Discrete groups acting on complex projective spaces e Seade 1 Jos 1 - - PowerPoint PPT Presentation

Discrete groups acting on complex projective spaces

Jos´ e Seade1

1Instituto de Matem´

aticas, Universidad Nacional Aut´

• noma de M´

exico.

LMS Singularity Day, Liverpool, G.B.,

March 29, 2016

Seade Discrete groups acting on complex projective spaces

Section 1: The classical case n = 1

• Recall: PSL(2, C) ∼

= AutHolCP1 ∼ = Conf+(S2) ∼ = Iso+(H3) ∼ = ∼ = { az+b

cz+d , a, b, c, d ∈ C, ad − bc = 1} .

• A Lie group, diffeomorphic (as manifold) to R3 × SO(3) .
• Its discrete subgroups are called Kleinian groups.
• These are either infinite or conjugate to a finite subgroup of

SU(2)

• If Γ ⊂ PSL(2, C) is finite, then VΓ := C2/Γ is a complex

analytic surface with an isolated singularity. Very interesting classical singularities:

• Du Val sings., Arnold’s modality 0, Rational double points, etc.
• These are all hypersurface singularities

Seade Discrete groups acting on complex projective spaces

For instance, if Γ is the binary icosahedral group, then VΓ is: {(z1, z2, z3) ∈ C3 | z2

1 + z3 2 + z5 3 = 0 } ,

and its link is Poincare’s homology 3-sphere. If Γ is the cyclic group of order r > 1, then VΓ is: {(z1, z2, z3) ∈ C3 | z2

1 + z2 2 + zr 3 = 0 } ,

and its link is a lens space L(r, 1), obtained by glueing two solid Tori S1 × D2 by a homeomorphism of their boundaries that takes a parallel into a curve of type (r, 1). This all goes back, essentially, to F . Klein’s work in the 19th Century. Focus today on infinite discrete subgroups .

Seade Discrete groups acting on complex projective spaces

Notice CP1 is compact, so every Γ-orbit has accumulation points: Definition The limit set of Γ ⊂ PSL(2, C) in CP1 is the set Λ of accumulation points of all orbits.

Seade Discrete groups acting on complex projective spaces

This has remarkable properties. Some basic ones are: Theorem

1

Λ is a closed invariant set.

2

If Λ has finite cardinality, then it consists of 1 or 2 points, and the group is called elementary. If we assume from now on that the group is non-elementary. Then:

3

The action on Λ is minimal (every orbit is dense)

4

Λ is the set of accumulation points of every orbit. And furthermore, if we are interested in complex geometry, then:

Seade Discrete groups acting on complex projective spaces

This has remarkable properties. Some basic ones are: Theorem

1

Λ is a closed invariant set.

2

If Λ has finite cardinality, then it consists of 1 or 2 points, and the group is called elementary. If we assume from now on that the group is non-elementary. Then:

3

The action on Λ is minimal (every orbit is dense)

4

Λ is the set of accumulation points of every orbit. And furthermore, if we are interested in complex geometry, then:

Seade Discrete groups acting on complex projective spaces

This has remarkable properties. Some basic ones are: Theorem

1

Λ is a closed invariant set.

2

If Λ has finite cardinality, then it consists of 1 or 2 points, and the group is called elementary. If we assume from now on that the group is non-elementary. Then:

3

The action on Λ is minimal (every orbit is dense)

4

Λ is the set of accumulation points of every orbit. And furthermore, if we are interested in complex geometry, then:

Seade Discrete groups acting on complex projective spaces

This has remarkable properties. Some basic ones are: Theorem

1

Λ is a closed invariant set.

2

If Λ has finite cardinality, then it consists of 1 or 2 points, and the group is called elementary. If we assume from now on that the group is non-elementary. Then:

3

The action on Λ is minimal (every orbit is dense)

4

Λ is the set of accumulation points of every orbit. And furthermore, if we are interested in complex geometry, then:

Seade Discrete groups acting on complex projective spaces

This has remarkable properties. Some basic ones are: Theorem

1

Λ is a closed invariant set.

2

If Λ has finite cardinality, then it consists of 1 or 2 points, and the group is called elementary. If we assume from now on that the group is non-elementary. Then:

3

The action on Λ is minimal (every orbit is dense)

4

Λ is the set of accumulation points of every orbit. And furthermore, if we are interested in complex geometry, then:

Seade Discrete groups acting on complex projective spaces

Theorem

1

Its complement Ω is the set where the action is discontinuous.

2

Ω actually is the maximal set where the action is properly

• discontinuous. It is also the equicontinuity set of the action.

3

If Ω has finitely many connected components, then it has at most two (it can be empty).

4

The quotient Ω/Γ is a Riemann surface with a projective

• rbifold structure.

5

Every Riemann surface can be realized in this way (Koebe’s retrosection theorem)

6

If Γ is finitely generated, then Ω/Γ is of finite type (Sullivan), i.e. It has finitely many connected components, each being a compact Riemann surface minus finitely many points.

Seade Discrete groups acting on complex projective spaces

Theorem

1

Its complement Ω is the set where the action is discontinuous.

2

Ω actually is the maximal set where the action is properly

• discontinuous. It is also the equicontinuity set of the action.

3

If Ω has finitely many connected components, then it has at most two (it can be empty).

4

The quotient Ω/Γ is a Riemann surface with a projective

• rbifold structure.

5

Every Riemann surface can be realized in this way (Koebe’s retrosection theorem)

6

If Γ is finitely generated, then Ω/Γ is of finite type (Sullivan), i.e. It has finitely many connected components, each being a compact Riemann surface minus finitely many points.

Seade Discrete groups acting on complex projective spaces

Theorem

1

Its complement Ω is the set where the action is discontinuous.

2

Ω actually is the maximal set where the action is properly

• discontinuous. It is also the equicontinuity set of the action.

3

If Ω has finitely many connected components, then it has at most two (it can be empty).

4

The quotient Ω/Γ is a Riemann surface with a projective

• rbifold structure.

5

Every Riemann surface can be realized in this way (Koebe’s retrosection theorem)

6

If Γ is finitely generated, then Ω/Γ is of finite type (Sullivan), i.e. It has finitely many connected components, each being a compact Riemann surface minus finitely many points.

Seade Discrete groups acting on complex projective spaces

Theorem

1

Its complement Ω is the set where the action is discontinuous.

2

Ω actually is the maximal set where the action is properly

• discontinuous. It is also the equicontinuity set of the action.

3

If Ω has finitely many connected components, then it has at most two (it can be empty).

4

The quotient Ω/Γ is a Riemann surface with a projective

• rbifold structure.

5

Every Riemann surface can be realized in this way (Koebe’s retrosection theorem)

6

If Γ is finitely generated, then Ω/Γ is of finite type (Sullivan), i.e. It has finitely many connected components, each being a compact Riemann surface minus finitely many points.

Seade Discrete groups acting on complex projective spaces

Theorem

1

Its complement Ω is the set where the action is discontinuous.

2

Ω actually is the maximal set where the action is properly

• discontinuous. It is also the equicontinuity set of the action.

3

If Ω has finitely many connected components, then it has at most two (it can be empty).

4

The quotient Ω/Γ is a Riemann surface with a projective

• rbifold structure.

5

Every Riemann surface can be realized in this way (Koebe’s retrosection theorem)

6

If Γ is finitely generated, then Ω/Γ is of finite type (Sullivan), i.e. It has finitely many connected components, each being a compact Riemann surface minus finitely many points.

Seade Discrete groups acting on complex projective spaces

Theorem

1

Its complement Ω is the set where the action is discontinuous.

2

Ω actually is the maximal set where the action is properly

• discontinuous. It is also the equicontinuity set of the action.

3

If Ω has finitely many connected components, then it has at most two (it can be empty).

4

The quotient Ω/Γ is a Riemann surface with a projective

• rbifold structure.

5

Every Riemann surface can be realized in this way (Koebe’s retrosection theorem)

6

If Γ is finitely generated, then Ω/Γ is of finite type (Sullivan), i.e. It has finitely many connected components, each being a compact Riemann surface minus finitely many points.

Seade Discrete groups acting on complex projective spaces

Theorem

1

Its complement Ω is the set where the action is discontinuous.

2

Ω actually is the maximal set where the action is properly

• discontinuous. It is also the equicontinuity set of the action.

3

If Ω has finitely many connected components, then it has at most two (it can be empty).

4

The quotient Ω/Γ is a Riemann surface with a projective

• rbifold structure.

5

Every Riemann surface can be realized in this way (Koebe’s retrosection theorem)

6

If Γ is finitely generated, then Ω/Γ is of finite type (Sullivan), i.e. It has finitely many connected components, each being a compact Riemann surface minus finitely many points.

Seade Discrete groups acting on complex projective spaces

Particularly interesting: Fuchsian groups: Discrete subgroups

• f PSL(2, R). Remarkable relations with singularities:

Let Γ be Fuchsian. Consider action on upper half plane H ⊂ C and extend it to tangent space TH ∼ = H × C via derivative: g · (z, w) → (g(z), g′(z) · w) The quotient TH/Γ is a complex surface; it contains the Riemann surface H/Γ as the image of zero-section. The singularities of TH/Γ are all contained in H/Γ, and this surface can be blown down analytically to get an analytic surface VΓ with a normal singularity: Dolgachev’s quotient conical canonical singularities Example: Arnold’s 14 exceptional unimodal singularities arise in this way. This includes the surface {z2

1 + z3 2 + z7 3 = 0}.

Seade Discrete groups acting on complex projective spaces

Remark PSL(2, R) ∼ = PU(1, 1), the group of transformations of CP1 that preserves ball B2 = {(z1 : z2) ∈ CP1 |z1|2 < |z2|2} which serves as model for complex hyperbolic 1-space H1

• C. Will be

relevant later. As noted before: CP1 ∼ = S2 = ∂H3

R and

PSL(2, C) ∼ = Conf+(S2) ∼ = Iso+(H3

R) .

Hence we have at least two essentially different ways of generalizing these to higher dimensions (one in red, other in lavender!!):

Seade Discrete groups acting on complex projective spaces

Higher dimensions can mean: Conf+(Sn) ∼ = Iso+(Hn+1). And, It also can mean PSL(n, C) ∼ = AutHolCPn+1. Entirely different (related) theories in higher dimensions!! In fact the former (conformal) is contained in the latter.

• Vast literature in conformal case. Outstanding results by

many people, e.g. Ahlfors, Bers, Maskit, Beardon, Thurston, Margulis, Sullivan, MacMullen, Kapovich, etc.

• The PSL(n, C) side still is in its childhood.

Seade Discrete groups acting on complex projective spaces

Remark Classically the term Kleinian referred to discrete subgroups of PSL(2, C) acting on Riemann sphere S2 with non-empty region

• f discontinuity Ω.

In this setting one has a splitting S2 = Λ ∪ Ω in two non-empty invariant sets: The limit set Λ is where the dynamics concentrates. The complement Ω is what we look for in complex geometry These observations gave rise to the following definition:

Seade Discrete groups acting on complex projective spaces

Definition (Verjovsky-Seade, ≈ 2000) A complex Kleinian group is a discrete subgroup of PSL(n, C) acting on CPn with a non-empty open invariant set where action is properly discontinuous THREE MAIN MOTIVATIONS:

• Complex Geometry: U open G-invariant subset of CPn

where action is properly discontinuous ⇒ quotient U/G is a variety (an orbifold) with a rich geometric structure (Projective structure)

• Holomorphic dynamics: Recall:

a) Richness of dynamics of classical kleinian groups; b) Sullivan’s dictionary: Iteration theory vs Kleinian groups

• Complex hyperbolic geometry

Seade Discrete groups acting on complex projective spaces

• Theory of Complex Kleinian groups is rich and fascinating; it

includes: Theory of discrete subgroups of PO(n, 1), i.e. Real hyperbolic (geometry) groups. Discrete subgroups of PU(n, 1), i.e. Complex hyperbolic (geometry) groups. Play the role of Fuchsian groups in higher dimensions. Discrete subgroups of Aff(n, C), and a lot more (Schottky groups, twistorial groups, etc.)

• Aim: make a systematic study of complex Kleinian groups.

Large and difficult program Plenty interesting questions!!

Seade Discrete groups acting on complex projective spaces

From now on I’ll speak about work with/by Juan-Pablo Navarrete and Angel Cano, two former students of mine, and a collaborator Waldemar Barrera.

• Some basic questions are

i) What is, or what ought to be, the limit set of a discrete subgroup G ⊂ PSL(n + 1, C)? ii) What kind of compact manifolds and orbifolds arise as quotients Ω/Γ with Γ ⊂ PSL(n + 1, C) acting properly discontinuously on Ω, an open subset in CPn? iii) Study the geometric and dynamical properties of these sets. iv) Are there interesting relations with singularities? In the rest of the talk I’ll give some answers to these questions.

Seade Discrete groups acting on complex projective spaces

Section 2: On the limit set First recall: Definition Let G be a discrete subgroup of Conf+(Sn) ∼ = Iso+(Hn+1) its limit set Λ is the subset of Sn ∼ = ∂Hn+1 of accumulation points of

• rbits of points in Hn+1.

For hyperbolic groups, this concept has all the nice properties

• ne may wish to have:
• It is a minimal closed invariant set.
• Its complement is the largest set where action is properly

discontinuous, and this coincides with equicontinuity region.

• It is the closure of set of fixed points of loxodromic elements.
• etc.

Seade Discrete groups acting on complex projective spaces

We could take this as definition for discrete groups of PSL(n, C): not a good concept in this setting if n > 2. In particular the action on the complement may not be properly discontinuous. ∃ examples where all above properties differ. For instance consider group in CP2 generated by: ˜ γ =   α1 α2 α3   , where |α1| < |α2| < |α3|.

• Each αi determines a fixed point ei in CP2.
• Orbits of points in CP2 accumulate at set {e1, e2, e3}. The

point {e1} is repelling point, {e2} a saddle, {e3} is an attractor.

• Yet, action on complement of these three points is neither

properly discontinuous nor equicontinuous.

• Equicontinuity region is not a maximal set where action is

properly discontinuous ....

Seade Discrete groups acting on complex projective spaces

Recall: Given a discrete group Γ ⊂ PSL(n + 1, C) : Definition (Equicontinuity region) Eq(Γ) = points x ∈ CPn for which there is an open neighbourhood U such that Γ |U is a normal family, i.e., every sequence of distinct elements has a subsequence which converges uniformly on compact sets in U. Definition Given a discrete group Γ ⊂ PSL(n + 1, C) define its Kulkarni limit set ΛKul to be the union Λ0 ∪ Λ2 where Λ0 is the closure of set of accumulation points of all orbits and Λ2 is the set of accumulation points of orbits of compact sets in CPn \ ΛKul. For n = 1 this coincides with usual definition. For n = 2 this is the good concept of limit set (which is false for n > 2). One has:

Seade Discrete groups acting on complex projective spaces

Recall: Given a discrete group Γ ⊂ PSL(n + 1, C) : Definition (Equicontinuity region) Eq(Γ) = points x ∈ CPn for which there is an open neighbourhood U such that Γ |U is a normal family, i.e., every sequence of distinct elements has a subsequence which converges uniformly on compact sets in U. Definition Given a discrete group Γ ⊂ PSL(n + 1, C) define its Kulkarni limit set ΛKul to be the union Λ0 ∪ Λ2 where Λ0 is the closure of set of accumulation points of all orbits and Λ2 is the set of accumulation points of orbits of compact sets in CPn \ ΛKul. For n = 1 this coincides with usual definition. For n = 2 this is the good concept of limit set (which is false for n > 2). One has:

Seade Discrete groups acting on complex projective spaces

Theorem Let Γ ⊂ PSL(3, C) be discrete and infinite. Then its limit set ΛKul may consist of :

• one projective line L ∼

= CP1.

• one line and one point (and in this case the group is virtually

cyclic).

• two lines, three lines or infinitely many lines.

Theorem If ΛKul has more than four lines in general position, then it has infinitely many lines in general position and in this case:

• Its complement is a complete Kobayashi hyperbolic space; it

coincides with Eq and is the largest set where the action is properly discontinuous.

• ΛKul is the closure of the repulsive invariant lines of

loxodromic elements.

• Its projective dual

ΛKul ⊂ Cˇ P2 is the set of accumulation points of the dual action, and it is a minimal set.

Seade Discrete groups acting on complex projective spaces

Theorem Let Γ ⊂ PSL(3, C) be discrete and infinite. Then its limit set ΛKul may consist of :

• one projective line L ∼

= CP1.

• one line and one point (and in this case the group is virtually

cyclic).

• two lines, three lines or infinitely many lines.

Theorem If ΛKul has more than four lines in general position, then it has infinitely many lines in general position and in this case:

• Its complement is a complete Kobayashi hyperbolic space; it

coincides with Eq and is the largest set where the action is properly discontinuous.

• ΛKul is the closure of the repulsive invariant lines of

loxodromic elements.

• Its projective dual

ΛKul ⊂ Cˇ P2 is the set of accumulation points of the dual action, and it is a minimal set.

Seade Discrete groups acting on complex projective spaces

Example (Cano-Parker-Seade; to appear in Asian J. Maths.) Let Γ be cofinite subgroup of Iso+H2

R.

First think of it embedded in PSL(3, C) as subgroup of PU(1, 1) ⊂ PU(2, 1) ⊂ PSL(3, C). Its action on CP2 preserves a line L ∼ = CP1 and a 4-ball H2

C.

Intersection L ∩ H2

C is H1 C and its boundary ∂H1 C is a circle. Let

P be the polar point of L. Then ΛKul is a cone; consists of all lines in CP2 joining P with a point in ∂H1

C.

Its complement ΩKul := CP2 \ ΛKul is a holomorphic C-bundle

• ver L \ partialH1

C, so it consists of two open 4-balls, glued

together along ΛKul. Now think of Γ embedded in PSL(3, C) as a subgroup of PSO(2, 1)PU(2, 1) ⊂ PSL(3, C).

Seade Discrete groups acting on complex projective spaces

Example Its action on CP2 now preserves a Lagrangian plane P ∼ = RP2 and the 4-ball H2

• C. P is a union

P = H2

R ∩ ∂H2 R ∩ M ,

where the latter is a M¨

• bis band.

The limit set ΛKul is now a union of infinitely many projective lines in general position: all lines tangent to ∂H2

C at a point in

∂H2

R.

The region of discontinuity ΩKul is now a union of three 4-balls.

Seade Discrete groups acting on complex projective spaces

Section 3 Orbit spaces Now look at ΩKul/Γ. In dimensions n > 2 the zoo is huge; so is for n = 2 for non-compact quotients. Focus on compact case. Definition Say that Γ ⊂ PSL(3, C) is quasi-cocompact if there exists a non-empty open invariant set U in CP2 where action is properly discontinuous and quotient U/Γ is compact. Following carries to the level of group actions a previous theorem by Kobayashi-Ochiai (strengthened by Klingberg) Theorem (Cano-Seade, Geo. Dedicata 2014) If Γ is quasi-cocompact, then it is either complex hyperbolic or virtually affine. We then gave: complete classification of the open sets that can appear as invariant sets in CP2 with compact quotient for some discrete group acting properly discontinuously. And we proved:

Seade Discrete groups acting on complex projective spaces

Section 3 Orbit spaces Now look at ΩKul/Γ. In dimensions n > 2 the zoo is huge; so is for n = 2 for non-compact quotients. Focus on compact case. Definition Say that Γ ⊂ PSL(3, C) is quasi-cocompact if there exists a non-empty open invariant set U in CP2 where action is properly discontinuous and quotient U/Γ is compact. Following carries to the level of group actions a previous theorem by Kobayashi-Ochiai (strengthened by Klingberg) Theorem (Cano-Seade, Geo. Dedicata 2014) If Γ is quasi-cocompact, then it is either complex hyperbolic or virtually affine. We then gave: complete classification of the open sets that can appear as invariant sets in CP2 with compact quotient for some discrete group acting properly discontinuously. And we proved:

Seade Discrete groups acting on complex projective spaces

Theorem Let Γ ⊂ PSL(3, C) be an infinite quasi-cocompact group. Then ∃ a finite covering SΓ − → ΩKul/Γ, ramified at the points in ΩKul/Γ with non-trivial isotropy, and SΓ is of the following type:

1

If ΩKul = C2, then SΓ is a complex torus S1 × S1 × S1 × S1

• r a primary Kodaira surface.

2

If ΩKul = C2 \ {0}, then SΓ is a complex torus or a primary Hopf surface.

3

If ΩKul = C∗ × C, then SΓ is to a complex torus.

4

If ΩKul = C∗ × C∗, then SΓ is to a complex torus.

5

If ΩKul = C∗ × (H+ ∪ H−), then SΓ is either M or M ⊔ M where M is a Inoue surface.

6

If ΩKul = D × C∗ ⇒ SΓ has countably many components and each compact component is an elliptic surface with an affine structure.

7

If ΩKul = H2

C, then SΓ is complex hyperbolic.

Seade Discrete groups acting on complex projective spaces

Theorem Let Γ ⊂ PSL(3, C) be an infinite quasi-cocompact group. Then ∃ a finite covering SΓ − → ΩKul/Γ, ramified at the points in ΩKul/Γ with non-trivial isotropy, and SΓ is of the following type:

1

If ΩKul = C2, then SΓ is a complex torus S1 × S1 × S1 × S1

• r a primary Kodaira surface.

2

If ΩKul = C2 \ {0}, then SΓ is a complex torus or a primary Hopf surface.

3

If ΩKul = C∗ × C, then SΓ is to a complex torus.

4

If ΩKul = C∗ × C∗, then SΓ is to a complex torus.

5

If ΩKul = C∗ × (H+ ∪ H−), then SΓ is either M or M ⊔ M where M is a Inoue surface.

6

If ΩKul = D × C∗ ⇒ SΓ has countably many components and each compact component is an elliptic surface with an affine structure.

7

If ΩKul = H2

C, then SΓ is complex hyperbolic.

Seade Discrete groups acting on complex projective spaces

Theorem Let Γ ⊂ PSL(3, C) be an infinite quasi-cocompact group. Then ∃ a finite covering SΓ − → ΩKul/Γ, ramified at the points in ΩKul/Γ with non-trivial isotropy, and SΓ is of the following type:

1

If ΩKul = C2, then SΓ is a complex torus S1 × S1 × S1 × S1

• r a primary Kodaira surface.

2

If ΩKul = C2 \ {0}, then SΓ is a complex torus or a primary Hopf surface.

3

If ΩKul = C∗ × C, then SΓ is to a complex torus.

4

If ΩKul = C∗ × C∗, then SΓ is to a complex torus.

5

If ΩKul = C∗ × (H+ ∪ H−), then SΓ is either M or M ⊔ M where M is a Inoue surface.

6

If ΩKul = D × C∗ ⇒ SΓ has countably many components and each compact component is an elliptic surface with an affine structure.

7

If ΩKul = H2

C, then SΓ is complex hyperbolic.

Seade Discrete groups acting on complex projective spaces

Theorem Let Γ ⊂ PSL(3, C) be an infinite quasi-cocompact group. Then ∃ a finite covering SΓ − → ΩKul/Γ, ramified at the points in ΩKul/Γ with non-trivial isotropy, and SΓ is of the following type:

1

If ΩKul = C2, then SΓ is a complex torus S1 × S1 × S1 × S1

• r a primary Kodaira surface.

2

If ΩKul = C2 \ {0}, then SΓ is a complex torus or a primary Hopf surface.

3

If ΩKul = C∗ × C, then SΓ is to a complex torus.

4

If ΩKul = C∗ × C∗, then SΓ is to a complex torus.

5

If ΩKul = C∗ × (H+ ∪ H−), then SΓ is either M or M ⊔ M where M is a Inoue surface.

6

If ΩKul = D × C∗ ⇒ SΓ has countably many components and each compact component is an elliptic surface with an affine structure.

7

If ΩKul = H2

C, then SΓ is complex hyperbolic.

Seade Discrete groups acting on complex projective spaces

Theorem Let Γ ⊂ PSL(3, C) be an infinite quasi-cocompact group. Then ∃ a finite covering SΓ − → ΩKul/Γ, ramified at the points in ΩKul/Γ with non-trivial isotropy, and SΓ is of the following type:

1

If ΩKul = C2, then SΓ is a complex torus S1 × S1 × S1 × S1

• r a primary Kodaira surface.

2

If ΩKul = C2 \ {0}, then SΓ is a complex torus or a primary Hopf surface.

3

If ΩKul = C∗ × C, then SΓ is to a complex torus.

4

If ΩKul = C∗ × C∗, then SΓ is to a complex torus.

5

If ΩKul = C∗ × (H+ ∪ H−), then SΓ is either M or M ⊔ M where M is a Inoue surface.

6

If ΩKul = D × C∗ ⇒ SΓ has countably many components and each compact component is an elliptic surface with an affine structure.

7

If ΩKul = H2

C, then SΓ is complex hyperbolic.

Seade Discrete groups acting on complex projective spaces

Theorem Let Γ ⊂ PSL(3, C) be an infinite quasi-cocompact group. Then ∃ a finite covering SΓ − → ΩKul/Γ, ramified at the points in ΩKul/Γ with non-trivial isotropy, and SΓ is of the following type:

1

If ΩKul = C2, then SΓ is a complex torus S1 × S1 × S1 × S1

• r a primary Kodaira surface.

2

If ΩKul = C2 \ {0}, then SΓ is a complex torus or a primary Hopf surface.

3

If ΩKul = C∗ × C, then SΓ is to a complex torus.

4

If ΩKul = C∗ × C∗, then SΓ is to a complex torus.

5

If ΩKul = C∗ × (H+ ∪ H−), then SΓ is either M or M ⊔ M where M is a Inoue surface.

6

If ΩKul = D × C∗ ⇒ SΓ has countably many components and each compact component is an elliptic surface with an affine structure.

7

If ΩKul = H2

C, then SΓ is complex hyperbolic.

Seade Discrete groups acting on complex projective spaces

Theorem Let Γ ⊂ PSL(3, C) be an infinite quasi-cocompact group. Then ∃ a finite covering SΓ − → ΩKul/Γ, ramified at the points in ΩKul/Γ with non-trivial isotropy, and SΓ is of the following type:

1

If ΩKul = C2, then SΓ is a complex torus S1 × S1 × S1 × S1

• r a primary Kodaira surface.

2

If ΩKul = C2 \ {0}, then SΓ is a complex torus or a primary Hopf surface.

3

If ΩKul = C∗ × C, then SΓ is to a complex torus.

4

If ΩKul = C∗ × C∗, then SΓ is to a complex torus.

5

If ΩKul = C∗ × (H+ ∪ H−), then SΓ is either M or M ⊔ M where M is a Inoue surface.

6

If ΩKul = D × C∗ ⇒ SΓ has countably many components and each compact component is an elliptic surface with an affine structure.

7

If ΩKul = H2

C, then SΓ is complex hyperbolic.

Seade Discrete groups acting on complex projective spaces

Theorem Let Γ ⊂ PSL(3, C) be an infinite quasi-cocompact group. Then ∃ a finite covering SΓ − → ΩKul/Γ, ramified at the points in ΩKul/Γ with non-trivial isotropy, and SΓ is of the following type:

1

If ΩKul = C2, then SΓ is a complex torus S1 × S1 × S1 × S1

• r a primary Kodaira surface.

2

If ΩKul = C2 \ {0}, then SΓ is a complex torus or a primary Hopf surface.

3

If ΩKul = C∗ × C, then SΓ is to a complex torus.

4

If ΩKul = C∗ × C∗, then SΓ is to a complex torus.

5

If ΩKul = C∗ × (H+ ∪ H−), then SΓ is either M or M ⊔ M where M is a Inoue surface.

6

If ΩKul = D × C∗ ⇒ SΓ has countably many components and each compact component is an elliptic surface with an affine structure.

7

If ΩKul = H2

C, then SΓ is complex hyperbolic.

Seade Discrete groups acting on complex projective spaces

Section 4: What about singularities? I have almost nothing to say to this respect, just few remarks. First thing is that quotients U/Γ as above are in general

• rbifolds, with a special type of singularities, and these orbifolds

can have be very interesting geometry and topology. For instance see my book with Cano and Navarrete: “Complex Kleinian groups” Progress in Mathematics vol. 303, Birkhauser (2012). What else? Recall Dolgachev’s quotient singularities:

Seade Discrete groups acting on complex projective spaces

• Start with cocompact Γ ⊂ PU(1, 1); it acts on CP1 preserving

a 2-ball D which serves as model for H1

C.

• Extend its action on H1

C to tangent space TH1 C via the

• derivative. Quotient TH1

C/Γ is an orbifold with singularities in

H1

C/Γ.

• Riemann surface H1

C/Γ can be blown down analytically. Get

complex analytic surface VΓ with a normal singularity, which is Gorenstein and quasi-homogeneous.

• One gets very interesting class of singularities in this way.

Is there a similar construction of ”Dolgachev singularities” in higher dimensions? Here is an idea that can be interesting to think about.

Seade Discrete groups acting on complex projective spaces

Start with cocompact Γ ⊂ PU(n, 1); it acts on CPn preserving a 2n-ball D which serves as model for Hn

C.

Quotient Hn

C/ is a complex projective orbifold.

The action of Γ on Hn

C extends to an action on its tangent space

THn

C ∼

= Hn

C × Cn via the derivative.

This induces an action on the anti-canonical bundle K∗(Hn

C) := ∧nTHn C.

The quotient K∗(Hn

C)/Γ contains the orbifold Hn C/Γ as the image

• f the zero-section.

Seade Discrete groups acting on complex projective spaces

Several natural questions arise. For instance: Question

1

Can we blow down Hn

C/Γ analytically in K∗Hn C to get a

normal singularity (VΓ, P)?

2

What can we say about these singularities? I believe these will be Gorenstein but not ICIS, and there is not much we know about the topology of higher dimensional singularities which are not ICIS. When the group Γ is torsion free, the link of the corresponding singularity is a complex hyperbolic manifold. This is a higher dimensional analogous of taking the tangent bundle of a Riemann surface of genus > 1 and blown down its zero section. and .....

Seade Discrete groups acting on complex projective spaces

Several natural questions arise. For instance: Question

1

Can we blow down Hn

C/Γ analytically in K∗Hn C to get a

normal singularity (VΓ, P)?

2

What can we say about these singularities? I believe these will be Gorenstein but not ICIS, and there is not much we know about the topology of higher dimensional singularities which are not ICIS. When the group Γ is torsion free, the link of the corresponding singularity is a complex hyperbolic manifold. This is a higher dimensional analogous of taking the tangent bundle of a Riemann surface of genus > 1 and blown down its zero section. and .....

Seade Discrete groups acting on complex projective spaces

Several natural questions arise. For instance: Question

1

Can we blow down Hn

C/Γ analytically in K∗Hn C to get a

normal singularity (VΓ, P)?

2

What can we say about these singularities? I believe these will be Gorenstein but not ICIS, and there is not much we know about the topology of higher dimensional singularities which are not ICIS. When the group Γ is torsion free, the link of the corresponding singularity is a complex hyperbolic manifold. This is a higher dimensional analogous of taking the tangent bundle of a Riemann surface of genus > 1 and blown down its zero section. and .....

Seade Discrete groups acting on complex projective spaces

Thank you very much indeed for your attention

Seade Discrete groups acting on complex projective spaces