Introduction Cohomology Algebra of X Resonance Varieties Formality of X Problems J.I. Cogolludo-Agustín The Cohomology Algebra of a Plane Curve

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Recent Progress on Topology of Plane Curves: A Quick Trip

Part II:

The Cohomology Algebra of a Plane Curve

José Ignacio COGOLLUDO-AGUSTÍN

Departamento de Matemáticas Universidad de Zaragoza

Branched Coverings in Tokyo - March 7-10, 2011

J.I. Cogolludo-Agustín The Cohomology Algebra of a Plane Curve

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Contents

1 Introduction

Settings and Results

The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

2 Cohomology Algebra ofX Weak Combinatorics

3 Resonance Varieties

4 Formality ofX

Max-Noether Fundamental Theorem Revisited

5 Problems

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Contents

1 Introduction

Settings and Results

The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

2 Cohomology Algebra ofX Weak Combinatorics

3 Resonance Varieties

4 Formality ofX

Max-Noether Fundamental Theorem Revisited

5 Problems

J.I. Cogolludo-Agustín The Cohomology Algebra of a Plane Curve

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Contents

1 Introduction

Settings and Results

The Line Arrangement Case

Log-resolution Logarithmic Forms Poincaré Residue Operators

2 Cohomology Algebra ofX Weak Combinatorics

3 Resonance Varieties

4 Formality ofX

Max-Noether Fundamental Theorem Revisited

5 Problems

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Contents

1 Introduction

Settings and Results

The Line Arrangement Case Log-resolution Logarithmic Forms

Poincaré Residue Operators

2 Cohomology Algebra ofX Weak Combinatorics

3 Resonance Varieties

4 Formality ofX

Max-Noether Fundamental Theorem Revisited

5 Problems

J.I. Cogolludo-Agustín The Cohomology Algebra of a Plane Curve

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Contents

1 Introduction

Settings and Results

The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

2 Cohomology Algebra ofX Weak Combinatorics

3 Resonance Varieties

4 Formality ofX

Max-Noether Fundamental Theorem Revisited

5 Problems

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Contents

1 Introduction

Settings and Results

The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

2 Cohomology Algebra ofX

Weak Combinatorics

3 Resonance Varieties

4 Formality ofX

Max-Noether Fundamental Theorem Revisited

5 Problems

J.I. Cogolludo-Agustín The Cohomology Algebra of a Plane Curve

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Contents

1 Introduction

Settings and Results

The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

2 Cohomology Algebra ofX Weak Combinatorics

3 Resonance Varieties

4 Formality ofX

Max-Noether Fundamental Theorem Revisited

5 Problems

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Contents

1 Introduction

Settings and Results

The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

2 Cohomology Algebra ofX Weak Combinatorics

3 Resonance Varieties

4 Formality ofX

Max-Noether Fundamental Theorem Revisited

5 Problems

J.I. Cogolludo-Agustín The Cohomology Algebra of a Plane Curve

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Contents

1 Introduction

Settings and Results

The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

2 Cohomology Algebra ofX Weak Combinatorics

3 Resonance Varieties

4 Formality ofX

Max-Noether Fundamental Theorem Revisited

5 Problems

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Contents

1 Introduction

Settings and Results

The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

2 Cohomology Algebra ofX Weak Combinatorics

3 Resonance Varieties

4 Formality ofX

Max-Noether Fundamental Theorem Revisited

5 Problems

J.I. Cogolludo-Agustín The Cohomology Algebra of a Plane Curve

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Contents

1 Introduction

Settings and Results

The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

2 Cohomology Algebra ofX Weak Combinatorics

3 Resonance Varieties

4 Formality ofX

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

Settings

C=C₀∪ C₁∪...∪ C_r ⊂P²

X :=P²\ C

H^∗(X) =H^∗(X;C)

Give a constructive description ofH^∗(X)by generators and relations, as well as describe the product.

WeakCombinatorial Invariants ofC.

Existence of anOrlik-Solomon-likealgebra. Prove Formality ofX.

J.I. Cogolludo-Agustín The Cohomology Algebra of a Plane Curve

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

Settings

C=C₀∪ C₁∪...∪ C_r ⊂P² X :=P²\ C

H^∗(X) =H^∗(X;C)

Give a constructive description ofH^∗(X)by generators and relations, as well as describe the product.

WeakCombinatorial Invariants ofC.

Existence of anOrlik-Solomon-likealgebra. Prove Formality ofX.

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

Settings

C=C₀∪ C₁∪...∪ C_r ⊂P² X :=P²\ C

H^∗(X) =H^∗(X;C)

Give a constructive description ofH^∗(X)by generators and relations, as well as describe the product.

WeakCombinatorial Invariants ofC.

Existence of anOrlik-Solomon-likealgebra. Prove Formality ofX.

J.I. Cogolludo-Agustín The Cohomology Algebra of a Plane Curve

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

Settings

C=C₀∪ C₁∪...∪ C_r ⊂P² X :=P²\ C

H^∗(X) =H^∗(X;C)

Give a constructive description ofH^∗(X)by generators and relations, as well as describe the product.

WeakCombinatorial Invariants ofC.

Existence of anOrlik-Solomon-likealgebra. Prove Formality ofX.

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

Settings

C=C₀∪ C₁∪...∪ C_r ⊂P² X :=P²\ C

H^∗(X) =H^∗(X;C)

Give a constructive description ofH^∗(X)by generators and relations, as well as describe the product.

WeakCombinatorial Invariants ofC.

Existence of anOrlik-Solomon-likealgebra. Prove Formality ofX.

J.I. Cogolludo-Agustín The Cohomology Algebra of a Plane Curve

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

Settings

C=C₀∪ C₁∪...∪ C_r ⊂P² X :=P²\ C

H^∗(X) =H^∗(X;C)

Give a constructive description ofH^∗(X)by generators and relations, as well as describe the product.

Prove Formality ofX.

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

Settings

C=C₀∪ C₁∪...∪ C_r ⊂P² X :=P²\ C

H^∗(X) =H^∗(X;C)

Give a constructive description ofH^∗(X)by generators and relations, as well as describe the product.

WeakCombinatorial Invariants ofC.

Existence of anOrlik-Solomon-likealgebra.

Prove Formality ofX.

J.I. Cogolludo-Agustín The Cohomology Algebra of a Plane Curve

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

The Line Arrangement Case

C=`<sub>0</sub>∪`₁∪...∪`<sub>r</sub> ⊂P<sup>2</sup>, where`_i is a line.

ConsiderX =C²\(`1∪...∪`r).

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

The Line Arrangement Case

C=`<sub>0</sub>∪`₁∪...∪`<sub>r</sub> ⊂P<sup>2</sup>, where`_i is a line.

ConsiderX =C²\(`1∪...∪`r).

J.I. Cogolludo-Agustín The Cohomology Algebra of a Plane Curve

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

The Line Arrangement Case

Theorem (Arnold, Brieskorn, Orlik-Solomon) The ring H^∗(X)is generated by H¹(X), that is, by:

σ_i := d`_i

`i

. A complete set of relations is given by:

σ_i∧σ_j+σ_j∧σ_k +σ_k∧σ_i =0,

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

The Line Arrangement Case

Note that whenever`<sub>i</sub>∩`_j∩`<sub>k</sub> 6=∅ ⇒`_k =a`<sub>i</sub>+b`_j

`i`j`k·σj∧σk =a`_i(d`j∧d`i)

`<sub>i</sub>`_j`<sub>k</sub>·σ<sub>k</sub>∧σ<sub>i</sub> =b`_j(d`<sub>j</sub>∧d`_i) Therefore,

`i`j`k ·(σj ∧σk +σk∧σi) =`k(d`j∧d`i) =−`<sub>i</sub>`j`k ·σi∧σj

J.I. Cogolludo-Agustín The Cohomology Algebra of a Plane Curve

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

The Line Arrangement Case

Note that whenever`<sub>i</sub>∩`_j∩`<sub>k</sub> 6=∅ ⇒`_k =a`<sub>i</sub>+b`_j

`i`j`k·σj∧σk =a`_i(d`j∧d`i)

`<sub>i</sub>`_j`<sub>k</sub>·σ<sub>k</sub>∧σ<sub>i</sub> =b`_j(d`<sub>j</sub>∧d`_i) Therefore,

`i`j`k ·(σj ∧σk +σk∧σi) =`k(d`j∧d`i) =−`<sub>i</sub>`j`k ·σi∧σj

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

The Line Arrangement Case

Note that whenever`<sub>i</sub>∩`_j∩`<sub>k</sub> 6=∅ ⇒`_k =a`<sub>i</sub>+b`_j

`i`j`k·σj∧σk =a`_i(d`j∧d`i)

`<sub>i</sub>`_j`<sub>k</sub>·σ<sub>k</sub>∧σ<sub>i</sub> =b`_j(d`<sub>j</sub>∧d`_i)

Therefore,

`i`j`k ·(σj ∧σk +σk∧σi) =`k(d`j∧d`i) =−`<sub>i</sub>`j`k ·σi∧σj

J.I. Cogolludo-Agustín The Cohomology Algebra of a Plane Curve

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

The Line Arrangement Case

Note that whenever`<sub>i</sub>∩`_j∩`<sub>k</sub> 6=∅ ⇒`_k =a`<sub>i</sub>+b`_j

`i`j`k·σj∧σk =a`_i(d`j∧d`i)

`<sub>i</sub>`_j`<sub>k</sub>·σ<sub>k</sub>∧σ<sub>i</sub> =b`_j(d`<sub>j</sub>∧d`_i) Therefore,

`i`j`k ·(σj ∧σk +σk∧σi) =`k(d`j∧d`i) =−`<sub>i</sub>`j`k ·σi∧σj

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

The General Case

However,

C=`<sub>0</sub>∪q, where`₀={z =0}andq:={z²=xy}.

H¹(X) =C

H²(X) =H₁(C) =C Therefore

∧²H¹(X)6=H²(X). In fact,

H²(X) =h ω

`₀q i_C, whereω :=zdx∧dy+xdy∧dz+ydz∧dx.

J.I. Cogolludo-Agustín The Cohomology Algebra of a Plane Curve

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

The General Case

However,

C=`<sub>0</sub>∪q, where`₀={z =0}andq:={z²=xy}.

H¹(X) =C

H²(X) =H₁(C) =C Therefore

∧²H¹(X)6=H²(X). In fact,

H²(X) =h ω

`₀q i_C, whereω :=zdx∧dy+xdy∧dz+ydz∧dx.

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

The General Case

However,

C=`<sub>0</sub>∪q, where`₀={z =0}andq:={z²=xy}.

H¹(X) =C

H²(X) =H₁(C) =C

Therefore

∧²H¹(X)6=H²(X). In fact,

H²(X) =h ω

`₀q i_C, whereω :=zdx∧dy+xdy∧dz+ydz∧dx.

J.I. Cogolludo-Agustín The Cohomology Algebra of a Plane Curve

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

The General Case

However,

C=`<sub>0</sub>∪q, where`₀={z =0}andq:={z²=xy}.

H¹(X) =C

H²(X) =H₁(C) =C Therefore

∧²H¹(X)6=H²(X).

In fact, H²(X) =h ω

`₀q i_C, whereω :=zdx∧dy+xdy∧dz+ydz∧dx.

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

The General Case

However,

C=`<sub>0</sub>∪q, where`₀={z =0}andq:={z²=xy}.

H¹(X) =C

H²(X) =H₁(C) =C Therefore

∧²H¹(X)6=H²(X).

In fact, H²(X) =h ω

`₀q i_C, whereω :=zdx∧dy+xdy∧dz+ydz∧dx.

J.I. Cogolludo-Agustín The Cohomology Algebra of a Plane Curve

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

Definitions

π : S → P²

∪ ∪

C¯ → C

(1)

Definition

The sheafπ∗E_S^∗(logC)¯ is the sheaf oflog-resolution logarithmic formsofCw.r.t.π.

Remark

The sheafπ∗E_S^∗(logC)¯ is independent of the resolution. Denote it byE^∗

P²(logC). E^∗

P²(logC)inherits a weight filtrationW∗.

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

Definitions

π : S → P²

∪ ∪

C¯ → C

(1)

Definition

The sheafπ∗E_S^∗(logC)¯ is the sheaf oflog-resolution logarithmic formsofCw.r.t.π.

Remark

The sheafπ∗E_S^∗(logC)¯ is independent of the resolution. Denote it byE^∗

P²(logC). E^∗

P²(logC)inherits a weight filtrationW∗.

J.I. Cogolludo-Agustín The Cohomology Algebra of a Plane Curve

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

Definitions

π : S → P²

∪ ∪

C¯ → C

(1)

Definition

The sheafπ∗E_S^∗(logC)¯ is the sheaf oflog-resolution logarithmic formsofCw.r.t.π.

Remark

The sheafπ∗E_S^∗(logC)¯ is independent of the resolution. Denote it byE^∗

P²(logC). E^∗

P²(logC)inherits a weight filtrationW∗.

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

Definitions

π : S → P²

∪ ∪

C¯ → C

(1)

Definition

The sheafπ∗E_S^∗(logC)¯ is the sheaf oflog-resolution logarithmic formsofCw.r.t.π.

Remark

The sheafπ∗E_S^∗(logC)¯ is independent of the resolution.

Denote it byE^∗

P²(logC). E^∗

P²(logC)inherits a weight filtrationW∗.

J.I. Cogolludo-Agustín The Cohomology Algebra of a Plane Curve

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

Definitions

π : S → P²

∪ ∪

C¯ → C

(1)

Definition

The sheafπ∗E_S^∗(logC)¯ is the sheaf oflog-resolution logarithmic formsofCw.r.t.π.

Remark

E^∗

P²(logC)inherits a weight filtrationW∗.

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

Definitions

π : S → P²

∪ ∪

C¯ → C

(1)

Definition

The sheafπ∗E_S^∗(logC)¯ is the sheaf oflog-resolution logarithmic formsofCw.r.t.π.

Remark

The sheafπ∗E_S^∗(logC)¯ is independent of the resolution.

Denote it byE^∗

P²(logC).

E^∗

P²(logC)inherits a weight filtrationW∗.

J.I. Cogolludo-Agustín The Cohomology Algebra of a Plane Curve

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

Hⁱ(P²;W_iE^∗

P²(logC)) k

Hⁱ(S;W_iE_S^∗(logC))¯

→ Hⁱ(S;W_i/W_i−1) ' H⁰( ¯C^[i]) k

Hⁱ(X)

Such a residue map will be denoted by Res^[i]. In more generality:

Hⁱ(P²;W_kE^∗

P²(logC)) ^Res

[i,k]

−→ H^i−k( ¯C^[k]).

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

Hⁱ(P²;W_iE^∗

P²(logC)) k

Hⁱ(S;W_iE_S^∗(logC))¯

→ Hⁱ(S;W_i/W_i−1) ' H⁰( ¯C^[i]) k

Hⁱ(X)

Such a residue map will be denoted by Res^[i]. In more generality:

Hⁱ(P²;W_kE^∗

P²(logC)) ^Res

[i,k]

−→ H^i−k( ¯C^[k]).

J.I. Cogolludo-Agustín The Cohomology Algebra of a Plane Curve

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

Hⁱ(P²;W_iE^∗

P²(logC)) k

Hⁱ(S;W_iE_S^∗(logC))¯

→ Hⁱ(S;W_i/W_i−1) ' H⁰( ¯C^[i])

k Hⁱ(X)

Such a residue map will be denoted by Res^[i]. In more generality:

Hⁱ(P²;W_kE^∗

P²(logC)) ^Res

[i,k]

−→ H^i−k( ¯C^[k]).

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

Hⁱ(P²;W_iE^∗

P²(logC)) k

Hⁱ(S;W_iE_S^∗(logC))¯ → Hⁱ(S;W_i/W_i−1)

' H⁰( ¯C^[i])

k Hⁱ(X)

Such a residue map will be denoted by Res^[i]. In more generality:

Hⁱ(P²;W_kE^∗

P²(logC)) ^Res

[i,k]

−→ H^i−k( ¯C^[k]).

J.I. Cogolludo-Agustín The Cohomology Algebra of a Plane Curve

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

Hⁱ(P²;W_iE^∗

P²(logC)) k

Hⁱ(S;W_iE_S^∗(logC))¯ → Hⁱ(S;W_i/W_i−1) ' H⁰( ¯C^[i]) k

Hⁱ(X)

Such a residue map will be denoted by Res^[i]. In more generality:

Hⁱ(P²;W_kE^∗

P²(logC)) ^Res

[i,k]

−→ H^i−k( ¯C^[k]).

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

Hⁱ(P²;W_iE^∗

P²(logC)) k

Hⁱ(S;W_iE_S^∗(logC))¯ → Hⁱ(S;W_i/W_i−1) ' H⁰( ¯C^[i]) k

Hⁱ(X)

Such a residue map will be denoted by Res^[i].

In more generality: Hⁱ(P²;W_kE^∗

P²(logC)) ^Res

[i,k]

−→ H^i−k( ¯C^[k]).

J.I. Cogolludo-Agustín The Cohomology Algebra of a Plane Curve

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

Hⁱ(P²;W_iE^∗

P²(logC)) k

Hⁱ(S;W_iE_S^∗(logC))¯ → Hⁱ(S;W_i/W_i−1) ' H⁰( ¯C^[i]) k

Hⁱ(X)

Such a residue map will be denoted by Res^[i]. In more generality:

Hⁱ(P²;W_kE^∗₂(logC)) ^Res

[i,k]

−→ H^i−k( ¯C^[k]).

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

Theorem (-,D.Matei)

Under the above conditions:

1 Res^[1,1]is injective.

2 Ifψ∈ E²(P²)(logC)is such thatRes^[2,2]ψ=0and Res^[2,1]ψ=0, thenψ=0.

J.I. Cogolludo-Agustín The Cohomology Algebra of a Plane Curve

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

Theorem (-,D.Matei)

Under the above conditions:

1 Res^[1,1]is injective.

2 Ifψ∈ E²(P²)(logC)is such thatRes^[2,2]ψ=0and Res^[2,1]ψ=0, thenψ=0.

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

Theorem (-,D.Matei)

Under the above conditions:

1 Res^[1,1]is injective.

2 Ifψ∈ E²(P²)(logC)is such thatRes^[2,2]ψ=0and Res^[2,1]ψ=0, thenψ=0.

J.I. Cogolludo-Agustín The Cohomology Algebra of a Plane Curve

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

Example

Considerf =y²−x³,C={f =0}, and the 2-form ^dx∧dy_f .

dx∧dy f

x=u1 y=u1v1

←− du₁∧dv₁ u₁(v₁²−u₁)

u1=u2v2 v1=v2

←− du₂∧dv₂ u₂v₂(v₂−u₂)

u2=u3v3 v2=v3

←− du₃∧dv₃ u₃v₃²(1−u₃) which isnot logarithmic.

However, ifψ=ϕ^dx∧dy_f , then ϕ∈(x,y)⇒ψ∈ E₀²(logC). Moreover, ifϕ∈(y)⇒

Res^[2,2]ψ

P =0 at allP ∈C¯^[1] infinitely near 0.

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

Example

Considerf =y²−x³,C={f =0}, and the 2-form ^dx∧dy_f . dx∧dy

f

x=u1 y=u1v1

←− du₁∧dv₁ u₁(v₁²−u₁)

u1=u2v2 v1=v2

←− du₂∧dv₂ u₂v₂(v₂−u₂)

u2=u3v3 v2=v3

←− du₃∧dv₃ u₃v₃²(1−u₃) which isnot logarithmic.

However, ifψ=ϕ^dx∧dy_f , then ϕ∈(x,y)⇒ψ∈ E₀²(logC). Moreover, ifϕ∈(y)⇒

Res^[2,2]ψ

P =0 at allP ∈C¯^[1] infinitely near 0.

J.I. Cogolludo-Agustín The Cohomology Algebra of a Plane Curve

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

Example

Considerf =y²−x³,C={f =0}, and the 2-form ^dx∧dy_f . dx∧dy

f

x=u1 y=u1v1

←− du₁∧dv₁ u₁(v₁²−u₁)

u1=u2v2 v1=v2

←− du₂∧dv₂ u₂v₂(v₂−u₂)

u2=u3v3 v2=v3

←− du₃∧dv₃ u₃v₃²(1−u₃) which isnot logarithmic.

However, ifψ=ϕ^dx∧dy_f , then ϕ∈(x,y)⇒ψ∈ E₀²(logC). Moreover, ifϕ∈(y)⇒

Res^[2,2]ψ

P =0 at allP ∈C¯^[1] infinitely near 0.

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

Example

Considerf =y²−x³,C={f =0}, and the 2-form ^dx∧dy_f . dx∧dy

f

x=u1 y=u1v1

←− du₁∧dv₁ u₁(v₁²−u₁)

u1=u2v2 v1=v2

←− du₂∧dv₂ u₂v₂(v₂−u₂)

u2=u3v3 v2=v3

←− du₃∧dv₃ u₃v₃²(1−u₃)

which isnot logarithmic. However, ifψ=ϕ^dx∧dy_f , then

ϕ∈(x,y)⇒ψ∈ E₀²(logC). Moreover, ifϕ∈(y)⇒

Res^[2,2]ψ

P =0 at allP ∈C¯^[1] infinitely near 0.

J.I. Cogolludo-Agustín The Cohomology Algebra of a Plane Curve

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

Example

Considerf =y²−x³,C={f =0}, and the 2-form ^dx∧dy_f . dx∧dy

f

x=u1 y=u1v1

←− du₁∧dv₁ u₁(v₁²−u₁)

u1=u2v2 v1=v2

←− du₂∧dv₂ u₂v₂(v₂−u₂)

u2=u3v3 v2=v3

←− du₃∧dv₃ u₃v₃²(1−u₃) which isnot logarithmic.

However, ifψ=ϕ^dx∧dy_f , then ϕ∈(x,y)⇒ψ∈ E₀²(logC). Moreover, ifϕ∈(y)⇒

Res^[2,2]ψ

P =0 at allP ∈C¯^[1] infinitely near 0.

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

Example

Considerf =y²−x³,C={f =0}, and the 2-form ^dx∧dy_f . dx∧dy

f

x=u1 y=u1v1

←− du₁∧dv₁ u₁(v₁²−u₁)

u1=u2v2 v1=v2

←− du₂∧dv₂ u₂v₂(v₂−u₂)

u2=u3v3 v2=v3

←− du₃∧dv₃ u₃v₃²(1−u₃) which isnot logarithmic.

However, ifψ=ϕ^dx∧dy_f , then

ϕ∈(x,y)⇒ψ∈ E₀²(logC). Moreover, ifϕ∈(y)⇒

Res^[2,2]ψ

P =0 at allP ∈C¯^[1] infinitely near 0.

J.I. Cogolludo-Agustín The Cohomology Algebra of a Plane Curve

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

Example

Considerf =y²−x³,C={f =0}, and the 2-form ^dx∧dy_f . dx∧dy

f

x=u1 y=u1v1

←− du₁∧dv₁ u₁(v₁²−u₁)

u1=u2v2 v1=v2

←− du₂∧dv₂ u₂v₂(v₂−u₂)

u2=u3v3 v2=v3

←− du₃∧dv₃ u₃v₃²(1−u₃) which isnot logarithmic.

However, ifψ=ϕ^dx∧dy_f , then

2

Moreover, ifϕ∈(y)⇒

Res^[2,2]ψ

P =0 at allP ∈C¯^[1] infinitely near 0.

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Settings and Results The Line Arrangement Case Log-resolution Logarithmic Forms Poincaré Residue Operators

Example

Considerf =y²−x³,C={f =0}, and the 2-form ^dx∧dy_f . dx∧dy

f

x=u1 y=u1v1

←− du₁∧dv₁ u₁(v₁²−u₁)

u1=u2v2 v1=v2

←− du₂∧dv₂ u₂v₂(v₂−u₂)

u2=u3v3 v2=v3

←− du₃∧dv₃ u₃v₃²(1−u₃) which isnot logarithmic.

However, ifψ=ϕ^dx∧dy_f , then ϕ∈(x,y)⇒ψ∈ E₀²(logC).

Moreover, ifϕ∈(y)⇒

Res^[2,2]ψ

P =0 at allP ∈C¯^[1]

infinitely near 0.

J.I. Cogolludo-Agustín The Cohomology Algebra of a Plane Curve

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Weak Combinatorics

Theorem

The following is a presentation of H^∗(X):

Generators in degree 1:σ_i, i =1, ...,r ,

Generators in degree 2:

ψ_P^δ1,δ2, P ∈ C_i∩ C_j, δ₁∈∆_P(C_i), δ₂∈∆_P(C_j) ψ∞^i,ki, i=1, ...,r,k_i =1, ...,d_i−1 η^i,si,η¯^i,si, i=1, ...,r,s_i =1, ...,g_i. Relations:

ψ^δ_P^1,δ2 =−ψ_P^δ2,δ1 ψ_P^δ1,δ2 +ψ^δ_P^2,δ3+ψ^δ_P^3,δ1 =0

for any P ∈ C_i∩ C_j∩ C_k andδ₁∈∆_P(C_i),δ₂∈∆_P(C_j), δ₃∈∆_P(C_k).

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Weak Combinatorics

Theorem

The following is a presentation of H^∗(X):

Generators in degree 1: σ_i, i =1, ...,r , Generators in degree 2:

ψ_P^δ1,δ2, P ∈ C_i∩ C_j, δ₁∈∆_P(C_i), δ₂∈∆_P(C_j) ψ∞^i,ki, i=1, ...,r,k_i =1, ...,d_i−1 η^i,si,η¯^i,si, i=1, ...,r,s_i =1, ...,g_i.

Relations:

ψ^δ_P^1,δ2 =−ψ_P^δ2,δ1 ψ_P^δ1,δ2 +ψ^δ_P^2,δ3+ψ^δ_P^3,δ1 =0

for any P ∈ C_i∩ C_j∩ C_k andδ₁∈∆_P(C_i),δ₂∈∆_P(C_j), δ₃∈∆_P(C_k).

J.I. Cogolludo-Agustín The Cohomology Algebra of a Plane Curve

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Weak Combinatorics

Theorem

The following is a presentation of H^∗(X):

Generators in degree 1: σ_i, i =1, ...,r , Generators in degree 2:

ψ_P^δ1,δ2, P ∈ C_i∩ C_j, δ₁∈∆_P(C_i), δ₂∈∆_P(C_j) ψ∞^i,ki, i=1, ...,r,k_i =1, ...,d_i−1 η^i,si,η¯^i,si, i=1, ...,r,s_i =1, ...,g_i. Relations:

ψ^δ_P^1,δ2 =−ψ_P^δ2,δ1

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Weak Combinatorics

Theorem

The following is a presentation of H^∗(X):

σ_i,ψ^δ_P^1,δ2, ψ∞^i,ki, ψ^s_iⁱ,ψ¯^s_iⁱ,

ψ^δ_P^1,δ2 =−ψ_P^δ2,δ1 ψ_P^δ1,δ2 +ψ^δ_P^2,δ3+ψ^δ_P^3,δ1 =0

Product:

σ_i∧σ_j = X

P∈Ci∩Cj

µ_P(δ₁, δ₂)ψ_P^δ1,δ2+d_i

dj−1

X

kj=1

ψ∞^j,kj −d_j

di−1

X

ki=1

ψ_∞^i,ki.

J.I. Cogolludo-Agustín The Cohomology Algebra of a Plane Curve

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Weak Combinatorics

Remark

Note that from the given presentation one can deduce that H^∗(X)only depends on the following invariants ofC:

({1, ...,r},S =SingC,{∆_P}_P∈S,{φ_P}_P∈S,{µ_P}_P∈S) such an ordered set of invariants ofCwill be referred to as the Weak Combinatorics ofC.

Hence Theorem

The cohomology algebra of X only depends on its weak combinatorics.

Introduction Cohomology Algebra ofX Resonance Varieties Formality ofX Problems

Weak Combinatorics

Remark

Note that from the given presentation one can deduce that H^∗(X)only depends on the following invariants ofC:

({1, ...,r},S =SingC,{∆_P}_P∈S,{φ_P}_P∈S,{µ_P}_P∈S) such an ordered set of invariants ofCwill be referred to as the Weak Combinatorics ofC.

Hence Theorem

The cohomology algebra of X only depends on its weak combinatorics.

J.I. Cogolludo-Agustín The Cohomology Algebra of a Plane Curve