arXiv:1602.06107v2 [math.AG] 3 Dec 2019 icvrdb .Hrerc 1]aon 96 asta h c the that says 1956, around [15] Hirzebruch F. by discovered ..A overview. An 0.0. Contents opc emta ymti space symmetric Hermitian compact naLegroup Lie a in ig ihpootoaiycntn en h odgenus Todd the being constant proportionality with ring, domain Principle usa eeainfrteipeetto fteGoa Com Global the of th implementation within the prepared for Federation was Russian article The 15-51-50045-a). project anbd fti okwsdn nAgs 04 sapr for“ our of part a as 2014, August in done was work this of body Main and (Kakenhi), Society Mathematical Japan from (26887009) NATMRHCFRSO ML EGTFRFK RJCIEPLA PROJECTIVE FAKE FOR WEIGHT SMALL OF FORMS AUTOMORPHIC ON .Atmrhcfrso ml egt w roso vanishin of proofs two weight: small charac of References of forms computation Automorphic weight: large of 5. forms Automorphic centr linearizations, divisors liftings, 4. rigid bundles, and Tautological weight canonical covers of 3. abelian forms for Automorphic numbers Betti odd 2. of Vanishing 1. Introduction 2010 Date e od n phrases. and words Key .K a upre yWrdPeirItrainlResearch International Premier World by supported was K. I. (Simons- Foundation Simons the by part in supported was G. S. ewudlk otakCR,Uiest el td iTrento di Studi degli Università CIRM, thank to like would We eebr4 2019. 4, December : ahmtc ujc Classification. Subject Mathematics D ietosotpof fti ttmn n hwaylct o acyclicity show and least statement at suc this with any of th that proofs if short conjectured unique two we give is Earlier and otherwise). exists unique, root not such planes projective fake On eosrt aihn fodBtinmesfracaso abe of class well. a as for bundle numbers finite, Betti a lin odd by some of linearized vanishing if be demonstrate that should idea it then the group, from abelian stems novelty The groups. finite Abstract. − 1,1,1,2] Let 26]. 16, 17, [15, := K G \ − G Let . ntepoetv ln hr sauiu ui oto h can the of root cubic unique a is there plane projective the On 9 − uoopim.Smlryt u ale okw mlysimp employ we work earlier our to Similarly automorphisms. nti okw aeaohrse oad h categorificatio the towards step another take we work this In othat so , aepoetv lns uoopi om,ecpinlco exceptional forms, automorphic planes, projective Fake G EGYGLI,IY AZEAO,ADEGN SHINDER EVGENY AND KARZHEMANOV, ILYA GALKIN, SERGEY + eacmatgopta hrstesm complexification same the shares that group compact a be X X eacmatcmlxmnfl nfrie yabuddHermi bounded a by uniformized manifold complex compact a be = D F − 42,3N5 14F05. 32N15, 14J29, / := = Γ D K + \ = G Introduction K − rmwr fasbiygatdt h S yteGvrmn o Government the by HSE the to granted subsidy a of framework e / \ Γ yteRsinAaei xelnePoet“ 100”. – “5 Project Excellence Academic Russian the by eiieesPorm(S rn,poet1-1003date 14-21-00053 project grant, (RSF Program petitiveness G where , non-abelian + ntaie(P) ET aa,Gati-i o Scientific for Grant-in-Aid Japan, MEXT, (WPI), Initiative a 1 letninadlclzto 7 localization and extension al n al PUfrhsiaiyadecletwrigcondi working excellent and hospitality for IPMU Kavli and , ulto dual eerhi ar”a IM rno(Italy). Trento CIRM, at Pairs” in Research := U elwhp,teDnsyFudto n FR(research RFBR and Foundation Dynasty the fellowship), IUM oeohrln ude ntefk rjcieplanes projective fake the on bundles line other some f ui otms eaylc ntepeetnt we note present the In acyclic. be must root cubic h incvr,aduealnaiaino naxlayline auxiliary an of linearization a use and covers, lian assof lasses r r o3trindvsr aduulyeit,but exists, usually (and divisors 3-torsion no are ere χ Γ udei o-ierzbewt epc oafinite a to respect with non-linearizable is bundle e ( X, ≃ esneggop o h eodpof ealso we proof, second the For group. Heisenberg , D g ters π O − 1 X ( X ntesneo .Cra.Armral fact, remarkable A Cartan. E. of sense the in X ) nparticular, In . ) and erpeetto hoyfrnon-abelian for theory representation le llections. and nclbnl n hsro sacyclic. is root this and bundle onical F K r rprinli hmsbordism Thom’s in proportional are stemxmlcmatsubgroup compact maximal the is of n X izbuhProportionality Hirzebruch and G c F with aetesm slope same the have G insymmetric tian − osdra Consider . 11.08.14). d NES Research tions. the f 12 11 6 5 1 9 d 2 R c2c1− ν := d , with non-zero denominator because KX and KF are ample. For example, all compact manifolds R c1 − d d 2 uniformizable by the d-dimensional complex ball B := (z1,...,zd) C zi < 1 are proportional to the { ∈ P | | } d d 1) complex projective space P in this sense, and have slope ν = 2(d+1) , whereas all compact manifolds uniformizable 1 d 1 d 1 by the poly-disc (B ) are proportional to the Segre variety (P ) and have slope ν = 2 . From his Proportionality Principle and the Hirzebruch – Riemann – Roch theorem Hirzebruch deduced that χ(X,ω⊗w)= a χ(F,ω⊗w) for all X · F integers w. 2) One computes the Euler characteristic χ(X,L) of any line bundle L Pic X, numerically proportional ∈ to K (that is L⊗q ω⊗p for w := p Q), in a similar way. Let us say then that a global section 0 = s H0(X,L) X ≃ X q ∈ 6 ∈ is a modular form of large weight if w > 1, a modular form of canonical weight if w = 1, and a modular form of small weight (or theta-characteristic) if 0
The preceding discussion was summarized in [16] (cf. [16, 17]) as follows:
Given a cohomology theory H• over Q, admitting the Poincaré duality and Künneth formula, there exists an injective ring homomorphism H•(F, Q) H•(X, Q). Moreover, this homomorphism is → given by an element K H•(X F ) via p p∗K for the projections p onto X and F , respectively. ∈ × 1∗ 2 i In [31], D. Mumford gave an ingenious construction of a smooth complex projective surface with K ample, K2 =9, pg = q = 0. All such surfaces are now known under the name of fake projective planes. They have been recently classified into 100 isomorphism classes by G. Prasad, S.-K. Yeung [32] and D. Cartwright, T. Steger [8]. According to [36] universal cover of any fake projective plane is the complex ball and the papers [32, 8, 9] explicitly describe all subgroups in the automorphism group of the ball which are the fundamental groups of fake projective planes (cf. [18] and [19]). However, these surfaces are poorly understood from the algebro-geometric perspective, since the uniformization maps (both complex and, as in the Mumford’s case, 2-adic) are highly transcendental. Most notably, the S. Bloch’s conjecture on zero-cycles [5] is not established yet for any fake projective plane, and we refer to [2] for a compatible result for the Mumford’s surface. 4)
Earlier we have initiated the study of fake projective planes from the homological algebra perspective (see [13, 12]). Namely, for P2 the corresponding bounded derived category of coherent sheaves has a semi-orthogonal decomposi- tion, Db(P2)= O, O(1), O(2) , as was shown by A. Beilinson in [3]. The “easy” part of his argument was in checking h i that the line bundles O(1) and O(2) are acyclic, and also that any line bundle on P2 is exceptional. All these results follow from Serre’s computation. In turn, the “hard” part consisted of checking that O, O(1), O(2) actually generate Db(P2), which is equivalent to the existence of Beilinson’s spectral sequence — resolution for the structure
1) On the other hand, for any d-dimensional compact X with ample KX S.-T. Yau demonstrated in [36] (as a corollary from the Aubin – Yau construction of the Kähler – Einstein metric) that the slope of X is bounded by the slope of Pd, and the equality holds iff X is uniformizable by Bd. 2) d ⊗w O d−w(d+1) Note that for X = B /Γ this gives χ(X,ωX )= χ(X, X ) d . 3) To keep our notation compatible with the classical one-dimensional case one could also say that s ∈ H0(X,L) has weight w(n + 1) ∈ Z. 4) Recently L. Borisov and J. Keum [6] have given the first explicit algebro-geometric construction of a fake projective plane. 2 sheaf of the diagonal in P2 P2 with terms of the form A ⊠ B (hence, in particular, it corresponds to an object × in Db(P2 P2)). However, as follows from [12], for fake projective planes one can not construct a full exceptional × collection this way. But still one can define analogues of O(1), O(2) for some of these surfaces (see Section 3 below) and try to establish the “easy part” for them. Then exceptionality of line bundles is equivalent to the vanishing of 0,1 0,2 h and h , which is clear due to q = pg =0, while acyclicity is not at all obvious.
We are going to treat acyclicity problem, more generally, in the context of ball quotients and modular forms. Our argument proceeds by proving the absence of modular forms of small weights on complex balls (compare with [17, 35]). Some categorical aspects of this approach are discussed in Remark 0.3 at the end of this section.
0.1. Structure of the paper. In Section 1, we prove Lemma 1.1 about vanishing of some Betti numbers b2l+1(X) for a class of compact Kähler manifolds X, which includes the abelian covers of fake projective spaces, minifolds (see [13]), and many other examples.
Further, in Section 2 we show how the vanishing b1(X)=0 implies regularity of the universal abelian covers of fake projective planes, and thus we deduce the uniqueness of modular forms of canonical weight in Theorem 2.2. This result is a priori: its proof only uses the definition of fake projective planes but not their classification.
The rest of the paper studies fake projective planes S with a faithful action of the automorphism group AS of order 9. ≥
Section 3 sets up the equivariant scene. We recall the construction of tautological line bundle OS(1) on S and discuss how its existence and linearizability (with respect to the automorphism group) are related to the liftings of the fundamental group of S to SU(2, 1). We also show that a non-linearizable faithful action of the abelian group (Z/3)2 gives rise to a linearizable non-faithful action of its central extension G — the Weyl – Heisenberg group
H3. Finally, we recall the holomorphic Lefschetz-type formula of Atiyah – Bott, which is an indispensable tool for explicit computations with equivariant sheaves.
In Section 4, we compute the characters of spaces of modular forms of large weight considered as G-representations (cf. 3.4 below), and in Section 5 we deduce the absence of modular forms of small weight from the computations for large weight.
0.2. Methods, arguments, and relation with other works. The proof of Lemma 1.1 recycles a trick from the proofs of [11, Lemma A.4], [34, Lemma 4], [14, Proposition 1.2], [25, Lemma 2.14] (note that it can be generalized to actions of abelian groups on Hodge – Lefschetz – Frobenius super-commutative algebras).
Our lemma is complementary to various vanishing results and has a peculiar domain of applicability that includes some ramified and unramified abelian covers. One can obtain even stronger results (by using variants of Lefschetz hyperplane theorem) in the case of totally ramified cyclic covers. For example, applying Nori’s variant of weak Lefschetz theorem, V. Kharlamov and V. Kulikov prove the following:
Theorem (see [24, Proposition 1]). If f : Y X is a finite cyclic cover of smooth complex surfaces, totally −→ ramified over a smooth irreducible curve B X with B2 > 0, then the induced homomorphism f : π (Y ) π (X) ⊂ ∗ 1 → 1 is an isomorphism. 5)
5) Similar result extends to higher-dimensional X and Y . 3 Apart from Lemma 1.1 we know of two other approaches which in some cases prove the vanishing of b1 for ball quotients: these are the theorem of J. Rogawski [33, 4] for congruence subgroups and D. Kazhdan’s property (T) [20], as used originally by D. Mumford [31], M.-N. Ishida [18], and subsequently by N. Fakhruddin [10].
Theorem (Rogawski’s vanishing, see [33, Theorem 15. 3. 1], [4]). Let Π be a torsion-free cocompact arithmetic subgroup in PU(2, 1) of type II (see [25, 37]). For S = B/Π, if Π is a congruence subgroup, then b1(S)=0 and the Picard number of S equals 1.
We had sketched the proof of Lemma 1.1 back in 2014 (although we did not use it in the first preprint version of this article because we thought that Rogawski’s vanishing would suffice). We thank an anonymous who has kindly pointed out to us that it is not a priori clear whether the fundamental groups of the abelian covers (or of fake projective planes themselves), which we consider, are congruence subgroups (in fact it is likely that most of these fundamental groups are not congruence).
In turn, the use of Kazhdan’s property (T) is even more intricate — the fundamental group π (S) PU(2, 1) 1 ⊂ itself never has this property, however for 2-adically uniformized fake projective planes abelianization H (S, Z)= π (S)/[π (S), π (S)] is isomorphic to the abelianization Γ /[Γ , Γ ] of a 2-adic lattice Γ P GL (Q ), 1 1 1 1 2 2 2 2 ⊂ 3 2 which satisfies property (T). In the approaches of [31, 18, 10], Kazhdan’s property still has to be used, for there are no alternative ways to see that the 2-adic construction produces surfaces with b2 =1.
In particular, Fakhruddin considers fake projective planes which admit 2-adic uniformization with a torsion-free covering group, and after the formulation of [10, Proposition 3.1] he writes:
It seems reasonable to expect that a similar result holds for all line bundles on all fake projective planes.
Our Lemma 2.1 and Theorem 2.2 confirm that the claim (2) in loc. cit. is indeed true for all fake projective planes. We also note that part (3) there is obvious, whereas part (1) corresponds to the wishful vanishing of theta- characteristics, which is discussed in the second part of our paper. Here the linearization of auxiliary line bundles on S enters the game.
Recall that all fake projective planes with at least 9 automorphisms fall into the six cases represented in Table A below (cf. [9] and [12, Section 6]). There one denotes by Π the fundamental group of S, so that S = B/Π for the unit ball B CP2, and N(Π) denotes the normalizer of Π in PU(2, 1). ⊂ One of the principle observations is that the group Π lifts to SU(2, 1). The lifting produces a line bundle O (1) Pic S such that O (3) := O (1)⊗3 ω — the canonical sheaf of S. Moreover, the preimage S ∈ S S ≃ S N^(Π) SU(2, 1) of N(Π) PU(2, 1) acts fiberwise-linearly on the total space Tot O (1) S, which pro- ⊂ ⊂ S −→ vides a natural linearization for OS(1) (and consequently for all other OS(k)). Furthermore, the action of the group Π N^(Π) is trivial, so that the graded algebra H0(S, O (k)) is endowed with the structure of a G-module, ⊂ M S k∈Z where G := N^(Π)/Π. In the same way one obtains the structure of a G-module on H0(S, O (k) ε) for any choice S ⊗ of G-linearization of an A -invariant torsion line bundle ε Pic0 S (cf. 3.2 below). For k 4, the G-character of S ∈ ≥ H0(S, O (k) ε) can be computed via Hirzebruch proportionality, Kodaira vanishing and Atiyah – Bott holomorphic S ⊗ Lefschetz formula (this is done in Section 4). 4 It turns out that for k 4, 5 and G = G or G = H the vector spaces H0(S, O (k) ε) are direct sums of irre- ∈{ } 21 3 S ⊗ ducible 3-dimensional G-representations. Furthermore, multiplication by 0 = s H0(S,ω ε) (cf. Theorem 2.2) 6 ε ∈ S ⊗ is a monomorphism of G-representations, and it is easy to compute that H0(S, O (2) ε)=0 (see Section 5). Note S ⊗ that Theorem 5.2 and the arguments in the proof of [7, Proposition 3.10] imply that the linear system 2K gives | S| a bicanonical embedding of S (cf. Theorem 3.8 in loc. cit). It also complements the results in [7, Section 4]. Related results are obtained by different methods in [10, 22, 28, 7].
Remark 0.3. When the metric on the Hermitian line bundle OS(1) is concerned one may replace S with its universal 2 cover B P . Then O (1) can be identified (locally) with OP2 (2) ωP2 (cf. 3.2 below). In other words, the global ⊂ S ⊗ sections of OS(1) are those of OP2 (2), “twisted by a functional determinant” (compare with [16, 14]), which suggests 0 2 2 a natural identification H (S, O (1)) = H (P , OP2 ( 2)) (= 0). Similar considerations apply to O (2). Furthermore, S − S since O( 1) are cubic roots of K (one for S and P2) and K admits a natural trivialization (holomorphic volume), ± B i i 2 ∗ which we fix, one may propose a duality H (S, O (j)) H (P , OP2 ( j)) to hold in general, where j Z/3, i 0. S ≃ − ∈ ≥ 6) More to the point, we expect that there exists an object K Db(S P2) (a “Beilinson – Green – Penrose kernel”), ∈ × which implements the asserted duality (cf. the discussion in 0.0). This categorical counterpart of the Proportionality Principle will be developed elsewhere. It would be interesting, though, to compare our present considerations with the twistor correspondence between (Green) functions on differentiable 4-manifolds and sheaves on the respective twistor spaces, as in e.g. [30].
l or C p T1 N #Π AS H1(S, Z) Π lifts? H1(S/AS, Z) N(Π) lifts? Q(√ 7) 2 21 3 G (Z/2)4 yes Z/2 yes − ∅ 21 7 21 4 G (Z/2)3 yes 0 yes { } 21 C 2 21 1 G (Z/2)6 yes 0 yes 20 ∅ 21 C 2 9 6 (Z/3Z)2 Z/14 yes Z/2 no 2 ∅ 3 9 1 (Z/3Z)2 Z/7 yes 0 no { } C 3 9 1 (Z/3Z)2 Z/26 Z/2 yes 0 no 18 ∅ × Table A
1. Vanishing of odd Betti numbers for abelian covers
Lemma 1.1. Let A : N be an action of a finite abelian group A on a compact Kähler manifold N of dimen- sion n = dimR N . If the A-invariant forms of type (k, k) satisfy dim Hk,k(N)A = 1, for an odd k n , then 2 ≤ 2 bk(N)= bk−2(N)= bk−4(N)= ... = b1(N)=0.
Proof. Recall that the space Hl,l(N, R)= Hl,l(N, C) H2l(N, R) consists of vectors in Hl,l(N, C), invariant under ∩ l,l l,l 1,1 A the complex conjugation, and that H (N, C) = H (N, R) R C. Let h H (N, R) be an A-invariant Kähler ⊗ ∈ class (e.g. any A-averaged Kähler class). Note that the class of hl is a non-zero element in Hl,l(N, R)A, 0 l n, ≤ ≤ k p,q l and the Hodge – Lefschetz decomposition H (N, Q) Q C = H (N) h is A-invariant. Then from ⊗ ⊕p+q+2l=k prim ∧ the assumption dim Hk,k(N)A =1 we get Hl,l(N, R)A = Rhl for all l k. ≤
6) i 2 2−i 2 The trace map is obtained from H (P , O 2 (−j)) ≃ H (P , O 2 (j) ⊗ ω 2 ), restriction (local) (O 2 (j) ⊗ ω 2 ) = O (−j) ⊗ ω , P P P P P S S S 2 and the fact that H (S,ωS) ≃ C. 5 p,q Further, since A is abelian, any of its complex representations Hprim(N, C) has a non-zero eigenvector α iff Hp,q = 0. In this case, there is a character χ : A C∗ such that a∗(α) = χ(a) α for all a A, and more- prim 6 → · ∈ −1 p+q over χ(a) = χ(a) because A is finite. Now, as the class h is real, the A-representations Hprim(N) are defined over R, and we have a∗(¯α) = χ(a) α¯. This implies that β := iα α¯ belongs to Hp+q,p+q(N, R)A. Hence we get · ∧ β = C hp+q for some C R. But then C2h2(p+q) = β2 = iα α¯ iα α¯ = (α α) (¯α α¯)=0 (the latter · ∈ ∧ ∧ ∧ ∧ ∧ ∧ equality follows from super-commutativity of the dot-product and because p + q is odd). So C = 0, β = 0, and thus iα α¯ hn−(p+q) = 0. The latter vanishing and the Hodge – Riemann relations imply that α = 0, that is R ∧ ∧ k Hprim(N)=0, and hence bk(N)= bk−2(N). The rest follows by induction.
Example 1.2. Consider the quotient f : N N/A =: M. The pullback f ∗ and the trace f give an identification −→ ∗ of H2k(N, Q)A with H2k(M, Q). Thus the assumption dim Hk,k(N)A = 1 translates into dim Hk,k(M)=1. The case k = 1 can be applied when M is a (fake) projective plane or any other variety with b2 = 1 — it says in particular that all smooth finite abelian covers of M (possibly ramified) have finite first homology.
We do not know any straightforward way to relax the condition for the group A to be abelian. Note for instance that the permutation group Σ , d 1, acts on the Cartesian power Ed of an elliptic curve E, with quotient being d+1 ≥ the projective space Pd.
2. Automorphic forms of canonical weight and rigid divisors
Let f : S′ S be the universal unramified abelian cover of a fake projective plane S. −→ ′ 1 ′ O Lemma 2.1. The surface S is regular (i.e. dim H (S , S′ )=0).
′ Proof. By Example 1.2, this is a particular case of Lemma 1.1 applied to N = S , A = H1(S, Z), with M = N/A = S and H1,1(S′)A = H1,1(S) R. ≃
Let ε be a non-trivial torsion line bundle on S.
Theorem 2.2. The linear system ω ε consists of a unique divisor D . More precisely, we have | S ⊗ | ε H1(S,ω ε)= H2(S,ω ε)=0 and H0(S,ω ε)= Cs , with (s ) = D . S ⊗ S ⊗ S ⊗ ε ε 0 ε
Proof. All abelian covers are constructed as relative spectra, S′ = Spec ε, so that f O = ε. S M ∗ S′ M ε∈Pic0 S ε∈Pic0 S Then Lemma 2.1 and Leray spectral sequence give
0= H1(S′, O )= H1(S,f O )= H1(S,ε). S′ ∗ S′ M ε∈Pic0 S Thus H1(S,ε)=0 for all ε Pic0 S. This vanishing combined with Serre duality Hk(S,ω ε) = H2−k(S,ε∗)∗ ∈ S ⊗ yields h0(S,ω ε)+ h0(S,ε∗)= h0(S,ω ε)+ h0(S,ε∗) h1(S,ε∗)= χ(S,ε∗)=1. S ⊗ S ⊗ − Finally, since h0(S,ε∗)=0 for all torsion line bundles O = ε Pic0 S, we deduce that h0(S,ω ε)=1. S 6 ∈ S ⊗
This concludes the description of all spaces of modular forms of canonical weight. The rest of this section gathers some immediate corollaries from the uniqueness of divisors of canonical weight. 6 Corollary 2.3. Define Θ1 := L Pic1 S h0(L) =0 for Pic1 S := L Pic S c (L)= c (O (1)) = 1 . Then + { ∈ 6 } { ∈ 1 1 S } we have:
(1) All divisors of canonical weight are irreducible Θ1 = . ⇐⇒ + ∅ 1 (2) |Θ+|+2 H (S, Z) 1. 3 ≤ | 1 |− Corollary 2.4. For all line bundles L Pic S, O = ε Pic0 S, tensor multiplication by s gives a monomorphism ∈ S 6 ∈ ε s : H0(S,L) H0(S,L ω ε). Also, if both L and ε happen to be A -invariant, then s is a monomorphism ⊗ ε → ⊗ S ⊗ S ⊗ ε of G-representations (cf. 0.2).
3. Tautological bundles, liftings, linearizations, central extension and localization
We begin by recalling the next
Lemma 3.1 (see [12, Lemma 2.1]). Let S be a fake projective plane with no 3-torsion in H1(S, Z). Then there O O exists a unique (ample) line bundle S(1) such that ωS ∼= S(3).
3.2. Linearizations of line bundles. Let S and OS(1) be as in Lemma 3.1. We will assume in what follows that A = G or (Z/3)2. This implies that one has a lifting of the fundamental group Π PU(2, 1) to SU(2, 1) (see S 21 ⊂ Table A above). Fix such a lifting r : Π ֒ SU(2, 1) and consider the central extension →
(3.3) 1 Z/3 SU(2, 1) PU(2, 1) 1. → → → →
Note that the embedding r is unique because H1(S, Z) = Π/[Π, Π] does not have 3-torsion in our case.
3 3 We thus get a linear action of r(Π) on C . In particular, both Bl C = Tot OP2 ( 1) and its restriction to the ball 0 − B P2 are preserved by r(Π), so that we get the equality ⊂
Tot O (1) = (Tot OP2 ( 1) )/r(Π) S − B
(cf. [27, 8.9]). Further, since there is a natural identification
0 ∗ ∗ Pic S = Hom(H1(S, Z), C ) = Hom(Π, C ), every torsion line bundle ε Pic0 S corresponds to a character χ : Π C∗. One may twist the fiberwise r(Π)-action ∈ ε → O on Tot P2 ( 1) by χ (we will refer to this modified action as r(Π) ε ) and obtain − ε χ O O 7) Tot (1) ε = (Tot P2 ( 1) )/r(Π) ε . S ⊗ − B χ
Observe that according to Table A there always exists such ε =0. This table also shows that in two cases one can 6 choose ε to be AS -invariant.
7) All these matters are treated extensively in [27, Ch. 8] 7 3.4. The central extension. Note that A = N(Π)/Π for the normalizer N(Π) PU(2, 1) of Π. Then (3.3) S ⊂ yields a central extension 1 Z/3 G A 1 → → → S → for G := N^(Π)/r(Π) and the preimage N^(Π) SU(2, 1) of N(Π). Further, the preceding construction of O (1) is ⊂ S N^(Π)-equivariant by the A -invariance of O (1), which gives a linear G-action on all the spaces H0(S, O (k)), k Z. S S S ∈ Similarly, if the torsion line bundle ε is A -invariant, we get a linear G-action on all H0(S, O (k) ε). S S ⊗
Recall next that when AS = G21, the line bundle OS(1) is AS-linearizable (i.e. the group AS lifts to G and the 0 corresponding extension splits), and so the spaces H (S, OS (k)) are some linear AS-representations in this case (see Table A or [12, Lemma 2.2]). The same holds for all H0(S, O (k) ε) and any A -invariant ε. S ⊗ S 2 In turn, if AS = (Z/3) , then the extension G of this AS does not split (see Table A), i.e. G coincides with the Heisenberg group H of order 27. Again the G-action on H0(S, O (k)) (resp. on H0(S, O (k) ε)) is linear here. 3 S S ⊗ Remark 3.5. Let ξ, η G = H be two elements that map to order 3 generators of A = (Z/3)2. Then their com- ∈ 3 S mutator [ξ, η] generates the center Z/3 G, and one obtains the following irreducible 3-dimensional (Schrödinger) ⊂ representation of G: 2π√ 1 ξ : x ω−ix , η : x x (i Z/3, ω := e 3− ), i 7→ i i 7→ i+1 ∈ 3 with xi forming a basis in C (this representation, together with its complex conjugate, are the only 3-dimensional irreducible representations of H3). Observe at this point that according to the above discussion [ξ, η] acts trivially on S and via the fiberwise scaling by ω on O (1). Furthermore, given any A -invariant ε Pic0 S, the commutator S S ∈ [ξ, η] acts non-trivially on O (k) ε whenever k is coprime to 3. S ⊗
3.6. Localization. Let S be a compact complex manifold. For a holomorphic endomorphism τ : S S and a −→ linearized coherent sheaf F Cohτ S (i.e. a coherent sheaf F Coh S equipped with a morphism φ : τ ∗F F ) ∈ ∈ → consider the compositions Hk(φ) τ ∗ : Hk(S, F ) Hk(S, F ) of pullbacks τ ∗ : Hk(S, F ) Hk(S, τ ∗F ) with ◦ → → Hk(φ): Hk(S, τ ∗F ) Hk(S, F ). Define the Lefschetz number → L(τ,F,φ) := ( 1)k Tr(Hk(φ) τ ∗ : Hk(S, F )). X − ◦ k
Theorem 3.7 (Woods Hole formula, see [1, Theorem 2]). If the graphs of τ and IdS (the diagonal) intersect transversally in S S, then × Tr(τ(P ): F (P )) L(τ,F,φ)= , X det(1 dτ(P ), ΩS (P )) f(P )=P − where dτ : τ ∗Ω Ω is the differential (canonical linearization of the cotangent bundle). S → S
Explicitly, for dim S =2 we get: det(1 dτ(P )) = (1 α (P ))(1 α (P )), where α (P ) are the eigenvalues of the − − 1 − 2 j Jacobian matrix Jτ .
3.8. Let us conclude this section by recalling well-known supplementary results.
It follows from the Riemann – Roch formula that (k 1)(k 2) χ(O (k) ε)= − − S ⊗ 2 for all k Z and ε Pic0 S. ∈ ∈ 8 In particular, we get H0(S, O (k) ε) C(k−1)(k−2)/2 when k 4, since S ⊗ ≃ ≥ Hi(S, O (k) ε)= Hi(S,ω O (k 3) ε)=0 S ⊗ S ⊗ S − ⊗ for all i> 0 by Kodaira vanishing.
4. Automorphic forms of large weight: computation of characters
Let τ A be any element of order 3 with a faithful action on S. Choose any linearization φ : τ ∗O (1) O (1) ∈ S S → S (cf. 3.2).
Lemma 4.1. The automorphism τ has only three fixed points, in the Woods Hole formula (see Theorem 3.7) all denominators coincide (and are equal to 3), and all numerators are three distinct 3rd roots of unity. In other words, one can choose a numbering of the fixed points Pi and weights αj (Pi) in such a way that the following holds:
α (P )= ωj for all i, j; • j i w := Tr(τ(P ): O (1) )= ωi for all i. • i i S Pi Moreover, for any τ-invariant line bundle ε Pic0 S of order m coprime to 3, we can choose a linearization such ∈ ik 8) that Tr τ =1. In particular, Tr τ O l = ω for all i,k,l. |εPi |( S (k)⊗ε )Pi
Proof. The claim about Pi and αj (Pi) follows from [21, Proposition 3.1].
0 O C3 3 We have V := H (S, S(4)) ∼= (see 3.8). Let v1, v2, v3 be the eigen values of τ acting on V . Then vi =1 for all i and Tr τ = v + v + v . At the same time, as follows from 3.6, we have V 1 2 3
w4 + w4 + w4 w + w + w Tr τ = 1 2 3 = 1 2 3 . V (1 ω)(1 ω2) 3 − − The latter can be equal to v1 + v2 + v3 only when all vi (resp. all wi) are pairwise distinct (so that both sums are zero). This is due to the fact that the sum of three 3rd roots of unity has the norm 0, √3, 3 and is zero iff all ∈{ } roots are distinct.
3 Finally, since (m, 3)=1, we may replace ε by ε , so that the action of τ on the closed fibers εPi is trivial. The last assertion about O (k) εl is evident. S ⊗
Fix some ε (Pic0 S)AS . Recall that the group G from 3.4 acts linearly on all spaces V := H0(S, O (k) ε). ∈ S ⊗ Proposition 4.2. Let G = H . Then for k 4 the following holds (cf. Table B below): 3 ≥ k k for k 0 mod3, we have V = V C[(Z/3)2]a as G-representations, where a := ( 1) and V C • ≡ 0 ⊕ 6 3 − 0 ≃ (resp. C[(Z/3)2]) is the trivial (resp. regular) representation;
for k 1 mod3, we have V = V⊕(k−1)(k−2)/6 as G-representations, where V is an irreducible 3- • ≡ 3 3 dimensional representation of H3; ⊕(k−1)(k−2)/6 for k 2 mod3, we have V = V as G-representations, where V is the complex conjugate to • ≡ 3 3 V3 above.
8) We are using the notation εi := ε⊗i. 9 4 k mod 3 0 1 2 ≤ ⊕(k−1)(k−2)/6 H0(S, O (k) ε) V C[(Z/3)2]a V⊕(k−1)(k−2)/6 V S ⊗ 0 ⊕ 3 3 Table B
Proof. Suppose that k 0 mod3. Then, since every element in H has order 3, applying Lemma 4.1 to any ≡ 3 non-central τ G we obtain (1 α (P ))(1 α (P ))=3 for all i and Tr τ =1 (cf. 3.6). ∈ − 1 i − 2 i V
Further, the element [ξ, η] G from Remark 3.5 acts trivially on O (k) (via scaling by ωk = 1). Also, since the ∈ S order of any ε = 0 is coprime to 3 (see Table A) and ε is flat (see its construction in 3.2), from Lemma 4.1 6 we find that [ξ, η] acts trivially on O (k) ε, hence on V as well. This implies that the G-action on V factors S ⊗ through that of its quotient (Z/3)2. Then the claimed decomposition V = V C[(Z/3)2]a follows from the fact 0 ⊕ that 1+9a = dim V = Tr[ξ, η] and that Tr τ =1 for all non-central τ G. V V ∈
Let now k 1 mod3 (resp. k 2 mod3). Then it follows from Remark 3.5 that [ξ, η] scales all vectors in ≡ ≡ V = H0(S, O (k) ε) = C(k−1)(k−2)/2 by ωk =1. Furthermore, since Tr ξ =0=Tr η by Lemma 4.1 and 3.6, S ⊗ ∼ 6 V V all irreducible summands of V are faithful G-representations, and hence they are isomorphic to V3 (resp. to V3). This concludes the proof.
Proposition 4.3. Let G A = G . Then for k 4 and V = H0(S, O (k) ε), we have the following equality ⊃ S 21 ≥ S ⊗ of (virtual) G-representations V = C[G]⊕ak U ⊕ k for some a Z expressed in terms of dim V , where U depends only on k mod 21 and is explicitly given in the k ∈ k table below, with rows (resp. columns) being enumerated by k mod 3 (resp. k mod 7)
0 1 2 3 4 5 6 ⊕2 0 C V V C V V C C V⊕2 V C (V V )⊕2 C V V C 3 ⊕ 3 ⊕ 3 ⊕ 3 ⊕ 3 ⊕ 3 ⊕ 3 ⊕ 3 ⊕ 3 ⊕ 3 ⊕ 1 or 2 ( V ) ( V ) 0 0 ( V ) ( V ) V V V V − 3 ⊕ − 3 − 3 ⊕ − 3 3 3 ⊕ 3 3 Proof. From 3.6 we obtain ζ6k ζ5k ζ3k Tr σ = + + . V (1 ζ)(1 ζ3) (1 ζ2)(1 ζ6) (1 ζ4)(1 ζ5) − − − − − − 2π√ 1 Here ζ := e 7− and σ G is an element of order 7. The value Tr σ depends only on k mod 7 and by the ∈ 21 V direct computation we obtain the following table:
k 0 1 2 3 4 5 6 Tr σ 1 0 0 1 ¯b 1 b V −
(Here b := ζ + ζ2 + ζ4 and ¯b = 1 b = ζ3 + ζ5 + ζ6.) − − Let τ G be an element of order 3 such that G = σ, τ . Recall the character table for the group G (see e.g. ∈ 21 21 h i 21 the proof of [12, Lemma 4.2]):
10 1 Tr σ Tr σ3 Tr τ Tr τ 2 C 1 1 1 1 1
V1 1 1 1 ω ω
V1 1 1 1 ω ω
V3 3 b b 0 0
V3 3 b b 0 0
Here “—” signifies, as usual, the complex conjugation and Vi are irreducible i-dimensional representations of G21.
Now from Lemma 4.1 and the preceding tables we get the claimed options for V . This concludes the proof.
5. Automorphic forms of small weight: two proofs of vanishing
Lemma 5.1. Let L PicG X be a G-linearized holomorphic line bundle on a compact complex variety X equipped ∈ with an action of a finite group G : X (not necessarily faithful). Assume that H0(X,L⊗2) is an irreducible 3- dimensional G-representation. Then
(1) the G-representation H0(X,L) is irreducible, of dimension either 0 or 2;
(2) if G has odd order, then H0(X,L)=0.
Proof. The proofs of [12, Lemma 4.2] and [27, Lemma 15. 6. 2] show that h0(X,L) 2. Thus, if the statement (1) is ≤ false, then H0(X,L)G =0 and therefore (S2H0(X,L))G =0. But loc. cit. also show that the natural homomorphism 6 6 of G-modules S2H0(X,L) H0(X,L⊗2) is an embedding and this contradicts the irreducibility assumption. To → prove (2) simply observe that irreducible representations of odd order groups have odd degree.
We retain the earlier notation.
Theorem 5.2. Let S be a fake projective plane with the automorphism group A of order at least 9 and ε (Pic S)AS S ∈ be an A -invariant torsion line bundle. Then H0(S, O (2) ε)=0. S S ⊗
Proof. (old proof)
Propositions 4.2 and 4.3 imply that H0(S, O (4) ε) is an irreducible 3-dimensional G-representation. By Lemma 5.1 S ⊗ we conclude that H0(S, O (2) ε)=0 — exactly as in the proof of [12, Theorem 1.3]. S ⊗
Proof. (new proof) comes with the extra assumption ε =0 (cf. the end of 3.2). 6 Suppose that H0(S, O (2) ε) =0. There is a natural homomorphism of G-modules S ⊗ 6 H0(S,ω ε∗) H0(S, O (2) ε) H0(S, O (5)) S ⊗ ⊗ S ⊗ → S and Theorem 2.2 implies that H0(S, O (2) ε) is a non-trivial subrepresentation in H0(S, O (5)) of dimension 2. S ⊗ S ≤ 2 But the latter contradicts Proposition 4.2 (for AS = (Z/3) ) or Proposition 4.3 (for AS = G21). Thus Theorem 5.2 follows in this case.
Similarly, the G-homomorphism
H0(S,ω ε) H0(S, O (2)) H0(S, O (5) ε) S ⊗ ⊗ S → S ⊗ 11 gives contradiction with either Proposition 4.2 or Proposition 4.3, provided that H0(S, O (2)) =0. This concludes S 6 the new proof.
References
1. Michael Atiyah, Raoul Bott: A Lefschetz fixed point formula for elliptic differential operators, Bull. Amer. Math. Soc., 72 (1966), 245 – 250. 2. Rebecca Barlow: Zero-cycles on Mumford’s surface, Mathematical Proceedings of the Cambridge Philosophical Society, 126 (1999), no. 3, 505 – 510. doi:10.1017/S0305004198003442 3. Alexander Beilinson: Coherent sheaves on Pn and problems in linear algebra, Funktsional. Anal. i Prilozhen., 12 (1978), no. 3, 68 – 69; Funct. Anal. Appl., 12 (1978), no. 3, 214 – 216. 4. Don Blasius, Jonathan Rogawski: Cohomology of congruence subgroups of SU(2, 1)p and Hodge cycles on some special complex hyperbolic surfaces, In: Regulators in Analysis, Geometry and Number Theory (Eds. A. Resnikov and N. Schappacher), Progress In Mathematics 171, Birkhauser 1999. 5. Spencer Bloch: Lectures on algebraic cycles, Duke University Mathematics Series. IV (1980). Durham, North Carolina: Duke University, Mathematics Department. 6. Lev Borisov, Jonghae Keum: Explicit equations of a fake projective plane, arXiv:1802.06333. 7. Gennaro Di Brino, Luca Di Cerbo: Exceptional collections and the bicanonical map of Keum’s fake projective planes, Commun. Contemp. Math. 20 (2018), no. 1, 1650066, 13 pp. Also arXiv:1603.04378. 8. Donald Cartwright, Tim Steger: Enumeration of the 50 fake projective planes, C. R. Acad. Sci. Paris, Ser. I, 348 (2010), 11 – 13. doi:10.1016/j.crma.2009.11.016 9. Donald Cartwright, Tim Steger: http://www.maths.usyd.edu.au/u/donaldc/fakeprojectiveplanes/ 10. Najmuddin Fakhruddin: Exceptional collections on 2-adically uniformised fake projective planes, Mathematical Research Letters, 22 (2015), no. 1, 43 – 57. Also arXiv:1310.3020. 11. Sergey Galkin, Hiroshi Iritani: Gamma conjecture via mirror symmetry, Advanced Studies in Pure Mathematics: Primitive forms and related subjects, 83 (2019), 53 – 113. To appear. Also arXiv:1508.00719. 12. Sergey Galkin, Ludmil Katzarkov, Anton Mellit, Evgeny Shinder: Derived categories of Keum’s fake projective planes, Advances in Mathematics, 278 (2015), 238 – 253. doi:10.1016/j.aim.2015.03.001 13. Sergey Galkin, Ludmil Katzarkov, Anton Mellit, Evgeny Shinder: Minifolds and phantoms, arXiv:1305.4549. 14. Claus Hertling, Yuri Manin, Constantin Teleman: An update on semisimple quantum cohomology and F-manifolds, Proceedings of the Steklov Institute of Mathematics, 264 (2009), no. 1, 62 – 69. Also arXiv:0803.2769. 15. Friedrich Hirzebruch: Automorphe Formen und der Satz von Riemann – Roch, Symposium internacional de topología algebraica (International symposium on algebraic topology): Universidad Nacional Autónoma de México and UNESCO, Mexico City, (1958), 129 – 144. 16. Friedrich Hirzebruch: Elliptische Differentialoperatoren auf Mannigfaltigkeiten, Festschr. Gedächtnisfeier K. Weierstrass, West- deutscher Verlag, Cologne, (1966), 583 – 608. Uspehi Mat. Nauk, 23 (1968), no. 1 (139), 191 – 209. 17. Mikio Ise: Generalized automorphic forms and certain holomorphic vector bundles, Amer. J. Math., 86 (1964), no. 1, 70 – 108. doi:10.2307/2373036 18. Masa-Nori Ishida: An elliptic surface covered by Mumford’s fake projective plane, Tohoku Math. J., 40 (1988), no. 3, 367 – 396. 19. Fumiharu Kato, Hiroyuki Ochiai: Arithmetic structure of CMSZ fake projective planes, J. Algebra, 305 (2006), no. 2, 1166 – 1185. Also arXiv:math/0006223. 20. David A. Kazhdan: Connection of the dual space of a group with the structure of its close subgroups, Funktsional. Anal. i Prilozhen., 1 (1967), no. 1, 71 – 74; Funct. Anal. Appl., 1 (1967), no. 1, 63 – 65. 21. Jonghae Keum: Quotients of fake projective planes, Geom. Topol., 12 (2008), no. 4, 2497 – 2515. Also arXiv:0802.3435. 22. Jonghae Keum: A vanishing theorem on fake projective planes with enough automorphisms, Trans. Amer. Math. Soc., 369 (2017), no. 10, 7067 – 7083. Also arXiv:math/1407.7632v3. 23. Vyacheslav Kharlamov, Viktor Kulikov: On real structures on rigid surfaces, Izv. RAN. Ser. Mat., 66 (2002), no. 1, 133 – 152; Izv. Math., 66 (2002), no. 1, 133 – 150. Also arXiv:0101098. 24. Vyacheslav Kharlamov, Viktor Kulikov: On numerically pluricanonical cyclic coverings, Izv. RAN. Ser. Mat., 78 (2014), no. 5, 143 – 166; Izv. Math., 78 (2014), no. 5, 986 – 1005. Also arXiv:1308.0516. 12 25. Bruno Klingler: Sur la rigidité de certains groupes fonndamentaux, l’arithméticité des réseaux hyperboliques complexes, et les ‘faux plans projectifs’, Invent. Math., 153 (2003), 105 – 143. doi:10.1007/s00222-002-0283-2 26. Toshiyuki Kobayashi, Kaoru Ono: Note on Hirzebruch’s proportionality principle, Jour. Fac. Sci. Univ. Tokyo, 37 (1990), no. 1, 71 – 87. 27. János Kollár: Shafarevich maps and automorphic forms, M. B. Porter Lectures, Princeton Univ. Press, Princeton, NJ, 1995. 28. Ching-Jui Lai, Sai-Kee Yeung: Exceptional collection of objects on some fake projective planes, available at http://www.math.purdue.edu/~yeung/papers/FPP-van.pdf 29. Robert Langlands: The Dimension of Spaces of Automorphic Forms, Amer. J. Math., 85 (1963), no. 1, 99 – 125. doi:10.2307/2373189 30. Claude LeBrun: Curvature functionals, optimal metrics, and the differential topology of 4-manifolds, Different faces of geometry, 199 – 256, Int. Math. Ser. (N. Y.), 3, Kluwer/Plenum, New York, 2004. doi:10.1007/b115003 2 31. David Mumford: An algebraic surface with K ample, K = 9,pg = q = 0, Amer. J. Math., 101 (1979), no. 1, 233 – 244. doi:10.2307/2373947 32. Gopal Prasad, Sai-Kee Yeung: Fake projective planes, Invent. Math., 168 (2007), 321 – 370; 182 (2010), 213 – 227. Also arXiv:math/0512115v5, arXiv:math/0906.4932v3. 33. Jonathan Rogawski: Automorphic Representations of Unitary Groups in Three Variables, Annals of Math. Studies: Princeton University Press, 123 (1990). 34. Matthew Stover: ERRATUM AND ADDENDUM: “PROPERTY (FA) AND LATTICES IN SU(2,1)”, International Journal of Algebra and Computation, 23 (2013), no. 7, 1783 – 1787. doi:10.1142/S0218196713920033 35. Rained Weissauer: Vektorwertige Siegelsche Modulformen kleinen Gewichtes, J. Reine Angew. Math., 343 (1983), 184 – 202. 36. Shing-Tung Yau: Calabi’s conjecture and some new results in algebraic geometry, Proceedings of the National Academy of Sciences, 74 (1977), no. 5, 1798 – 1799. doi:10.1073/pnas.74.5.1798 37. Sai-Kee Yeung: Integrality and arithmeticity of co-compact lattices corresponding to certain complex two-ball quotients of Picard number one, Asian J. Math., 8 (2004), 107 – 130.
HSE University, Russian Federation
Independent University of Moscow, Russian Federation
E-mail address: [email protected]
Kavli IPMU (WPI), The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Chiba, Japan
Current address: Moscow Institute of Physics and Technology, Institutskij pereulok, 9, Dolgoprudnyi, Russia
E-mail address: [email protected]
School of Mathematics and Statistics, University of Sheffield, The Hicks Building, Hounsfield Road, Sheffield, S3 7RH, United Kingdom
E-mail address: [email protected]
13