# On the syzygies of ample line bundles
on fake projective planes

Sui-Chung Ng and Sai-Kee Yeung  S.-C. Ng, School of Mathematical Sciences,
Shanghai Key Laboratory of PMMP, East China Normal University, Minhang
District, Shanghai, China  S.-K. Yeung, Mathematics Department, Purdue
University, West Lafayette, IN 47907 USA<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Our goal is to study the syzygies of the projective embeddings defined by
ample line bundles on a fake projective plane $S$. The syzygies are studied in
terms of the property $N_{p}$. For various kinds of ample line bundles, we
give explicit lower bounds for their powers above which the property $N_{p}$
is satisfied.

Key words: fake projective planes, pluricanonical mappings, syzygies
AMS 2010 Mathematics subject classification: 13D02, 14C20, 14J29
The first author was partially supported by National Natural Science
Foundation of China (Grant No. 11501205) and Science and Technology Commission
of Shanghai Municipality (Grant No. 13dz2260400). The second author was
partially supported by a grant from the National Science Foundation, USA

##  1\. Introduction

The study of very-ampleness, projective normality and syzygy of ample line
bundles on a surface is natural and has a long history. The main purpose of
this paper is to give some detailed estimates on fake projective planes.

There are two natural questions for a projective algebraic manifold $M$ of
complex dimension $n$ equipped with an ample line bundle $L$. The first is to
study the smallest multiple of $L$ which gives a projective embedding of the
manifold, and the second is to study the syzygy relative to such a canonical
embedding. For the first question, the Fujita Conjecture states that
$K_{M}\otimes L^{k}$ is very ample if $k\geqslant n+2$. For the second
question, there is the Mukai Conjecture which states that $K_{M}\otimes
L^{n+2+p}$ satisfies the $N_{p}$ property. For surfaces, Fujita Conjecture is
proved by Reider [R]. The Mukai Conjecture is mostly open.

Fake projective planes are interesting surfaces to study among smooth surfaces
of general type since they have the smallest Euler number (viz., $3$) among
all such surfaces and have been classified in [PY] and [CS]. However, there is
still no good way to describe such surfaces directly from classical algebraic
geometry. For a fake projective plane $S$, $K_{S}^{3}$ is very ample from a
well-known result of Bombieri and also Reider [R]. We also know that $K_{S}$
has no sections. Hence a natural question is whether $K_{S}^{2}$ is very ample
or not, a case at the borderline for which Reider’s result [R] cannot be
applied. This is not settled, even though there are some examples which are
true (cf. [Y2] and the references therein). Hence one would expect that sharp
result for the syzygy of $S$ may be difficult to achieve. The interest about
syzygy is that it would provide information about the pluricanonical ring of
$S$ which may eventually lead to explicit descriptions of a fake projective
plane, an interesting problem which is widely open. As a consequence of the
current work, it turns out that we have pretty good information about the
syzygy, cf. Corollary 1, even though it is still short of Mukai Conjecture.
The results of this paper are close to the optimal one with the currently
available techniques. The difficulty to achieve a sharper result is related to
the difficulty in getting the very-ampleness of $K_{S}^{2}$, which in turn is
related to the existence of exceptional collection of objects, cf. Conjecture
1 in §5. Sharper estimates are obtained for some special fake projective
planes as explained in Theorem 9 and the remark following it in §5.

The syzygies of a given projective embedding can be described using the so-
called property $N_{p}$, where $p$ is a non-negative integer. For the details
of its definition, we refer the reader to [EL]. Here, we simply recall that
the property $N_{0}$ corresponds to projective normality and the property
$N_{1}$ corresponds to projective normality together with the condition that
the homogeneous ideal of the embedded subvariety is generated by quadratic
polynomials.

The present paper gives effective results for the lower bounds of the powers
of a given ample line bundle on a fake projective plane satisfying the
property $N_{p}$. The reader may refer to [Rém] or [Y1] for basic geometric
facts about fake projective planes.

For general study of syzygies related to surfaces of general type, the reader
can see, for example, the works of Banagere-Hanumanthu [BH], Gallego-
Purnaprajna [GP], Hwang-To [HT], Purnaprajna [P] and the very recent work of
Niu [N]. Here we remark that unlike the previous results found in the
literature for surfaces of general type, which are pertaining to base-point-
free line bundles and are expressed in terms of some sort of “regularity”
related the relevant line bundles, we established in this paper explicit and
concrete numbers for a given arbitrary ample line bundle, which is not
necessarily base-point-free, and in particular, including the the canonical
bundle (which has no non-trivial section).

Our first main result is regarding an arbitrary ample line bundle on a fake
projective plane.

######  Theorem 1.

Let $A$ and $N$ be line bundles on a fake projective plane $S$ such that $A$
is ample and $N$ is nef. Then, $A^{6(p+3)}\otimes N$ satisfies the property
$N_{p}$.

It follows easily from the definition that any fake projective plane $S$ is of
Picard number 1. Moreover, it is known (see for example [Y1]) that the
canonical line bundle can be written as $K_{S}=G^{3}\otimes\tau$, where $G$ is
an ample generator $G$ of Pic$(S)$ and $\tau$ is some torsion line bundle.
Thus, we have the following immediate corollary of Theorem 1.

######  Corollary 1.

Let $S$ be a fake projective plane. Then $K^{m}_{S}$ satisfies the property
$N_{p}$ if $m\geq 2p+6$. In particular, $K^{m}_{S}$ is projectively normal if
$m\geq 6$.

Remark. We have also obtained results for smaller values of $m$, but those
require various additional efforts. In fact, we will see that $K^{2p+2}_{S}$
(resp. $K^{2p+4}_{S}$) also satisfies the property $N_{p}$ for $p\geq 1$
(resp. $p\geq 0$), as a consequence of Theorem 7 in Section 3. In particular,
$K^{4}_{S}$ is also projectively normal. Furthermore, motivated by the
existence problem for exceptional collections for objects in the derived
category of coherent sheaves on a fake projective plane $S$, if we assume that
for any ample generator $G$ of Pic$(S)$ and any torsion line bundle $\delta$,
we have $h^{0}(S,G^{2}\otimes\delta)=0$, a vanishing statement which is a
variant of a conjecture mentioned in [GKMS], then we will be able show that
that $K^{2p+5}_{S}$ also satisfies the property $N_{p}$ for $p\geq 0$ (Section
5). Such conjecture is known to be true for fake projective planes with more
than three automorphisms. As an intermediate step of the proof, we also
obtained some vanishing or almost vanishing results concerning the torsion
line bundles of a fake projective plane, which are of some independent
interests.

######  Proposition 1.

Let $\delta$ be a torsion line bundle on a fake projective plane $S$ and $G$
an ample generator of $\mbox{Pic}(S)$. Then
(a). $h^{1}(S,\delta)=0$;
(b). $h^{0}(S,G\otimes\delta)\leq 1$, $h^{0}(S,G^{2}\otimes\delta)\leq 2$ and
$h^{0}(S,G^{3}\otimes\delta)\leq 1$.

Finally, we mention the following theorem for an arbitrary base-point-free
line bundle on a fake projective plane, which will be established in Section
4.

######  Theorem 2.

Let $S$ be a fake projective plane and $B$ be a base-point-free line bundle on
$S$. Then, $B^{m}$ is projectively normal if $m\geq 2$. Moreover, for $p\geq
1$, $B^{m}$ satisfies the property $N_{p}$ if $m\geq p+1$.

This follows from a vanishing result on the unramified cover of each fake
projective plane defined by the commutator subgroup of its fundamental group.

The results in this paper are established through the well known cohomology
criterion given by Green [G]. The details of Green’s criterion will be
recalled in Section 2. To show that the cohomology groups appearing in Green’s
theorem vanish, our main tool is the classical result of Castelnuovo-Mumford
on the surjectivity of certain tensor product maps on various spaces of global
sections, which has been used already in the literature, in particular in the
work of Gallego-Purnaprajna [GP] for surfaces under various assumptions and
for the properpty $N_{p}$, $p\geqslant 1$. Our goal is to provide as sharp an
estimate as possible for fake projective planes, especially for lower values
of $p$. Careful analysis is needed to make sure that there is no numerical gap
in the range of possible powers of the line bundle satisfying the theorems of
this paper, (see the remark before Theorem 6). We have also made an effort to
make the the presentation reasonably self-contained.

##  2\. Cohomological criterion for the property $N_{p}$

We will first recall in this section the cohomological criterion for the
property $N_{p}$ obtained by Green [G].

Let $X$ be a complex projective manifold and $L$ be an ample and base-point-
free line bundle on $X$. Denote by $\mathcal{H}^{0}(X,L)$ the trivial vector
bundle on $X$ whose fibers are $H^{0}(X,L)$. Then we have the following exact
sequence of vector bundles

$0\rightarrow M_{L}\rightarrow\mathcal{H}^{0}(X,L)\rightarrow L\rightarrow 0,$

where the fiber of $M_{L}$ at $x\in X$ is the subspace in $H^{0}(X,L)$
consisting of the sections vanishing at $x$. Using the theory of Koszul
cohomology, Green [G] obtained the following

######  Theorem 3 (Green).

The line bundle $L$ satisfies the property $N_{p}$ if

$H^{1}(X,\wedge^{r}M_{L}\otimes L^{s})=0$

for $0\leq r\leq p+1$ and $s\geq 1$.

Over a field of characteristic zero, $\wedge^{r}M_{L}\subset M_{L}^{\otimes
r}$ is a direct summand and therefore it is a common practice to simply verify
the vanishing of $H^{1}(X,M_{L}^{\otimes r}\otimes L^{s})$ for studying the
property $N_{p}$. The main tool for us to achieve this is the following
theorem, pertaining to the so-called Castelnuovo-Mumford regularity (cf. [L]).

######  Theorem 4 (Castelnuovo-Mumford).

Let $E$ be a base-point-free line bundle on a projective variety $M$ and
$\mathcal{F}$ be a coherent sheaf on $M$. Suppose that
$H^{i}(M,\mathcal{F}\otimes E^{-i})=0$ for $i\geqslant 1$. Then the
multiplication mapping

$H^{0}(M,\mathcal{F}\otimes E^{j})\otimes H^{0}(M,E)\rightarrow
H^{0}(M,\mathcal{F}\otimes E^{j+1})$

is surjective for all $j\geqslant 0$.

##  3\. Vanishing theorems and proofs of the main theorems

In what follows, we let $S$ be a fake projective plane. We recall some known
facts about fake projective planes, which can be found in the survey articles
by Rémy [Rém] and Yeung [Y1]. First of all, the Picard number of $S$ is 1 and
we will fix an ample generator $G$ of $\textrm{Pic}(S)$. We have then $G\cdot
G=1$. Moreover, if $K_{S}$ denotes the canonical line bundle, then
$K_{S}=G^{3}\otimes\tau$, for some torsion line bundle $\tau$.

######  Lemma 1.

Let $A$ be an ample line bundle and $\sigma$ be a torsion line bunde on $S$.
Then, $H^{1}(S,A^{k}\otimes\sigma)=H^{2}(S,A^{k}\otimes\sigma)=0$ for $k\geq
4$.

###### Proof.

Since $A$ is ample and the Picard number of $S$ is 1, we can write
$A=G^{m}\otimes\epsilon$ for some $m\in\mathbb{N}^{+}$ and some torsion line
bundle $\epsilon$. Then, for $i=1,2$ and $k\geq 4$,

$H^{i}(S,A^{k}\otimes\sigma)=H^{i}(S,K_{S}\otimes
G^{mk-3}\otimes\tau^{-1}\otimes\epsilon^{k}\otimes\sigma)=0$

by Kodaira Vanishing Theorem as $G^{mk-3}$ is ample. ∎

######  Proposition 2.

Let $B$ be an ample line bundle on $S$ such that $B\otimes K_{S}^{-2}$ is nef,
or equivalently, $B=G^{m}\otimes\sigma$ for some $m\geq 6$ and some torsion
line bundle $\sigma$. Then $B^{k}$ is base-point-free for $k\geq 1$ and
$H^{1}(S,B^{k})=H^{2}(S,B^{k})=0$ for $k\geq 0$.

###### Proof.

By hypotheses, we can write $B^{k}=K_{S}\otimes L$, where
$L=G^{m^{\prime}}\otimes\sigma^{\prime}$ for some $m^{\prime}\geq 3$ and some
torsion line bundle $\sigma^{\prime}$. Since $L\cdot L=(m^{\prime})^{2}\geq
9$, using the classical results of Reider [Rei], we see that $B^{k}$ is base-
point-free. For $k\geq 1$, the vanishing of $H^{1}(S,B^{k})$ and
$H^{2}(S,B^{k})$ follows directly from Lemma 1, while for $k=0$, it follows
from the fact that $H^{1}(S,\mathcal{O}_{S})=0$ and
$H^{2}(S,\mathcal{O}_{S})\cong H^{0}(S,K_{S})=0$ for any fake projective
plane. ∎

In the rest of this section, we will establish two vanishing theorems for the
cohomology groups appearing in Green’s criterion for the property $N_{p}$. As
mentioned before, our main tool is the theorem of Castelnuovo-Mumford (Theorem
4). The basic ideas behind our methods are taken from the work of Gallego-
Purnaprajna [GP]. For base-point-free line bundles on $S$, we streamline the
arguments in [GP] and adapt to the present context for fake projective planes.
Geometric properties of fake projective planes are utilized to make sure that
conditions for the Castelnuovo-Mumford theorem are satisfied.

From Proposition 2, if $B$ is an ample line bundle such that $B\otimes
K_{S}^{-2}$ is nef, then $B^{k}$ is base-point-free for $k\geq 1$. Thus, we
have following exact sequence of vector bundles

$0\rightarrow M_{B^{k}}\rightarrow\mathcal{H}^{0}(S,B^{k})\rightarrow
B^{k}\rightarrow 0.$

Tensoring with $M^{\otimes(r-1)}_{B^{k}}\otimes B^{\ell}$ for $r\geq 1$ and
$\ell\geq 0$, we have

$0\rightarrow M^{\otimes r}_{B^{k}}\otimes
B^{\ell}\rightarrow\mathcal{H}^{0}(S,B^{k})\otimes
M^{\otimes(r-1)}_{B^{k}}\otimes B^{\ell}\rightarrow
M^{\otimes(r-1)}_{B^{k}}\otimes B^{k+\ell}\rightarrow
0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\star)$

for $k,r\geq 1$ and $\ell\geq 0$.

######  Theorem 5.

Let $B$ be an ample line bundle on a fake projective plane $S$ such that
$B\otimes K_{S}^{-2}$ is nef. Let $k\geq 1$ and $\ell\geq 0$. Then,

(i) $H^{1}(S,M_{B^{k}}\otimes B^{\ell})=0$ if $k+\ell\geq 3$;
(ii) $H^{1}(S,M^{\otimes r}_{B^{k}}\otimes B^{\ell})=0$ if $k+\ell\geq r+2$
and $\ell\geq r\geq 2$;
(iii) $H^{2}(S,M^{\otimes r}_{B^{k}}\otimes B^{\ell})=0$ if $\ell+1\geq r\geq
1$.

###### Proof.

We first show that $H^{1}(S,M_{B^{k}}\otimes B^{\ell})=0$ if $k+\ell\geq 3$.
In order to simplify the notations, in the proof we will skip the reference to
$S$ when writing the cohomology groups.

From the long exact sequence associated to $(\star)$, we get the exact
sequence

$\begin{matrix}\cdots\rightarrow H^{0}(\mathcal{H}^{0}(S,B^{k})\otimes
B^{\ell})\rightarrow H^{0}(B^{k+\ell})\\\ \rightarrow H^{1}(M_{B^{k}}\otimes
B^{\ell})\rightarrow H^{1}(\mathcal{H}^{0}(S,B^{k})\otimes
B^{\ell})\rightarrow\cdots\end{matrix}$

As $H^{1}(\mathcal{H}^{0}(S,B^{k})\otimes B^{\ell})\cong H^{0}(B^{k})\otimes
H^{1}(B^{\ell})$, it vanishes by Proposition 2. Thus, $H^{1}(M_{B^{k}}\otimes
B^{\ell})$ vanishes if the map $H^{0}(\mathcal{H}^{0}(S,B^{k})\otimes
B^{\ell})\rightarrow H^{0}(B^{k+\ell})$ is surjective. Equivalently, we need
the surjectivity for the map

$H^{0}(B^{k})\otimes H^{0}(B^{\ell})\rightarrow
H^{0}(B^{k+\ell})\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\sharp)$

since $H^{0}(\mathcal{H}^{0}(S,B^{k})\otimes B^{\ell})\cong
H^{0}(B^{k})\otimes H^{0}(B^{\ell})$. Consider the natural maps

$H^{0}(B)\otimes\cdots\otimes H^{0}(B)\otimes
H^{0}(B^{\ell})\overset{p}{\longrightarrow}H^{0}(B^{k})\otimes
H^{0}(B^{\ell})\overset{q}{\longrightarrow}H^{0}(B^{k+\ell}).$

We see that it suffices to have the surjectivity for $q\circ p$, which is in
turn implied by the surjectivity of all the mappings

$H^{0}(B)\otimes H^{0}(B^{m-1+\ell})\rightarrow H^{0}(B^{m+\ell})$

for $1\leq m\leq k$. We now apply the result of Castelnuovo-Mumford (Theorem
4). Thus, the above maps are surjective if

$H^{1}(B^{\ell-1})=H^{2}(B^{\ell-2})=0.$

By Proposition 2, these are satisfied if $\ell\geq 2$. (Here we remark that
all the higher cohomology groups appeared in Castelnuovo-Mumford Theorem
vanish because ${\mbox{dim}}(S)=2$.) Thus, we see that for $k\geq 1$,
$H^{1}(S,M_{B^{k}}\otimes B^{\ell})=0$ if $\ell\geq 2$. Since the roles of $k$
and $\ell$ are symmetric for the mapping in $(\sharp)$, it follows that for
$k\geq 2$, we also have $H^{1}(S,M_{B^{k}}\otimes B^{\ell})=0$ for $\ell=1$.
We hence deduce that $H^{1}(S,M_{B^{k}}\otimes B^{\ell})=0$ if $k+\ell\geq 3$.
Note that we can also allow $\ell=0$ here as the map in $(\sharp)$ is
surjective for $\ell=0$. We have thus proven $(i)$.

To prove $(ii)$ and $(iii)$, we will need to first establish
$H^{2}(M_{B^{k}}\otimes B^{\ell})=0$ for $\ell\geq 0$ and $H^{2}(M^{\otimes
2}_{B^{k}}\otimes B^{\ell})=0$ for $\ell\geq 1$. We proceed further along the
long exact sequence associated to $(\star)$,

$\begin{matrix}\cdots\rightarrow H^{1}(M^{\otimes(r-1)}_{B^{k}}\otimes
B^{k+\ell})\\\ \rightarrow H^{2}(M^{\otimes r}_{B^{k}}\otimes
B^{\ell})\rightarrow H^{2}(\mathcal{H}^{0}(S,B^{k})\otimes
M^{\otimes(r-1)}_{B^{k}}\otimes B^{\ell})\rightarrow\cdots\end{matrix}$

Here in the sequence, for $r=1$, we see that the first term
$H^{1}(B^{k+\ell})$ is zero by Proposition 2 and for the last term, we have
$H^{2}(\mathcal{H}^{0}(S,B^{k})\otimes B^{\ell})\cong H^{0}(B^{k})\otimes
H^{2}(B^{\ell})=0$. It now follows that $H^{2}(M_{B^{k}}\otimes B^{\ell})=0$
for $\ell\geq 0$. For $r=2$, the first term $H^{1}(M_{B^{k}}\otimes
B^{k+\ell})$ vanishes if $2k+\ell\geq 3$ by $(i)$. For the last term,

$H^{2}(\mathcal{H}^{0}(S,B^{k})\otimes M_{B^{k}}\otimes B^{\ell})\cong
H^{0}(B^{k})\otimes H^{2}(M_{B^{k}}\otimes B^{\ell})=0$

if $\ell\geq 0$. Therefore, we have $H^{2}(M^{\otimes 2}_{B^{k}}\otimes
B^{\ell})=0$ if $\ell\geq 1$.

Now we prove the $(ii)$ and $(iii)$ for $r\geq 2$ by induction on $r$.

Consider now the case $r=2$. From the long exact sequence associated to
$(\star)$ above, we have the exact sequence

$\begin{matrix}\cdots\rightarrow H^{0}(\mathcal{H}^{0}(S,B^{k})\otimes
M_{B^{k}}\otimes B^{\ell})\rightarrow H^{0}(M_{B^{k}}\otimes B^{k+\ell})\\\
\rightarrow H^{1}(M^{\otimes 2}_{B^{k}}\otimes B^{\ell})\rightarrow
H^{1}(\mathcal{H}^{0}(S,B^{k})\otimes M_{B^{k}}\otimes
B^{\ell})\rightarrow\cdots\end{matrix}$

The last term

$H^{1}(\mathcal{H}^{0}(S,B^{k})\otimes M_{B^{k}}\otimes B^{\ell})\cong
H^{0}(B^{k})\otimes H^{1}(M_{B^{k}}\otimes B^{\ell})$

vanishes when $k+\ell\geq 3$. Hence, for $k+\ell\geq 3$, the cohomology
$H^{1}(M^{\otimes 2}_{B^{k}}\otimes B^{\ell})$ vanishes if

$H^{0}(\mathcal{H}^{0}(S,B^{k})\otimes M_{B^{k}}\otimes B^{\ell})\rightarrow
H^{0}(M_{B^{k}}\otimes B^{k+\ell})$

is surjective, which is equivalent to the surjectivity of

$H^{0}(B^{k})\otimes H^{0}(M_{B^{k}}\otimes B^{\ell})\rightarrow
H^{0}(M_{B^{k}}\otimes B^{k+\ell})$

since

$H^{0}(\mathcal{H}^{0}(S,B^{k})\otimes M_{B^{k}}\otimes B^{\ell})\cong
H^{0}(B^{k})\otimes H^{0}(M_{B^{k}}\otimes B^{\ell}).$

From the Castelnuovo-Mumford Theorem again, as in the case for $r=1$, the
above map is surjective if we have

$H^{1}(M_{B^{k}}\otimes B^{\ell-1})=H^{2}(M_{B^{k}}\otimes B^{\ell-2})=0.$

By what we have proven previously, these are satisfied if $\ell\geq 2$ and
$k+\ell\geq 4$.

Suppose now $r\geq 3$. From $(\star)$, we have the exact sequence

$\begin{matrix}\cdots\rightarrow H^{0}(\mathcal{H}^{0}(S,B^{k})\otimes
M^{\otimes(r-1)}_{B^{k}}\otimes B^{\ell})\rightarrow
H^{0}(M^{\otimes(r-1)}_{B^{k}}\otimes B^{k+\ell})\\\ \rightarrow
H^{1}(M^{\otimes r}_{B^{k}}\otimes B^{\ell})\rightarrow
H^{1}(\mathcal{H}^{0}(S,B^{k})\otimes M^{\otimes(r-1)}_{B^{k}}\otimes
B^{\ell})\rightarrow\cdots\end{matrix}$

By the induction hypothesis, the last term

$H^{1}(\mathcal{H}^{0}(S,B^{k})\otimes M^{\otimes(r-1)}_{B^{k}}\otimes
B^{\ell})\cong H^{0}(B^{k})\otimes H^{1}(M^{\otimes(r-1)}_{B^{k}}\otimes
B^{\ell})$

vanishes when $k+\ell\geq r$ and $\ell\geq r$. Hence, under the same
conditions on $k,\ell,r$, the cohomology $H^{1}(M^{\otimes r}_{B^{k}}\otimes
B^{\ell})$ vanishes if

$H^{0}(\mathcal{H}^{0}(S,B^{k})\otimes M^{\otimes(r-1)}_{B^{k}}\otimes
B^{\ell})\rightarrow H^{0}(M^{\otimes(r-1)}_{B^{k}}\otimes B^{k+\ell})$

is surjective, which is equivalent to the surjectivity of

$H^{0}(B^{k})\otimes H^{0}(M^{\otimes(r-1)}_{B^{k}}\otimes
B^{\ell})\rightarrow H^{0}(M^{\otimes(r-1)}_{B^{k}}\otimes B^{k+\ell})$

since

$H^{0}(\mathcal{H}^{0}(S,B^{k})\otimes M^{\otimes(r-1)}_{B^{k}}\otimes
B^{\ell})\cong H^{0}(B^{k})\otimes H^{0}(M^{\otimes(r-1)}_{B^{k}}\otimes
B^{\ell}).$

We argue similarly with the Castelnuovo-Mumford Theorem. Thus, the above map
is surjective if we have

$H^{1}(M^{\otimes(r-1)}_{B^{k}}\otimes
B^{\ell-1})=H^{2}(M^{\otimes(r-1)}_{B^{k}}\otimes B^{\ell-2})=0.$

The induction hypothesis implies that these conditions are satisfied if
$k+(\ell-1)\geq(r-1)+2$ and $\ell-1\geq r-1$ and $\ell-2\geq(r-1)-1$. In
particular, it suffices to have $k+\ell\geq r+2$ and $\ell\geq r$.

It remains to consider $H^{2}(M^{\otimes r}_{B^{k}}\otimes B^{\ell})$. From
the long exact sequence associated to $(\star)$, we get the exact sequence

$\begin{matrix}\cdots\rightarrow H^{1}(M^{\otimes(r-1)}_{B^{k}}\otimes
B^{k+\ell})\\\ \rightarrow H^{2}(M^{\otimes r}_{B^{k}}\otimes
B^{\ell})\rightarrow H^{2}(\mathcal{H}^{0}(S,B^{k})\otimes
M^{\otimes(r-1)}_{B^{k}}\otimes B^{\ell})\rightarrow\cdots\end{matrix}$

By the induction hypothesis, we have $H^{1}(M^{\otimes(r-1)}_{B^{k}}\otimes
B^{k+\ell})=0$ for $2k+\ell\geq r+1$ and $k+\ell\geq r-1$. Also by the
induction hypothesis,

$H^{2}(\mathcal{H}^{0}(S,B^{k})\otimes M^{\otimes(r-1)}_{B^{k}}\otimes
B^{\ell})\cong H^{0}(B^{k})\otimes H^{2}(M^{\otimes(r-1)}_{B^{k}}\otimes
B^{\ell})=0$

if $\ell+1\geq r-1$. Combining all these, it now follows that
$H^{2}(M^{\otimes r}_{B^{k}}\otimes B^{\ell})=0$ if $\ell+1\geq r$. The proof
is now complete. ∎

Remark. As in the proof of Proposition 2, by applying the result of Reider on
a fake projective plane $S$, it follow easily that for any ample line bundle
$A$, its power $A^{k}$ is base-point-free for $k\geq 6$ and hence the previous
theorem will also imply certain vanishing statements for $A$. However, if we
just simply do the direct translation in this way, there will be “gaps”
between the powers of $A$ in which we do not know whether the analogous
cohomology groups vanish or not.

In view of the remark above, for the purpose of obtaining a lower bound, above
which every power of an arbitrary ample line bundle on $S$ satisfies the
property $N_{p}$, we improve upon the proof of Theorem 5 and obtain the
following

######  Theorem 6.

Let $A$ be an ample line bundle and $\sigma$, $\tau$ be torsion line bundles
on a fake projective plane $S$. For $r\geq 1$ and $k\geq 12$, we have
$H^{1}(S,M^{\otimes r}_{A^{k}\otimes\sigma}\otimes A^{\ell}\otimes\tau)=0$ if
$\ell\geq 6(r+2)$, and $H^{2}(S,M^{\otimes r}_{A^{k}\otimes\sigma}\otimes
A^{\ell}\otimes\tau)=0$ if $\ell\geq 6r-2$.

###### Proof.

The structure of the proof will be similar to that of Theorem 5 and thus we
will be brief on certain arguments. We will again prove by induction on $r$.

Suppose $r=1$. When $\ell\geq 18$, as $H^{1}(A^{\ell}\otimes\tau)=0$, the
cohomology $H^{1}(M_{A^{k}\otimes\sigma}\otimes A^{\ell}\otimes\tau)$ vanishes
if the map

$H^{0}(A^{k}\otimes\sigma)\otimes H^{0}(A^{\ell}\otimes\tau)\rightarrow
H^{0}(A^{k+\ell}\otimes\sigma\otimes\tau)$

is surjective.

Since $k\geq 12$, we can write
$A^{k}\otimes\sigma=A^{k_{1}}\otimes\cdots\otimes A^{k_{p}}\otimes\sigma$ for
some $p\geq 2$ such that $6\leq k_{1}\leq k_{2}\leq\cdots\leq k_{p}$. When
$k=12$ or $13$, such decomposition is unique. If $14\leq k\leq 17$, we choose
$k_{1}=7$ and hence $7\leq k_{2}\leq 10$. If $k\geq 18$, we can always choose
our $k_{j}$ such that $6=k_{1}\leq\cdots\leq k_{p}\leq 9$ for some $p\geq 3$.
Hence, for any $k\geq 12$, we can always choose $k_{j}$ such that $6\leq
k_{1}\leq 7$ and $k_{1}\leq\cdots\leq k_{p}\leq 10$.

By considering the natural maps

$H^{0}(A^{k_{1}})\otimes\cdots\otimes H^{0}(A^{k_{p}}\otimes\sigma)\otimes
H^{0}(A^{\ell}\otimes\tau)\longrightarrow H^{0}(A^{k}\otimes\sigma)\otimes
H^{0}(A^{\ell}\otimes\tau)\longrightarrow
H^{0}(A^{k+\ell}\otimes\sigma\otimes\tau),$

we see that it suffices to have the surjectivity for the naturally defined
mappings

$\varphi_{j}:H^{0}(A^{k_{j}})\otimes
H^{0}(A^{k_{1}+\cdots+k_{j-1}+\ell}\otimes\tau)\rightarrow
H^{0}(A^{k_{1}+\cdots+k_{j}+\ell}\otimes\tau)$

for $1\leq j\leq p-1$ and also

$\varphi_{p}:H^{0}(A^{k_{p}}\otimes\sigma)\otimes
H^{0}(A^{k_{1}+\cdots+k_{p-1}+\ell}\otimes\tau)\rightarrow
H^{0}(A^{k+\ell}\otimes\sigma\otimes\tau).$

Since $k_{j}\geq 6$, every $A_{k_{j}}$ and also $A_{k_{p}}\otimes\sigma$ are
base-point-free as in the proof of Proposition 2. Thus, we can apply the
result of Castelnuovo-Mumford and it says that $\varphi_{1}$ is surjective if

$H^{1}(A^{\ell-k_{1}}\otimes\tau)=H^{2}(A^{\ell-2k_{1}}\otimes\tau)=0.$

By Lemma 1, these are true if $\ell\geq 18$ since $6\leq k_{1}\leq 7$.
Similarly, for $2\leq j\leq p-1$, $\varphi_{j}$ is surjective if

$H^{1}(A^{k_{1}+\cdots+k_{j-1}+\ell-
k_{j}}\otimes\tau)=H^{2}(A^{k_{1}+\cdots+k_{j-1}+\ell-2k_{j}}\otimes\tau)=0.$

These hold true if $\ell\geq 18$ since $k_{1}\geq 6$ and $k_{j}\leq 10$. The
surjectivity for $\varphi_{p}$ can be seen in the same manner.

For $H^{2}(M_{A^{k}\otimes\sigma}\otimes A^{\ell}\otimes\tau)$, it vanishes
for $\ell\geq 4$, which can be seen easily from the long exact sequence
associated to $(\star)$ for $r=1$,

$\begin{matrix}\cdots\rightarrow
H^{1}(A^{k+\ell}\otimes\sigma\otimes\tau)\rightarrow
H^{2}(M_{A^{k}\otimes\sigma}\otimes A^{\ell}\otimes\tau)\rightarrow
H^{2}(\mathcal{H}^{0}(S,A^{k}\otimes\sigma)\otimes
A^{\ell}\otimes\tau)\rightarrow\cdots\end{matrix}$

The case for $r=1$ is now settled.

Suppose now $r\geq 2$. We first consider $H^{1}(M^{\otimes r}_{A^{k}}\otimes
A^{\ell})$. We have the exact sequence

$\begin{matrix}\cdots\rightarrow
H^{0}(\mathcal{H}^{0}(S,A^{k}\otimes\sigma)\otimes
M^{\otimes(r-1)}_{A^{k}\otimes\sigma}\otimes A^{\ell}\otimes\tau)\rightarrow
H^{0}(M^{\otimes(r-1)}_{A^{k}\otimes\sigma}\otimes
A^{k+\ell}\otimes\sigma\otimes\tau)\\\ \rightarrow H^{1}(M^{\otimes
r}_{A^{k}\otimes\sigma}\otimes A^{\ell}\otimes\tau)\rightarrow
H^{1}(\mathcal{H}^{0}(S,A^{k}\otimes\sigma)\otimes
M^{\otimes(r-1)}_{A^{k}\otimes\sigma}\otimes
A^{\ell}\otimes\tau)\rightarrow\cdots\end{matrix}$

By the induction hypothesis, the last term

$H^{1}(\mathcal{H}^{0}(S,A^{k}\otimes\sigma)\otimes
M^{\otimes(r-1)}_{A^{k}\otimes\sigma}\otimes A^{\ell}\otimes\tau)\cong
H^{0}(A^{k}\otimes\sigma)\otimes
H^{1}(M^{\otimes(r-1)}_{A^{k}\otimes\sigma}\otimes A^{\ell}\otimes\tau)$

vanishes when $\ell\geq 6(r+1)$. Hence, for $\ell\geq 6(r+1)$, the cohomology
group $H^{1}(M^{\otimes r}_{A^{k}\otimes\sigma}\otimes A^{\ell}\otimes\tau)$
vanishes if

$H^{0}(A^{k}\otimes\sigma)\otimes
H^{0}(M^{\otimes(r-1)}_{A^{k}\otimes\sigma}\otimes
A^{\ell}\otimes\tau)\rightarrow
H^{0}(M^{\otimes(r-1)}_{A^{k}\otimes\sigma}\otimes
A^{k+\ell}\otimes\sigma\otimes\tau)$

is surjective. As in the case $r=1$, we can write
$A^{k}\otimes\sigma=A^{k_{1}}\otimes\cdots\otimes A^{k_{p}}\otimes\sigma$ for
some $p\geq 2$ such that $k_{j}\geq 6$ for $1\leq j\leq p$. But for $r\geq 2$,
we choose our $k_{j}$ such that $6=k_{1}\leq k_{2}\leq\cdots\leq k_{p}\leq
11$.

As before, it suffices to have the surjectivity for

$\phi_{j}:H^{0}(A^{k_{j}})\otimes
H^{0}(M^{\otimes(r-1)}_{A^{k}\otimes\sigma}\otimes
A^{k_{1}+\cdots+k_{j-1}+\ell}\otimes\tau)\rightarrow
H^{0}(M^{\otimes(r-1)}_{A^{k}\otimes\sigma}\otimes
A^{k_{1}+\cdots+k_{j}+\ell}\otimes\tau)$

for $1\leq j\leq p-1$ and also

$\phi_{p}:H^{0}(A^{k_{p}}\otimes\sigma)\otimes
H^{0}(M^{\otimes(r-1)}_{A^{k}\otimes\sigma}\otimes
A^{k_{1}+\cdots+k_{p-1}+\ell}\otimes\tau)\rightarrow
H^{0}(M^{\otimes(r-1)}_{A^{k}\otimes\sigma}\otimes
A^{k+\ell}\otimes\sigma\otimes\tau).$

By Castelnuovo-Mumford Theorem, $\phi_{1}$ is surjective if

$H^{1}(M^{\otimes(r-1)}_{A^{k}\otimes\sigma}\otimes A^{\ell-
k_{1}}\otimes\tau)=H^{2}(M^{\otimes(r-1)}_{A^{k}\otimes\sigma}\otimes
A^{\ell-2k_{1}}\otimes\tau)=0.$

The induction hypothesis implies that these conditions are satisfied if $\ell-
k_{1}=\ell-6\geq 6(r+1)$ and $\ell-2k_{1}=\ell-12\geq 6(r-1)-2$, which are
both true if $\ell\geq 6(r+2)$.

For $2\leq j\leq p-1$, $\phi_{j}$ is surjective if

$H^{1}(M^{\otimes(r-1)}_{A^{k}\otimes\sigma}\otimes
A^{k_{1}+\cdots+k_{j-1}+\ell-
k_{j}}\otimes\tau)=H^{2}(M^{\otimes(r-1)}_{A^{k}\otimes\sigma}\otimes
A^{k_{1}+\cdots+k_{j-1}+\ell-2k_{j}}\otimes\tau)=0.$

By simple arithmetics, together with the induction hypothesis, it follows that
these are satisfied if $\ell-5\geq 6(r+1)$ and $\ell-16\geq 6(r-1)-2$. In
particular, it is more than sufficient if we have $\ell\geq 6(r+2)$. The
surjectivity for $\phi_{p}$ is checked similarly.

It remains to check that $H^{2}(M^{\otimes r}_{A^{k}\otimes\sigma}\otimes
A^{\ell}\otimes\tau)=0$ when $\ell\geq 6r-2$. From the exact sequence

$\begin{matrix}\cdots\rightarrow
H^{1}(M^{\otimes(r-1)}_{A^{k}\otimes\sigma}\otimes
A^{k+\ell}\otimes\sigma\otimes\tau)\\\ \rightarrow H^{2}(M^{\otimes
r}_{A^{k}\otimes\sigma}\otimes A^{\ell}\otimes\tau)\rightarrow
H^{2}(\mathcal{H}^{0}(S,A^{k}\otimes\sigma)\otimes
M^{\otimes(r-1)}_{A^{k}\otimes\sigma}\otimes
A^{\ell}\otimes\tau)\rightarrow\cdots,\end{matrix}$

we have, by the induction hypothesis,
$H^{1}(M^{\otimes(r-1)}_{A^{k}\otimes\sigma}\otimes
A^{k+\ell}\otimes\sigma\otimes\tau)=0$ for if $k+\ell\geq 6(r+1)$. Since
$k\geq 12$, the inequality is satisfied when $\ell\geq 6r-2$. Similarly,

$H^{2}(\mathcal{H}^{0}(S,A^{k}\otimes\sigma)\otimes
M^{\otimes(r-1)}_{A^{k}\otimes\sigma}\otimes A^{\ell}\otimes\tau)\cong
H^{0}(A^{k}\otimes\sigma)\otimes
H^{2}(M^{\otimes(r-1)}_{A^{k}\otimes\sigma}\otimes A^{\ell}\otimes\tau)=0$

if $\ell\geq 6(r-1)-2$. Thus, the second term $H^{2}(M^{\otimes
r}_{A^{k}}\otimes A^{\ell})$ in the exact sequence above is zero when
$\ell\geq 6r-2$. The proof is now complete. ∎

We are now ready to obtain the final result of this section and also give the
proof of Theorem 1.

######  Theorem 7.

Let $B$ be an ample line bundle on a fake projective plane $S$ such that
$B\otimes K_{S}^{-2}$ is nef, where $K_{S}$ is the canonical line bundle.
Then, $B^{m}$ is projectively normal (i.e. satisfies the property $N_{0}$) if
$m\geq 2$. More generally, for $p\geq 1$, $B^{m}$ satisfies the property
$N_{p}$ if $m\geq p+1$.

###### Proof.

As remark earlier, if $B\otimes K_{S}^{-2}$ is nef, then $B^{m}$ is base-
point-free for $m\geq 1$. Suppose $m\geq 2$. For $B^{m}$ to satisfy the
property $N_{0}$, by Green’s criterion, it suffices to have
$H^{1}(S,B^{ms})=H^{1}(S,M_{B^{m}}\otimes B^{ms})=0$ for $s\geq 1$. These
follow respectively from Proposition 2 and Theorem 5.

Suppose now $p\geq 1$ and $m\geq p+1$. To see that $B^{m}$ satisfies the
property $N_{p}$, we just need to check $H^{1}(S,M^{\otimes r}_{B^{m}}\otimes
B^{ms})=0$ for $0\leq r\leq p+1$ and $s\geq 1$. When $r=0$, it again follows
from Proposition 2. For $r=1$, it follows from Theorem 5 $(i)$ since $m+ms\geq
2m\geq 2(p+1)\geq 4$. Finally, for $2\leq r\leq p+1$, we have

$m+ms\geq 2m\geq 2(p+1)\geq 2r\geq r+2,$

and $ms\geq p+1\geq r\geq 2$ and hence the result follows from Theorem 5
$(ii)$. ∎

######  Corollary 2.

In addition to the results for pluricanonical bundles given by Corollary 1, we
also have $K^{2p+2}_{S}$ satisfies the property $N_{p}$ for $p\geq 1$. In
particular, $K^{2p+4}_{S}$ satisfies the property $N_{p}$ for $p\geq 0$.

Remark. For an ample and base-point-free line bundle $B$ on a complex
projective surface of general type, Gallego-Purnaprajna [GP] proved that
$B^{m}$ satisfies the property $N_{1}$ if $m\geq 3$; and for $p\geq 2$,
$B^{m}$ satisfies the property $N_{p}$ if $m\geq p+1$. The latter result
contains in particular our results for $p\geq 2$ in Theorem 7 since the line
bundles satisfying the hypotheses of Theorem 7 are ample and base-point-free.

###### Proof of Theorem 1.

Let $m$ be a positive integer such that $m\geq 6(p+3)$. We fix an ample
generator $G$ of Pic$(S)$ and let $A$ and $N$ be an ample and a nef line
bundle on $S$ respectively. To show that $A^{m}\otimes N$ satisfies the
property $N_{p}$, we will verify for $0\leq r\leq p+1$ and $s\geq 1$,
$H^{1}(S,M^{\otimes r}_{A^{m}\otimes N}\otimes(A^{m}\otimes N)^{s})=0$ .

We have $A^{m}\otimes N=G^{k}\otimes\sigma$, for some positive integer $k\geq
6(p+3)$ and some torsion line bundle $\sigma$. Thus, $H^{1}(S,(A^{m}\otimes
N)^{s})=H^{1}(S,G^{ks}\otimes\sigma^{s})=0$ for $s\geq 1$ by Lemma 1. Next,
for $1\leq r\leq p+1$ and $s\geq 1$,

$H^{1}(S,M^{\otimes r}_{A^{m}\otimes N}\otimes(A^{m}\otimes
N)^{s})=H^{1}(S,M^{\otimes r}_{G^{k}\otimes\sigma}\otimes
G^{ks}\otimes\sigma^{s})=0$

by Theorem 6 since $k\geq 18$ and $ks\geq k\geq 6(p+3)\geq 6(r+2)$. ∎

##  4\. Syzygies for base-point-free line bundles

Let $S$ be a fake projective plane. There is a one-to-one correspondence
between set of torsion elements of $H_{1}(S,{\mathbb{Z}})$ and the set of
torsion line bundles on $S$, as given by the Universal Coefficient Theorem
(cf. [LY]). Moreover, a torsion line bundle $\delta$ corresponds to a
surjective representation
$\rho_{\delta}:\pi_{1}(S)\rightarrow\mathbb{Z}/d\mathbb{Z}$, where
$d\in\mathbb{N}^{+}$ is the order of $\delta$. The representation
$\rho_{\delta}$ necessarily descends to a representation
$\rho^{\flat}_{\delta}:H_{1}(S,\mathbb{Z})\cong\pi_{1}(S)/[\pi_{1}(S),\pi_{1}(S)]\rightarrow\mathbb{Z}/d\mathbb{Z}$.
If we let $C_{\delta}:=\ker(\rho^{\flat}_{\delta})$, then
$H_{1}(S,\mathbb{Z})/C_{\delta}\cong\mathbb{Z}/d\mathbb{Z}$.

######  Lemma 2.

Let $M$ be a compact complex manifold with $b_{1}(M)=0$ and $\delta$ be a
torsion line bundle on $M$ of order $d$. Let $M_{\delta}$ be the finite
unramified covering of $M$ defined by the kernel of the representation
$\rho_{\delta}:\pi_{1}(M)\rightarrow\mathbb{Z}/d\mathbb{Z}$ associated to
$\delta$. Then $h^{1}(M,\delta)=0$ if $b_{1}(M_{\delta})=0$.

###### Proof.

Let $\pi:M_{\delta}\rightarrow M$ be the finite unramified covering map. Then
$\pi^{*}\delta$ is the trivial line bundle on $M_{\delta}$ and we have (see
[BHPV, p.55])

$\pi_{*}({\mathcal{O}}_{M_{\delta}})=\bigoplus^{d-1}_{j=0}{\mathcal{O}}_{M}(\delta^{j}).$

Hence,

$\frac{1}{2}b_{1}(M_{\delta})=h^{1}(M_{\delta},{\mathcal{O}}_{M_{\delta}})=h^{1}(M,\pi_{*}({\mathcal{O}}_{M_{\delta}}))=\sum^{d-1}_{j=0}h^{1}(M,\delta^{j}).$

The lemma now follows. ∎

Let $S$ be a fake projective plane and $\Pi\subset PU(2,1)$ be the lattice
associated to $S$, which is isomorphic to the fundamental group $\pi_{1}(S)$.
By definition, $b_{1}(S)=0$ as $S$ is a fake projective plane and it follows
that $H_{1}(S,{\mathbb{Z}})\cong\Pi/[\Pi,\Pi]$ is a finite abelian group. Let
$\rho:\Pi\rightarrow\Pi/[\Pi,\Pi]$ be the canonical projection. Let $C$ be a
subgroup of $\Pi/[\Pi,\Pi]$ and
$p_{C}:\Pi/[\Pi,\Pi]\rightarrow(\Pi/[\Pi,\Pi])/C$ be the natural projection
and $\rho_{C}=p_{C}\circ\rho$. Let $S_{C}=B_{{\mathbb{C}}}^{2}/\ker(\rho_{C})$
be the finite unramified covering of $S$ associated to $\ker(\rho_{C})$, where
$B_{{\mathbb{C}}}^{2}\subset\mathbb{C}^{2}$ is the unit ball.

######  Theorem 8.

For every subgroup $C$ of $\Pi/[\Pi,\Pi]$, we have $b_{1}(S_{C})=0$.

###### Proof.

Let $S_{0}$ be the unramified covering of $S$ corresponding to the lattice
$[\Pi,\Pi]$. As $\ker(\rho)=[\Pi,\Pi]$ and $\ker(\rho)\subset\ker(\rho_{C})$,
it is clear that $S_{0}$ is a finite unramified covering of $S_{C}$, which is
also a holomorphic isometric covering with respect to the canonical metrics
induced from $B_{{\mathbb{C}}}^{2}$. In particular, any harmonic one form on
$B_{{\mathbb{C}}}^{2}/\ker(\rho_{C})$ can be pulled back to
$B_{{\mathbb{C}}}^{2}/\ker(\rho)$ and hence $b_{1}(S_{0})\geqslant
b_{1}(S_{C})$ from Poincaré Duality. Hence it suffices to check that
$b_{1}(S_{0})=0$. Equivalently, we must check that the abelianization
$\Pi^{\prime}/[\Pi^{\prime},\Pi^{\prime}]$ of $\Pi^{\prime}=[\Pi,\Pi]$ is
finite. This was kindly verified for us by Donald Cartwright using Magma, as
follows. There are (see [CS]) exactly 50 groups $\Pi$’s which are fundamental
groups of fake projective planes $S$. Their names are listed in the file
`registerofgps.txt` in the weblink of [CS]). Each is a subgroup of a maximal
arithmetic subgroup $\bar{\Gamma}$ of $PU(2,1)$. Explicit generators and
relations for these $\bar{\Gamma}$’s are given in various files of the
weblink. For example, see `C20p2/gpc20p2generators_reducesyntax.txt` for the
generators and relations for the three $\bar{\Gamma}$’s named
$(C_{20},p=2,\emptyset)$, $(C_{20},p=2,\\{3+\\})$ and $(C_{20},p=2,\\{3-\\})$.
Abstract presentations of these groups $\bar{\Gamma}$ are given in the file
barGammapresentations.txt of that weblink, where for each of the fake
projective planes $S$ named in registerofgps.txt, generators
$s_{1},s_{2},\ldots$ are given for its fundamental group $\Pi$.

In the file finite-cover-b1.txt, for each of these 50 $\Pi$’s, we exhibit the
subgroup $\Pi^{\prime}=[\Pi,\Pi]$ of $\Pi$, then verify that
$\Pi^{\prime}/[\Pi^{\prime},\Pi^{\prime}]$ is finite using Magma’s
AbelianQuotientInvariants routine.

The subgroup $\Pi^{\prime}$ was in each case found as follows. Using Magma’s
Rewrite routine, we first find a presentation of $\Pi$ from that of
$\bar{\Gamma}$. Let $n=|\Pi/[\Pi,\Pi]|=|H_{1}(S,{\mathbb{Z}})|$. Using Magma’s
LowIndexNormalSubgroups$(\Pi,n)$ routine, we list the normal subgroups of
$\Pi$ of index at most $n$. Then we find the unique subgroup in this
(sometimes lengthy) list which has index exactly $n$, and contains all
commutators $s_{i}^{-1}s_{j}^{-1}s_{i}s_{j}$. ∎

As discussed at the beginning of this section, for each torsion line bundle
$\delta$ of order $d$ on a fake projective plane $S=B^{2}_{{\mathbb{C}}}/\Pi$,
there is a subgroup $C_{\delta}<H_{1}(S,\mathbb{Z})$ such that
$H_{1}(S,\mathbb{Z})/C_{\delta}\cong\mathbb{Z}/d\mathbb{Z}$ and the
representation of $\pi_{1}(S)$ associated to $\delta$ is just the composition
$\pi_{1}(S)\rightarrow H_{1}(S,\mathbb{Z})\rightarrow
H_{1}(S,\mathbb{Z})/C_{\delta}$. With the identifications $\pi_{1}(S)\cong\Pi$
and
$H_{1}(S,\mathbb{Z})\cong\pi_{1}(S)/[\pi_{1}(S),\pi_{1}(S)]\cong\Pi/[\Pi,\Pi]$,
the finite unramified cover $M_{\delta}$ associated to $\delta$ described in
Lemma 2 is just $S_{C}$ in Theorem 8 when $C=C_{\delta}$. Thus, by combining
Lemma 2 and Theorem 8, we have

######  Corollary 3.

Let $S$ be a fake projective plane and $\delta$ be a torsion line bundle on
$S$. Then $h^{1}(S,\delta)=0$.

The following lemma is a consequence of a well known theorem of Remmert-Van de
Ven [RV, p.155]. One may also see [K, 15.6.2].

######  Lemma 3.

Let $M$ be a complex projective manifold and $L$ be a holomorphic line bundle
on $M$ such that $h^{0}(M,L)\geq 1$. Then $h^{0}(M,L^{2})\geq 2h^{0}(M,L)-1$.

######  Proposition 3.

Let $S$ be a fake projective plane and $G$ be an ample generator of
$\textrm{Pic}(S)$. For any torsion line bundle $\delta$ on $S$,
$h^{0}(S,G\otimes\delta)\leq 1$, $h^{0}(S,G^{2}\otimes\delta)\leq 2$ and
$h^{0}(S,G^{3}\otimes\delta)\leq 1$.

###### Proof.

Since $K_{S}=G^{3}\otimes\tau$ for some torsion line bundle $\tau$, using
Riemann-Roch Formula with Kodaira Vanishing Theorem, it is easy to get that
$h^{0}(G^{4}\otimes\delta^{2})=3$ for any torsion line bundle $\delta$. It now
follows from Lemma 3 that $h^{0}(S,G^{2}\otimes\delta)\leq 2$, which in turn
implies $h^{0}(S,G\otimes\delta)\leq 1$.

For $h^{0}(S,G^{3}\otimes\delta)$, the result follows again from Riemann-Roch
Formula, together with Corollary 3 and the fact that $K_{S}=G^{3}\otimes\tau$.
∎

###### Proof of Proposition 1.

This is now a consequence of Corollary 3 and Proposition 3. ∎

We are now ready to prove Theorem 2.

###### Proof of Theorem 2.

Let $G$ be an ample generator of $\textrm{Pic}(S)$. As the Picard number of
$S$ is 1, $B$ must be ample, and we can write $B=G^{k}\otimes\sigma$, for some
$k\geq 1$ and some torsion line bundle $\sigma$. As $B$ is base-point-free, it
is clear from Proposition 3 that $k\neq 1,3$. We claim that $k\neq 2$ neither.
Suppose otherwise that $k=2$. Then by Proposition 3, $h^{0}(S,B)\leq 2$. Since
$B$ is base-point-free, we have in fact $h^{0}(S,B)=2$. But $S$ is of Picard
number 1, the zero divisors of two linearly independent sections of
$\Gamma(S,B)$ must have non-trivial intersection, which contradicts base-
point-freeness. Hence, $k\geq 4$. Finally, as Lemma 1 is true whenever $k\geq
4$, we conclude that Proposition 2, Theorem 5 and Theorem 7 are all true for
any base-point-free line bundle. ∎

##  5\. Refinement with a conjecture on vanishing cohomology

Motivated by the existence problem for exceptional collections for objects in
the derived category of coherent sheaves on a fake projective plane $S$, we
formulate the following conjecture, which is a variant of Conjecture 3.1 in
[GKMS] and Conjecture 2 in [LY].

######  Conjecture 1.

Let $S$ be a fake projective plane and $G$ be an ample generator of Pic$(S)$.
Then, $H^{0}(S,G^{2}\otimes\sigma)=0$ for any torsion line bundle $\sigma$.

The conjecture above implies a stronger version of Lemma 1, as follows:

######  Proposition 4.

If Conjecture 1 holds for a fake projective plane $S$, then for any ample
generator $G$ and torsion $\sigma$ of Pic$(S)$, we have
$H^{1}(S,G^{k}\otimes\sigma)=0$ for $k\geq 1$ and
$H^{2}(S,G^{k}\otimes\sigma)=0$ for $k=1,2$, or $k\geq 4$.

###### Proof.

If $H^{0}(S,G^{2}\otimes\sigma)=0$ for any torsion $\sigma$, then we also have
$H^{0}(S,G\otimes\sigma)=0$ for any torsion $\sigma$. Since
$K_{S}=G^{3}\otimes\tau$ for some torsion $\tau$, using Serre Duality, we get

$H^{2}(S,G\otimes\sigma)=H^{2}(S,G^{2}\otimes\sigma)=0$

for any torsion $\sigma$. Now, by Riemann-Roch formula, it follows that

$H^{1}(S,G\otimes\sigma)=H^{1}(S,G^{2}\otimes\sigma)=0$

for any torsion $\sigma$.

Finally, $H^{1}(S,G^{3}\otimes\sigma)=H^{1}(S,\tau\otimes\sigma^{-1})=0$ by
Serre Duality and Corollary 3.

(Note that $H^{2}(S,G^{3}\otimes\sigma)=H^{0}(S,\tau\otimes\sigma^{-1})$ is
zero if and only if $\sigma\neq\tau$.) ∎

With the help of Proposition 4, we are now able to show that for a fake
projective plane $S$ satisfying Conjecture 1, its pluricanonical line bundle
$K^{2p+5}_{S}$, which is missed in Corollary 1, also satisfies the property
$N_{p}$. The essential case is the projective normality (i.e. when $p=0$),
which we have singled it out in the following theorem.

######  Theorem 9.

Let $S$ be a fake projective plane satisfying Conjecture 1. Then $K_{S}^{m}$
is projectively normal for $m\geq 4$.

###### Proof.

By Corollary 1, it remains to prove that $K^{5}_{S}$ is projectively normal.
Since $K_{S}=G^{3}\otimes\tau$, for some torsion $\tau$, using Green’s
criterion (for the property $N_{0}$), we just need to show that for any
torsion line bundles $\sigma$, $\delta$,
$H^{1}(S,M_{G^{15}\otimes\sigma}\otimes G^{\ell}\otimes\delta)=0$ for
$\ell=15s$, where $s\in\mathbb{N}^{+}$. As in the proof of Theorem 6, after
decomposing as $G^{15}\otimes\sigma=G^{7}\otimes G^{8}\otimes\sigma$, it
sufficies to verify that the tensor product maps

$H^{0}(G^{7})\otimes H^{0}(G^{\ell}\otimes\delta)\rightarrow
H^{0}(G^{7+\ell}\otimes\sigma\otimes\delta),$
$H^{0}(G^{8}\otimes\sigma)\otimes H^{0}(G^{7+\ell}\otimes\delta)\rightarrow
H^{0}(G^{15+\ell}\otimes\sigma\otimes\delta)$

are surjective for $\ell=15s$, $s\in\mathbb{N}^{+}$. Using Castelnuovo-Mumford
Theorem, it is sufficient to have, respectively,

$H^{1}(G^{\ell-7}\otimes\delta)=H^{2}(G^{\ell-14}\otimes\delta)=0$

and

$H^{1}(G^{7+\ell-8}\otimes\sigma\otimes\delta)=H^{2}(G^{7+\ell-16}\otimes\sigma\otimes\delta)=0.$

By Proposition 4, these conditions hold for $\ell=15s$, $s\in\mathbb{N}^{+}$.
∎

Remark. The above argument implies that $K_{S}^{m}$ is projective normal for
$m\geq 4$ for fake projective planes with cardinality of automorphism group
$|\mbox{Aut}(S)|>3$, using the facts, cf. [LY] and the references therein,
that $K_{S}=G^{3}$ for some ample line bundle $G$ with $H^{0}(S,G^{2})=0$ and
letting $\delta$ and $\sigma$ be trivial in the above argument.

After settling projective normality, we can now prove that the general
property $N_{p}$ holds for $K^{2p+5}_{S}$. We will see that for $p\geq 1$, the
argument does not require Conjecture 1. The major step is the following
vanishing theorem tailored for the purpose.

######  Proposition 5.

Let $S$ be a fake projective plane. Let $G$ be an ample generator and
$\sigma$, $\delta$ be torsions in Pic$(S)$. For $k\geq 18$ and $r\geq 1$, we
have $H^{1}(S,M^{\otimes r}_{G^{k}\otimes\sigma}\otimes
G^{\ell}\otimes\delta)=0$ if $\ell\geq 6(r+1)$.

###### Proof.

We will prove by induction on $r$. The proof is similar to that of Theorem 6
and will be brief.

Suppose $r=1$. We observe that $H^{1}(M_{G^{k}\otimes\sigma}\otimes
G^{\ell}\otimes\delta)=0$ if $k\geq 18$ and $\ell\geq 12$. To see this, we
just need to note that at the beginning of the proof in Theorem 6, the roles
of $k$ and $\ell$ are symmetric and thus we can exchange $k$ and $\ell$ in the
vanishing statement for $r=1$ in Theorem 6. The case for $r=1$ is now settled.

Suppose $r=2$. Since $k\geq 18$, we have a decomposition
$G^{k}\otimes\sigma=G^{k_{1}}\otimes\cdots\otimes G^{k_{p}}\otimes\sigma$ for
some $p\geq 3$ and $6=k_{1}\leq\cdots\leq k_{p}\leq 9$.

To show that $H^{1}(M^{\otimes r}_{G^{k}\otimes\sigma}\otimes
G^{\ell}\otimes\delta)=0$, it suffices to verify that
$H^{1}(M^{\otimes{r-1}}_{G^{k}\otimes\sigma}\otimes G^{\ell}\otimes\delta)=0$
and the surjectivity for the following mappings:

$\varphi_{j}:H^{0}(G^{k_{j}})\otimes
H^{0}(M^{\otimes{r-1}}_{G^{k}\otimes\sigma}\otimes
G^{k_{1}+\cdots+k_{j-1}+\ell}\otimes\delta)\rightarrow
H^{0}(M^{\otimes{r-1}}_{G^{k}\otimes\sigma}\otimes
G^{k_{1}+\cdots+k_{j}+\ell}\otimes\delta)$

for $1\leq j\leq p-1$ and

$\varphi_{p}:H^{0}(G^{k_{p}}\otimes\sigma)\otimes
H^{0}(M^{\otimes{r-1}}_{G^{k}\otimes\sigma}\otimes
G^{k_{1}+\cdots+k_{p-1}+\ell}\otimes\delta)\rightarrow
H^{0}(M^{\otimes{r-1}}_{G^{k}\otimes\sigma}\otimes
G^{k+\ell}\otimes\sigma\otimes\delta).$

Now, together with Castelnuovo-Mumford Theorem, we conclude that the
cohomology $H^{1}(S,M^{\otimes r}_{G^{k}\otimes\sigma}\otimes
G^{\ell}\otimes\delta)$ vanishes if we have

$H^{1}(M^{\otimes{r-1}}_{G^{k}\otimes\sigma}\otimes G^{\ell}\otimes\delta)=0,$
$H^{1}(M^{\otimes{r-1}}_{G^{k}\otimes\sigma}\otimes G^{\ell-
k_{1}}\otimes\delta)=H^{2}(M^{\otimes{r-1}}_{G^{k}\otimes\sigma}\otimes
G^{\ell-2k_{1}}\otimes\delta)=0,$
$H^{1}(M^{\otimes{r-1}}_{G^{k}\otimes\sigma}\otimes
G^{k_{1}+\cdots+k_{j-1}+\ell-
k_{j}}\otimes\delta)=H^{2}(M^{\otimes{r-1}}_{G^{k}\otimes\sigma}\otimes
G^{k_{1}+\cdots+k_{j-1}+\ell-2k_{j}}\otimes\delta)=0$

for $2\leq j\leq p-1$ and

$H^{1}(M^{\otimes{r-1}}_{G^{k}\otimes\sigma}\otimes
G^{k+\ell-2k_{p}}\otimes\sigma\otimes\delta)=H^{2}(M^{\otimes{r-1}}_{G^{k}\otimes\sigma}\otimes
G^{k+\ell-3k_{p}}\otimes\sigma\otimes\delta)=0.$

These all hold if $k\geq 18$ and $\ell-6\geq 6r$, by the induction hypothesis
and Theorem 6 (for the vanishing of $H^{2}$). The proof is now complete. ∎

######  Theorem 10.

Let $S$ be a fake projective plane satisfying Conjecture 1. Then $K^{m}_{S}$
satisfies the property $N_{p}$ if $m\geq 2p+4$.

###### Proof.

By Corollary 1, it remains to check that $K^{2p+5}_{S}$ satisfies the property
$N_{p}$. We have aleady shown that $K^{5}_{S}$ satisfies the property $N_{0}$
in Theorem 9 and we now let $p\geq 1$. Let $L_{p}=K^{2p+5}_{S}$. According to
Green’s criterion, we just need to check that

$H^{1}(S,M^{\otimes r}_{L_{p}}\otimes L_{p}^{s})=0$

for $0\leq r\leq p+1$ and $s\geq 1$.

Let $G$ be an ample generator of Pic$(S)$ and then we can write

$L_{p}=K^{2p+5}_{S}=G^{6p+15}\otimes\delta$

for some torsion line bundle $\delta$. For $p\geq 1$, the desired vanishing
statement follows from Proposition 5 since $6p+15>18$ and

$(6p+15)s\geq 6p+15>6(p+2)\geq 6(r+1).$

∎

Acknowledgement: It is a pleasure for the authors to thank Donald Cartwright
for providing computer data files for the proof of Theorem 8 which leads to
Theorem 2. They would also like to thank Lawrence Ein and Jun-Muk Hwang for
fruitful discussions and providing very useful references.

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