Fake Projective Planes

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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

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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

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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_CH⁰(C,Ω¹_C) = ¹₂dim_QH1(C,Q).

g(C) =0⇐⇒C∼=P¹∼= (Riemann sphere) =C∪ {∞}.

In dimension>1, many invariants: Hodge numbers, Betti numbers h^i,j(X) =dimH^j(X,Ωⁱ_X), b_i(X) :=dimHⁱ(X,Q).

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

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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_CH⁰(C,Ω¹_C) = ¹₂dim_QH1(C,Q).

g(C) =0⇐⇒C∼=P¹∼= (Riemann sphere) =C∪ {∞}.

In dimension>1, many invariants: Hodge numbers, Betti numbers h^i,j(X) =dimH^j(X,Ωⁱ_X), b_i(X) :=dimHⁱ(X,Q).

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

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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) :=dimHⁿ(X,O_X) =dimH⁰(X,Ωⁿ_X) =h^0,n(X) =h^n,0(X), q(X) :=dimH¹(X,O_X) =dimH⁰(X,Ω¹_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."

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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) :=dimHⁿ(X,O_X) =dimH⁰(X,Ωⁿ_X) =h^0,n(X) =h^n,0(X), q(X) :=dimH¹(X,O_X) =dimH⁰(X,Ω¹_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."

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Smooth Algebraic Surfaces with p

_g

= q = 0

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

P², rational ruled surfaces;

Enriques surfaces;

properly elliptic surfaces withpg=q=0;

surfaces of general type withpg=0 (these haveK²=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

P²;

fake projective planes (= surfaces of general type withpg =0,K²=9). Remark.FPPs are ball quotients, so not simply connected.

ExoticP²does NOT exist in complex geometry.

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Smooth Algebraic Surfaces with p

_g

= q = 0

Enriques surfaces;

P²;

fake projective planes (= surfaces of general type withpg =0,K²=9).

Remark.FPPs are ball quotients, so not simply connected. ExoticP²does NOT exist in complex geometry.

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Smooth Algebraic Surfaces with p

_g

= q = 0

Enriques surfaces;

P²;

fake projective planes (= surfaces of general type withpg =0,K²=9).

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

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Q-homology Projective Planes

Q -homology P

²

Definition

A normal projective surfaceSis called aQ-homologyP²ifbi(S) =bi(P²)for alli, i.e.b1=b3=0,b0=b2=b4=1.

IfSis smooth, thenS=P²or afake projective plane.

IfShasA1-singularities only, thenS∼= (w²=xy)⊂P³. IfShasA₂-singularities only, thenShas 3A₂or 4A₂and S∼=P²/Gor FPP/G, whereG∼=Z/3 or(Z/3)².

Any cubic surface inP³with 3A₂is isom. to(w³=xyz).

IfShasA₁orA₂-singularities only,S=P²(1,2,3)or one of the above. In this talk, we assumeShas at worstquotientsingularities.

ThenSis aQ-homologyP²ifb₂(S) =1. For a minimal resolutionS⁰→S,

p_g(S⁰) =q(S⁰) =0.

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Q -homology P

²

Definition

IfShasA₁orA₂-singularities only,S=P²(1,2,3)or one of the above. In this talk, we assumeShas at worstquotientsingularities.

ThenSis aQ-homologyP²ifb₂(S) =1. For a minimal resolutionS⁰→S,

p_g(S⁰) =q(S⁰) =0.

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Q -homology P

²

Definition

IfShasA₁orA₂-singularities only,S=P²(1,2,3)or one of the above.

In this talk, we assumeShas at worstquotientsingularities. ThenSis aQ-homologyP²ifb₂(S) =1.

For a minimal resolutionS⁰→S,

p_g(S⁰) =q(S⁰) =0.

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Q -homology P

²

Definition

In this talk, we assumeShas at worstquotientsingularities.

ThenSis aQ-homologyP²ifb₂(S) =1.

p_g(S⁰) =q(S⁰) =0.

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Q -homology P

²

Definition

In this talk, we assumeShas at worstquotientsingularities.

ThenSis aQ-homologyP²ifb₂(S) =1.

pg(S⁰) =q(S⁰) =0.

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Trichotomy: K

_S

= ample, −ample, num. trivial

LetSbe aQ-homP²with quotient singularities.

−K_S is ample

log del Pezzo surfaces of Picard number 1, e.g.P²/G,P²(a,b,c),. . .

κ(S⁰) =−∞.

K_Sis numerically trivial.

log Enriques surfaces of Picard number 1.

κ(S⁰) =−∞,0.

K_Sis ample.

e.g. all quotients of fake projective planes,

suitable contraction of a suitable blowup ofP², some Enriques surface,. . . κ(S⁰) =−∞,0,1,2.

Problem

Classify allQ-homologyP²’s with quotient singularities.

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The Maximum Number of Quotient Singularities

Question

How many singular points on S, aQ-homologyP²with quotient singularities?

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

1

3K_S²≤eorb(S) :=e(S)− X

p∈Sing(S)

1− 1

|π₁(L_p)|

.

(Keel-McKernan for−K nef)

0≤eorb(S).

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

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

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The Maximum Number of Quotient Singularities

Question

1

p∈Sing(S)

1− 1

|π₁(L_p)|

.

0≤eorb(S).

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The Maximum Number of Quotient Singularities

Question

1

p∈Sing(S)

1− 1

|π₁(L_p)|

.

0≤eorb(S).

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Theorem (D.Hwang-Keum, JAG 2011)

Let S be aQ-homologyP²with quotient singularities. Then|Sing(S)| ≤4 except the following case:

S has5singular points of type3A1+2A3, and its minimal resolution S⁰is an Enriques surface.

Corollary

EveryZ-homologyP²with quotient singularities has at most4singular points.

Remark

(1) EveryZ-cohomologyP²with quotient singularities has at most1singular point. If it has, then the singularity is of type E8[Bindschadler-Brenton, 1984]. (2)Q-homologyP²withrational singularitiesmay havearbitrarily many singularities, no bound.

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Theorem (D.Hwang-Keum, JAG 2011)

Let S be aQ-homologyP²with quotient singularities. Then|Sing(S)| ≤4 except the following case:

S has5singular points of type3A1+2A3, and its minimal resolution S⁰is an Enriques surface.

Corollary

EveryZ-homologyP²with quotient singularities has at most4singular points.

Remark

(1) EveryZ-cohomologyP²with quotient singularities has at most1singular point. If it has, then the singularity is of type E8[Bindschadler-Brenton, 1984].

(2)Q-homologyP²withrational singularitiesmay havearbitrarily many singularities, no bound.

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Montgomery-Yang Problem

C

^∞

-action of S

¹

on S

^m

S¹⊂Diff(S^m).

The identity element 1∈S¹acts identically onS^m.

Each diffeomorphismg ∈S¹is homotopic to the identity map 1S^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.

Assumem=2n−1odd. Definition

AC^∞-action ofS¹onS²ⁿ⁻¹

S¹×S²ⁿ⁻¹→S²ⁿ⁻¹

is called apseudofreeS¹-actiononS²ⁿ⁻¹if it is free except for finitely many orbits (whose isotropy groupsZ/a₁, . . . ,Z/a_k have pairwise prime orders).

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C

^∞

-action of S

¹

on S

^m

S¹⊂Diff(S^m).

The identity element 1∈S¹acts identically onS^m.

Each diffeomorphismg ∈S¹is homotopic to the identity map 1S^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.Assumem=2n−1odd.

Definition

AC^∞-action ofS¹onS²ⁿ⁻¹

S¹×S²ⁿ⁻¹→S²ⁿ⁻¹

is called apseudofreeS¹-actiononS²ⁿ⁻¹if it is free except for finitely many orbits (whose isotropy groupsZ/a₁, . . . ,Z/a_k have pairwise prime orders).

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Pseudofree S

¹

-action on S

²ⁿ⁻¹

Example (Linear actions)

S²ⁿ⁻¹={(z₁,z₂, ...,z_n)∈Cⁿ:|z₁|²+|z₂|²+...+|z_n|²=1} ⊂Cⁿ S¹={λ∈C:|λ|=1} ⊂C.

Positive integersa₁, ...,a_npairwise prime.

S¹×S²ⁿ⁻¹→S²ⁿ⁻¹

(λ,(z1,z2, ...,zn))→(λâ¹z1, λâ²z2, ..., λâⁿzn).

In thislinear action

S²ⁿ⁻¹/S¹∼=CPⁿ⁻¹(a₁,a₂, ...,a_n).

The orbit of thei-th coordinate pointei ∈S²ⁿ⁻¹is exceptional iffai ≥2.

The orbit of a non-coordinate point ofS²ⁿ⁻¹is NOT exceptional.

This action has at mostnexceptional orbits.

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Pseudofree S

¹

-action on S

²ⁿ⁻¹

Forn=2 Seifert (1932) showed that each pseudo-freeS¹-action onS³is 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 S¹-action on a homotopyS⁷whose 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-freeS¹-action onS⁵has at most3exceptional orbits.

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

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Pseudo-freeS¹-actions on a manifoldΣhave been studied in terms of the orbit spaceΣ/S¹.

The orbit spaceX =S⁵/S¹of such an action is a 4-manifold with isolated singularities whose neighborhoods are cones over lens spacesS³/Zai

corresponding to the exceptional orbits of theS¹-action.

Easy to check thatX is simply connected andH₂(X,Z)has rank 1 and intersection matrix(1/a₁a₂· · ·a_k).

An exceptional orbit with isotropy typeZ/ahas an equivariant tubular neighborhood which may be identified withC×C×S¹with aS¹-action

λ·(z,w,u) = (λ^rz, λ^sw, λ^au) wherer andsare relatively prime toa.

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Pseudo-freeS¹-actions on a manifoldΣhave been studied in terms of the orbit spaceΣ/S¹.

The orbit spaceX =S⁵/S¹of such an action is a 4-manifold with isolated singularities whose neighborhoods are cones over lens spacesS³/Zai

corresponding to the exceptional orbits of theS¹-action.

Easy to check thatX is simply connected andH₂(X,Z)has rank 1 and intersection matrix(1/a₁a₂· · ·a_k).

An exceptional orbit with isotropy typeZ/ahas an equivariant tubular neighborhood which may be identified withC×C×S¹with aS¹-action

λ·(z,w,u) = (λ^rz, λ^sw, λ^au) whererandsare relatively prime toa.

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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-freeS¹-actions onQ-homology 5-spheresΣwith H₁(Σ,Z) =0.

2 Compact differentiable4-manifolds M with boundary such that

1 ∂M=S

i

Liis a disjoint union of lens spaces Li=S³/Z^ai,

2 the ai’s are pairwise prime,

3 H1(M,Z) =0,

4 H2(M,Z)∼=Z.

Furthermore,Σis diffeomorphic toS⁵iffπ₁(M) =1.

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Algebraic Montgomery-Yang Problem

This is the M-Y Problem whenS⁵/S¹attains a structure of a normal projective surface.

Conjecture (J. Kollár)

Let S be aQ-homologyP²with at worst quotient singularities. Ifπ₁(S⁰) ={1}, then S has at most3singular points.

What if the conditionπ₁(S⁰) ={1}is replaced by the weaker condition H1(S⁰,Z) =0?

There are infinitely many examplesSwith H1(S⁰,Z) =0,π₁(S⁰)6={1},|Sing(S)|=4.

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

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Algebraic Montgomery-Yang Problem

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Algebraic Montgomery-Yang Problem

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Example (coming from Brieskorn’s classification of surface singularities) I_m⊂GL(2,C)the 2m-ary icosahedral groupI_m=Z2m.A₅.

1→Z2m→I_m→ A₅⊂PSL(2,C) I_macts onC². This action extends naturally toP². Then

S:=P²/Im

is aZ-homologyP²with−K_S ample, Shas 4 quotient singularities:

one non-cyclic singularity of typeIm(the image ofO∈C²), and

3 cyclic singularities of order 2,3,5 (on the image of the line at infinity), π₁(S⁰) =A₅, henceH₁(S⁰,Z) =0.

Call these surfacesBrieskorn quotients.

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Progress on Algebraic Montgomery-Yang Problem

Theorem (D.Hwang-Keum, MathAnn 2011)

Let S be aQ-homologyP²with quotient singularities, not all cyclic, such that π₁(S⁰) ={1}. Then|Sing(S)| ≤3.

More precisely

Let S be aQ-homologyP²with4or more quotient singularities, not all cyclic, such that H1(S⁰,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-homologyP²with cyclic singularities such that H₁(S⁰,Z) =0. If either S is not rational or−K_S is ample, then|Sing(S)| ≤3.

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Progress on Algebraic Montgomery-Yang Problem

Let S be aQ-homologyP²with quotient singularities, not all cyclic, such that π₁(S⁰) ={1}. Then|Sing(S)| ≤3.

More precisely

Let S be aQ-homologyP²with4or more quotient singularities, not all cyclic, such that H1(S⁰,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-homologyP²with cyclic singularities such that H₁(S⁰,Z) =0. If either S is not rational or−K is ample, then|Sing(S)| ≤3.

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The Remaining Case of Algebraic M-Y Problem:

Sis aQ-homologyP²satisfying (1)Shas cyclic singularities only, (2)Sis a rational surface withK_Sample.

π^∗K_S=K_S⁰ +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.

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The Remaining Case of Algebraic M-Y Problem:

|Sing(S)|=2.

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

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The Remaining Case of Algebraic M-Y Problem:

|Sing(S)|=2.

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

Problem

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The Remaining Case of Algebraic M-Y Problem:

|Sing(S)|=2.

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

Problem

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

No examples known yet.

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Kollár’s examples

Y =Y(a₁,a₂,a₃,a₄) := (x₁â¹x₂+x₂â²x₃+x₃â³x₄+x₄â⁴x₁=0) inP(w₁,w₂,w₃,w₄).Y has 4 singularities, two each on

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

ContractingC₁andC₂we getX(a₁,a₂,a₃,a₄), aQ-homologyP²with 2 singularities

[2, . . . ,2

| {z }

a4−1

,a₃,a₁,2, . . . ,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 upP²,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.

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More examples

can be obtained by blowing upP²many 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-homologyP²which is a rational surface S with K_Sample and with|Sing(S)|=4?

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More examples

can be obtained by blowing upP²many 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-homologyP²which is a rational surface S with K_Sample and with|Sing(S)|=4?

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Symplectic Montgomery-Yang Problem

This is the M-Y Problem whenS⁵/S¹attains 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? c₁²≤4c2

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Symplectic Montgomery-Yang Problem

This is the M-Y Problem whenS⁵/S¹attains 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?

c₁²≤4c2

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Fake Projective Planes

A compact complex surface with the same Betti numbers asP²is called a fake projective planeif it is not biholomorphic toP².

A FPP has ample canonical divisorK, so it is a smooth proper (geometrically connected) surface of general type withpg=0 andK²=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.

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Fake Projective Planes

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.

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Fake Projective Planes

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.

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Fake Projective Planes

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.

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FPP’s have Chern numbersc₁²=3c₂=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).

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Key ingredients:

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Key ingredients:

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Key ingredients:

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Interesting problems on fake projective planes:

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

Explicit equations

Bloch conjecture on zero cycles Modular forms

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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 inP⁹with 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 ∼=G₂₁=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.

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Explicit equations of a Fake Projective Plane

−7].

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

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Explicit equations of a Fake Projective Plane

−7].

Conjugating equations we get the complex conjugate of the surface.

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

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Explicit equations of a Fake Projective Plane

−7].

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Explicit equations of a Fake Projective Plane

−7].

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B² Pd²_fake X −→^π P³

& . & . &

P²_fake Y P¹

& . & . P²_fake/Z7 P¹

(1)

B²is the complex 2-ball.P²_fakeis our FPP.

Y →P¹is a(2,4)-elliptic surface with oneI9-fibre and three 4-sections.

X →P¹is 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 →P³. The image is a sextic surface, highly singular.

Its equation is computed explicitly using the elliptic fibration structureX →P¹.

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& . & . &

P²_fake Y P¹

& . & . P²_fake/Z7 P¹

(1)

π:X →P³.

The image is a sextic surface, highly singular.

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& . & . &

P²_fake Y P¹

& . & . P²_fake/Z7 P¹

(1)

π:X →P³. The image is a sextic surface, highly singular.

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84 Equations of the fake projective plane

eq1=U1U2U3+ (1−i√

7)(U₃²U4+U₁²U5+U₂²U6) + (10−2i√

7)U4U5U6

eq₂= (−3+i

√

7)U₀³+ (7+i

√

7)(−2U₁U₂U₃+U₇U₈U₉−8U₄U₅U₆) +8U₀(U₁U₄+U₂U₅+U₃U₆) + (6+2i√

7)U₀(U₁U₇+U₂U₈+U₃U₉) eq₃= (11−i

√

7)U₀³+128U₄U₅U₆−(18+10i√

7)U₇U₈U₉ +64(U₂U₄²+U₃U₅²+U₁U₆²) + (−14−6i√

7)U₀(U₁U₇+U₂U₈+U₃U₉) +8(1+i

√

7)(U₁²U8+U₂²U9+U₃²U7−2U1U2U3) eq4=−(1+i√

7)U0U3(4U6+U9) +8(U1U2U3+U1U6U9+U5U7U9) +16(U5U6U7−U₁²U5−U3U₅²)

eq₅=g₃(eq₄) eq₆=g₃²(eq₄)

...

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On the coordinates(U0:U1:U2:U3:U4:U5:U6:U7:U8:U9)ofP⁹ g7:= (U0:ζ⁶U1:ζ⁵U2:ζ³U3:ζU4:ζ²U5:ζ⁴U6:ζU7:ζ²U8:ζ⁴U9)

g3:= (U0:U2:U3:U1:U5:U6:U4:U8:U9:U7) whereζ=ζ₇is the primitive 7-th root of 1.

It can be verified that the variety

Z ⊂P⁹

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

h⁰(Z,O_Z(k)) = 1

2(6k−1)(6k−2) =18k²−9k+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 18k²−9k+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.

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Z ⊂P⁹

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

h⁰(Z,O_Z(k)) = 1

2(6k−1)(6k−2) =18k²−9k+1, k ≥0. Smoothness ofZ is a subtle problem.

(63)

Z ⊂P⁹

−7=16modp.

Magma calculates the Hilbert series ofZ h⁰(Z,O_Z(k)) = 1

2(6k−1)(6k−2) =18k²−9k+1, k ≥0.

Smoothness ofZ is a subtle problem.

(64)

Z ⊂P⁹

−7=16modp.

Magma calculates the Hilbert series ofZ h⁰(Z,O_Z(k)) = 1

2(6k−1)(6k−2) =18k²−9k+1, k ≥0.

Smoothness ofZ is a subtle problem.

By adding suitably chosen 3 minors to the ideal of 84 cubics, the Hilbert polynomial drops from 18k²−9k+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.

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

D²=36,DKZ =18, χ(Z,O_Z) =1.

In part,Z P².

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 overC. Since all the sheavesO(−k)are acyclic, we see that

h¹(Z,O_Z) =h²(Z,O_Z) =0. Macaulay also calculates (again working modulo 263)

χ(Z,2K_Z) =10. This impliesK_Z²=9. ThusZ is a FPP.

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

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

D²=36,DKZ =18, χ(Z,O_Z) =1.

In part,Z P².

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 overC.

Since all the sheavesO(−k)are acyclic, we see that h¹(Z,O_Z) =h²(Z,O_Z) =0.

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

This impliesK_Z²=9. ThusZ is a FPP.

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

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