Fake projective planes, automorphic forms, exceptional collections
aa r X i v : . [ m a t h . AG ] F e b Acyclicity of non-linearizable line bundles on fake projective planes
SERGEY GALKIN, ILYA KARZHEMANOV, AND EVGENY SHINDER
Аннотация.
On the projective plane there is a unique cubic root of the canonical bundle and this root is acyclic.On fake projective planes such root exists and is unique if there are no 3-torsion divisors (and usually exists butnot unique otherwise). Earlier we conjectured that any such cubic root (assuming it exists) must be acyclic. In thepresent note we give a new short proof of this statement and show acyclicity of some other line bundles on thosefake projective planes with at least automorphisms. Similarly to our earlier work we employ simple representationtheory for non-abelian finite groups. The novelty stems from the idea that if some line bundle is non-linearizablewith respect to a finite abelian group, then it should be linearized by a finite (non-abelian) Heisenberg group. Ourargument also exploits J. Rogawski’s vanishing theorem and the linearization of an auxiliary line bundle. This work was done in August 2014 as a part of our “Research in Pairs” at CIRM, Trento (Italy).1.
Introduction
In [10] Mumford gave an ingenious construction of a smooth complex algebraic surface with K ample, K = 9 , p g = q = 0 . All such surfaces are now known under the name of fake projective planes . They have been recentlyclassified into isomorphism classes by Prasad – Yeung [11] and Cartwright – Steger [4]. Universal cover of anyfake projective plane is the complex ball and papers [11, 4, 5] describe explicitly all subgroups in the automorphismgroup of the ball which are fundamental groups of the fake projective planes.However, these surfaces are poorly understood from the algebro-geometric perspective, since the uniformizationmaps (both complex and, as in Mumford’s case, -adic) are highly transcendental, and so far not a single fakeprojective plane has been constructed geometrically. Most notably the Bloch’s conjecture on zero-cycles (see [3])for the fake projective planes is not established yet.Earlier we have initiated the study of fake projective planes from the homological algebra perspective (see [6]).Namely, for P the corresponding bounded derived category of coherent sheaves has a semiorthogonal decomposition, D b ( P ) = hO , O (1) , O (2) i , as was shown by A. Beilinson in [2]. The “easy” part of his argument was in checkingthat the line bundles O (1) and O (2) are acyclic, and that any line bundle on P is exceptional. All these resultsfollow from Serre’s computation. In turn, the “hard” part consisted of checking that O , O (1) , O (2) actually generate D b ( P ) . But for fake projective planes, according to [6], one can not construct a full exceptional collection this way.Anyhow, one still can define an analogue of O (1) , O (2) for some of these planes (cf. below) and try to establishthe “easy part” for them. Then exceptionality of line bundles is equivalent to the vanishing of h , and h , , whichis clear, while acyclicity is not at all obvious. MS 2010 classification : 14J29, 32N15, 14F05.
Key words : fake projective planes, automorphic forms, exceptional collections. e are going to treat acyclicity problem, more generally, in the context of ball quotients and modular forms.We hope that our argument might be useful for proving the absence of modular forms of small weights on complexballs (compare with [13]). In the present paper we study those fake projective planes S whose group of automorphisms A S has orderat least . All these surfaces fall into the six cases represented in Table A below (cf. [5] and [6, Section 6]). Thereone denotes by Π the fundamental group of S , so that S = B / Π for the unit ball B ⊂ CP , and N (Π) denotes thenormalizer of Π in P U (2 , .One of the principle observations is that the group Π lifts to SU (2 , . The lifting produces a line bundle O S (1) ∈ Pic S such that O S (3) := O S (1) ⊗ ≃ ω S , the canonical sheaf of S . Moreover, the preimage ^ N (Π) ⊂ SU (2 , of N (Π) ⊂ P U (2 , acts fiberwise-linearly on the total space Tot O S (1) −→ S , which provides a natural linearizationfor O S (1) (and consequently for all O S ( k ) ). Furthermore, the action of the group Π ⊂ ^ N (Π) is trivial. Thus allvector spaces H ( S, O S ( k )) , k ∈ Z , are endowed with the structure of G -modules, where G := ^ N (Π) / Π . In the sameway one obtains the structure of G -module on H ( S, O S ( k ) ⊗ ε ) for any A S -invariant torsion line bundle ε ∈ Pic S (cf. below).Another observation is the use of vanishing result due to J. Rogawski (see Theorem 3.2 below), which we applyto show that h ( S, ω S ⊗ ε ) = 1 whenever ε = 0 . Twisting H ( S, O S (2) ⊗ ε ) (resp. H ( S, O S (2)) ) by the globalsection of ω S ⊗ ε allows us to prove our main result: Theorem 1.3.
Let S be a fake projective plane with at least nine automorphisms. Then H ( S, O S (2) ⊗ ε ) = H ( S, O S (2)) = 0 for any non-trivial A S -invariant torsion line bundle ε ∈ Pic S . Let us also point out that one can prove Theorem 1.3 directly by employing the arguments from [6] and the onlyfact that the group G is non-abelian (see for details). l or C p T N A S H ( S, Z ) Π lifts? H ( S/A S , Z ) N (Π) lifts? Q ( √−
7) 2 ∅
21 3 G ( Z / yes Z / yes { }
21 4 G ( Z / yes yes C ∅
21 1 G ( Z / yes yes C ∅ Z / Z ) Z / yes Z / no { } Z / Z ) Z / yes no C ∅ Z / Z ) Z / × Z / yes noTable A2. Preliminaries
We begin by recalling the next
Lemma 2.2 (see [6, Lemma 2.1]) . Let S be a fake projective plane with no -torsion in H ( S, Z ) . Then there existsa unique (ample) line bundle O S (1) such that ω S ∼ = O S (3) . et S and O S (1) be as in Lemma 2.2. We will assume for what follows that A S = G or ( Z / . This implies thatone has a lifting of the fundamental group Π ⊂ P U (2 , to SU (2 , (see Table A). Fix a lifting r : Π ֒ → SU (2 , and consider the central extension(2.3) → µ → SU (2 , → P U (2 , → (here µ denotes the cyclic group of order ). Note that since H ( S, Z ) = Π / [Π , Π] does not contain a -torsion inour case, the embedding r is unique .We thus get a linear action of r (Π) on C . In particular, both Bl C = Tot O P ( − and its restriction to theball B ⊂ P are preserved by r (Π) , so that we get the equalityTot O S (1) = ( Tot O P ( − (cid:12)(cid:12) B ) /r (Π) (cf. [9, 8.9]). Further, since there is a natural identification Pic S = Hom (Π , C ∗ ) = Hom ( H ( S, Z ) , C ∗ ) , every torsion line bundle ε ∈ Pic S corresponds to a character χ ε : Π → C ∗ . One may twist the fiberwise r (Π) -actionon Tot O P ( − by χ ε (we will refer to this modified action as r (Π) χ ε ) and obtainTot O S (1) ⊗ ε = ( Tot O P ( − (cid:12)(cid:12) B ) /r (Π) χ ε . Observe that according to Table A there always exists such ε = 0 . This table also shows that in two cases onecan choose ε to be A S -invariant. Note that A S = N (Π) / Π for the normalizer N (Π) ⊂ P U (2 , of Π . Then (2.3) yields a central extension → µ → G → A S → for G := ^ N (Π) /r (Π) and the preimage ^ N (Π) ⊂ SU (2 , of N (Π) . Further, the previous construction of O S (1) is ^ N (Π) -equivariant by the A S -invariance of O S (1) , which gives a linear G -action on all the spaces H ( S, O S ( k )) , k ∈ Z . Similarly, if the torsion bundle ε is A S -invariant, we get a linear G -action on all H ( S, O S ( k ) ⊗ ε ) .Recall next that when A S = G , the bundle O S (1) is A S -linearizable (i. e. the group A S lifts to G and thecorresponding extension splits), and so the spaces H ( S, O S ( k )) are some linear A S -representations in this case (seeTable A or [6, Lemma 2.2]). The same holds for all H ( S, O S ( k ) ⊗ ε ) and any A S -invariant ε .In turn, if A S = ( Z / , then extension G of this A S does not split (see Table A), i. e. G coincides with the Heisenberg group H of order . Again the G -action on H ( S, O S ( k )) (resp. on H ( S, O S ( k ) ⊗ ε ) ) is linear here. Remark . Let ξ, η ∈ G = H be two elements that map to order generators of A S = ( Z / . Then theircommutator [ ξ, η ] generates the center µ ⊂ G and one obtains the following irreducible -dimensional (Schr¨odinger)representation of G : ξ : x i ω − i x i , η : x i x i +1 ( i ∈ Z / , ω := e π √− ) , with x i forming a basis in C . (This representation, together with its complex conjugate, are the only -dimensionalirreducible representations of H .) Observe at this point that according to the discussion above, [ ξ, η ] acts on S All these matters are extensively treated in [9, Ch. 8] rivially and on O S (1) via fiberwise-scaling by ω . Furthermore, given any A S -invariant ε ∈ Pic S , the commutator [ ξ, η ] acts non-trivially on O S ( k ) ⊗ ε whenever k is coprime to . Let us conclude this section by recalling two supplementary results.Firstly, it follows from the Riemann – Roch formula that χ ( O S ( k ) ⊗ ε ) = ( k − k − for all k ∈ Z and ε ∈ Pic S . In particular, we get H ( S, O S ( k ) ⊗ ε ) ≃ C ( k − k − / when k > because H i ( S, O S ( k ) ⊗ ε ) = H i ( S, ω S ⊗ O S ( k − ⊗ ε ) = 0 for all i > by Kodaira vanishing.Secondly, for every τ ∈ A S with only isolated set of fixed points P , . . . , P N and any line bundle L ∈ Pic S satisfying τ ∗ L ∼ = L , one has the Holomorphic Lefschetz Fixed Point Formula (see [1, Theorem 2]): X i =0 ( − i Tr τ (cid:12)(cid:12) H i ( S,L ) = N X i =1 Tr τ | L Pi (1 − α ( P i ))(1 − α ( P i )) , where α ( P i ) , α ( P i ) are the eigenvalues of τ acting on the tangent spaces T S,P i to S at P i (resp. L P i are the closedfibers of L at P i ). 3. Automorphic forms of higher weight
We retain the notation of Section 2. Fix some A S -invariant torsion line bundle ε .Recall that fundamental group Π of S is a torsion-free type II arithmetic subgroup in P U (2 , (see [8, 14]).The same applies to any finite index subgroup of Π . In particular, for the unramified cyclic covering φ : S ′ −→ S associated with ε one has [Π : π ( S ′ )] < ∞ and the following important result takes place (without any assumptionon A S and invariance of ε ): Theorem 3.2 (see [12, Theorem 15. 3. 1]) . The surface S ′ is regular. That is rank H ( S ′ , Q ) := q ( S ′ ) = 0 . We have S ′ = Spec S m − M i =0 ε i ! , where m is the order of ε ∈ Pic S , and hence φ ∗ O S ′ = m − M i =0 ε i . From Leray spectral sequence and Theorem 3.2 we deduce q ( S ′ ) = h ( S, O S ′ ) = h ( S, φ ∗ O S ′ ) = m − M i =0 h ( S, ε i ) . Thus h ( S, ε i ) = 0 for all i ∈ Z . Lemma 3.3. h ( S, ω S ⊗ ε i ) = 1 for all i not divisible by m . We are using the notation ε i := ε ⊗ i . roof. Indeed, we have χ ( S, ω S ⊗ ε i ) = 1 (see ) and also h ( S, ω S ⊗ ε i ) = h ( S, ε i ) = 0 = h ( S, ε i ) = h ( S, ε − i ) = h ( S, ω S ⊗ ε i ) by Serre duality, which gives h ( S, ω S ⊗ ε i ) = χ ( S, ω S ⊗ ε i ) = 1 . (cid:3) Let τ ∈ A S be any element of order with a faithful action on S . Choose any linearization of the action of τ on Tot O S (1) . Lemma 3.5.
The automorphism τ has only three fixed points, and in the holomorphic fixed point formula alldenominators coincide (and are equal to ) and all numerators are three distinct rd roots of unity. In other words,one can choose a numbering of the fixed points P i and weights α j ( P i ) in such a way that the following holds: q α j ( P i ) = ω j for all i, j ; q w i := Tr τ | O S (1) Pi = ω i for all i .Moreover, for any τ -invariant ε ∈ Pic S of order m coprime to we can choose a linearization such that Tr τ | ε Pi =1 . In particular, Tr τ | ( O S ( k ) ⊗ ε l ) Pi = ω ik for all i, k, l .Proof. The claim about P i and α j ( P i ) follows from [7, Proposition 3.1].Further, we have V := H ( S, O S (4)) ∼ = C (see ). Let v , v , v be the eigen values of τ acting on V .Then v i = 1 for all i and Tr τ (cid:12)(cid:12) V = v + v + v . At the same time, as follows from definitions and (with N = 3 , L = O S (4) ), we have Tr τ (cid:12)(cid:12) V = w + w + w (1 − ω )(1 − ω ) = w + w + w The latter can be equal to v + v + v only when all v i (resp. all w i ) are pairwise distinct (so that both sums arezero). This is due to the fact that the sum of three rd roots of unity has the norm ∈ { , √ , } and is zero iff allroots are distinct.Finally, since ( m,
3) = 1 , we may replace ε by ε , so that the action of τ on the closed fibers ε P i is trivial. Thelast assertion about O S ( k ) ⊗ ε l is evident. (cid:3) Recall that the group G from acts linearly on all spaces V := H ( S, O S ( k ) ⊗ ε ) . Proposition 3.6.
Let G = H . Then for k > the following holds (cf. Table B below): q for k ≡ we have V = V ⊕ C [( Z / ] a as G -representations, where a := k k − and V ≃ C (resp. C [( Z / ] ) is the trivial (resp. regular) representation; q for k ≡ we have V = V ⊕ ( k − k − / as G -representations, where V is an irreducible -dimensionalrepresentation of H ; q for k ≡ we have V = V ⊕ ( k − k − / as G -representations, where V is the complex conjugate to V above. k mod 3 0 1 2 H ( S, O S ( k ) ⊗ ε ) V ⊕ C [( Z / ] a V ⊕ ( k − k − / V ⊕ ( k − k − / Table B
Proof.
Suppose that k ≡ . Then, since every element in H has order , applying Lemma 3.5 to anynon-central τ ∈ G we obtain (1 − α ( P i ))(1 − α ( P i )) = 3 for all i and Tr τ (cid:12)(cid:12) V = 1 (cf. ).Further, the element [ ξ, η ] ∈ G from Remark 2.5 acts trivially on O S ( k ) (via scaling by ω k = 1 ). Also, sincethe order of any ε = 0 is coprime to (see Table A) and ε is flat (cf. its construction in ), from Lemma 3.5we find that [ ξ, η ] acts trivially on O S ( k ) ⊗ ε , hence on V as well. This implies that the G -action on V factorsthrough that of its quotient ( Z / . Then the claimed decomposition V = V ⊕ C [( Z / ] a follows from the factthat a = dim V = Tr [ ξ, η ] (cid:12)(cid:12) V and that Tr τ (cid:12)(cid:12) V = 1 for all non-central τ ∈ G .Let now k ≡ (resp. k ≡ ). Then it follows from Remark 2.5 that [ ξ, η ] scales all vectors in V = H ( S, O S ( k ) ⊗ ε ) ∼ = C ( k − k − / by ω k = 1 . Furthermore, since Tr ξ (cid:12)(cid:12) V = 0 = Tr η (cid:12)(cid:12) V according to Lemma 3.5and , all irreducible summands of V are faithful G -representations, hence isomorphic to V (resp. to V ). Thisconcludes the proof. (cid:3) Proposition 3.7.
Let G ⊃ A S = G . Then for k > and V = H ( S, O S ( k ) ⊗ ε ) we have the following equalityof (virtual) G -representations V = C [ G ] ⊕ a k ⊕ U k for some a k ∈ Z expressed in terms of dim V , where U k depends only on k mod 21 and is explicitly given in thetable below, with rows (resp. columns) being enumerated by k mod 3 (resp. k mod 7 ) C V ⊕ V ⊕ C V ⊕ V ⊕ C C V ⊕ ⊕ V ⊕ C ( V ⊕ V ) ⊕ ⊕ C V ⊕ V ⊕ ⊕ C or −V ) ⊕ ( −V ) 0 0 ( −V ) ⊕ ( −V ) V V ⊕ V V Proof.
From we obtainTr σ (cid:12)(cid:12) V = ζ k (1 − ζ )(1 − ζ ) + ζ k (1 − ζ )(1 − ζ ) + ζ k (1 − ζ )(1 − ζ ) Here ζ := e π √− and σ ∈ G is an element of order . The value Tr σ (cid:12)(cid:12) V depends only on k mod 7 and by directcomputation we obtain the following table: k Tr σ (cid:12)(cid:12) V b − b (Here b := ζ + ζ + ζ and ¯ b = − − b = ζ + ζ + ζ .)Let τ ∈ G be an element of order such that G = h σ, τ i . Recall the character table for the group G (seee. g. the proof of Lemma 4.2 in [6]): Tr σ Tr σ Tr τ Tr τ C V ω ω V ω ω V b b V b b (Here “ — ” signifies, as usual, the complex conjugation and V i are irreducible i -dimensional representations of G .)Now from Lemma 3.5 (cf. ) and the previous tables we get the claimed options for V . This concludes theproof. (cid:3) Proof of Theorem 1.3
We retain the earlier notation.Suppose that G = H and H ( S, O S (2) ⊗ ε ) = 0 . Let also ε = 0 (cf. the end of ). Consider the naturalhomomorphism of G -modules H ( S, O S (2) ⊗ ε ) ⊗ H ( S, O S (2) ⊗ ε ) → H ( S, O S (4) ⊗ ε ) . Since h ( S, O S (4) ⊗ ε ) = χ ( S, O S (4) ⊗ ε ) = 3 (see ), from [9, Lemma 15. 6. 2] we obtain h ( S, O S (2) ⊗ ε ) (cf. the proof of Lemma 4.2 in [6]).On the other hand, there is natural homomorphism of G -modules H ( S, ω S ⊗ ε − ) ⊗ H ( S, O S (2) ⊗ ε ) → H ( S, O S (5)) , and so Lemma 3.3 implies that H ( S, O S (2) ⊗ ε ) is a non-trivial subrepresentation in H ( S, O S (5)) of dimension . But the latter contradicts Proposition 3.6 and Theorem 1.3 follows in this case.Similarly, consider the natural homomorphism of G -modules H ( S, O S (2)) ⊗ H ( S, O S (2)) → H ( S, O S (4)) , where again h ( S, O S (2)) . Then the G -homomorphism H ( S, ω S ⊗ ε ) ⊗ H ( S, O S (2)) → H ( S, O S (5) ⊗ ε ) gives contradiction with Proposition 3.6 whenever H ( S, O S (2)) = 0 . This concludes the proof of Theorem 1.3 for A S = ( Z / . Finally, the case of A S = G is literally the same, with Proposition 3.7 used instead. Alternatively, consider the natural homomorphism of G -modules for any G as in : S H ( S, O S (2) ⊗ ε ) → H ( S, O S (4)) , where ε is -torsion (cf. Table A) and the case ε = 0 is also allowed. Again we have h ( S, O S (2) ⊗ ε ) . Then,applying Propositions 3.6 and 3.7 we conclude that H ( S, O S (2) ⊗ ε ) = 0 , exactly as in the proof of [6, Theorem1.3]. This is another proof of Theorem 1.3. писок литературы [1] M. F. Atiyah, R. Bott: A Lefschetz fixed point formula for elliptic differential operators , Bull. Amer. Math. Soc. (1966), 245 –250.[2] A. A. Be˘ılinson, Coherent sheaves on P n and problems in linear algebra, Funktsional. Anal. i Prilozhen. (1978), no. 3, 68 – 69.[3] Spencer Bloch: Lectures on algebraic cycles , Duke University Mathematics Series. IV (1980). Durham, North Carolina: DukeUniversity, Mathematics Department.[4] Donald Cartwright, Tim Steger:
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Sergey Galkin
National Research University Higher School of Economics (HSE)Faculty of Mathematics and Laboratory of Algebraic GeometryVavilova Str. 7, Moscow, 117312, RussiaIndependent University of MoscowBolshoy Vlasyevskiy Pereulok 11, Moscow, 119002, Russiae-mail:
Ilya Karzhemanov
Kavli IPMU (WPI), The University of Tokyo5-1-5 Kashiwanoha, Kashiwa, 277-8583, Chiba, Japane-mail: [email protected]
Evgeny Shinder