## Fake Projective Planes

JongHae Keum

(Korea Institute for Advanced Study)

A Celebration of the 75th Birthday of Gopal Prasad ICTS, Bangalore, online

30 July 2020

Outline

1 Algebraic curves and surfaces

2 Q-homology Projective Planes

3 Montgomery-Yang Problem

4 Algebraic Montgomery-Yang Problem

5 Fake Projective Planes

6 Fake Projective Spaces

Algebraic curves and surfaces

## Classify algebraic varieties up to connected moduli

Nonsingular projective algebraic curves/C(compact Riemann surfaces) are classified by the “ mighty"genus

g(C) :=(the number of “holes" ofC)=dim_{C}H^{0}(C,Ω^{1}_{C}) = ^{1}_{2}dim_{Q}H1(C,Q).

g(C) =0⇐⇒C∼=**P**^{1}∼= (Riemann sphere) =C∪ {∞}.

In dimension>1, many invariants: Hodge numbers, Betti numbers
h^{i,j}(X) =dimH^{j}(X,Ω^{i}_{X}), b_{i}(X) :=dimH^{i}(X,Q).

Given Hodge numbers (and even fixing fundamental group), hard to describe the moduli, in general.

Algebraic curves and surfaces

Classify algebraic varieties up to connected moduli

Nonsingular projective algebraic curves/C(compact Riemann surfaces) are classified by the “ mighty"genus

g(C) :=(the number of “holes" ofC)=dim_{C}H^{0}(C,Ω^{1}_{C}) = ^{1}_{2}dim_{Q}H1(C,Q).

g(C) =0⇐⇒C∼=**P**^{1}∼= (Riemann sphere) =C∪ {∞}.

In dimension>1, many invariants: Hodge numbers, Betti numbers
h^{i,j}(X) =dimH^{j}(X,Ω^{i}_{X}), b_{i}(X) :=dimH^{i}(X,Q).

Given Hodge numbers (and even fixing fundamental group), hard to describe the moduli, in general.

Algebraic curves and surfaces

## Smooth Algebraic Surfaces with p

_{g}

= q = 0

Long history :Castelnuovo’s rationality criterion, Severi conjecture, ...

Here, the geometric genus and the irregularity

p_{g}(X) :=dimH^{n}(X,O_{X}) =dimH^{0}(X,Ω^{n}_{X}) =h^{0,n}(X) =h^{n,0}(X),
q(X) :=dimH^{1}(X,O_{X}) =dimH^{0}(X,Ω^{1}_{X}) =h^{0,1}(X) =h^{1,0}(X).

Max Nöther(1844-1921)said [in the book ofFederigo Enriques(1871-1946)] :

"Algebraic curves are created by god, algebraic surfaces are created by devil."

Algebraic curves and surfaces

Smooth Algebraic Surfaces with p

_{g}

= q = 0

Long history :Castelnuovo’s rationality criterion, Severi conjecture, ...

Here, the geometric genus and the irregularity

p_{g}(X) :=dimH^{n}(X,O_{X}) =dimH^{0}(X,Ω^{n}_{X}) =h^{0,n}(X) =h^{n,0}(X),
q(X) :=dimH^{1}(X,O_{X}) =dimH^{0}(X,Ω^{1}_{X}) =h^{0,1}(X) =h^{1,0}(X).

Max Nöther(1844-1921)said [in the book ofFederigo Enriques(1871-1946)] :

"Algebraic curves are created by god, algebraic surfaces are created by devil."

Algebraic curves and surfaces

## Smooth Algebraic Surfaces with p

_{g}

= q = 0

Enriques-Kodaira classificationof algebraic surfaces (1940’s):

**P**^{2}, rational ruled surfaces;

Enriques surfaces;

properly elliptic surfaces withpg=q=0;

surfaces of general type withpg=0 (these haveK^{2}=1,2, . . . ,9);

blow-ups of the above surfaces.

Smooth algebraic surfaces with minimal invariants, that is, with b1=b3=0, b0=b2=b4=1(⇒pg =q =0) are

**P**^{2};

fake projective planes (= surfaces of general type withpg =0,K^{2}=9).
**Remark.**FPPs are ball quotients, so not simply connected.

Exotic**P**^{2}does NOT exist in complex geometry.

Algebraic curves and surfaces

## Smooth Algebraic Surfaces with p

_{g}

= q = 0

Enriques-Kodaira classificationof algebraic surfaces (1940’s):

**P**^{2}, rational ruled surfaces;

Enriques surfaces;

properly elliptic surfaces withpg=q=0;

surfaces of general type withpg=0 (these haveK^{2}=1,2, . . . ,9);

blow-ups of the above surfaces.

Smooth algebraic surfaces with minimal invariants, that is, with b1=b3=0, b0=b2=b4=1(⇒pg =q =0) are

**P**^{2};

fake projective planes (= surfaces of general type withpg =0,K^{2}=9).

**Remark.**FPPs are ball quotients, so not simply connected.
Exotic**P**^{2}does NOT exist in complex geometry.

Algebraic curves and surfaces

## Smooth Algebraic Surfaces with p

_{g}

= q = 0

Enriques-Kodaira classificationof algebraic surfaces (1940’s):

**P**^{2}, rational ruled surfaces;

Enriques surfaces;

properly elliptic surfaces withpg=q=0;

surfaces of general type withpg=0 (these haveK^{2}=1,2, . . . ,9);

blow-ups of the above surfaces.

Smooth algebraic surfaces with minimal invariants, that is, with b1=b3=0, b0=b2=b4=1(⇒pg =q =0) are

**P**^{2};

fake projective planes (= surfaces of general type withpg =0,K^{2}=9).

**Remark.**FPPs are ball quotients, so not simply connected.

Q-homology Projective Planes

## Q -homology P

^{2}

Definition

A normal projective surfaceSis called aQ-homology**P**^{2}ifbi(S) =bi(**P**^{2})for
alli, i.e.b1=b3=0,b0=b2=b4=1.

IfSis smooth, thenS=**P**^{2}or afake projective plane.

IfShasA1-singularities only, thenS∼= (w^{2}=xy)⊂**P**^{3}.
IfShasA_{2}-singularities only, thenShas 3A_{2}or 4A_{2}and
S∼=**P**^{2}/Gor FPP/G, whereG∼=Z/3 or(Z/3)^{2}.

Any cubic surface in**P**^{3}with 3A_{2}is isom. to(w^{3}=xyz).

IfShasA_{1}orA_{2}-singularities only,S=**P**^{2}(1,2,3)or one of the above.
In this talk, we assumeShas at worstquotientsingularities.

ThenSis aQ-homology**P**^{2}ifb_{2}(S) =1.
For a minimal resolutionS^{0}→S,

p_{g}(S^{0}) =q(S^{0}) =0.

Q-homology Projective Planes

## Q -homology P

^{2}

Definition

A normal projective surfaceSis called aQ-homology**P**^{2}ifbi(S) =bi(**P**^{2})for
alli, i.e.b1=b3=0,b0=b2=b4=1.

IfSis smooth, thenS=**P**^{2}or afake projective plane.

IfShasA1-singularities only, thenS∼= (w^{2}=xy)⊂**P**^{3}.
IfShasA_{2}-singularities only, thenShas 3A_{2}or 4A_{2}and
S∼=**P**^{2}/Gor FPP/G, whereG∼=Z/3 or(Z/3)^{2}.

Any cubic surface in**P**^{3}with 3A_{2}is isom. to(w^{3}=xyz).

IfShasA_{1}orA_{2}-singularities only,S=**P**^{2}(1,2,3)or one of the above.
In this talk, we assumeShas at worstquotientsingularities.

ThenSis aQ-homology**P**^{2}ifb_{2}(S) =1.
For a minimal resolutionS^{0}→S,

p_{g}(S^{0}) =q(S^{0}) =0.

Q-homology Projective Planes

## Q -homology P

^{2}

Definition

A normal projective surfaceSis called aQ-homology**P**^{2}ifbi(S) =bi(**P**^{2})for
alli, i.e.b1=b3=0,b0=b2=b4=1.

IfSis smooth, thenS=**P**^{2}or afake projective plane.

IfShasA1-singularities only, thenS∼= (w^{2}=xy)⊂**P**^{3}.
IfShasA_{2}-singularities only, thenShas 3A_{2}or 4A_{2}and
S∼=**P**^{2}/Gor FPP/G, whereG∼=Z/3 or(Z/3)^{2}.

Any cubic surface in**P**^{3}with 3A_{2}is isom. to(w^{3}=xyz).

IfShasA_{1}orA_{2}-singularities only,S=**P**^{2}(1,2,3)or one of the above.

In this talk, we assumeShas at worstquotientsingularities.
ThenSis aQ-homology**P**^{2}ifb_{2}(S) =1.

For a minimal resolutionS^{0}→S,

p_{g}(S^{0}) =q(S^{0}) =0.

Q-homology Projective Planes

## Q -homology P

^{2}

Definition

**P**^{2}ifbi(S) =bi(**P**^{2})for
alli, i.e.b1=b3=0,b0=b2=b4=1.

IfSis smooth, thenS=**P**^{2}or afake projective plane.

^{2}=xy)⊂**P**^{3}.
IfShasA_{2}-singularities only, thenShas 3A_{2}or 4A_{2}and
S∼=**P**^{2}/Gor FPP/G, whereG∼=Z/3 or(Z/3)^{2}.

Any cubic surface in**P**^{3}with 3A_{2}is isom. to(w^{3}=xyz).

IfShasA_{1}orA_{2}-singularities only,S=**P**^{2}(1,2,3)or one of the above.

In this talk, we assumeShas at worstquotientsingularities.

ThenSis aQ-homology**P**^{2}ifb_{2}(S) =1.

For a minimal resolutionS^{0}→S,

p_{g}(S^{0}) =q(S^{0}) =0.

Q-homology Projective Planes

## Q -homology P

^{2}

Definition

**P**^{2}ifbi(S) =bi(**P**^{2})for
alli, i.e.b1=b3=0,b0=b2=b4=1.

IfSis smooth, thenS=**P**^{2}or afake projective plane.

^{2}=xy)⊂**P**^{3}.
IfShasA_{2}-singularities only, thenShas 3A_{2}or 4A_{2}and
S∼=**P**^{2}/Gor FPP/G, whereG∼=Z/3 or(Z/3)^{2}.

Any cubic surface in**P**^{3}with 3A_{2}is isom. to(w^{3}=xyz).

IfShasA_{1}orA_{2}-singularities only,S=**P**^{2}(1,2,3)or one of the above.

In this talk, we assumeShas at worstquotientsingularities.

ThenSis aQ-homology**P**^{2}ifb_{2}(S) =1.

For a minimal resolutionS^{0}→S,

pg(S^{0}) =q(S^{0}) =0.

Q-homology Projective Planes

## Trichotomy: K

_{S}

= ample, −ample, num. trivial

LetSbe aQ-hom**P**^{2}with quotient singularities.

−K_{S} is ample

log del Pezzo surfaces of Picard number 1,
e.g.**P**^{2}/G,**P**^{2}(a,b,c),. . .

κ(S^{0}) =−∞.

K_{S}is numerically trivial.

log Enriques surfaces of Picard number 1.

κ(S^{0}) =−∞,0.

K_{S}is ample.

e.g. all quotients of fake projective planes,

suitable contraction of a suitable blowup of**P**^{2}, some Enriques surface,. . .
κ(S^{0}) =−∞,0,1,2.

Problem

Classify allQ-homology**P**^{2}’s with quotient singularities.

Q-homology Projective Planes

## The Maximum Number of Quotient Singularities

Question

How many singular points on S, aQ-homology**P**^{2}with quotient singularities?

|Sing(S)| ≤5 bythe orbifold Bogomolov-Miyaoka-Yau inequality (Sakai, Miyaoka, Megyesi forK nef)

1

3K_{S}^{2}≤eorb(S) :=e(S)− X

p∈Sing(S)

1− 1

|π_{1}(L_{p})|

.

(Keel-McKernan for−K nef)

0≤eorb(S).

Many examples with|Sing(S)| ≤4 (cf. Brenton, 1977)
If−K_{S}is ample,|Sing(S)| ≤4 (Belousov, 2008).

The case with|Sing(S)|=5 were classified by Hwang-Keum.

Q-homology Projective Planes

## The Maximum Number of Quotient Singularities

Question

How many singular points on S, aQ-homology**P**^{2}with quotient singularities?

|Sing(S)| ≤5 bythe orbifold Bogomolov-Miyaoka-Yau inequality (Sakai, Miyaoka, Megyesi forK nef)

1

3K_{S}^{2}≤eorb(S) :=e(S)− X

p∈Sing(S)

1− 1

|π_{1}(L_{p})|

.

(Keel-McKernan for−K nef)

0≤eorb(S).

Many examples with|Sing(S)| ≤4 (cf. Brenton, 1977)
If−K_{S}is ample,|Sing(S)| ≤4 (Belousov, 2008).

The case with|Sing(S)|=5 were classified by Hwang-Keum.

Q-homology Projective Planes

## The Maximum Number of Quotient Singularities

Question

How many singular points on S, aQ-homology**P**^{2}with quotient singularities?

|Sing(S)| ≤5 bythe orbifold Bogomolov-Miyaoka-Yau inequality (Sakai, Miyaoka, Megyesi forK nef)

1

3K_{S}^{2}≤eorb(S) :=e(S)− X

p∈Sing(S)

1− 1

|π_{1}(L_{p})|

.

(Keel-McKernan for−K nef)

0≤eorb(S).

Many examples with|Sing(S)| ≤4 (cf. Brenton, 1977)
If−K_{S}is ample,|Sing(S)| ≤4 (Belousov, 2008).

The case with|Sing(S)|=5 were classified by Hwang-Keum.

Q-homology Projective Planes

Theorem (D.Hwang-Keum, JAG 2011)

Let S be aQ-homology**P**^{2}with quotient singularities. Then|Sing(S)| ≤4
except the following case:

S has5singular points of type3A1+2A3, and its minimal resolution S^{0}is an
Enriques surface.

Corollary

EveryZ-homology**P**^{2}with quotient singularities has at most4singular points.

Remark

(1) EveryZ-cohomology**P**^{2}with quotient singularities has at most1singular
point. If it has, then the singularity is of type E8[Bindschadler-Brenton, 1984].
(2)Q-homology**P**^{2}withrational singularitiesmay havearbitrarily many
singularities, no bound.

Q-homology Projective Planes

Theorem (D.Hwang-Keum, JAG 2011)

Let S be aQ-homology**P**^{2}with quotient singularities. Then|Sing(S)| ≤4
except the following case:

S has5singular points of type3A1+2A3, and its minimal resolution S^{0}is an
Enriques surface.

Corollary

EveryZ-homology**P**^{2}with quotient singularities has at most4singular points.

Remark

(1) EveryZ-cohomology**P**^{2}with quotient singularities has at most1singular
point. If it has, then the singularity is of type E8[Bindschadler-Brenton, 1984].

(2)Q-homology**P**^{2}withrational singularitiesmay havearbitrarily many
singularities, no bound.

Montgomery-Yang Problem

C

^{∞}

## -action of S

^{1}

on S

^{m}

**S**^{1}⊂Diff(**S**^{m}).

The identity element 1∈**S**^{1}acts identically on**S**^{m}.

Each diffeomorphismg ∈**S**^{1}is homotopic to the identity map 1**S**^{m}.
By Lefschetz Fixed Point Formula,

e(Fix(g)) =e(Fix(1)) =e(**S**^{m}).

Ifmis even, thene(**S**^{m}) =2 and such an action has a fixed point, so the
foliation by circles degenerates.

**Assume**m=2n−1**odd.**
Definition

AC^{∞}-action of**S**^{1}on**S**^{2n−1}

**S**^{1}×**S**^{2n−1}→**S**^{2n−1}

is called apseudofree**S**^{1}-actionon**S**^{2n−1}if it is free except for finitely many
orbits (whose isotropy groupsZ/a_{1}, . . . ,Z/a_{k} have pairwise prime orders).

Montgomery-Yang Problem

C

^{∞}

## -action of S

^{1}

on S

^{m}

**S**^{1}⊂Diff(**S**^{m}).

The identity element 1∈**S**^{1}acts identically on**S**^{m}.

Each diffeomorphismg ∈**S**^{1}is homotopic to the identity map 1**S**^{m}.
By Lefschetz Fixed Point Formula,

e(Fix(g)) =e(Fix(1)) =e(**S**^{m}).

Ifmis even, thene(**S**^{m}) =2 and such an action has a fixed point, so the
foliation by circles degenerates.**Assume**m=2n−1**odd.**

Definition

AC^{∞}-action of**S**^{1}on**S**^{2n−1}

**S**^{1}×**S**^{2n−1}→**S**^{2n−1}

is called apseudofree**S**^{1}-actionon**S**^{2n−1}if it is free except for finitely many
orbits (whose isotropy groupsZ/a_{1}, . . . ,Z/a_{k} have pairwise prime orders).

Montgomery-Yang Problem

## Pseudofree S

^{1}

-action on S

^{2n−1}

Example (Linear actions)

**S**^{2n−1}={(z_{1},z_{2}, ...,z_{n})∈C^{n}:|z_{1}|^{2}+|z_{2}|^{2}+...+|z_{n}|^{2}=1} ⊂C^{n}
**S**^{1}={λ∈C:|λ|=1} ⊂C.

Positive integersa_{1}, ...,a_{n}pairwise prime.

**S**^{1}×**S**^{2n−1}→**S**^{2n−1}

(λ,(z1,z2, ...,zn))→(λ^{a}^{1}z1, λ^{a}^{2}z2, ..., λ^{a}^{n}zn).

In thislinear action

**S**^{2n−1}/**S**^{1}∼=CP^{n−1}(a_{1},a_{2}, ...,a_{n}).

The orbit of thei-th coordinate pointei ∈**S**^{2n−1}is exceptional iffai ≥2.

The orbit of a non-coordinate point of**S**^{2n−1}is NOT exceptional.

This action has at mostnexceptional orbits.

Montgomery-Yang Problem

## Pseudofree S

^{1}

-action on S

^{2n−1}

Forn=2 Seifert (1932) showed that each pseudo-free**S**^{1}-action on**S**^{3}is
linear and hence has at most 2 exceptional orbits.

Forn=4 Montgomery-Yang (1971) showed that given arbitrary collection
of pairwise prime positive integersa1, . . . ,ak, there is a pseudofree
**S**^{1}-action on a homotopy**S**^{7}whose exceptional orbits have exactly those
orders.

Petrie (1974) generalised the above M-Y for alln≥5.

Conjecture (Montgomery-Yang problem, Fintushel-Stern 1987)
A pseudo-free**S**^{1}-action on**S**^{5}has at most3exceptional orbits.

This problem is wide open. F-S withdrew their paper [O(2)-actions on the 5-sphere, Invent. Math. 1987].

Montgomery-Yang Problem

Pseudo-free**S**^{1}-actions on a manifoldΣhave been studied in terms of
the orbit spaceΣ/**S**^{1}.

The orbit spaceX =**S**^{5}/**S**^{1}of such an action is a 4-manifold with isolated
singularities whose neighborhoods are cones over lens spacesS^{3}/Zai

corresponding to the exceptional orbits of the**S**^{1}-action.

Easy to check thatX is simply connected andH_{2}(X,Z)has rank 1 and
intersection matrix(1/a_{1}a_{2}· · ·a_{k}).

An exceptional orbit with isotropy typeZ/ahas an equivariant tubular
neighborhood which may be identified withC×C×**S**^{1}with a**S**^{1}-action

λ·(z,w,u) = (λ^{r}z, λ^{s}w, λ^{a}u)
wherer andsare relatively prime toa.

Montgomery-Yang Problem

Pseudo-free**S**^{1}-actions on a manifoldΣhave been studied in terms of
the orbit spaceΣ/**S**^{1}.

The orbit spaceX =**S**^{5}/**S**^{1}of such an action is a 4-manifold with isolated
singularities whose neighborhoods are cones over lens spacesS^{3}/Zai

corresponding to the exceptional orbits of the**S**^{1}-action.

Easy to check thatX is simply connected andH_{2}(X,Z)has rank 1 and
intersection matrix(1/a_{1}a_{2}· · ·a_{k}).

An exceptional orbit with isotropy typeZ/ahas an equivariant tubular
neighborhood which may be identified withC×C×**S**^{1}with a**S**^{1}-action

λ·(z,w,u) = (λ^{r}z, λ^{s}w, λ^{a}u)
whererandsare relatively prime toa.

Montgomery-Yang Problem

The following 1-1 correspondence was known to Montgomery-Yang, Fintushel-Stern, and revisited by Kollár(2005).

Theorem

There is a one-to-one correspondence between:

1 Pseudo-free**S**^{1}-actions onQ-homology 5-spheresΣwith H_{1}(Σ,Z) =0.

2 Compact differentiable4-manifolds M with boundary such that

1 ∂M=S

i

Liis a disjoint union of lens spaces Li=S^{3}/Z^{a}i,

2 the ai’s are pairwise prime,

3 H1(M,Z) =0,

4 H2(M,Z)∼=Z.

Furthermore,Σis diffeomorphic to**S**^{5}iffπ_{1}(M) =1.

Algebraic Montgomery-Yang Problem

## Algebraic Montgomery-Yang Problem

This is the M-Y Problem when**S**^{5}/**S**^{1}attains a structure of a normal projective
surface.

Conjecture (J. Kollár)

Let S be aQ-homology**P**^{2}with at worst quotient singularities. Ifπ_{1}(S^{0}) ={1},
then S has at most3singular points.

What if the conditionπ_{1}(S^{0}) ={1}is replaced by the weaker condition
H1(S^{0},Z) =0?

There are infinitely many examplesSwith
H1(S^{0},Z) =0,π_{1}(S^{0})6={1},|Sing(S)|=4.

These examples obtained from the classification of surface quotient singularities [E. Brieskorn, Invent. Math. 1968].

Algebraic Montgomery-Yang Problem

## Algebraic Montgomery-Yang Problem

This is the M-Y Problem when**S**^{5}/**S**^{1}attains a structure of a normal projective
surface.

Conjecture (J. Kollár)

Let S be aQ-homology**P**^{2}with at worst quotient singularities. Ifπ_{1}(S^{0}) ={1},
then S has at most3singular points.

What if the conditionπ_{1}(S^{0}) ={1}is replaced by the weaker condition
H1(S^{0},Z) =0?

There are infinitely many examplesSwith
H1(S^{0},Z) =0,π_{1}(S^{0})6={1},|Sing(S)|=4.

These examples obtained from the classification of surface quotient singularities [E. Brieskorn, Invent. Math. 1968].

Algebraic Montgomery-Yang Problem

## Algebraic Montgomery-Yang Problem

This is the M-Y Problem when**S**^{5}/**S**^{1}attains a structure of a normal projective
surface.

Conjecture (J. Kollár)

Let S be aQ-homology**P**^{2}with at worst quotient singularities. Ifπ_{1}(S^{0}) ={1},
then S has at most3singular points.

What if the conditionπ_{1}(S^{0}) ={1}is replaced by the weaker condition
H1(S^{0},Z) =0?

There are infinitely many examplesSwith
H1(S^{0},Z) =0,π_{1}(S^{0})6={1},|Sing(S)|=4.

These examples obtained from the classification of surface quotient singularities [E. Brieskorn, Invent. Math. 1968].

Algebraic Montgomery-Yang Problem

Example (coming from Brieskorn’s classification of surface singularities)
I_{m}⊂GL(2,C)the 2m-ary icosahedral groupI_{m}=Z2m.A_{5}.

1→Z2m→I_{m}→ A_{5}⊂PSL(2,C)
I_{m}acts onC^{2}. This action extends naturally to**P**^{2}. Then

S:=**P**^{2}/Im

is aZ-homology**P**^{2}with−K_{S} ample,
Shas 4 quotient singularities:

one non-cyclic singularity of typeIm(the image ofO∈C^{2}), and

3 cyclic singularities of order 2,3,5 (on the image of the line at infinity),
π_{1}(S^{0}) =A_{5}, henceH_{1}(S^{0},Z) =0.

Call these surfacesBrieskorn quotients.

Algebraic Montgomery-Yang Problem

## Progress on Algebraic Montgomery-Yang Problem

Theorem (D.Hwang-Keum, MathAnn 2011)

Let S be aQ-homology**P**^{2}with quotient singularities, not all cyclic, such that
π_{1}(S^{0}) ={1}. Then|Sing(S)| ≤3.

More precisely

Theorem (D.Hwang-Keum, MathAnn 2011)

Let S be aQ-homology**P**^{2}with4or more quotient singularities, not all cyclic,
such that H1(S^{0},Z) =0. Then S is isomorphic to a Brieskorn quotient.

More Progress on Algebraic Montgomery-Yang Problem: Theorem (D.Hwang-Keum, 2013, 2014)

Let S be aQ-homology**P**^{2}with cyclic singularities such that H_{1}(S^{0},Z) =0. If
either S is not rational or−K_{S} is ample, then|Sing(S)| ≤3.

Algebraic Montgomery-Yang Problem

## Progress on Algebraic Montgomery-Yang Problem

Theorem (D.Hwang-Keum, MathAnn 2011)

Let S be aQ-homology**P**^{2}with quotient singularities, not all cyclic, such that
π_{1}(S^{0}) ={1}. Then|Sing(S)| ≤3.

More precisely

Theorem (D.Hwang-Keum, MathAnn 2011)

Let S be aQ-homology**P**^{2}with4or more quotient singularities, not all cyclic,
such that H1(S^{0},Z) =0. Then S is isomorphic to a Brieskorn quotient.

More Progress on Algebraic Montgomery-Yang Problem:

Theorem (D.Hwang-Keum, 2013, 2014)

Let S be aQ-homology**P**^{2}with cyclic singularities such that H_{1}(S^{0},Z) =0. If
either S is not rational or−K is ample, then|Sing(S)| ≤3.

Algebraic Montgomery-Yang Problem

## The Remaining Case of Algebraic M-Y Problem:

Sis aQ-homology**P**^{2}satisfying
(1)Shas cyclic singularities only,
(2)Sis a rational surface withK_{S}ample.

π^{∗}K_{S}=K_{S}^{0} +X
D_{p}.

There are such surfaces. Examples given by Keel and Mckernan (Mem. AMS 1999),

Kollár (Pure Appl. Math. Q. 2008) — an infinite series of examples with

|Sing(S)|=2.

D. Hwang and Keum (Proc. AMS 2012) — infinite series of examples with

|Sing(S)|=1,2,3. Problem

Are there such surfaces S with|Sing(S)|=4? No examples known yet.

Algebraic Montgomery-Yang Problem

## The Remaining Case of Algebraic M-Y Problem:

Sis aQ-homology**P**^{2}satisfying
(1)Shas cyclic singularities only,
(2)Sis a rational surface withK_{S}ample.

π^{∗}K_{S}=K_{S}^{0} +X
D_{p}.

There are such surfaces. Examples given by Keel and Mckernan (Mem. AMS 1999),

Kollár (Pure Appl. Math. Q. 2008) — an infinite series of examples with

|Sing(S)|=2.

D. Hwang and Keum (Proc. AMS 2012) — infinite series of examples with

|Sing(S)|=1,2,3. Problem

Are there such surfaces S with|Sing(S)|=4? No examples known yet.

Algebraic Montgomery-Yang Problem

## The Remaining Case of Algebraic M-Y Problem:

Sis aQ-homology**P**^{2}satisfying
(1)Shas cyclic singularities only,
(2)Sis a rational surface withK_{S}ample.

π^{∗}K_{S}=K_{S}^{0} +X
D_{p}.

There are such surfaces. Examples given by Keel and Mckernan (Mem. AMS 1999),

Kollár (Pure Appl. Math. Q. 2008) — an infinite series of examples with

|Sing(S)|=2.

D. Hwang and Keum (Proc. AMS 2012) — infinite series of examples with

|Sing(S)|=1,2,3.

Problem

Are there such surfaces S with|Sing(S)|=4? No examples known yet.

Algebraic Montgomery-Yang Problem

## The Remaining Case of Algebraic M-Y Problem:

**P**^{2}satisfying
(1)Shas cyclic singularities only,
(2)Sis a rational surface withK_{S}ample.

π^{∗}K_{S}=K_{S}^{0} +X
D_{p}.

There are such surfaces. Examples given by Keel and Mckernan (Mem. AMS 1999),

Kollár (Pure Appl. Math. Q. 2008) — an infinite series of examples with

|Sing(S)|=2.

D. Hwang and Keum (Proc. AMS 2012) — infinite series of examples with

|Sing(S)|=1,2,3.

Problem

Are there such surfaces S with|Sing(S)|=4?

No examples known yet.

Algebraic Montgomery-Yang Problem

## Kollár’s examples

Y =Y(a_{1},a_{2},a_{3},a_{4}) := (x_{1}^{a}^{1}x_{2}+x_{2}^{a}^{2}x_{3}+x_{3}^{a}^{3}x_{4}+x_{4}^{a}^{4}x_{1}=0)
in**P**(w_{1},w_{2},w_{3},w_{4}).Y has 4 singularities, two each on

C1:= (x1=x3=0), C2:= (x2=x4=0).

ContractingC_{1}andC_{2}we getX(a_{1},a_{2},a_{3},a_{4}), aQ-homology**P**^{2}with 2
singularities

[2, . . . ,2

| {z }

a4−1

,a_{3},a_{1},2, . . . ,2

| {z }

a2−1

]

[2, . . . ,2

| {z }

a3−1

,a2,a4,2, . . . ,2

| {z }

a1−1

].

KX is ample iffP

aj >12 andai ≥3 for alli.

X can be obtained by blowing up**P**^{2},P

a_{j} times inside 4 lines, then
contracting all negative curves with self-intersection≤ −2 (Hwang-Keum
2012, also Urzua-Yanez 2016). The number of such curves isP

a_{j}.

Algebraic Montgomery-Yang Problem

## More examples

can be obtained by blowing up**P**^{2}many times

(1) inside the union of 3 lines and a conic (total degree 5), then contracting all negative curves with self-intersection≤ −2

=⇒infinite series of examples with|Sing(S)|=2,3;

(2) inside the union of 4 lines and a nodal cubic (total degree 7), then contracting all negative curves with self-intersection≤ −2

=⇒infinite series of examples with|Sing(S)|=1.

Problem

Are there anyQ-homology**P**^{2}which is a rational surface S with K_{S}ample and
with|Sing(S)|=4?

Algebraic Montgomery-Yang Problem

## More examples

can be obtained by blowing up**P**^{2}many times

(1) inside the union of 3 lines and a conic (total degree 5), then contracting all negative curves with self-intersection≤ −2

=⇒infinite series of examples with|Sing(S)|=2,3;

(2) inside the union of 4 lines and a nodal cubic (total degree 7), then contracting all negative curves with self-intersection≤ −2

=⇒infinite series of examples with|Sing(S)|=1.

Problem

Are there anyQ-homology**P**^{2}which is a rational surface S with K_{S}ample and
with|Sing(S)|=4?

Algebraic Montgomery-Yang Problem

## Symplectic Montgomery-Yang Problem

This is the M-Y Problem when**S**^{5}/**S**^{1}attains a structure of a symplectic
orbifold,

i.e. away from its quotient singularities, a symplectic 4-manifold.

Question

Bogomolov inequality holds for symplectic compact 4-manifolds?
c_{1}^{2}≤4c2

Algebraic Montgomery-Yang Problem

Symplectic Montgomery-Yang Problem

This is the M-Y Problem when**S**^{5}/**S**^{1}attains a structure of a symplectic
orbifold,

i.e. away from its quotient singularities, a symplectic 4-manifold.

Question

Bogomolov inequality holds for symplectic compact 4-manifolds?

c_{1}^{2}≤4c2

Fake Projective Planes

## Fake Projective Planes

A compact complex surface with the same Betti numbers as**P**^{2}is called a
fake projective planeif it is not biholomorphic to**P**^{2}.

A FPP has ample canonical divisorK, so it is a smooth proper (geometrically
connected) surface of general type withpg=0 andK^{2}=9 (this definition
extends to arbitrary characteristic.)

The existence of a FPP was first proved byMumford (1979)based on the theory of 2-adic uniformization, and later two more examples byIshida-Kato (1998)in this abstract method.

Keum (2006)gave a construction of a FPP with an order 7 automorphism, which is birational to an order 7 cyclic cover of aDolgachev surface. Keum FPP and Mumford FPP belong tothe same class, in the sense that both fundamental groups are contained in the same maximal arithmetic subgroup of PU(2,1), the isometry group of the complex 2-ball.

Fake Projective Planes

## Fake Projective Planes

A compact complex surface with the same Betti numbers as**P**^{2}is called a
fake projective planeif it is not biholomorphic to**P**^{2}.

A FPP has ample canonical divisorK, so it is a smooth proper (geometrically
connected) surface of general type withpg=0 andK^{2}=9 (this definition
extends to arbitrary characteristic.)

The existence of a FPP was first proved byMumford (1979)based on the theory of 2-adic uniformization, and later two more examples byIshida-Kato (1998)in this abstract method.

Keum (2006)gave a construction of a FPP with an order 7 automorphism, which is birational to an order 7 cyclic cover of aDolgachev surface. Keum FPP and Mumford FPP belong tothe same class, in the sense that both fundamental groups are contained in the same maximal arithmetic subgroup of PU(2,1), the isometry group of the complex 2-ball.

Fake Projective Planes

## Fake Projective Planes

A compact complex surface with the same Betti numbers as**P**^{2}is called a
fake projective planeif it is not biholomorphic to**P**^{2}.

A FPP has ample canonical divisorK, so it is a smooth proper (geometrically
connected) surface of general type withpg=0 andK^{2}=9 (this definition
extends to arbitrary characteristic.)

The existence of a FPP was first proved byMumford (1979)based on the theory of 2-adic uniformization, and later two more examples byIshida-Kato (1998)in this abstract method.

Keum (2006)gave a construction of a FPP with an order 7 automorphism, which is birational to an order 7 cyclic cover of aDolgachev surface.

Keum FPP and Mumford FPP belong tothe same class, in the sense that both fundamental groups are contained in the same maximal arithmetic subgroup of PU(2,1), the isometry group of the complex 2-ball.

Fake Projective Planes

## Fake Projective Planes

**P**^{2}is called a
fake projective planeif it is not biholomorphic to**P**^{2}.

^{2}=9 (this definition
extends to arbitrary characteristic.)

Keum (2006)gave a construction of a FPP with an order 7 automorphism, which is birational to an order 7 cyclic cover of aDolgachev surface.

Keum FPP and Mumford FPP belong tothe same class, in the sense that both fundamental groups are contained in the same maximal arithmetic subgroup of PU(2,1), the isometry group of the complex 2-ball.

Fake Projective Planes

FPP’s have Chern numbersc_{1}^{2}=3c_{2}=9 and are complex 2-ball quotients by
Aubin (1976) and Yau (1977). Such ball quotients are strongly rigid by

Mostow’s rigidity theorem (1973), that is, determined by fundamental group up to holomorphic or anti-holomorphic isomorphism.

FPP’s come in complex conjugate pairs byKharlamov-Kulikov (2002)and have been classified as quotients of the two-dimensional complex ball by explicitly written co-compact torsion-free arithmetic subgroups ofPU(2,1)by Prasad-Yeung (2007, 2010) and Cartwright-Steger (2010).

Key ingredients:

The arithmeticity of their fundamental groupsKlingler (2003) The volume formulaPrasad (1989)

There are exactly100fake projective planes total, corresponding to50distinct fundamental groupsCartwright-Steger (2010).

Fake Projective Planes

FPP’s have Chern numbersc_{1}^{2}=3c_{2}=9 and are complex 2-ball quotients by
Aubin (1976) and Yau (1977). Such ball quotients are strongly rigid by

Mostow’s rigidity theorem (1973), that is, determined by fundamental group up to holomorphic or anti-holomorphic isomorphism.

FPP’s come in complex conjugate pairs byKharlamov-Kulikov (2002)and have been classified as quotients of the two-dimensional complex ball by explicitly written co-compact torsion-free arithmetic subgroups ofPU(2,1)by Prasad-Yeung (2007, 2010) and Cartwright-Steger (2010).

Key ingredients:

The arithmeticity of their fundamental groupsKlingler (2003) The volume formulaPrasad (1989)

There are exactly100fake projective planes total, corresponding to50distinct fundamental groupsCartwright-Steger (2010).

Fake Projective Planes

FPP’s have Chern numbersc_{1}^{2}=3c_{2}=9 and are complex 2-ball quotients by
Aubin (1976) and Yau (1977). Such ball quotients are strongly rigid by

Mostow’s rigidity theorem (1973), that is, determined by fundamental group up to holomorphic or anti-holomorphic isomorphism.

FPP’s come in complex conjugate pairs byKharlamov-Kulikov (2002)and have been classified as quotients of the two-dimensional complex ball by explicitly written co-compact torsion-free arithmetic subgroups ofPU(2,1)by Prasad-Yeung (2007, 2010) and Cartwright-Steger (2010).

Key ingredients:

The arithmeticity of their fundamental groupsKlingler (2003) The volume formulaPrasad (1989)

There are exactly100fake projective planes total, corresponding to50distinct fundamental groupsCartwright-Steger (2010).

Fake Projective Planes

_{1}^{2}=3c_{2}=9 and are complex 2-ball quotients by
Aubin (1976) and Yau (1977). Such ball quotients are strongly rigid by

Key ingredients:

The arithmeticity of their fundamental groupsKlingler (2003) The volume formulaPrasad (1989)

Fake Projective Planes

Interesting problems on fake projective planes:

Exceptional collections inD^{b}(coh(X))
Bicanonical map

Explicit equations

Bloch conjecture on zero cycles Modular forms

Fake Projective Planes

## Explicit equations of a Fake Projective Plane

It has long been of great interest since Mumford to find equations of an FPP.

WithLev Borisov (Duke M.J. 2020), we find equations of a conjugate pair of fake projective planes by using the geometry of the quotients of such FPP [Keum, 2008].

The equations are given explicitly as84 cubics in**P**^{9}with coefficients in the
fieldQ[√

−7].

Conjugating equations we get the complex conjugate of the surface. This pair has themost geometric symmetriesamong the 50 pairs, in the sense that

(i)Aut ∼=G_{21}=Z7:Z3, the largest (Keum’s FPPs);

(ii) theZ7-quotient has a smooth model of a(2,4)-elliptic surface, not simply connected.

The universal double cover of this elliptic surface is an(1,2)-elliptic surface, has the same Hodge numbers as K3, but Kodaira dimension 1.

Fake Projective Planes

## Explicit equations of a Fake Projective Plane

It has long been of great interest since Mumford to find equations of an FPP.

WithLev Borisov (Duke M.J. 2020), we find equations of a conjugate pair of fake projective planes by using the geometry of the quotients of such FPP [Keum, 2008].

The equations are given explicitly as84 cubics in**P**^{9}with coefficients in the
fieldQ[√

−7].

Conjugating equations we get the complex conjugate of the surface. This pair has themost geometric symmetriesamong the 50 pairs, in the sense that

(i)Aut ∼=G_{21}=Z7:Z3, the largest (Keum’s FPPs);

(ii) theZ7-quotient has a smooth model of a(2,4)-elliptic surface, not simply connected.

The universal double cover of this elliptic surface is an(1,2)-elliptic surface, has the same Hodge numbers as K3, but Kodaira dimension 1.

Fake Projective Planes

## Explicit equations of a Fake Projective Plane

It has long been of great interest since Mumford to find equations of an FPP.

WithLev Borisov (Duke M.J. 2020), we find equations of a conjugate pair of fake projective planes by using the geometry of the quotients of such FPP [Keum, 2008].

The equations are given explicitly as84 cubics in**P**^{9}with coefficients in the
fieldQ[√

−7].

Conjugating equations we get the complex conjugate of the surface.

This pair has themost geometric symmetriesamong the 50 pairs, in the sense that

(i)Aut ∼=G_{21}=Z7:Z3, the largest (Keum’s FPPs);

(ii) theZ7-quotient has a smooth model of a(2,4)-elliptic surface, not simply connected.

The universal double cover of this elliptic surface is an(1,2)-elliptic surface, has the same Hodge numbers as K3, but Kodaira dimension 1.

Fake Projective Planes

## Explicit equations of a Fake Projective Plane

It has long been of great interest since Mumford to find equations of an FPP.

The equations are given explicitly as84 cubics in**P**^{9}with coefficients in the
fieldQ[√

−7].

Conjugating equations we get the complex conjugate of the surface.

This pair has themost geometric symmetriesamong the 50 pairs, in the sense that

(i)Aut ∼=G_{21}=Z7:Z3, the largest (Keum’s FPPs);

(ii) theZ7-quotient has a smooth model of a(2,4)-elliptic surface, not simply connected.

Fake Projective Planes

## Explicit equations of a Fake Projective Plane

It has long been of great interest since Mumford to find equations of an FPP.

The equations are given explicitly as84 cubics in**P**^{9}with coefficients in the
fieldQ[√

−7].

Conjugating equations we get the complex conjugate of the surface.

This pair has themost geometric symmetriesamong the 50 pairs, in the sense that

(i)Aut ∼=G_{21}=Z7:Z3, the largest (Keum’s FPPs);

(ii) theZ7-quotient has a smooth model of a(2,4)-elliptic surface, not simply connected.

Fake Projective Planes

B^{2} **P**d^{2}_{fake} X −→^{π} **P**^{3}

& . & . &

**P**^{2}_{fake} Y **P**^{1}

& . & .
**P**^{2}_{fake}/Z**7** **P**^{1}

(1)

B^{2}is the complex 2-ball.**P**^{2}_{fake}is our FPP.

Y →**P**^{1}is a(2,4)-elliptic surface with oneI9-fibre and three 4-sections.

X →**P**^{1}is an(1,2)-elliptic surface with twoI9-fibres and six 2-sections.

Using these 24 smooth rational curves onX we find a linear system which gives a birational map

π:X →**P**^{3}.
The image is a sextic surface, highly singular.

Its equation is computed explicitly using the elliptic fibration structureX →**P**^{1}.

Fake Projective Planes

B^{2} **P**d^{2}_{fake} X −→^{π} **P**^{3}

& . & . &

**P**^{2}_{fake} Y **P**^{1}

& . & .
**P**^{2}_{fake}/Z**7** **P**^{1}

(1)

B^{2}is the complex 2-ball.**P**^{2}_{fake}is our FPP.

Y →**P**^{1}is a(2,4)-elliptic surface with oneI9-fibre and three 4-sections.

X →**P**^{1}is an(1,2)-elliptic surface with twoI9-fibres and six 2-sections.

Using these 24 smooth rational curves onX we find a linear system which gives a birational map

π:X →**P**^{3}.

The image is a sextic surface, highly singular.

Its equation is computed explicitly using the elliptic fibration structureX →**P**^{1}.

Fake Projective Planes

B^{2} **P**d^{2}_{fake} X −→^{π} **P**^{3}

& . & . &

**P**^{2}_{fake} Y **P**^{1}

& . & .
**P**^{2}_{fake}/Z**7** **P**^{1}

(1)

B^{2}is the complex 2-ball.**P**^{2}_{fake}is our FPP.

Y →**P**^{1}is a(2,4)-elliptic surface with oneI9-fibre and three 4-sections.

X →**P**^{1}is an(1,2)-elliptic surface with twoI9-fibres and six 2-sections.

Using these 24 smooth rational curves onX we find a linear system which gives a birational map

π:X →**P**^{3}.
The image is a sextic surface, highly singular.

Its equation is computed explicitly using the elliptic fibration structureX →**P**^{1}.

Fake Projective Planes

## 84 Equations of the fake projective plane

eq1=U1U2U3+ (1−i√

7)(U_{3}^{2}U4+U_{1}^{2}U5+U_{2}^{2}U6) + (10−2i√

7)U4U5U6

eq_{2}= (−3+i

√

7)U_{0}^{3}+ (7+i

√

7)(−2U_{1}U_{2}U_{3}+U_{7}U_{8}U_{9}−8U_{4}U_{5}U_{6})
+8U_{0}(U_{1}U_{4}+U_{2}U_{5}+U_{3}U_{6}) + (6+2i√

7)U_{0}(U_{1}U_{7}+U_{2}U_{8}+U_{3}U_{9})
eq_{3}= (11−i

√

7)U_{0}^{3}+128U_{4}U_{5}U_{6}−(18+10i√

7)U_{7}U_{8}U_{9}
+64(U_{2}U_{4}^{2}+U_{3}U_{5}^{2}+U_{1}U_{6}^{2}) + (−14−6i√

7)U_{0}(U_{1}U_{7}+U_{2}U_{8}+U_{3}U_{9})
+8(1+i

√

7)(U_{1}^{2}U8+U_{2}^{2}U9+U_{3}^{2}U7−2U1U2U3)
eq4=−(1+i√

7)U0U3(4U6+U9) +8(U1U2U3+U1U6U9+U5U7U9)
+16(U5U6U7−U_{1}^{2}U5−U3U_{5}^{2})

eq_{5}=g_{3}(eq_{4})
eq_{6}=g_{3}^{2}(eq_{4})

...

Fake Projective Planes

On the coordinates(U0:U1:U2:U3:U4:U5:U6:U7:U8:U9)of**P**^{9}
g7:= (U0:ζ^{6}U1:ζ^{5}U2:ζ^{3}U3:ζU4:ζ^{2}U5:ζ^{4}U6:ζU7:ζ^{2}U8:ζ^{4}U9)

g3:= (U0:U2:U3:U1:U5:U6:U4:U8:U9:U7)
whereζ=ζ_{7}is the primitive 7-th root of 1.

It can be verified that the variety

Z ⊂**P**^{9}

defined by the 84 equations is indeed a FPP. Use Magma and Macaulay 2. Take a primep=263. Then√

−7=16modp. Magma calculates the Hilbert series ofZ

h^{0}(Z,O_{Z}(k)) = 1

2(6k−1)(6k−2) =18k^{2}−9k+1, k ≥0.
Smoothness ofZ is a subtle problem.

The 84×10 Jacobian matrix has too many 7×7 minors.

By adding suitably chosen 3 minors to the ideal of 84 cubics, the Hilbert
polynomial drops from 18k^{2}−9k+1 to linear, then to constant, then to 0.
If the equations generate the ring modulo 263, then they also generate it with
exact coefficients.

Fake Projective Planes

On the coordinates(U0:U1:U2:U3:U4:U5:U6:U7:U8:U9)of**P**^{9}
g7:= (U0:ζ^{6}U1:ζ^{5}U2:ζ^{3}U3:ζU4:ζ^{2}U5:ζ^{4}U6:ζU7:ζ^{2}U8:ζ^{4}U9)

g3:= (U0:U2:U3:U1:U5:U6:U4:U8:U9:U7)
whereζ=ζ_{7}is the primitive 7-th root of 1.

It can be verified that the variety

Z ⊂**P**^{9}

defined by the 84 equations is indeed a FPP. Use Magma and Macaulay 2.

Take a primep=263. Then√

−7=16modp. Magma calculates the Hilbert series ofZ

h^{0}(Z,O_{Z}(k)) = 1

2(6k−1)(6k−2) =18k^{2}−9k+1, k ≥0.
Smoothness ofZ is a subtle problem.

The 84×10 Jacobian matrix has too many 7×7 minors.

By adding suitably chosen 3 minors to the ideal of 84 cubics, the Hilbert
polynomial drops from 18k^{2}−9k+1 to linear, then to constant, then to 0.
If the equations generate the ring modulo 263, then they also generate it with
exact coefficients.

Fake Projective Planes

On the coordinates(U0:U1:U2:U3:U4:U5:U6:U7:U8:U9)of**P**^{9}
g7:= (U0:ζ^{6}U1:ζ^{5}U2:ζ^{3}U3:ζU4:ζ^{2}U5:ζ^{4}U6:ζU7:ζ^{2}U8:ζ^{4}U9)

g3:= (U0:U2:U3:U1:U5:U6:U4:U8:U9:U7)
whereζ=ζ_{7}is the primitive 7-th root of 1.

It can be verified that the variety

Z ⊂**P**^{9}

defined by the 84 equations is indeed a FPP. Use Magma and Macaulay 2.

Take a primep=263. Then√

−7=16modp.

Magma calculates the Hilbert series ofZ
h^{0}(Z,O_{Z}(k)) = 1

2(6k−1)(6k−2) =18k^{2}−9k+1, k ≥0.

Smoothness ofZ is a subtle problem.

The 84×10 Jacobian matrix has too many 7×7 minors.

By adding suitably chosen 3 minors to the ideal of 84 cubics, the Hilbert
polynomial drops from 18k^{2}−9k+1 to linear, then to constant, then to 0.
If the equations generate the ring modulo 263, then they also generate it with
exact coefficients.

Fake Projective Planes

**P**^{9}
g7:= (U0:ζ^{6}U1:ζ^{5}U2:ζ^{3}U3:ζU4:ζ^{2}U5:ζ^{4}U6:ζU7:ζ^{2}U8:ζ^{4}U9)

g3:= (U0:U2:U3:U1:U5:U6:U4:U8:U9:U7)
whereζ=ζ_{7}is the primitive 7-th root of 1.

It can be verified that the variety

Z ⊂**P**^{9}

defined by the 84 equations is indeed a FPP. Use Magma and Macaulay 2.

Take a primep=263. Then√

−7=16modp.

Magma calculates the Hilbert series ofZ
h^{0}(Z,O_{Z}(k)) = 1

2(6k−1)(6k−2) =18k^{2}−9k+1, k ≥0.

Smoothness ofZ is a subtle problem.

The 84×10 Jacobian matrix has too many 7×7 minors.

By adding suitably chosen 3 minors to the ideal of 84 cubics, the Hilbert
polynomial drops from 18k^{2}−9k+1 to linear, then to constant, then to 0.

If the equations generate the ring modulo 263, then they also generate it with exact coefficients.

Fake Projective Planes

ThusZ is a smooth surface with a very ample divisor classD=O_{Z}(1). From
the Hilbert polynomial we see that

D^{2}=36,DKZ =18, χ(Z,O_{Z}) =1.

In part,Z **P**^{2}.

Macaulay 2 calculates the projective resolution ofO_{Z} as

0→ O(−9)^{⊕28}→ O(−8)^{⊕189} → O(−7)^{⊕540}→ O(−6)^{⊕840}

→ O(−5)^{⊕756}→ O(−4)^{⊕378}→ O(−3)^{⊕84}→ O → O_{Z} →0.
By semicontinuity, the resolution is of the same shape over**C**.
Since all the sheavesO(−k)are acyclic, we see that

h^{1}(Z,O_{Z}) =h^{2}(Z,O_{Z}) =0.
Macaulay also calculates (again working modulo 263)

χ(Z,2K_{Z}) =10.
This impliesK_{Z}^{2}=9. ThusZ is a FPP.

Z can be further identified with the FPP which we started with.

Fake Projective Planes

ThusZ is a smooth surface with a very ample divisor classD=O_{Z}(1). From
the Hilbert polynomial we see that

D^{2}=36,DKZ =18, χ(Z,O_{Z}) =1.

In part,Z **P**^{2}.

Macaulay 2 calculates the projective resolution ofO_{Z} as

0→ O(−9)^{⊕28}→ O(−8)^{⊕189} → O(−7)^{⊕540}→ O(−6)^{⊕840}

→ O(−5)^{⊕756}→ O(−4)^{⊕378}→ O(−3)^{⊕84}→ O → O_{Z} →0.

By semicontinuity, the resolution is of the same shape over**C**.

Since all the sheavesO(−k)are acyclic, we see that
h^{1}(Z,O_{Z}) =h^{2}(Z,O_{Z}) =0.

Macaulay also calculates (again working modulo 263)
χ(Z,2K_{Z}) =10.

This impliesK_{Z}^{2}=9. ThusZ is a FPP.

Z can be further identified with the FPP which we started with.

Fake Projective Planes