IPMU 13-0102
MINIFOLDS AND PHANTOMSSERGEY GALKIN, LUDMIL KATZARKOV, ANTON MELLIT, EVGENY SHINDER
Abstract. A minifold is a smooth projective n-dimensional variety such that its bounded derived
category of coherent sheaves admits a semi-orthogonal decomposition into a collection of n + 1
exceptional objects. In this paper we classify minifolds of dimension n 6 4.
We conjecture that the derived category of fake projective spaces have a similar semi-orthogonal
decomposition into a collection of n+1 exceptional objects and a category with vanishing Hochschildhomology. We prove this for fake projective planes with non-abelian automorphism group (such as
Keum’s surface). Then by passing to equivariant categories we construct new examples of phantom
categories with both Hochschild homology and Grothendieck group vanishing.

1. Introduction.
The question of homological characterization of projective spaces goes back to Severi, and thepioneering work of Hirzebruch–Kodaira [28]. Beautiful results have been obtained by Kobayashi–
Ochiai [41], Yau [62], Fujita [24], Libgober–Wood [49].
Among smooth projective varieties of given dimension projective spaces have the smallest cohomology groups. We call a smooth projective variety a Q-homology projective space if it has the same
Hodge numbers as a projective space. Any odd-dimensional quadric is an example of Q-homology
projective space. We call an n-dimensional Q-homology projective space X of general type a fake⊗lprojective space if in addition it has the same “Hilbert polynomial” as Pn : χ(X, ωX) = χ(Pn , ωP⊗ln )for all l ∈ Z. Any fake projective plane is simply a Q-homology plane of general type, since Hodge
numbers of a surface determine its Hilbert polynomial. On the level of realizations over C, e.g.
from the point of view of the Hodge structure, fake projective spaces are identical to projective
spaces, however the study of their K-theory, motive or derived category meets cohomologicalsubtleties.
The first example of a fake projective plane was constructed by Mumford [52] using p-adic
uniformization developed by Drinfeld [21] and Mustafin [53]. From the point of view of complex
geometry fake projective planes have been studied by Aubin [3] and Yau [62], who proved thatany such surface S is uniformized by a complex ball, hence by Mostow’s rigidity theorem S is
determined by its fundamental group π1 (S) uniquely up to complex conjugation; Kharlamov–
Kulikov [39] shown that the conjugate surfaces are distinct (not biholomorphic). Further Klingler
Date: October 17, 2013.
This work was partially supported by World Premier International Research Center Initiative (WPI Initiative),
MEXT, Japan, Grant-in-Aid for Scientific Research (10554503) from Japan Society for Promotion of Science, andAG Laboratory NRU-HSE, RF government grant, ag. 11.G34.31.0023. S. G. and L. K. were funded by grants NSF
DMS0600800, NSF FRG DMS-0652633, NSF FRG DMS-0854977, NSF DMS-0854977, NSF DMS-0901330, grants
FWF P 24572-N25 and FWF P20778, and an ERC grant — GEMIS. E. S. has been supported by the Max-PlanckInstitut für Mathematik, SFB 45 Bonn-Essen-Mainz grant and the Hausdorff Center, Bonn. S. G. and A. M. would
like to thank ICTP for inviting them to the “School and Conference on Modular Forms and Mock Modular Forms
and their Applications in Arithmetic, Geometry and Physics” to Trieste in March 2011, where this collaboration
started, and IPMU for A. M.’s visit in November 2011, when the first version of this paper was completed. Partof this work was done in June 2013 during the visit of S. G. to Johannes Gutenberg Universität Mainz, funded by
SFB 45.
1

[40] and Yeung [63] proved that π1 (S) is a torsion-free cocompact arithmetic subgroup of P U (2, 1).
Finally such groups have been classified by Cartwright–Steger [18] and Prasad–Yeung [57]: there
are 50 explicit subgroups and so all fake projective planes fit into 100 isomorphism classes.
Fake projective fourspaces were introduced and studied by Prasad and Yeung in [58].
In this paper we take a different perspective that started with a seminal discovery of full exceptional collections by Beilinson [6], Kapranov [32], Bondal and Orlov [15] with Bondal’s students
Kuznetsov, Razin, Samokhin (see [45]): they found out that all known to them examples of Fano
Q-homology projective spaces admit a full exceptional collection of vector bundles. They put a conjecture that gives a homological characterization of projective spaces based on derived categories,
and in this paper we prove it in Theorem 1.1(3).
We call an n-dimensional smooth complex projective variety a minifold if it has a full exceptional collection of minimal possible length n + 1 in its bounded derived category of coherent
sheaves. A minifold is necessarily a Q-homology projective space. Projective spaces and odddimensional quadrics are examples of minifolds [6, 32].
It follows from work of Bondal, Bondal–Polishchuk and Positselski [12, 16, 56], that if a minifold
X is not Fano then all full exceptional collections on it are not strict and consist not of puresheaves. In fact it is expected that all minifolds are Fano.
The novelty of this paper is the following main theorem, which gives a classification of minifolds
in dimension less than or equal to 4 (with one-dimensional case being trivial).
Theorem 1.1. 1) The only two-dimensional minifold is P2 .
2) The minifolds of dimension 3 are: the projective space P3 the quadric Q3 , the del Pezzo
quintic threefold V5 , and a six-dimensional family of Fano threefolds V22 .
3) The only four-dimensional Fano minifold is P4 .
In Section 2 we recall the necessary definitions and facts. In particular in Proposition 2.1 we recall
that varieties admitting full exceptional collections have Tate motives with rational coefficients [51]and outline a straightforward proof of that fact. Section 2 finishes with the proof of Theorem 1.1.
We also show that except for P4 the only possible minifolds of dimension 4 are non-arithmetic
fake projective fourfolds, which presumably do not exist [65] (paragraph 4 and section 8.4).
In fact, study of minifolds is closely related to study of the fake projective spaces. The reason
is that fake projective spaces sometimes admit exceptional collections of the appropriate length
but these collections fail to be full. In this case the orthogonal to such a collection is a so-called
phantom.
More precisely, we call an admissible non-zero subcategory A of a derived category of coherent
sheaves an H-phantom if HH∗ (A) = 0 and a K-phantom if K0 (A) = 0.
In Section 3 we formulate a conjecture that under some mild conditions fake projective n-spaces
admit non-full exceptional collections of length n + 1 and thus have H-phantoms in their derived
categories (Conjecture 3.1 and its Corollary 3.2). We prove this conjecture for fake projective
planes admitting an action of the non-abelian group G21 of order 21.
Theorem 1.2. Let S be one of the six fake projective planes with automorphism group of order
21. Then KS = O(3) for a unique line bundle O(1) on S. Furthermore O, O(−1), O(−2) is an
exceptional collection on S.
Most of Section 3 deals with the proof of this Theorem, which relies on the holomorphic Lefschetz
fixed point formula applied to the three fixed points of an element of order 7 as in [38].
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It follows from Theorem 1.2 that Db (S) has an H-phantom subcategory AS . We show thatbthis H-phantom descends to an H-phantom AG
S in the equivariant derived categories DG (S) for
any G ⊂ G21 . In particular, when G = Z/3 this gives yet new examples of surfaces having an
H-phantom in their derived categories (fake cubic surfaces: see Remark 3.8).
Finally, in four cases G ⊃ Z/7 and surface S/G is simply-connected, then we show that the
H-phantom AG
S is also a K-phantom (Proposition 3.10).
In the Appendix we give a table of arithmetic subgroups Π ⊂ P SU (2, 1) giving rise to fakeprojective planes and the corresponding automorphism and first homology groups. These results
are taken from the computations of Steger and Cartwright [19].
It took a long way for the paper to take its present form. We would like to thank our friends
and colleagues with whom we had fruitful discussions on the topic.
We thank Denis Auroux, Alexey Bondal, Paul Bressler, Alessio Corti, Igor Dolgachev, Alexander
Efimov, Sergey Gorchinskiy, Jeremiah Heller, Daniel Huybrechts, Umut Isik, Yujiro Kawamata,
Maxim Kontsevich, Viktor Kulikov, Alexander Kuznetsov, Serge Lvovski, Dmitri Orlov, Dmitri
Panov, Tony Pantev, Yuri Prokhorov, Kyoji Saito, Konstantin Shramov, Duco van Straten, Misha
Verbitsky, Vadim Vologodsky, and Alexander Voronov for their useful suggestions, references andcareful proofreading. We thank Donald Cartwright, Philippe Eyssidieux and Bruno Klingler, Gopal
Prasad, Sai Kee Yeung, for answering our questions about fake projective planes and fourspaces.
While this paper was in preparation, Najmuddin Fakhruddin gave a proof of Conjecture 3.1
for those fake projective planes that admit 2-adic uniformization [23], in particular for Mumford’s
fake projective plane. These cases are disjoint from the ones that satisfy the assumptions of ourTheorem 1.2. An idea of a proof in the case of Mumford’s fake projective plane similar to thatof [23] was also hinted to us by Dolgachev. Independently it was sketched to us by van Straten
in June 2013: van Straten and Spandau has an unpublished work circa 2001, where based on
description of Ishida [29] they construct the reduction modulo two of the 2-canonical image of
Mumford’s fake projective plane as an image of P2 (F2 ) by an explicit 10-dimensional linear systemof octics (plane curves of degree 8), one then checks that these octics are not divisible by 3 asWeil divisors. Our proof of Theorem 1.2 is indebted to discussions with Panov in June 2012 (hesuggested us to exploit extra symmetries of Keum’s surface) and with Dolgachev in June 2013
(then we looked for higher-dimensional irreducible representations of non-abelian groups). The
rest of the ideas of the paper is from 2011.
2. Minifolds
An exceptional collection of length r on a smooth projective variety X/C is a sequence of objects
E1 , . . . Er in the bounded derived category of coherent sheaves Db (X) such that Hom(Ej , Ei [k]) = 0
for all j > i and k ∈ Z, and moreover each object Ei is exceptional, that is spaces Hom(Ei , Ei [k])vanish for all k except for one-dimensional spaces Hom(Ei , Ei ). An exceptional collection is calledfull if the smallest triangulated subcategory which contains it, coincides with Db (X).
Proposition 2.1. Assume that X admits a full exceptional collection of length r. Then:(1) The Chow motive of X with rational coefficients is a direct sum of r Tate motives Lj . In
particular, all cohomology classes on X are algebraic. (For the definition and properties of Chow
motives see [50].)P(2) H p,q (X) = 0 for p 6= q and χ(X) = hp,p (X) = r.
(3) Pic(X) is a free abelian group of finite rank. Moreover the first Chern class map gives anisomorphism Pic(X) ∼= H 2 (X, Z).
3

(4) H1 (X, Z) = 0.
(5) The Grothendieck group K0 (X) = K0 (Db (X)) is free of rank r and the bilinear Euler pairingX
χ(E, F ) =(−1)i dim Hom(E, F [i])iis non-degenerate and unimodular. Classes [Ei ] of exceptional objects form a semi-orthonormal
basis in K0 (X). (By a semiorthonormal basis we mean a basis (ei )ni=0 such that χ(ej , ei ) = 0, j > iand χ(ei , ei ) = 1.)
Proof. Most of the claims are well-known. (1) is proved in [51] using the language of noncommutative motives and in [27] using K-motives. We give a direct proof of (1) using the ideas
developed in [55] (see also [7]) for the sake of completeness.
First observe that the structure sheaf of the diagonal O∆ in the derived category Db (X × X) liesin the full triangulated subcategory generated by the objects p∗1 F1 ⊗ p∗2 F2 . This can be deduced
from the standard fact that if E1 , . . . , Er is a full exceptional collection on X, then p∗1 Ei ⊗ p∗2 Ej
forms a full exceptional collection on X × X [13](see Lemma 3.4.1 and note that for the category
generated by an exceptional collection the notions of generator and strong generator coincide, sotaking direct summands is not necessary), [9], [59].
It follows that the class of the diagonal [O∆ ] ∈ K0 (X × X) has a decomposition
X(2.1)[O∆ ] =p∗1 [Fj ] · p∗2 [Gj ],jfor some sheaves Fj , Gj on X.
Applying the Chern character to (2.1) and using the Grothendieck-Riemann-Roch formulach(O∆ ) = [∆] · p∗2 td(X)we obtain an analogous decomposition for the class of the diagonal [∆] ∈ CH ∗ (X × X)Q :
X
(2.2)
[∆] =
p∗1 αj · p∗2 β j ,jfor some classes αj , βj ∈ CH ∗ (X)Q . We may assume that αj are homogenous, say αj ∈ CH aj (X)Qand hence βj ∈ CH dim(X)−aj (X)Q .
We claim that the set {αj } spans CH ∗ (X)Q . Indeed for any α ∈ CH ∗ (X)Q we have
α = p1∗ ([∆] · p∗2 α) =X= p1∗ ((
p∗1 αj · p∗2 β j ) · p∗2 α) =
j

(2.3)

X
= p1∗ (
p∗1 αj · p∗2 (β j · α)) =j=X

αj · p1∗ (p∗2 (β j · α)) =j

X=
hβ j , αiαj .
jHere we use the notation hα, βi for the bilinear form deg(α · β).
We may assume that {αj } are linearly independent, that is form a homogeneous basis ofCH ∗ (X)Q . From the formula (2.3) we see that {βj } is a dual basis.
4

We now define an isomorphism M (X) ∼= ⊕j Laj . By definition of the morphisms in the categoryof Chow motives we have
Hom(La , M (X)) = CH a (M )Hom(M (X), La ) = CH dim(X)−a (M )Therefore the set {αj } determines a morphism of motivesΦ : ⊕j Laj → M (X)and the {βj } determines a morphism in the opposite direction
Ψ : M (X) → ⊕j Laj .
The composition Ψ ◦ Φ is equal to identity due to the fact that {αj } and {βj } are dual bases.
The composition Φ ◦ Ψ is equal to identity because of the decomposition (2.2).
By taking Hodge realization (1) implies (2). Alternatively, we can deduce (2) from Hochschild–
Kostant–Rosenberg theorem
HHi (Db (X)) ∼= ⊕p−q=i H p,q (X)and additivity of Hochschild homology for semiorthogonal decompositions [37, 46]: if C = hA, Bi
then HHi (C) = HHi (A) ⊕ HHi (B).
The fact that Pic(X) is free follows from (5) and Lemma 2.2 below. The isomorphism Pic(X) ∼=2
H (X, Z) comes from the exponential long exact sequence and (2). Pic(X) is of finite rank since
it is isomorphic to H 2 (X, Z).
To prove (4) note that the Universal Coefficient Theorem implies that we have a non-canonicalisomorphism
H 2 (X, Z) ∼= Zrk ⊕ H1 (X, Z)tors
which by (3) implies that H1 (X, Z) must be torsion-free as well. On the other hand h1,0 (X) = 0
and hence H1 (X, Z) = 0.
(5) follows easily from definitions.

Lemma 2.2. Let X be a smooth algebraic variety such that K0 (X) has no p-torsion. Then Pic(X)
has no p-torsion.
Proof. We prove that if Pic(X) has p-torsion, then the same is true for K0 (X).
Let L be a line bundle on X such that L⊗p ∼= OX . Let N = [L] − 1 ∈ K0 (X); then N
is nilpotent. Indeed N being of rank zero, sits in the first term F 1 K0 (X) of the topologicalfiltration on K0 (X) ([25], Example 15.1.5). The topological filtration is multiplicative; therefore
N dim(X)+1 ∈ F dim(X)+1 K0 (X) = 0.
Let k be the smallest positive integer such that N k = 0. If k = 1, that is N = 0 and [L] = 1 ∈K0 (X), then L ∼= OX since to F 1 X/F 2 X ∼= Pic(X) by [25], Example 15.3.6.
We assume now that k > 2. We have[L] = 1 + N.
Taking p-th tensor power of both sides we obtain
1 = 1 + pN + N 2 α, α ∈ K0 (X)
0 = pN + N 2 α
and after multiplying by N k−2 :pN k−1 = 0,
5

so that N k−1 is nontrivial p-torsion class in K0 (X). 
Remark 2.3. In fact if we assume X to be a compact Kähler manifold with a full exceptionalb(X) of complexes of OX -modules with boundedcollection in the analytic derived category Dan
2,0coherent cohomology, one can show that H (X, C) = 0, so that the Kähler cone is open in
H 2 (X, R) = H 1,1 (X, C) ∩ H 1,1 (X, C) hence it has non-trivial intersection with H 2 (X, Z).
Then the Kodaira embedding theorem implies that X is projective.
Definition 2.4. We call a smooth projective complex variety of dimension n admitting a full
exceptional collection of length n + 1 a minifold.
It follows from Proposition 2.1 (2), that n + 1 is the minimal number of objects in such acollection, and the term ”minifold” originates from here. Minifolds have the same Hodge numbers
as projective spaces. By results of Beilinson [6] and Kapranov [32], projective spaces and odddimensional quadrics are minifolds.
Lemma 2.5. Let X be a minifold. Thenq either X is a Fano variety i.e. the anticanonical line bundle ω ∨ = det TX is ampleXq or canonical line bundle ωX = det T ∗ is ample
X
In particular, the variety X is uniquely determined by Db (X).
Proof. We first note that ωX is not trivial, since h0 (ωX ) = hn,0 (X) = 0 by Proposition 2.1(2). By
Proposition 2.1(3), Pic(X) is torsion free; hence the class of ωX in Pic(X)Q ∼= H 2 (X, Q) = Q is∨
non-zero. Therefore either ωX or ωX is ample. Now the Bondal-Orlov [15] reconstruction theorem
implies the last statement. Remark 2.6. If we weaken the assumption from ”projective” to ”proper” in the definition ofa minifold, we still get the same class of varieties. Indeed, if X is a proper smooth variety ofdimension n with a full exceptional collection of length n + 1 we can still deduce that ωX or its
dual is ample, in particular that X is projective as follows.
From [49](Theorem 3) it follows that for a compact complex n-dimensional manifold, the Chern
number c1 cn−1 is determined by Hirzebruch χ-genera χy and hence by the Hodge numbers.
2
Thus we have c1 cn−1 [X] = c1 cn−1 [Pn ] = n(n+1)
6= 0. Since the Kleiman-Mori cone of effective
2
one-cycles modulo numerical equivalence N1 (X) ⊂ H 2 (X, R) is one dimensional (that is because
H 2 (X, R) itself is one dimensional by Proposition 2.1(2) which still holds under the assumptionthat X is proper), Kleiman’s criterion for ampleness implies that either ωX or its dual is ample.
The rest of this section is devoted to proof of Theorem 1.1. In view of Lemma 2.5, the proof
consists of classifying Fano minifolds and showing that there is no minifolds among varieties ofgeneral type.
We start in dimension 2. The only del Pezzo surface with Picard number one is a projective
plane.
On the other hand it is known that fake projective planes have non-vanishing torsion firsthomology group [57], Theorem 10.1. Hence by Proposition 2.1(4) there is no minifold of general
type of dimension 2.
Let us consider Fano threefolds. By Proposition 2.1(2) conditions b2 (X) = 1 and b3 (X) = 0
are necessary for a minifold. Such Fano threefolds were classified by Iskovskikh [31] into four6

deformation types: the projective space P3 , the quadric Q3 , the del Pezzo quintic threefold V5 ,and a family of Fano threefolds V22 .
All these varieties are known to admit an exceptional collection of length 4 by results of Beilinson,
Kapranov, Orlov and Kuznetsov respectively [6], [32], [54], [44].
It is easy to see that 3-dimensional Q-homology varieties of general type do not exist. Indeed KX
ample implies that c1 (X)3 is negative, but by Todd’s theorem c1 (X)c2 (X) = 24. This contradicts
to Yau’s inequality c1 (X)3 > 83 c2 (X)c1 (X) [62].
According to Wilson [61] and Yeung [64] there are three alternatives for a Q-homology projective
fourspace X: either X is P4 , or X is a fake projective fourspace, or X has Hilbert polynomial−lχ(ωX) = 1 + 25
l(l + 1)(3l2 + 3l + 2) and Chern numbers [c41 , c2 c21 , c22 , c1 c3 , c4 ] = [225, 150, 100, 50, 5].
8
In what follows the varieties of the latter type are named Wilson’s fourfolds.
There are some known examples of fake projective fourfolds, but it is not known whether any
Wilson’s fourfold actually exist.
In what follows we show that (possibly non-existent) Wilson’s fourfolds do not satisfy conditionsof Proposition 2.1(5), and hence do not admit a full exceptional collection. In order to do that we
relate the Grothendieck group of a minifold to its Hilbert polynomial.
We need a simple Lemma from linear algebra.
PnjLemma 2.7. Let P (x) =j=0 pj x ∈ K[x] be a polynomial of degree 6 n with coefficientsin a field K of characteristic zero and let AP be the (n + 1) × (n + 1)-matrix with coefficients
ai,j = P (j − i). Then we havedet(AP ) = (n! pn )n+1 .
In particular the matrix AP is non-degenerate if and only if deg P = n.
Proof. It suffices to prove the statement for algberaic closure K̄ of K, we thus assume K to bealgebraically closed.
We first prove that
(2.4)

det(AP ) = 0 ⇐⇒ pn = 0.

Indeed if deg(P (x)) < n, then n+1 polynomials P (x), P (x+1), . . . , P (x+n) are linearly dependent
which makes the columns of AP linearly dependent, thus det(AP ) = 0. On the other hand, it iseasy to see that if deg(P (x)) = n, thenP (x), P (x + 1), . . . , P (x + n)
form a basis of the space of polynomials of degree 6 n, and AP is a matrix of an invertible linear
transformation P 7→ (P (0), P (−1), . . . , P (−n)) ∈ K n+1 in this basis, hence det(AP ) 6= 0.
Let F (p0 , p1 , . . . , pn ) = det(AP ). Since the entries of the matrix AP are linear forms in
p0 , p1 , . . . , pn , it follows that F is homogeneous in pi ’s of degree n + 1. Then (2.4) says thatthe support of the degree n + 1 hypersurface F = 0 in Pn is contained in the hyperplane pn = 0.
Therefore(2.5)

F (p0 , p1 , . . . , pn ) = Cn · pn+1nfor some constant Cn ∈ K. In particular det(AP ) takes the same value Cn for any monic polynomial
P (x) of degree n.
Let P0 (x) = (x + 1) · (x + 2) · · · · · (x + n). Then the matrix AP0 is uppertriangular with all
diagonal entries equal to n!:(2.6)

Cn = det(AP0 ) = (n!)n+1
7

The result now is the combination of (2.5) and (2.6) Proposition 2.8. Let X be a minifold. Let O(1) = det(TX ) be the anticanonical bundle, deg(X)be the anticanonical degree c1 (X)n and PX (k) = χ(O(k)) be the Hilbert polynomial. Consider asublattice Λ ⊂ K0 (X) spanned by[O], [O(1)], . . . , [O(n)].
Then the Euler pairing restricted to Λ is non-degenerate, that is classes [O], [O(1)], . . . , [O(n)]are linearly independent in K0 (X) and Λ is a sublattice in K0 (X) of full rank. Furthermore, Λ1] and hence modulo any prime p that does
admits a semi-orthonormal basis over the ring Z[ deg(X)not divide deg(X).
Proof. Let AX denote the matrix of the pairing on Λ, that is a matrix with entries ai,j =χ(O(i), O(j)) = PX (j − i).
We apply Lemma 2.7 to P = PX , the Hilbert polynomial. Its top coefficient is equal to pn =deg(X); therefore det(AX ) = deg(X) 6= 0 is the anticanonical degree and the pairing on Λ isn!
non-degenerate.
The inclusion Λ ⊂ K0 (X) becomes an isomorphism after inverting det(AX ) = deg(X). Indeed
let ej , j = 0, . . . n be a basis in K0 (X) and writeX[O(i)] =Gj,i ej , 0 6 i 6 n.
The matrix G−t AX G−1 is unimodular, hence deg(X) = det(G)2 , and after inverting deg(X), G
becomes invertible.
Since K0 (X) admits a semiorthonormal basis by assumption and Proposition 2.1(5), the same1
holds for Λ ⊗ Z[ deg(X)]Let PX be the Hilbert polynomial of Wilson’s fourfolds and AX be the 5 × 5-matrix (AX )i,j =PX (j − i). Consider their residues modulo two: AX = AX mod 2, (AX )i,j = P X (j − i).
In the Proof of Proposition 2.8 we showed that the determinant of matrix AX equals deg(X)n+1 =2255 = 1510 , hence the assumption that X is a minifold would imply that AX admits asemiorthonormal basis modulo all primes p 6= 3, 5, in particular this would imply that AX has
a semiorthonormal basis.
Entries of AX and AX are determined by values P (n) for 0 6 n 6 4 (that we tabulate) and
Serre duality P (n) = P (−1 − n):
n
0 1
2
3
4
P (n)
1 51 376 1426 3876
P (n) mod 2 1 1
0
0
0


1 1 0 0 0
1 1 1 0 0 


.
1
1
1
1
0
AX = 
0 1 1 1 1 
0 0 1 1 1
The following Lemma gives a contradiction, from which we see that a Wilson fourfold X cannot be a minifold.
Lemma 2.9. Let (u, v) 7→ ut AX v be the bilinear form on a vector space V = F52 given by thematrix AX . There is no basis e1 , e2 , e3 , e4 , e5 of V such that (ei , ej ) = 0 for i > j and (ei , ei ) = 1.
8

Proof. We begin by making a few remarks.
−1t(1) Let S := AX AX be an automorphism of V . In fact S is induced by the Serre functor
SX = ⊗ωX [dim X] on Db (X) [12, 14]. S satisfies (u, v) = (v, Su) for all u, v, so it preserves
AX , i.e. (u, v) = (Su, Sv), equivalently S t AX S = AX . We have

1 1 0 0 0
0 0 1 0 0



S=
0 0 0 1 0
1 0 0 0 1
1 0 0 0 0
and S has order 8 because the value of P (n) mod 2 depends only on n mod 8.
(2) There are precisely 12 vectors x such that (x, x) = 1. Indeed (x, x) = 1 if and only if
the point x does not lie on quadric Q = {x|(x, x) = 0}. The quadric Q has a unique
singular point in P(V ) so it has 19 points over F2 and its complement has 12 points. These
twelve points form two orbits under the action of S. One orbit of length 8 is generated by
a1 := (1, 0, 0, 0, 0)t , another orbit of length 4 is generated by b1 := (1, 0, 1, 0, 0)t .
(3) If a basis e1 , e2 , . . . , e5 is semi-orthonormal, then for each i (1 6 i 6 4) the basis obtained byreplacing ei , ei+1 with ei+1 , ei + ei+1 (ei , ei+1 ) is also semi-orthonormal. This transformation
corresponds to mutations of exceptional collections [12, 14].
Denote ai = S i−1 a1 , bi = S i−1 b1 and c = (a1 , . . . , a8 , b1 , . . . , b4 ). The following matrix has (ci , cj )on position i, j:
1 1 1 0 0 0 0 1 1 0 0 1
 1 1 1 1 0 0 0 0 1 1 0 0  0 1 1 1 1 0 0 0 0 1 1 0  0 0 1 1 1 1 0 0 0 0 1 1  0 0 0 1 1 1 1 0 1 0 0 1  0 0 0 0 1 1 1 1 1 1 0 0  1 0 0 0 0 1 1 1 0 1 1 0  1 1 0 0 0 0 1 1 0 0 1 1  0 1 1 0 0 1 1 0 1 1 1 1  0 0 1 1 0 0 1 1 1 1 1 1  1 0 0 1 1 0 0 1 1 1 1 1 
1 1 0 0 1 1 0 0 1 1 1 1
Assume there exists a semi-orthonormal basis. Then all of its vectors must be from the set{ai } ∪ {bi }. Since there are only 4 vectors in {bi }, at least one of the basis vectors must be from{ai }. Applying S if necessary we may assume that this vector is a1 . Applying the transformation(3) we can obtain a semi-orthonormal basis with a1 on the first position.
Any remaining basis vector x must satisfy (x, a1 ) = 0. Looking at the first column of the matrixof (ci , cj ) we see that the remaining basis vectors must be from the set {a3 , a4 , a5 , a6 , b1 , b2 }. Let
x be the second basis vector. Then any vector y out of the remaining 3 basis vectors must satisfy(y, x) = 0. However, trying for x each of the {a3 , a4 , a5 , a6 , b1 , b2 } we see that there are only 2
choices remaining for y. This is a contradiction. We also can prove that there is no minifolds among arithmetic fake projective fourspaces. This
goes similarly to dimension 2 case: Prasad and Yeung proved that for an arithmetic fake projective
9

fourspace the first homology group H1 (X, Z) is non-zero [58], Theorem 4. Therefore by Proposition
2.1(3) these fourfolds are not minifolds.
3. Phantoms in fake projective spaces
Fake projective spaces seem to be very similar and yet very different from ordinary projective
spaces. We propose the following conjecture.
Conjecture 3.1. Assume that X is an n-dimensional fake projective space with canonical classdivisible by (n + 1). Then for some choice of O(1) such that ωX = O(n + 1), the sequence
O, O(−1), . . . , O(−n)
is an exceptional collection on X.
We call an non-zero admissible subcategory A ⊂ Db (X) an H-phantom if HH q (A) = 0 and a
K-phantom if K0 (A) = 0.
Corollary 3.2. Fake projective spaces as in Conjecture admit an H-phantom admissible subcategories in their derived categories Db (X).
Proof. Assume that O, O(−1), . . . , O(−n) is an exceptional collection, and consider its right orthogonal A. By results of Bondal and Kapranov [12, 14] the category A is admissible, and thus
we have a semi-orthogonal decomposition:
Db (X) = hO, O(−1), . . . , O(−n), Ai .
Note that this exceptional collection could not be full at least for two reasons:q if O(i) would be a full collection then by [16](Theorem 3.4) or [56](see proof of main
theorem) manifold X would be Fano, which contradicts to general type assumption,
q use Corollary 4.6 and Proposition 4.7 of [47]: by Kodaira vanishing for i < j space
Extk (O(−i), O(−j)) vanish unless k = n, so relative height of any two objects in a helix
O(i) equals n, thus pseudoheight of the collection coincides with its height and is equal ton − 1, hence Hochschild cohomology HH 0 (A) = HH 0 (X) 6= 0 for n > 1.
Finally, Hochschild homology is additive for semi-orthogonal decompositions (cf the alternative
proof of 2.1(2)), so dim HH q (A) = 0 that is A is an H-phantom. Remark 3.3. 1. A statement analogous to Conjecture 3.1 holds for some fake del Pezzo surfacesof degrees one [10, 11], six [1] and eight [26, 48]. Here we add degrees three (Remark 3.8) and nine
(Theorem 1.2).
2. Fake projective planes with properties as in Conjecture 3.1 are constructed in [57], 10.4.
Choose O(1) such that O(3) = ωX . Then by the Riemann-Roch theorem the Hilbert polynomial
is given by(k − 1)(k − 2)χ(O(k)) =.
2
Therefore the collection E q = (O, O(−1), O(−2)) is at least numerically exceptional, that isχ(Ej , Ei ) = 0, j > i.
In addition we have
H 0 (S, O(1)) = H 0 (S, O(3)) = 0.
10

Furthermore it follows from Serre duality that a necessary and sufficient condition for E q to be
exceptional is vanishing of the space of the global sections H 0 (S, O(2)). It is not hard to see thatfor all fake projective planes h0 (S, O(2)) 6 2 (cf end of the Proof of Theorem 1.2).
3. More generally our definition of an n-dimensional fake projective space includes that its
Hilbert polynomial is the same as that of a Pn . It follows that if we assume ωX = O(n + 1), thenwe have(k − 1)(k − 2) . . . (k − n)χ(O(k)) = (−1)n,n!
so that k = 1, . . . , n are the roots of χ, and the collection
O, O(−1), . . . , O(−n)
is numerically exceptional.
4. G.Prasad and S.-K. Yeung informed us that the assumption ωX = O(5) is known to be true
for the four arithmetic fake projective fourspaces constructed in [58].
We now prove Theorem 1.2, which shows that conjecture 3.1 holds for fake projective planes
admitting an action of the non-abelian group G21 of order 21.
According to the Table given in the Appendix there are 6 such surfaces: there are three relevantgroups in the table and there are two complex conjugate surfaces for each group [39].
We first prove a general fact about fake projective planes.
Lemma 3.4. Let S be a fake projective plane with no 3-torsion in H1 (S, Z). Then there exists a
unique (ample) line bundle O(1) such that KS ∼= O(3).
Proof. First note that the torsion in Pic(S) = H 2 (S, Z) is isomorphic to H1 (S, Z) (cf Proof of
Proposition 2.1), hence Pic(S) has no 3-torsion by assumption.
By Poincare duality Pic(S)/tors ∼= H 2 (S, Z)/tors is a unimodular lattice, therefore there exists
an ample line bundle L with c1 (L)2 = 1. Now KS − 3c1 (L) ∈ Pic(S) is torsion which can be
uniquely divided by 3. 
Proof of Theorem 1.2. As follows from the classification of the fake projective planes by PrasadYeung and Cartwright–Steger, the order of the first homology group of the six fake projective
planes with automorphism group G21 is coprime to 3 (see the Table in the Appendix). Therefore
by Lemma 3.4 we have
KS = O(3).
for a unique line bundle O(1).
Recall that G21 = Aut(S) = N (Π)/Π where N (Π) is a normalizer of Π in P U (2, 1) and by [20]the embedding
N (Π) ⊂ P U (2, 1)
lifts to an embedding
N (Π) ⊂ SU (2, 1)
in all cases with G21 -action. Therefore OB (−1) admits a N (Π)-linearization and hence O(1) admitsa G21 -linearization, compatible with the natural G21 -linearization of KS . We will consider vector
spaces H ∗ (S, O(k)) as G21 -representations.
According to Remark 3.3(2), it suffices to show that H 0 (S, O(2)) = 0.
We now study the group G21 and its representation theory. By Sylow’s theorems G21 admits
a unique subgroup of order 7 and this subgroup is normal. We let σ denote a generator of11this subgroup. Let τ denote an element of G21 of order 3. Conjugating by τ gives rise to an
automorphism of Z/7 = hσi and we can choose τ so thatτ −1 στ = σ 2 .
Thus G21 is a semi-direct product of Z/7 and Z/3 and has a presentation
G21 = σ, τ | σ 7 = 1, τ 3 = 1, στ = τ σ 2 .
Using this presentation it is easy to check that there are five conjugacy classes of elements in
G21 :
{1}
{σ, σ 2 , σ 4 }{σ 3 , σ 5 , σ 6 }{τ σ k , k = 0, . . . , 6}
{τ 2 σ k , k = 0, . . . , 6}
and by basic representation theory there exist five irreducible representations of G21 . Let d1 , . . . , d5
be the dimensions of these representations. Basic representation theory also tells us that each didivides 21 and thatd21 + d22 + d23 + d24 + d25 = 21.
Considering different possibilities one finds the only combination (d1 , d2 , d3 , d4 , d5 ) = (1, 1, 1, 3, 3)
satisfying the conditions above.
It is not hard to check that the character table of G21 is the following one:
1 [σ] [σ 3 ] [τ ] [τ 2 ]
C 1 1
1
1
1
V1 1 11ωωV1 1 11ωω
V3 3 bb00
V3 3 bb00Here we use the notation:
2πi

ω=e 3

2πi

ξ=e 7
and

√
−1
+
−7
b = ξ + ξ2 + ξ4 =.
2Explicitly V1 and V1 are one-dimensional representations restricted from G21 / hσi = Z/3. V3and V3 are three-dimensional representations induced from Z/7: ρ : G21 → GL(V3 ) is given bymatrices




ξ
0 0 1
 ρ(τ ) =  1 0 0 ξ2
ρ(σ) = 4
ξ
0 1 0
and V3 is its complex conjugate.
Lemma 3.5. H 0 (S, O(4)) is a 3-dimensional irreducible representation of G21 (and thus is isomorphic to V3 or V3 ).
12

Proof. We show that the trace of an element σ ∈ G21 of order 7 acting on H 0 (S, O(4)) is equal
to b or b. This is sufficient since if H 0 (S, O(4)) were reducible it would have to be a sum of three
one-dimensional representations and the character table of G21 shows that in this case the traceof σ on H 0 (S, O(4)) would be equal to 3.
By [38], Proposition 2.4(4) σ has three fixed points P1 , P2 , P3 . Let τ be an element of order3. τ does not stabilize any of the Pi ’s, since a tangent space of a fixed point of G21 would givea faithful 2-dimensional representation of G21 which does not exist as is seen from its character
table.
Thus Pi ’s are cyclically permuted by τ . We reorder Pi ’s in such a way that
(3.1)

τ (Pi ) = Pi+1 mod 3 .

We apply the so-called Holomorphic Lefschetz Fixed Point Formula (Theorem 2 in [2]) to σ andline bundles O(k):
(3.2)

2
3
X
 X
p
(−1) T r σ H p (S,O(k)) =
p=0

i=1

T r(σ|O(k)Pi )(1 − α1 (Pi ))(1 − α2 (Pi ))where α1 (Pi ), α2 (Pi ) are inverse eigenvalues of σ on TPi :det(1 − tσ∗ |TPi ) = (1 − tα1 (Pi ))(1 − tα2 (Pi )).
αj (Pi ) are 7-th roots of unity. We let αj := αj (P1 ), j = 1, 2. Using (3.1) and commutation
relations in G21 we find that
αj (Pi+1 ) = αj (Pi )2
so thatαj (P1 ) = αj
αj (P2 ) = αj2
αj (P3 ) = αj4 .
To find the values of αj we apply (3.2) with k = 0:
(3.3)

1=

1
1
1
+
+
.
2
2
4
(1 − α1 )(1 − α2 ) (1 − α1 )(1 − α2 ) (1 − α1 )(1 − α24 )All αj (Pi ) are 7-th roots of unity and it turns out that up to renumbering the only possible valuesof αj (Pi ) which satisfy (3.3) are(α1 (P1 ), α2 (P1 )) = (ξ, ξ 3 )(α1 (P2 ), α2 (P2 )) = (ξ 2 , ξ 6 )(α1 (P3 ), α2 (P3 )) = (ξ 4 , ξ 5 )or their complex conjugate in which case we would get b instead of b for the trace below.
It follows that T r(σ|KS,Pi ) = T r(σ|O(3)Pi ) is equal to ξ 4 , ξ, ξ 2 for i = 1, 2, 3 respectively. Dividing
by 3 modulo 7 we see that T r(σ|O(k)Pi ) is equal to ξ 6k , ξ 5k , ξ 3k for i = 1, 2, 3 respectively.
We use (3.2) for k = 4 (note that H p (S, O(4)) = 0 for p > 0 by Kodaira vanishing):T r σ H 0 (S,O(4)) =

ξ6
ξ5
ξ3
+
+
=b
(1 − ξ)(1 − ξ 3 ) (1 − ξ 2 )(1 − ξ 6 ) (1 − ξ 4 )(1 − ξ 5 )
13

We are now ready to show that H 0 (S, O(2)) = 0. Let δ = h0 (S, O(2)). We know thath0 (S, O(4)) = 3, hence it follows from Lemma 3.6 applied to L = L0 = O(2) that δ 6 2. Therefore
as a representation of G21 the space H 0 (S, O(2)) is a sum of 1-dimensional representations and thesame is true for H 0 (S, O(2))⊗2 . Since H 0 (S, O(4)) is three-dimensional irreducible, this implies
that the natural morphism
H 0 (S, O(2))⊗2 → H 0 (S, O(4))
has to be zero by Schur’s Lemma. Now again by Lemma 3.6 H 0 (S, O(2)) = 0. This finishes theproof of Theorem 1.2. 
Lemma 3.6 (see [43](Lemma 15.6.2)). Let X be a normal and proper variety, L,L0 effective line
bundles on X. Let
φ : H 0 (X, L) ⊗ H 0 (X, L0 ) → H 0 (X, L ⊗ L0 )denote the natural map induced by multiplication. Thendim Im(φ) > h0 (X, L) + h0 (X, L0 ) − 1.
b(S) for various subgroups G ⊂ G21 . A good
We now consider equivariant derived categories DG
reference for equivariant derived categories and their semi-orthogonal decompositions is [22].
It is easy to see that

(3.4)

{O(−j) ⊗ V }j=0,1,2; V ∈IrrRep(G)b(S). We denote by AG
forms an exceptional collection in the equivariant derived category DG
S theright orthogonal to this collection.
It is easy to see that the category AG
S is non-zero. This follows from the Kuznetsov’s criterion(cf the second proof of Corollary 3.2) since the height of the exceptional collection equals n − 1.
We also notice that for any nonzero object A in AS the object
M
g∗A
g∈G

will have a natural G-linearization so will be a non-zero object in AG
S.
Proposition 3.7. Let S be a fake projective plane with automorphism group G21 . For any G ⊂G21 , AG
S is an H-phantom.
Proof. We denote by ZG the minimal resolution of S/G. The geometry of ZG has been carefully
studied by Keum [38]: if |G| = 7 or |G| = 21 then ZG is an elliptic surface of Kodaira dimension
κ(ZG ) = 1 (Dolgachev surface), if |G| = 3 then ZG is a surface of general type κ(ZG ) = 2. In eachbcase we compare the equivariant derived category DG(S) to Db (ZG ).
The stabilizers of the fixed points of G action are cyclic and we use [30] or [36] to obtain thesemi-orthogonal decompositionb
DG(S) ' Db (ZG ), E1 , . . . , ErG

where rG is the number of non-special characters of the stabilizers [30].
Note that pg (ZG ) = q(ZG ) = 0, therefore
dim HH∗ (Db (ZG )) = dim H ∗ (ZG , C) = χ(ZG ).

14

We list χ(ZG ) as well as other relevant invariants in the table:
G #IrrRep(G)Sing(S/G)rG χ(ZG ) κ(ZG )
1
1
∅
0
3
2
Z/3
3
3 × 13 (1, 2)
0
9
2
1
Z/7
7
3 × 7 (1, 3)
9
12
1
1
1
G21
5
3 × 3 (1, 2) + 7 (1, 3) 3
12
1
As already mentioned above rG is the sum of non-special characters of the stabilizers at fixed
points: 13 (1, 2) fixed points don’t contribute to rG whereas each 17 (1, 3) fixed point has 3 non-special
characters.
It follows from the table that in each case we have
3 · #IrrRep(G) = χ(ZG ) + rGb
This implies that the number of exceptional objects in (3.4) matches dim HH∗ (DG), and thereforeGin each case AS is an H-phantom. Remark 3.8. When G = Z/3, rG = 0 means thatb
DG(S) ' Db (ZG )in agreement with the derived McKay correspondence [33, 17] which is applicable since S/G has
A2 singularities. ZG is a fake cubic surface (pg (ZG ) = q(ZG ) = 0, b2 (ZG ) = 7) and the image ofthe exceptional collection (3.4) of 9 objects in Db (ZG ) has an H-phantom orthogonal.
Remark 3.9. One can give an alternative proof of Proposition 3.7 using orbifold cohomology.
Baranovsky [4] proved an analogue of Hochschild–Kostant–Rosenberg isomorphism for orbifolds.
b
His result implies that (total) Hochschild homology HH∗ (DG(S)) is isomorphic as a non-graded
vector space to the (total) orbifold cohomology
MM∗(S/G, C) =H ∗ (S g , C) G =H ∗ (S g , C)Z(g) .
Horb
g∈G

[g]∈G/G

Here S g is the fixed locus of g ∈ G, Z(g) is the centralizer, [g] is the conjugacy class of g, and (·)Gdenotes coinvariants. In our case the following two assumptions are satisfied:q group G acts trivially on H ∗ (S, C) (thanks to minimality of S),
q for each element g 6= 1 its fixed locus S g is a union of dim H ∗ (S, C) points (this is usuallyderived from Hirzebruch proportionality principle, see e.g. [39, 38]).
For the so-called main sector [g] = {id} we haveH ∗ (S, C)G = H ∗ (S, C) = C3 .
For each g 6= id the fixed locus S g consists of three points, so H ∗ (S g ) = C3 and the action ofZ(g) = hgi on it is trivial, thus each twisted sector is also 3-dimensional.
Taking the sum over all conjugacy classes [g] we obtainb∗dim HH∗ (DG(S)) = dim Horb
(S/G, C) = 3 × #IrrRep(G),which shows that HH∗ (AGS ) = 0.
15

Proposition 3.10. Let S be a fake projective plane with automorphism √group G21 . In the notationsof [57, 18] and the appendix assume that the class of S is either (Q( −7), p = 2, T1 = {7}) orb
C20 . Let G = Z/7 ⊂ G21 or G = G21 . Then the orthogonal to the collection (3.4) in DG(S) is a
K-phantom.
Proof. Let ΠG ⊂ Γ be the group generated by π1 (S) and G. By [5](0.4) the fundamental group
π1 (S/G) equals to ΠG /E where E ⊂ ΠG is the subgroup generated by elliptic elements of ΠGi.e. elements γ ∈ ΠG such that fixed locus B γ 6= ∅ is non-empty. Cartwright and Steger in[20] explicitly computed E and so π1 (S/G) for various subgroups Π ⊂ ΠG ⊂ Γ and it turns outthat in the cases under consideration the quotients S/G are simply connected. By a standard
argument (e.g. using Van Kampen’s theorem as in [5](0.5) or [60](Section 4.1), or more generally
see [42](Theorem 7.8.1)) the resolutions ZG are also simply-connected, in particular H1 (ZG , Z) = 0.
Then Pic(ZG ) = H 2 (ZG , Z) is finitely generated free abelian group. Keum shows in [38] that
Kodaira dimension κ(ZG ) = 1 (see also Ishida [29]). Thus Bloch conjecture for ZG is true by
Bloch–Kas–Lieberman [8], that is CH0 (ZG ) = Z. Now by Lemma 2.7 of [26] it follows thatb(S)).
K0 (ZG ) is a finitely generated free abelian group, and the same holds for K0G (S) = K0 (DG
The computation of Euler numbers shows thatbb(number of objects in (3.4)) = dim HH∗ (DG(S)) = rk K0 (DG(S)).

Finally, the additivity of the Grothendieck group implies that AG
S is a K-phantom. 
Appendix: automorphisms and first homology groups of fake projective planes
Recall that all fake projective planes S are quotients of a complex ball B ⊂ CP2 by a cocompact
torsion-free arithmetic subgroup Π = π1 (S) [57], [18], and each of the fifty possible groups Π
corresponds to a pair complex conjugate surfaces S and S which are not isomorphic to each other[39]. The first homology group H1 (S, Z) of S is isomorpic to the abelianisation of the Π/[Π, Π]and the automorphism group equals Aut(S) = N (Π)/Π, where N (Π) is the normaliser of Π (inmaximal arithmetic group Γ and hence in any group that contains it, in particular in P U (2, 1)).
We enhance the classification table of the fake projective planes given in [18] which is based on
GAP and Magma computer code and its output [19] with the automorphism group Aut(S) andthe first homology group H1 (S, Z), which we also take from [19].
In the table Γ is described using the following data: l is a totally complex quadractic extensionof a totally real field, p is a prime 2, 3 or 5, T1 is a set of prime numbers (possibly empty).
N is the index [Γ : Π] and suf. is the suffix (a, b, c, d, e or f ) of each group in [19]. G21 is the
non-abelian group of order 21. In the last column symbol [n1 , . . . , nk ] denotes the abelian group(Z/n1 Z) × · · · × (Z/nk Z).
Consider the quotient-map f : N (Π) → N (Π)/Π = Aut(S) and for a subgroup G ⊂ Aut(S) =N (Π)/Π let ΠG ⊂ N (Π) ⊂ Γ be the preimage ΠG = f −1 G. Line bundle OS (1) is G-linearisable⇐⇒ group ΠG lifts from P U (2, 1) to SU (2, 1). Computation of Cartwright and Steger [20] showsthat it holds for all S and G unless group Γ lies in classes C2 or C18 . Fundamental group of
the quotient-surface π1 (S/G) equals ΠG /E where E ⊂ ΠG is the subgroup generated by elliptic
elements (cf the proof of Proposition 3.10). All those groups for all S and G ⊂ Aut(S) were also
computed in [20]: surface S/G is simply-connected in twelve cases, including the four cases of
Proposition 3.10.

16

l or C√
Q( −1)p5√
Q( −2)

3√
Q( −7)

2√
Q( −15)

2

T1∅∅/{2I}{2}∅∅/{2I}
{2}
∅

N
3

#Π
2

3
3

1
2

3
21

1
3

{3}

3

2

{3, 7}

3

2

{7}

21

4

{5}
{5, 7}
∅

1
1
3

1
1
2

{3}

3

3

{3, 5}

3

3

{5}

3

2

√
Q( −23)

2

∅
{23}

1
1

1
1

C2

2

∅

9

6

C10

2

C18

3

{3}
∅
{17−}
∅
∅/{2I}
{2}

9
3
3
9
1
3

1
1
1
1
1
3

C20

2

∅
{3−}

21
3

1
2

{3+}

3

2

17

suf.
a
b/b
a
a
b/b
a
a
b
c
a
b
a
b
a
b
c
d
−
−
a
b
a
b
c
a
b
c
a
b
−
−

Aut(S)
Z/3Z
{1}
Z/3Z
Z/3Z
{1}
Z/3Z
Z/3Z
G21
{1}
Z/3Z
{1}
Z/3Z
{1}
G21
Z/3Z
Z/3Z
{1}
{1}
{1}
Z/3Z
{1}
Z/3Z
Z/3Z
Z/3Z
Z/3Z
Z/3Z
Z/3Z
Z/3Z
{1}
{1}
{1}

H1 (S, Z)[2, 4, 31][2, 3, 4, 4][4, 31][2, 2, 13][2, 2, 2, 2, 3][2, 2, 13][2, 7][2, 2, 2, 2][2, 2, 3, 7][2, 4, 7][2, 2, 3, 4][4, 7][2, 3, 4][2, 2, 2][2, 7][2, 2, 7][2, 2, 2, 3][2, 2, 9][2, 9][2, 2, 7][2, 2, 2, 9][2, 3, 7][2, 2, 2, 3][2, 3][3, 7][2, 2, 3][3][2, 7][2, 2, 9][2, 3, 7][3, 7]

a
b
c
d
f
g
−
−
−
a
b/d
a
b
c
−
a
b
a
b

(Z/3Z)2
Z/3Z
Z/3Z
Z/3Z
1
1
(Z/3Z)2
Z/3Z
Z/3Z(Z/3Z)2
1
Z/3Z
Z/3Z
Z/3Z
G21
Z/3Z
{1}
Z/3Z
{1}

[2, 7][2, 7, 9][2, 9][2, 9][2, 3, 3][2, 3, 3][7][2, 7][7][2, 2, 13][2, 3, 3][2, 3, 13][2, 3][2, 3][2, 2, 2, 2, 2, 2][4, 7][2, 3, 4][4, 7][2, 3, 4]References[1] Valery Alexeev, Dmitri Orlov: Derived categories of Burniat surfaces and exceptional collections,
arXiv:1208.4348, Math. Ann. (2013), 1–17.
[2] M.F.Atiyah, R.Bott: A Lefschetz fixed point formula for elliptic differential operators, Bull. Amer. Math. Soc.
72 1966 245–250.
[3] Thierry Aubin: Equations du type Monge-Amp‘ere sur les varietes kahleriennes compactes, C. R. Acad. Sci.
Paris Ser. A-B 283 (1976), no. 3, Aiii, A119–A121.
[4] Vladimir Baranovsky: Orbifold cohomology as periodic cyclic homology, arXiv:math/0206256, International
Journal of Mathematics 14 (08), 2003, 791–812.
[5] Rebecca Barlow: A simply connected surface of general type with pg = 0, Invent. Math. 79 (1985), 293–301.
[6] Alexander Beilinson: Coherent sheaves on Pn and problems in linear algebra, Funkcionalniy analiz i ego pril.
12 (1978), 68-69; English transl. in Functional Anal. Appl. 12 (1978), no. 3, 214–216.
[7] Marcello Bernardara, Michele Bolognesi: Categorical representability and intermediate Jacobians of Fano threefolds, arXiv:1103.3591[8] Spencer Bloch, Arnold Kas, David Lieberman: Zero cycles on surfaces with pg = 0, Compositio Mathematica,
33 no 2 (1976), p. 135–145.
[9] Christian Böhning:
Derived categories of coherent sheaves on rational homogeneous manifolds,
arXiv:math/0506429, Documenta Mathematica 11 (2006): 261–331.
[10] Christian Böhning, Hans-Christian Graf von Bothmer, Pawel Sosna: On the derived category of the classicalGodeaux surface, arXiv:1206.1830, Advances in Mathematics 243 (2013): 203–231.
[11] Christian Böhning, Hans-Christian Graf von Bothmer, Ludmil Katzarkov, Pawel Sosna: Determinantal Barlow
surfaces and phantom categories, arXiv:1210.0343[12] Alexey Bondal: Representations of associative algebras and coherent sheaves, Izv. Akad. Nauk SSSR Ser. Mat.,
53:1 (1989), 25–44[13] Alexey Bondal, Michel Van den Bergh: Generators and representability of functors in commutative and noncommutative geometry, arXiv:math/0204218, Moscow Math. J. 3 (2003), no. 1, 1–36, 258.
[14] Alexey Bondal, Mikhail Kapranov: Representable functors, Serre functors, and mutations, Izv. Akad. NaukSSSR Ser. Mat., 53:6 (1989), 1183–1205[15] Alexey Bondal, Dmitri Orlov: Reconstruction of a variety from the derived category and groups of autoequivalences, arXiv:alg-geom/9712029, Compositio Math. 125 (2001), no. 3, 327–344.
[16] Alexey Bondal, Alexander Polishchuk: Homological properties of associative algebras: the method of helices,
Russian Academy of Sciences. Izvestiya Mathematics, 1994, 42:2, 219–260.
[17] Tom Bridgeland, Alastair King, Miles Reid: Mukai implies McKay: the McKay correspondence as an equivalence of derived categories, arXiv:math/9908027, Journal of the American Mathematical Society, 14(3), 535–
554, 2001.
[18] Donald Cartwright, Tim Steger: Enumeration of the 50 fake projective planes, C. R. Acad. Sci. Paris, Ser. I
348 (2010), 11–13.
[19] Donald Cartwright, Tim Steger: http://www.maths.usyd.edu.au/u/donaldc/fakeprojectiveplanes/[20] Donald Cartwright: Private communication based on computer algebra computations of Dondal Cartwright and
Tim Steger[21] Vladimir Drinfeld: Coverings of p-adic symmetric domains, (Russian) Funkcional. Anal. i Prilozen. 10 (1976),
no. 2, 29–40.
[22] Alexey Elagin: Semi-orthogonal decompositions for derived categories of equivariant coherent sheaves, Izv.
Ross. Akad. Nauk Ser. Mat. 73 (2009), no. 5, 37–66; translation in Izv. Math. 73 (2009), no. 5, 893–920.
[23] Najmuddin Fakhruddin:
Exceptional collections on 2-adically uniformised fake projective planes,
arXiv:arXiv:1310.3020[24] Takao Fujita: On topological characterizations of complex projective spaces and affine linear spaces, Proc. Japan
Acad. Ser. A Math. Sci. Volume 56, Number 5 (1980), 231–234.
[25] William Fulton: Intersection theory (1984).
[26] Sergey Galkin, Evgeny Shinder: Exceptional collections of line bundles on the Beauville surface,arXiv:1210.3339, Advances in Mathematics 244 (2013): 1033–1050.
[27] Sergey Gorchinskiy, Dmitri Orlov:
Geometric Phantom Categories, arXiv:1209.6183, Publications
mathématiques de l’IHÉS (2013): 1–21.
18

[28] Friedrich Hirzebruch, Kunihiko Kodaira: On the complex projective spaces, J. Math. Pures Appl., IX Ser, 1957.
[29] Masa-Nori Ishida: An elliptic surface covered by Mumford’s fake projective plane, Tohoku Math. J. (2) 40(1988), no. 3, 367-396.
[30] Akira Ishii and Kazushi Ueda: The special McKay correspondence and exceptional collection, arXiv:1104.2381.
[31] Vasily Iskovskikh: Fano threefolds II, Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), no. 3, 506–549.
[32] Michail Kapranov: On the derived categories of coherent sheaves on some homogeneous spaces, Invent. Math.
92 (1988), no. 3, 479–508.
[33] Mikhail Kapranov, Eric Vasserot:
Kleinian singularities, derived categories and Hall algebras,
arXiv:math/9812016, Mathematische Annalen 316 (3), 565–576, 2000.
[34] Yujiro Kawamata: Log crepant birational maps and derived categories, arXiv:math/0311139, J. Math. Sci.
Univ. Tokyo 12 (2005), no. 2, 211-231.
[35] Yujiro Kawamata: Derived categories of toric varieties, arXiv:math/0503102, Michigan Math. J. 54 (2006),
no. 3, 517-535.
[36] Yujiro Kawamata: Derived categories of toric varieties II, arXiv:1201.3460[37] Bernhard Keller: Invariance and localization for cyclic homology of DG algebras, J. Pure Appl. Algebra 123(1998), no. 1–3, 223–273.
[38] Jonghae Keum: Quotients of fake projective planes, Geom. Topol. 12 (2008), no. 4, 2497–2515.
[39] Viatcheslav M. Kharlamov and Viktor S. Kulikov: On real structures on rigid surfaces (English) Izv. Math.
66, No. 1, 133–150 (2002); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 66, No. 1, 133–152 (2002),
arXiv:math/0101098.
[40] 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.
[41] Shoshichi Kobayashi, Takushiro Ochiai: Characterizations of complex projective spaces and hyperquadrics, J.
Math. Kyoto Univ., 13:31–47 , 1973.
[42] János Kollár: Shafarevich maps and plurigenera of algebraic varieties, Invent. Math. 113 (1993), 177–215.
[43] János Kollár: Shafarevich maps and automorphic forms, Princeton: Princeton University Press, 1995.
[44] Alexander Kuznetsov: An exceptional set of vector bundles on the varieties V22 , Vestnik Moskov. Univ. Ser. I
Mat. Mekh. 92(3), 41–44 (1996). Fano threefolds V22 , MPI preprint 1997-24.
[45] Alexander Kuznetsov: Hyperplane sections and derived categories, arXiv:math.AG/0503700, Izvestiya RAN:
Ser. Mat. 70:3 (2006) p. 23–128 (in Russian); translation in Izvestiya: Mathematics 70:3 (2006) p. 447–547.
[46] Alexander Kuznetsov: Hochschild homology and semiorthogonal decompositions, arXiv:0904.4330[47] Alexander Kuznetsov: Height of exceptional collections and Hochschild cohomology of quasiphantom categories,
arXiv:1211.4693, to appear in Crelle journal.
[48] Kyoung-Seog Lee: Derived categories of surfaces isogenous to a higher product, arXiv:1303.0541[49] Anatoly Libgober, John Wood: Uniqueness of the complex structure on Kaehler manifolds of certain homotopy
types, J. Differential Geometry, 32 (1990), 139–154.
[50] Ju.I. Manin, Correspondences, motifs and monoidal transformations, Mat. Sb. (N.S.) 77 (119) 1968 475–507.
[51] M. Marcoli, G.Tabuada: From exceptional collections to motivic decompositions via noncommutative motives,
arXiv:1202.6297, Journal für die reine und angewandte Mathematik (Crelles Journal) (2013).
[52] David Mumford: An algebraic surface with K ample, K 2 = 9, pg = q = 0, Amer. J. Math. 101 (1979), no. 1,
233–244.
[53] G.A. Mustafin: Non-Archimedean uniformization, Mat. Sb. (N.S.) 105(147) (1978), no. 2, 207–237, 287.
[54] Dmitri Orlov: Exceptional set of vector bundles on the variety V5 , Vestnik Moskov. Univ. Ser. I Mat. Mekh.
(5), 69–71 (1991).
[55] Dmitri Orlov: Derived categories of coherent sheaves and motives, arXiv:math/0512620, Russian Math. Surveys
60 (2005), 1242–1244.
b[56] Leonid Positselski: All strictly exceptional collections in Dcoh(Pm ) consist of vector bundles, arXiv:alggeom/9507014[57] Gopal Prasad, Sai-Kee Yeung: Fake projective planes, arXiv:math/0512115v5, arXiv:0906.4932v3. Invent.
Math. 168(2007), 321–370; 182(2010), 213–227.
[58] Gopal Prasad, Sai-Kee Yeung: Arithmetic fake projective spaces and arithmetic fake Grassmannians,
arXiv:math/0602144v4, American Journal of Mathematics, 131(2), 379–407.
[59] Alexander Samokhin: Some remarks on the derived categories of coherent sheaves on homogeneous spaces,
arXiv:math/0612800, Journal of the London Mathematical Society, 76(1), 122–134.
19

[60] Misha Verbitsky: Holomorphic symplectic geometry and orbifold singularities, arXiv:math/9903175, Asian J.
Math. 4, 2000, no. 3, 553–563.
[61] Pelham M.H. Wilson: On projective manifolds with the same rational cohomology as P4 , Conference on algebraic
varieties of small dimension (Turin, 1985). Rend. Sem. Mat. Univ. Politec. Torino 1986, Special Issue, 15–23(1987).
[62] Shing Tung Yau: Calabi’s conjecture and some new results in algebraic geometry, Proceedings of the National
Academy of Sciences of the United States of America (National Academy of Sciences) 74 (5): 1798–1799 (1977).
[63] Sai-Kee Yeung: Integrality and arithmeticity of co-compact lattices corresponding to certain complex two-ballquotients of Picard number one, Asian J. Math. 8 (2004), 107–130.
[64] Sai-Kee Yeung: Uniformization of fake projective four spaces, Acta Mathematica Vietnamica, 35, 1, 2010,
199–205.
[65] Sai-Kee Yeung: Classification of fake projective planes, Handbook of geometric analysis, No. 2, 391–431, Adv.
Lect. Math. (ALM), 13, Int. Press, Somerville, MA, 2010.

Sergey Galkin, National Research University Higher School of Economics
Sergey.Galkin@phystech.edu
Ludmil Katzarkov, University of Miami and University of Vienna
lkatzark@math.uci.edu
Anton Mellit, International Centre for Theoretical Physics
Mellit@gmail.com
Evgeny Shinder, University of Edinburgh
E.Shinder@ed.ac.uk

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