified and of degree 3, so a uniformizer in R is (up to aunit) the cube of a uniformizer in S , it follows that each irreducible component of f Z × R F has multiplicitydivisible by 3. Since specialisation commutes with proper pushforward [6, Proposition 20.3], the same holdsfor the irreducible components of Z × R F , thereby proving the claim.From the claim we see that all the irreducible components of C have multiplicity divisible by 3 in thecorresponding Weil divisor. By Lemma 3.2 below no such divisor exists. Thus, if d = 2, we must have h ( L ) = 0.If d = 1 and h ( L ) = 0, then h ( L ⊗ ) = 0. It follows from the d = 2 case already considered that this isnot possible.If d ∈ { , } , then L ⊗− ⊗ ω M K has degree 3 − d ∈ { , } , so it follows from the above and Serre dualitythat h ( L ) = 0. Since χ ( L ) = 0, it follows that we also have h ( L ) = 0.If d = 3 and L ∼ = ω M , then the statements follow since p g ( M ) = q ( M ) = 0. Otherwise, L = L ⊗ ω − M is anon-trivial torsion line bundle, so by Serre duality h ( L ) = 0. Since χ ( L ) = 1, it follows that h ( L ) >
0. Since L ∈ P , the line bundle L corresponds to a non-trivial homomorphism f from Γ into K × . Let π : M ′ → M K be the cover of M K corresponding to Ker( f ). Then π ∗ ( O M ′ ) is isomorphic to ⊕ e − i =0 ( L ) ⊗ i , where e is theorder of L in Pic( M K ). From the projection formula, π ∗ ( ω M ′ ) is isomorphic to ⊕ e − i =0 ω M ⊗ ( L ) ⊗ i . We have χ ( ω M ′ ) = eχ ( ω M ) and q ( M ′ ) = 0 by [12, p. 238], so p g ( M ′ ) = e −
1. It follows that P e − i =0 h ( ω M ⊗ ( L ) ⊗ i ) = e −
1. Since all summands except for i = 0 must be at least 1, it follows that they are all equal to 1. Inparticular, h ( ω M ⊗ L ) = h ( L ) = 1. Since χ ( L ) is also 1, it follows that h ( L ) = 0.The last statement follows from the previous claims if L is ample. If L is not ample, then ω M K ⊗ L − isample so the claim follows by Serre duality. (cid:3) Lemma 3.2.
There is no (Cartier) divisor C in the linear system on M associated to ω ⊗ M such that theassociated Weil divisor has multiplicity divisible by along each geometric component of its support.Proof. Suppose such a divisor C exists. Let ν : B → M be the normalisation morphism and π : B → P F the morphism blowing up all F -rational points of P F . The morphism ν identifies each exceptional curve E i with the strict transform of a F -rational line F i in P F in a way that depends on the particular fake XCEPTIONAL COLLECTIONS ON FAKE PROJECTIVE PLANES 5 projective plane under consideration. According to [12, p. 238], ν ∗ ( ω M ) is equal to O B ( π ∗ H − P i =1 E i ),where H is the class of a line in P F . It follows that e C := ν ∗ ( C ) = π ∗ ( C ) − P i =1 E i (as Cartier divisors)where C is a curve of degree 8 in P F passing through all the F -rational points and having multiplicity atleast 2 at each such point.If ν ( E i ) ⊂ M is contained in the support of C for some i , then both E i and F i must be containedin the support of e C . Moreover, since the multiplicity of this component is divisible by 3, the sum of themultiplicities of E i and F i in e C must be divisible by 3.First suppose that the support of C is not contained in the double point locus of M . Then C has acomponent which is not a rational line and the multiplicity of each such component must be be divisible by3; let C ′ be the union of all such components (with multiplicity). Since deg( C ) = 8, it follows that deg( C ′ )is 3 or 6. If deg( C ′ ) = 3, then C ′ must be a triple (rational) line which is not possible by assumption. Ifdeg( C ′ ) = 6 then the corresponding reduced curve must be a conic and C ′′ := C − C ′ is either a doubleline or a union of two distinct lines.Suppose C ′′ is a double line. Since C must contain all rational points it follows that C ′ must contain atleast 4 rational points. Since a smooth conic over F has only 3 rational points and a singular (irreducible)conic has only 1 rational point, it follows that C ′ must be a union of two rational lines which is a contradiction.Suppose C ′′ is a union of two distinct rational lines, say F i and F j . It then follows that both E i and E j must have multiplicity at least 2 in e C , so the corresponding points p i and p j in P F have multiplicity at least4 in C . Since the union of F i and F j contains 5 rational points, C ′ must be a smooth conic. Since a smoothconic contains 3 rational points, it follows that ( F i ∪ F j ) ∩ C ′ contains at most 1 rational point. Since F i and F j have multiplicity 1 in C , it follows that there is at most one point of multiplicity at least 4 on C ,a contradiction.It remains to consider the case that C is contained in the singular locus of M , so C is a union of rationallines (with multiplicity). Since every rational point must lie on C , C must have at least three irreduciblecomponents.If there are exactly three components, then there must be a (unique) point p contained in all of them sincethis is the only way that the union of three lines in P F can contain all rational points. Since deg( C ) = 8, C has multiplicity 8 at p , so the exceptional divisor E p has multiplicity 6 >
0, in C . It follows that thestrict transform of the corresponding line F p must also be in the support of e C , so F p must be one of the linesin the support of C and its multiplicity is divisible by 3. If the multiplicity is 3, then since deg( C ) = 8one of the other lines must also have multiplicity ≥
3. But then the 5 points on the union of these lines willhave multiplicity > C , so the corresponding exceptional divisors must have multiplicity > C . Butthis implies that C must contain the 5 rational lines corresponding to these exceptional divisors which is acontradiction. If the multiplicity of F p in C is 6 then both the other lines must have multiplicity one, butthen there would exist rational points on C of multiplicity 1 which is not possible. Thus C cannot havethree components.Suppose C has exactly 4 components. Since all rational points must lie on C , one sees that there isonly one such configuration (up to automorphisms of P F ) consisting of the union of all three lines passingthrough a distinguished point together with one other line F . It follows that exactly one rational point lieson 3 lines, 3 rational points lie on 2 lines each and the remaining 3 rational points on a single line each.These three lines must have multiplicty at least 2, since the multiplicity of any rational point on C mustbe at least 2. Furthermore, these three lines intersect F in distinct points, so the rational points p , p and p on this line have multiplicity > C . It follows that the exceptional divisors E p , E p and E p corresponding to these points must be contained in the support of e C , so the corresponding lines F p , F p and F p must be contained in the support of C . If F has multiplicity 2, then the multiplicity of C at all p i , i = 1 , , E p i in C is 2 so the multiplicity of each F p i in C must be congruent to 1 modulo 3. Since 2 F must have multiplicity 1 and so one of the other lines must have multiplicity 3. Butthen there are at least 5 points on C with multiplicity at least 3 which implies that C has at least 5 linesin its support, also a contradiction.Suppose that there are 5 lines in the support of C so there are exactly 5 rational points on C withmultiplicity >
2. The union of any three lines contains at least 6 rational points so at most 2 of the lines
NAJMUDDIN FAKHRUDDIN are multiple and the multiplicities must be one line of multiplicity 2 and another of multiplicity 3 or a singleline of multiplicity 4.There is a unique configuration of 5 lines up to automorphism. In such a configuration, there is one pointlying on a single line, 4 on two lines each and the remaining 2 lie on 3 lines. The line F containing the point p which lies on only one line must be multiple, since the multiplicity of each point must be at least 2. If themultiplicity is 4, then there is no other multiple component, so there are 4 points of multiplicity 2 which isnot possible.If the multiplicity of F is 2, then there is another component of multiplicity 3. Then there are still threerational points of multiplicity 2 which is not possible.If the multiplicity of F is 3, then there is another component of multiplicty 2. Then the multiplicities ofthe points are 2 , , , , , , , , , , , ,
4. Since thesum of the multiplicity of any exceptional divisor and the line corresponding to it is divisible by 3, it followsthat there must be 4 multiple lines which is a contradiction.Suppose there are 6 lines in the support of C . Then there are 3 collinear points which lie on 2 lines eachand the remaining 4 points lie on 3 lines each. Also, there must be exactly 6 rational points with multiplicityat least 3 on C . Thus, there are exactly two points in the 3 element set of points lying on only 2 linesand a line containing each point of multiplicity 2. The intersection point of these two lines will then havemultiplicity 5 and the two other points on them have multiplicity 4. Thus, the multiplicities of the rationalpoints on C must be 2 , , , , , , > C is the union of all 7 rational lines. Since each rationalpoint lies on 3 lines and exactly one of the lines, call it F , must be double, the 4 rational points not on F have multiplicity 3 on C . By the congruence argument as before, this implies that there must be 4 multiplelines, a contradiction. (cid:3) Exceptional collections
Let X be a smooth projective variety over a field K . A sequence of objects E , E , . . . , E n of D b ( X ), thebounded derived category of coherent sheaves on X , is called an exceptional collection if Hom( E j , E i [ k ])is non-zero for j ≥ i and k ∈ Z iff i = j and k = 0, in which case it is one dimensional. Galkin,Katzarkov, Mellit and Shinder have conjectured [7, Conjecture 3.1] that if X is an n -dimensional fakeprojective space over C such that the canonical bundle ω X has an ( n + 1)-th root O X ( − O X , O X ( − , . . . , O X ( − n ) form an exceptional collection. If X is a surface, they observe that toprove this it suffices to show that H ( X, O X (2)) = 0. This conjecture appears to be difficult to prove ingeneral, but the computations of the previous section lead to the following: Theorem 4.1. (a) Let M be a -adically uniformised fake projective plane over Q and let L , L ∈ P beline bundles of degree − and − . Then the sequence of line bundles O M K , L , L is an exceptionalcollection.(b) Let B = h O M K , L , L i , the subcategory of D b ( X ) generated by O M K , L and L and let A = B ⊥ .Then HH • ( A ) = 0 , i.e., A is a quasiphantom category. Moreover, the dimensions of the vector spaces HH t ( A ) , t ≥ , are given by the sequence , , , , , , , , . . . . In particular, the product of any twoelements of HH • ( A ) of positive degree is .Proof. For ease of notation, we denote O M K by L . Then Hom( L j , L i [ k ]) = H k ( M K , L i ⊗ L ⊗− j ). If j > i ,then the degree of L i ⊗ L ⊗− j is 1 or 2 so the cohomology groups vanish for all k by Proposition 3.1. If i = j ,then L i ⊗ L ⊗− j ∼ = O M K , and since p g ( M ) = q ( M ) = 0, (a) follows.The first part of (b) follows from the fact that the Betti numbers b i ( M ) are equal to 1 for i = 1 , , i together with Kuznetsov’s additivity theorem for Hochschild homology [10, Theorem 7.3].To compute the Hochschild cohomology of A we first compute that of D b ( X ). Recall, for example from[10], that this is given by HH t ( D b ( X )) = n M p =0 H t − p ( M, ∧ p T M ) . XCEPTIONAL COLLECTIONS ON FAKE PROJECTIVE PLANES 7
The only nonzero cohomology group of O M is in degree 0 where the dimension is 1. We have H ( M, T M ) = 0since M is of general type and H ( M, T M ) = 0 since M is (infinitesimally) rigid. By the Hirzebruch-Riemann-Roch theorem we then have h ( M, T M ) = χ ( T M ) = Z M ch ( T M ) · td ( T M )= Z M (2 + c ( T M ) + ( c ( T M ) − c ( T M )) / · (1 + c ( T M ) / c ( T M ) + c ( T M )) / Z M ( c ( T M ) + c ( T M )) / c ( T M ) / c ( T M ) − c ( T M )) / c ( T M ) = 9 and c ( T M ) = 3.The only non-zero cohomology group of ∧ T M = ω − M is in degree 2 and h ( M, ω − M ) = h ( M, ω M ) = 10by Proposition 3.1. Thus, the dimensions of the vector spaces HH t ( D b ( X )), for t ≥
0, are given by thesequence 1 , , , , , , , · · · .In the article [11], Kuznetsov defines the normal Hochschild cohomology of B in D b ( X ), denoted by N HH • ( B , D b ( X )), for B any admissible subcategory of D b ( X ) with X a smooth projective variety, whichsits in a distinguished triangle(4.1) N HH • ( B , D b ( X )) → HH • ( D b ( X )) → HH • ( A ) +1 −→ , where A = B ⊥ . Moreover, if B is generated by an exceptional collection E , E , . . . , E n then he constructsa spectral sequence converging to N HH • ( B , D b ( X )) whose E − p,q term is given by M ≤ a
10 = 30. • p = 1. Then dim( Ext ( E i , E j )) = is 3 if ( i, j ) = (1 , 2) or (2 , 3) and 6 if (1 , j ) = (1 , Ext ( E j , S − E i )) = 6 if ( i, j ) = (1 , 2) or (2 , 3) and 3 if ( i, j ) = (1 , 3) and all other groupsare 0. Thus, we must have q = 2 + 4 = 6 and each of the three possibilities for ( i, j ) contributes asummand of dimension 3 · · E − , ) = 3 · 18 = 54. • p = 2. We must have ( a , a , a ) = (1 , , q = 2 + 2 + 4 = 8 and dim( E − , ) = 3 · · E , , E − , and E − , are the only non-zero terms in the spectral sequence it follows that itdegenerates at E . Consequently, the dimension of N HH t ( B , D b ( M )), for t ≥ 0, are given by the sequence0 , , , , , , , , , . . . .From the computations of HH • ( D b ( M )) and N HH • ( B , D b ( M )) above, it follows from (4.1) that tocompute the dimensions of all HH t ( A ) it suffices to compute the rank of the map from HH ( B , D b ( M ))to HH ( D b ( M )). Since E = O M , it follows from the case B = h O X i , where X is any smooth projectivevariety, considered by Kuznetsov [10, Theorem 8.5], and the functoriality of restriction maps on Hochschildcohomology for admissible subcategories, that this map is surjective. (cid:3) We note that the dimensions of the Ext groups in our exceptional collection satisfy the same duality withrespect to those for the exceptional collection O P ( − , O P ( − , O P on P as discussed by Alexeev andOrlov [1, p. 757] in the case of Burniat surfaces.As mentioned in the introduction, Bloch’s conjecture on zero cycles is still not known for any fake projectiveplane. Based on standard motivic conjectures, we make the following NAJMUDDIN FAKHRUDDIN Conjecture 4.2. Let X be a smooth projective variety over a field k of characteristic zero. If A ⊂ D b ( X ) is an admissible subcategory with HH • ( A ) = 0 then K ( A ) is a torsion group. Although the usual motivic conjectures are notoriously intractable, we hope that the extra structure heremight make this more accessible. If true, together with Theorem 4.1 it would clearly imply Bloch’s conjecturefor the fake projective planes we have considered. Remark 4.3. Because of the existence of 2-torsion in P for two out of the three 2-adically uniformised fakeprojective planes, we get more exceptional collections in these cases than was conjectured in [7]. Moreover,this suggests that for general fake projective planes the condition on the existence of cube roots of thecanonical bundle might be unnecessary. Remark 4.4. Recently Allcock and Kato [2] have found a cocompact lattice Γ in PGL ( Q ) containingnon-trivial torsion such that b Ω / Γ is still a fake projective plane and have suggested that there might beother examples as well. Our methods do not immediately apply to their example, but the results of § p -adic uniformisation completely new methods would be needed. 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Mustafin , Non-Archimedean uniformization , Mat. Sb. (N.S.), 105(147) (1978), pp. 207–237, 287.[14] G. Prasad and S.-K. Yeung , Fake projective planes , Invent. Math., 168 (2007), pp. 321–370. School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India E-mail address ::