[PPT] - GIT characterizations of Harder-Narasimhan filtrations Alfonso PowerPoint Presentation

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GIT characterizations of Harder-Narasimhan filtrations

Alfonso Zamora

Instituto Superior Técnico Lisboa, Portugal

AMS-EMS-SPM Meeting Porto, June 2015

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Introduction Correspondence for sheaves Correspondence for other problems Further directions

Index

1

Introduction

2

Correspondence for sheaves Harder-Narasimhan filtration Gieseker construction of a moduli space Kempf theorem

3

Correspondence for other problems Holomorphic pairs Higgs sheaves Rank 2 tensors Quiver representations (G, h)-constellations

4

Further directions

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Introduction Correspondence for sheaves Correspondence for other problems Further directions

Classification problems in geometry

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Introduction Correspondence for sheaves Correspondence for other problems Further directions

Classification problems in geometry Want to classify geometric objects (A) up to equivalence relation (∼)

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Introduction Correspondence for sheaves Correspondence for other problems Further directions

Classification problems in geometry Want to classify geometric objects (A) up to equivalence relation (∼) Moduli space, a space whose points correspond to equivalence classes

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Introduction Correspondence for sheaves Correspondence for other problems Further directions

Classification problems in geometry Want to classify geometric objects (A) up to equivalence relation (∼) Moduli space, a space whose points correspond to equivalence classes Moduli functor is a contravariant functor F : Schk → Sets, F(S) set of equivalence classes of families parametrized by S. Triple (A, ∼, F) is a moduli problem.

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Introduction Correspondence for sheaves Correspondence for other problems Further directions

Classification problems in geometry Want to classify geometric objects (A) up to equivalence relation (∼) Moduli space, a space whose points correspond to equivalence classes Moduli functor is a contravariant functor F : Schk → Sets, F(S) set of equivalence classes of families parametrized by S. Triple (A, ∼, F) is a moduli problem. Solution of moduli problem (A, ∼, F) is existence of a moduli space M representing or corepresenting the functor F

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Introduction Correspondence for sheaves Correspondence for other problems Further directions

GIT constructions of moduli spaces

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Introduction Correspondence for sheaves Correspondence for other problems Further directions

GIT constructions of moduli spaces Different Aut(A), A ∈ A ⇒ Notion of stability

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Introduction Correspondence for sheaves Correspondence for other problems Further directions

GIT constructions of moduli spaces Different Aut(A), A ∈ A ⇒ Notion of stability Semistable objects A ∈ A (with no structure)

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Introduction Correspondence for sheaves Correspondence for other problems Further directions

GIT constructions of moduli spaces Different Aut(A), A ∈ A ⇒ Notion of stability Semistable objects A ∈ A (with no structure) Add extra data (A, d) ∈ Q (with structure)

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Introduction Correspondence for sheaves Correspondence for other problems Further directions

GIT constructions of moduli spaces Different Aut(A), A ∈ A ⇒ Notion of stability Semistable objects A ∈ A (with no structure) Add extra data (A, d) ∈ Q (with structure) Data added turns out to be group action: (A, d1) G ∼ (A, d2)

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Introduction Correspondence for sheaves Correspondence for other problems Further directions

GIT constructions of moduli spaces Different Aut(A), A ∈ A ⇒ Notion of stability Semistable objects A ∈ A (with no structure) Add extra data (A, d) ∈ Q (with structure) Data added turns out to be group action: (A, d1) G ∼ (A, d2) Want to take the quotient Q/G. Have to remove some

rbits (GIT-unstables) and identify others (S-equivalence)

⇒ Geometric Invariant Theory (GIT) [Mumford]

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Introduction Correspondence for sheaves Correspondence for other problems Further directions

GIT constructions of moduli spaces Different Aut(A), A ∈ A ⇒ Notion of stability Semistable objects A ∈ A (with no structure) Add extra data (A, d) ∈ Q (with structure) Data added turns out to be group action: (A, d1) G ∼ (A, d2) Want to take the quotient Q/G. Have to remove some

rbits (GIT-unstables) and identify others (S-equivalence)

⇒ Geometric Invariant Theory (GIT) [Mumford] Show stability ⇔ GIT stability to get the moduli space of S-equivalence classes as the GIT quotient Qss/ /G.

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Introduction Correspondence for sheaves Correspondence for other problems Further directions

GIT constructions of moduli spaces Different Aut(A), A ∈ A ⇒ Notion of stability Semistable objects A ∈ A (with no structure) Add extra data (A, d) ∈ Q (with structure) Data added turns out to be group action: (A, d1) G ∼ (A, d2) Want to take the quotient Q/G. Have to remove some

rbits (GIT-unstables) and identify others (S-equivalence)

⇒ Geometric Invariant Theory (GIT) [Mumford] Show stability ⇔ GIT stability to get the moduli space of S-equivalence classes as the GIT quotient Qss/ /G. For unstable objects ⇒ Harder-Narasimhan filtration

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Introduction Correspondence for sheaves Correspondence for other problems Further directions

GIT constructions of moduli spaces Different Aut(A), A ∈ A ⇒ Notion of stability Semistable objects A ∈ A (with no structure) Add extra data (A, d) ∈ Q (with structure) Data added turns out to be group action: (A, d1) G ∼ (A, d2) Want to take the quotient Q/G. Have to remove some

rbits (GIT-unstables) and identify others (S-equivalence)

⇒ Geometric Invariant Theory (GIT) [Mumford] Show stability ⇔ GIT stability to get the moduli space of S-equivalence classes as the GIT quotient Qss/ /G. For unstable objects ⇒ Harder-Narasimhan filtration For GIT-unstable orbits ⇒ Maximal 1-parameter subgroups

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Introduction Correspondence for sheaves Correspondence for other problems Further directions

GIT constructions of moduli spaces Different Aut(A), A ∈ A ⇒ Notion of stability Semistable objects A ∈ A (with no structure) Add extra data (A, d) ∈ Q (with structure) Data added turns out to be group action: (A, d1) G ∼ (A, d2) Want to take the quotient Q/G. Have to remove some

rbits (GIT-unstables) and identify others (S-equivalence)

⇒ Geometric Invariant Theory (GIT) [Mumford] Show stability ⇔ GIT stability to get the moduli space of S-equivalence classes as the GIT quotient Qss/ /G. For unstable objects ⇒ Harder-Narasimhan filtration For GIT-unstable orbits ⇒ Maximal 1-parameter subgroups Correspondence between 2 notions of maximal unstability which appear on GIT constructions of moduli spaces

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Introduction Correspondence for sheaves Correspondence for other problems Further directions

Stability for sheaves

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Introduction Correspondence for sheaves Correspondence for other problems Further directions

Stability for sheaves (X, OX(1)) polarized smooth complex projective variety of dim n. Fix P, degree n polynomial and consider E → X torsion free coherent sheaves with PE = P

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Introduction Correspondence for sheaves Correspondence for other problems Further directions

Stability for sheaves (X, OX(1)) polarized smooth complex projective variety of dim n. Fix P, degree n polynomial and consider E → X torsion free coherent sheaves with PE = P Observation: If X is Riemann surface, holomorphic vector bundles E of rank r and degree d

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Introduction Correspondence for sheaves Correspondence for other problems Further directions

Stability for sheaves (X, OX(1)) polarized smooth complex projective variety of dim n. Fix P, degree n polynomial and consider E → X torsion free coherent sheaves with PE = P Observation: If X is Riemann surface, holomorphic vector bundles E of rank r and degree d E(m) = E ⊗ OX(m) , PE(m) = χ(E(m)) = n

i=0(−1)ihi(E(m))

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Introduction Correspondence for sheaves Correspondence for other problems Further directions

Stability for sheaves (X, OX(1)) polarized smooth complex projective variety of dim n. Fix P, degree n polynomial and consider E → X torsion free coherent sheaves with PE = P Observation: If X is Riemann surface, holomorphic vector bundles E of rank r and degree d E(m) = E ⊗ OX(m) , PE(m) = χ(E(m)) = n

i=0(−1)ihi(E(m))

Definition [Gieseker] E is semistable if ∀F E, PF

rk F ≤ PE rk E . If not E is unstable

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Introduction Correspondence for sheaves Correspondence for other problems Further directions

Stability for sheaves (X, OX(1)) polarized smooth complex projective variety of dim n. Fix P, degree n polynomial and consider E → X torsion free coherent sheaves with PE = P Observation: If X is Riemann surface, holomorphic vector bundles E of rank r and degree d E(m) = E ⊗ OX(m) , PE(m) = χ(E(m)) = n

i=0(−1)ihi(E(m))

Definition [Gieseker] E is semistable if ∀F E, PF

rk F ≤ PE rk E . If not E is unstable

If dimCX = 1, PE(m) = rm + d + r(1 − g), µ(E) = deg E

rk E = d r

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Introduction Correspondence for sheaves Correspondence for other problems Further directions

Stability for sheaves (X, OX(1)) polarized smooth complex projective variety of dim n. Fix P, degree n polynomial and consider E → X torsion free coherent sheaves with PE = P Observation: If X is Riemann surface, holomorphic vector bundles E of rank r and degree d E(m) = E ⊗ OX(m) , PE(m) = χ(E(m)) = n

i=0(−1)ihi(E(m))

Definition [Gieseker] E is semistable if ∀F E, PF

rk F ≤ PE rk E . If not E is unstable

If dimCX = 1, PE(m) = rm + d + r(1 − g), µ(E) = deg E

rk E = d r

E is semistable if ∀F E, µ(F) ≤ µ(E). If not E is unstable

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Harder-Narasimhan filtration

Harder-Narasimhan filtration If E is unstable

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Harder-Narasimhan filtration

Harder-Narasimhan filtration If E is unstable ∃! Harder-Narasimhan filtration 0 ⊂ E1 ⊂ E2 ⊂ . . . ⊂ Et ⊂ E verifying

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Harder-Narasimhan filtration

Harder-Narasimhan filtration If E is unstable ∃! Harder-Narasimhan filtration 0 ⊂ E1 ⊂ E2 ⊂ . . . ⊂ Et ⊂ E verifying

1

P1

E

rk E1 > P2

E

rk E2 > . . . > Pt+1

E

rk Et+1

2

Ei := Ei/Ei−1 are semistable

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Harder-Narasimhan filtration

Harder-Narasimhan filtration If E is unstable ∃! Harder-Narasimhan filtration 0 ⊂ E1 ⊂ E2 ⊂ . . . ⊂ Et ⊂ E verifying

1

P1

E

rk E1 > P2

E

rk E2 > . . . > Pt+1

E

rk Et+1

2

Ei := Ei/Ei−1 are semistable Idea to construct (for v. bundles over Riemann surfaces) E unstable ⇒ ∃ E′ of rank r ′ < r and degree d′, 0 E′ E such that µ(E′) = d′

r′ > µ(E) = deg E rk E

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Harder-Narasimhan filtration

Harder-Narasimhan filtration If E is unstable ∃! Harder-Narasimhan filtration 0 ⊂ E1 ⊂ E2 ⊂ . . . ⊂ Et ⊂ E verifying

1

P1

E

rk E1 > P2

E

rk E2 > . . . > Pt+1

E

rk Et+1

2

Ei := Ei/Ei−1 are semistable Idea to construct (for v. bundles over Riemann surfaces) E unstable ⇒ ∃ E′ of rank r ′ < r and degree d′, 0 E′ E such that µ(E′) = d′

r′ > µ(E) = deg E rk E

Choose unique E1 with maximal slope µ(E1) > µ(E) and maximal rank among those of same slope Call it the maximal destabilizing subbundle of E

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Harder-Narasimhan filtration

Construction of the Harder-Narasimhan filtration Let F = E/E1

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Harder-Narasimhan filtration

Construction of the Harder-Narasimhan filtration Let F = E/E1 If F semistable ⇒ Harder-Narasimhan filtration is 0 ⊂ E1 ⊂ E

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Harder-Narasimhan filtration

Construction of the Harder-Narasimhan filtration Let F = E/E1 If F semistable ⇒ Harder-Narasimhan filtration is 0 ⊂ E1 ⊂ E If not, ∃ 0 F1 F as before and so on. Finally...

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Harder-Narasimhan filtration

Construction of the Harder-Narasimhan filtration Let F = E/E1 If F semistable ⇒ Harder-Narasimhan filtration is 0 ⊂ E1 ⊂ E If not, ∃ 0 F1 F as before and so on. Finally... Harder-Narasimhan filtration 0 ⊂ E1 ⊂ E2 ⊂ · · · ⊂ Et ⊂ Et+1 = E verifying:

1

µ(E1) > µ(E2) > µ(E3) > ... > µ(Et) > µ(Et+1)

2

Ei := Ei/Ei−1 is semistable

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Gieseker construction of a moduli space

Gieseker construction of a moduli space [Maruyama] ∃m ∈ Z such that all E semistable are m-regular (i.e. semistable sheaves are a bounded family)

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Gieseker construction of a moduli space

Gieseker construction of a moduli space [Maruyama] ∃m ∈ Z such that all E semistable are m-regular (i.e. semistable sheaves are a bounded family) Choose an isomorphism (extra data) V

g

≃ H0(E(m))

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Gieseker construction of a moduli space

Gieseker construction of a moduli space [Maruyama] ∃m ∈ Z such that all E semistable are m-regular (i.e. semistable sheaves are a bounded family) Choose an isomorphism (extra data) V

g

≃ H0(E(m)) {V ⊗ OX(−m) ։ E} = QuotP SL(V)

X

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Gieseker construction of a moduli space

Gieseker construction of a moduli space [Maruyama] ∃m ∈ Z such that all E semistable are m-regular (i.e. semistable sheaves are a bounded family) Choose an isomorphism (extra data) V

g

≃ H0(E(m)) {V ⊗ OX(−m) ։ E} = QuotP SL(V)

X

q = (E, g) ∈ QuotP

X ι

→ (PN)SL(V)

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Gieseker construction of a moduli space

Gieseker construction of a moduli space [Maruyama] ∃m ∈ Z such that all E semistable are m-regular (i.e. semistable sheaves are a bounded family) Choose an isomorphism (extra data) V

g

≃ H0(E(m)) {V ⊗ OX(−m) ։ E} = QuotP SL(V)

X

q = (E, g) ∈ QuotP

X ι

→ (PN)SL(V) Let Z = im ι,

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Gieseker construction of a moduli space

Gieseker construction of a moduli space [Maruyama] ∃m ∈ Z such that all E semistable are m-regular (i.e. semistable sheaves are a bounded family) Choose an isomorphism (extra data) V

g

≃ H0(E(m)) {V ⊗ OX(−m) ։ E} = QuotP SL(V)

X

q = (E, g) ∈ QuotP

X ι

→ (PN)SL(V) Let Z = im ι, prove that GIT-stability = stability and

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Gieseker construction of a moduli space

Gieseker construction of a moduli space [Maruyama] ∃m ∈ Z such that all E semistable are m-regular (i.e. semistable sheaves are a bounded family) Choose an isomorphism (extra data) V

g

≃ H0(E(m)) {V ⊗ OX(−m) ։ E} = QuotP SL(V)

X

q = (E, g) ∈ QuotP

X ι

→ (PN)SL(V) Let Z = im ι, prove that GIT-stability = stability and ⇒ Z ss/ /SL(V)

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Gieseker construction of a moduli space

Gieseker construction of a moduli space [Maruyama] ∃m ∈ Z such that all E semistable are m-regular (i.e. semistable sheaves are a bounded family) Choose an isomorphism (extra data) V

g

≃ H0(E(m)) {V ⊗ OX(−m) ։ E} = QuotP SL(V)

X

q = (E, g) ∈ QuotP

X ι

→ (PN)SL(V) Let Z = im ι, prove that GIT-stability = stability and ⇒ Z ss/ /SL(V) = MP

X, moduli space

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Gieseker construction of a moduli space

Characterization of GIT stability Calculating invariants is very complicated

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Gieseker construction of a moduli space

Characterization of GIT stability Calculating invariants is very complicated Use 1-parameter subgroups (measure if 0 ∈ SL(V) · q)

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Gieseker construction of a moduli space

Characterization of GIT stability Calculating invariants is very complicated Use 1-parameter subgroups (measure if 0 ∈ SL(V) · q) Γ : C∗ → SL(V) t → diag

tΓ1, . . . , tΓ1, tΓ2, . . . , tΓ2, . . . , tΓt+1, . . . , tΓt+1

(convention: Γ1 < Γ2 < . . . < Γt+1)

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Gieseker construction of a moduli space

Characterization of GIT stability Calculating invariants is very complicated Use 1-parameter subgroups (measure if 0 ∈ SL(V) · q) Γ : C∗ → SL(V) t → diag

tΓ1, . . . , tΓ1, tΓ2, . . . , tΓ2, . . . , tΓt+1, . . . , tΓt+1

(convention: Γ1 < Γ2 < . . . < Γt+1) Γ defines a weighted filtration (V•, n•) 0 ⊂

n1

V1⊂

n2

V2⊂ . . . ⊂

nt

Vt⊂ Vt+1 = V , ni = Γi+1−Γi

dim V

> 0

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Gieseker construction of a moduli space

Characterization of GIT stability Calculating invariants is very complicated Use 1-parameter subgroups (measure if 0 ∈ SL(V) · q) Γ : C∗ → SL(V) t → diag

tΓ1, . . . , tΓ1, tΓ2, . . . , tΓ2, . . . , tΓt+1, . . . , tΓt+1

(convention: Γ1 < Γ2 < . . . < Γt+1) Γ defines a weighted filtration (V•, n•) 0 ⊂

n1

V1⊂

n2

V2⊂ . . . ⊂

nt

Vt⊂ Vt+1 = V , ni = Γi+1−Γi

dim V

> 0 Proposition [Hilbert-Mumford criterion] q is GIT-unstable if ∃ Γ s.t. limt→0 Γ(t) · q = 0 ⇔

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Gieseker construction of a moduli space

Characterization of GIT stability Calculating invariants is very complicated Use 1-parameter subgroups (measure if 0 ∈ SL(V) · q) Γ : C∗ → SL(V) t → diag

tΓ1, . . . , tΓ1, tΓ2, . . . , tΓ2, . . . , tΓt+1, . . . , tΓt+1

(convention: Γ1 < Γ2 < . . . < Γt+1) Γ defines a weighted filtration (V•, n•) 0 ⊂

n1

V1⊂

n2

V2⊂ . . . ⊂

nt

Vt⊂ Vt+1 = V , ni = Γi+1−Γi

dim V

> 0 Proposition [Hilbert-Mumford criterion] q is GIT-unstable if ∃ Γ s.t. limt→0 Γ(t) · q = 0 ⇔ if ∃(V•, n•)

ni(r dim Vi − ri dim V) > 0 (measure of unstability)

where ri = rk Ei, Vi ⊗ OX ։ Ei(m)

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Kempf theorem

Kempf theorem

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Kempf theorem

Kempf theorem Take E unstable ⇒ q GIT-unstable

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Kempf theorem

Kempf theorem Take E unstable ⇒ q GIT-unstable There exists a way of maximally destabilize q in the sense

f GIT?

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Kempf theorem

Kempf theorem Take E unstable ⇒ q GIT-unstable There exists a way of maximally destabilize q in the sense

f GIT?

Kempf ⇒

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Kempf theorem

Kempf theorem Take E unstable ⇒ q GIT-unstable There exists a way of maximally destabilize q in the sense

f GIT?

Kempf ⇒ If q is GIT-unstable ∃! (V•, n•) (up to scalar) (or ∃! Γ up to scalar and conjugation by an element of the parabolic subgroup) such that

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Kempf theorem

Kempf theorem Take E unstable ⇒ q GIT-unstable There exists a way of maximally destabilize q in the sense

f GIT?

Kempf ⇒ If q is GIT-unstable ∃! (V•, n•) (up to scalar) (or ∃! Γ up to scalar and conjugation by an element of the parabolic subgroup) such that Kempf function: ni(r dim Vi − ri dim V) dim V iΓ2

i

achieves its maximum

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Kempf theorem

Kempf theorem Take E unstable ⇒ q GIT-unstable There exists a way of maximally destabilize q in the sense

f GIT?

Kempf ⇒ If q is GIT-unstable ∃! (V•, n•) (up to scalar) (or ∃! Γ up to scalar and conjugation by an element of the parabolic subgroup) such that Kempf function: ni(r dim Vi − ri dim V) dim V iΓ2

i

achieves its maximum Denominator is Γ, norm in the space of 1-parameter subgroups measuring its velocity

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Kempf theorem 11/ 17

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Kempf theorem

0 ⊂ V1 ⊂ V2 ⊂ · · · ⊂ Vt ⊂ Vt+1 = V ≃ H0(E(m)), ni > 0, m-Kempf filtration of V maximizes the Kempf function

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Kempf theorem

0 ⊂ V1 ⊂ V2 ⊂ · · · ⊂ Vt ⊂ Vt+1 = V ≃ H0(E(m)), ni > 0, m-Kempf filtration of V maximizes the Kempf function

evaluating

= ⇒ 0 ⊂ Em

1 ⊂ Em 2 ⊂ · · · ⊂ Em t

⊂ Em

t+1 = E, ni > 0,

m-Kempf filtration of E

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Kempf theorem

0 ⊂ V1 ⊂ V2 ⊂ · · · ⊂ Vt ⊂ Vt+1 = V ≃ H0(E(m)), ni > 0, m-Kempf filtration of V maximizes the Kempf function

evaluating

= ⇒ 0 ⊂ Em

1 ⊂ Em 2 ⊂ · · · ⊂ Em t

⊂ Em

t+1 = E, ni > 0,

m-Kempf filtration of E Natural question Is Kempf filtration = Harder-Narasimhan filtration?

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Kempf theorem

0 ⊂ V1 ⊂ V2 ⊂ · · · ⊂ Vt ⊂ Vt+1 = V ≃ H0(E(m)), ni > 0, m-Kempf filtration of V maximizes the Kempf function

evaluating

= ⇒ 0 ⊂ Em

1 ⊂ Em 2 ⊂ · · · ⊂ Em t

⊂ Em

t+1 = E, ni > 0,

m-Kempf filtration of E Natural question Is Kempf filtration = Harder-Narasimhan filtration? Theorem [Gómez-Sols-Z.]

1

∃m′ ≫ 0 such that for every m ≥ m′ all m-Kempf filtrations are equal independently of m

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Kempf theorem

0 ⊂ V1 ⊂ V2 ⊂ · · · ⊂ Vt ⊂ Vt+1 = V ≃ H0(E(m)), ni > 0, m-Kempf filtration of V maximizes the Kempf function

evaluating

= ⇒ 0 ⊂ Em

1 ⊂ Em 2 ⊂ · · · ⊂ Em t

⊂ Em

t+1 = E, ni > 0,

m-Kempf filtration of E Natural question Is Kempf filtration = Harder-Narasimhan filtration? Theorem [Gómez-Sols-Z.]

1

∃m′ ≫ 0 such that for every m ≥ m′ all m-Kempf filtrations are equal independently of m

2

Kempf filtration = Harder-Narasimhan filtration

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Holomorphic pairs

A holomorphic pair is (E, ϕ : E → OX)

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Holomorphic pairs

A holomorphic pair is (E, ϕ : E → OX) Definition [Bradlow - García-Prada, Huybrechts - Lehn] A pair is δ-semistable if for every subpair (F, ϕ|F) (E, ϕ) PF − δε(F) rk F ≤ PE − δ rk E , ε(F) = 1 si ϕ|F = 0 and 0 otherwise

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Holomorphic pairs

A holomorphic pair is (E, ϕ : E → OX) Definition [Bradlow - García-Prada, Huybrechts - Lehn] A pair is δ-semistable if for every subpair (F, ϕ|F) (E, ϕ) PF − δε(F) rk F ≤ PE − δ rk E , ε(F) = 1 si ϕ|F = 0 and 0 otherwise Take (E, ϕ) unstable ⇒ q GIT unstable, for each m ∈ Z, ⇒ ∃! m-Kempf filtration (V•, n•) GIT maximally destabilizing

evaluating

= ⇒ (E•, ϕ|E•) ⊂ (E, ϕ), m-Kempf filtration of E

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Holomorphic pairs

A holomorphic pair is (E, ϕ : E → OX) Definition [Bradlow - García-Prada, Huybrechts - Lehn] A pair is δ-semistable if for every subpair (F, ϕ|F) (E, ϕ) PF − δε(F) rk F ≤ PE − δ rk E , ε(F) = 1 si ϕ|F = 0 and 0 otherwise Take (E, ϕ) unstable ⇒ q GIT unstable, for each m ∈ Z, ⇒ ∃! m-Kempf filtration (V•, n•) GIT maximally destabilizing

evaluating

= ⇒ (E•, ϕ|E•) ⊂ (E, ϕ), m-Kempf filtration of E Theorem [Gómez-Sols-Z.] The m-Kempf filtrations do not depend on m for m ≥ m′ and coincide with the Harder-Narasimhan filtration 0 ⊂ (E1, ϕ|E1) ⊂ (E2, ϕ|E2) ⊂ . . . ⊂ (Et, ϕ|Et) ⊂ (E, ϕ) of a δ-unstable holomorphic pair (E, ϕ).

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Higgs sheaves 13/ 17

SLIDE 66

Introduction Correspondence for sheaves Correspondence for other problems Further directions Higgs sheaves

A Higgs sheaf is a pair (E, φ : E → E ⊗ Ω1

X), φ ∧ φ = 0

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Introduction Correspondence for sheaves Correspondence for other problems Further directions Higgs sheaves

A Higgs sheaf is a pair (E, φ : E → E ⊗ Ω1

X), φ ∧ φ = 0

Definition [Hitchin, Simpson] A Higgs sheaf is semistable if ∀ F E φ-invariant, it is

PF rk F ≤ PE rk E

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SLIDE 68

Introduction Correspondence for sheaves Correspondence for other problems Further directions Higgs sheaves

A Higgs sheaf is a pair (E, φ : E → E ⊗ Ω1

X), φ ∧ φ = 0

Definition [Hitchin, Simpson] A Higgs sheaf is semistable if ∀ F E φ-invariant, it is

PF rk F ≤ PE rk E

Higgs sheaves (E, φ)

Simpson

⇐ ⇒ E → Z = P(T ∗X ⊕ O) E of pure dimension = dim X, E = π∗E, π : T ∗X → X.

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SLIDE 69

Introduction Correspondence for sheaves Correspondence for other problems Further directions Higgs sheaves

A Higgs sheaf is a pair (E, φ : E → E ⊗ Ω1

X), φ ∧ φ = 0

Definition [Hitchin, Simpson] A Higgs sheaf is semistable if ∀ F E φ-invariant, it is

PF rk F ≤ PE rk E

Higgs sheaves (E, φ)

Simpson

⇐ ⇒ E → Z = P(T ∗X ⊕ O) E of pure dimension = dim X, E = π∗E, π : T ∗X → X. Take (E, φ) unstable ⇒ q GIT-unstable, for each m ∈ Z, ⇒ ∃! m-Kempf filtration (V•, n•) GIT maximally destabilizing

evaluating

= ⇒ E• ⊂ E, m-Kempf filtration of E

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SLIDE 70

Introduction Correspondence for sheaves Correspondence for other problems Further directions Higgs sheaves

A Higgs sheaf is a pair (E, φ : E → E ⊗ Ω1

X), φ ∧ φ = 0

Definition [Hitchin, Simpson] A Higgs sheaf is semistable if ∀ F E φ-invariant, it is

PF rk F ≤ PE rk E

Higgs sheaves (E, φ)

Simpson

⇐ ⇒ E → Z = P(T ∗X ⊕ O) E of pure dimension = dim X, E = π∗E, π : T ∗X → X. Take (E, φ) unstable ⇒ q GIT-unstable, for each m ∈ Z, ⇒ ∃! m-Kempf filtration (V•, n•) GIT maximally destabilizing

evaluating

= ⇒ E• ⊂ E, m-Kempf filtration of E Theorem [Z.] The m-Kempf filtrations do not depend on m for m ≥ m′ and 0 ⊂ π∗E1 ⊂ π∗E2 ⊂ . . . ⊂ π∗Et ⊂ π∗Et+1 = E coincide with the Harder-Narasimhan filtration of the Higgs sheaf (E, φ).

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SLIDE 71

Introduction Correspondence for sheaves Correspondence for other problems Further directions Rank 2 tensors 14/ 17

SLIDE 72

Introduction Correspondence for sheaves Correspondence for other problems Further directions Rank 2 tensors

Rank 2 tensor: (E, ϕ :

s times

E ⊗ · · · ⊗ E −

→ OX), rk E = 2

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SLIDE 73

Introduction Correspondence for sheaves Correspondence for other problems Further directions Rank 2 tensors

Rank 2 tensor: (E, ϕ :

s times

E ⊗ · · · ⊗ E −

→ OX), rk E = 2 Definition [Gómez-Sols] (E, ϕ) is δ-semistable if ∀ (L, ϕ|L) (E, ϕ), PL − δε(L) ≤ PE−δs

2

, where ε(L) = max ♯ times L can appear in a non vanishing restriction ϕ|L⊗t⊗E⊗s−t

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SLIDE 74

Introduction Correspondence for sheaves Correspondence for other problems Further directions Rank 2 tensors

Rank 2 tensor: (E, ϕ :

s times

E ⊗ · · · ⊗ E −

→ OX), rk E = 2 Definition [Gómez-Sols] (E, ϕ) is δ-semistable if ∀ (L, ϕ|L) (E, ϕ), PL − δε(L) ≤ PE−δs

2

, where ε(L) = max ♯ times L can appear in a non vanishing restriction ϕ|L⊗t⊗E⊗s−t (E, ϕ) δ-unstable ⇒ q GIT-unstable ⇒ ∃! m-Kempf filtration (E•, ϕ|E•) ⊂ (E, ϕ)

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SLIDE 75

Introduction Correspondence for sheaves Correspondence for other problems Further directions Rank 2 tensors

Rank 2 tensor: (E, ϕ :

s times

E ⊗ · · · ⊗ E −

→ OX), rk E = 2 Definition [Gómez-Sols] (E, ϕ) is δ-semistable if ∀ (L, ϕ|L) (E, ϕ), PL − δε(L) ≤ PE−δs

2

, where ε(L) = max ♯ times L can appear in a non vanishing restriction ϕ|L⊗t⊗E⊗s−t (E, ϕ) δ-unstable ⇒ q GIT-unstable ⇒ ∃! m-Kempf filtration (E•, ϕ|E•) ⊂ (E, ϕ) Theorem [Z.] The m′-Kempf filtrations do not depend on m′, for m′ ≥ m.

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SLIDE 76

Introduction Correspondence for sheaves Correspondence for other problems Further directions Rank 2 tensors

Rank 2 tensor: (E, ϕ :

s times

E ⊗ · · · ⊗ E −

→ OX), rk E = 2 Definition [Gómez-Sols] (E, ϕ) is δ-semistable if ∀ (L, ϕ|L) (E, ϕ), PL − δε(L) ≤ PE−δs

2

, where ε(L) = max ♯ times L can appear in a non vanishing restriction ϕ|L⊗t⊗E⊗s−t (E, ϕ) δ-unstable ⇒ q GIT-unstable ⇒ ∃! m-Kempf filtration (E•, ϕ|E•) ⊂ (E, ϕ) Theorem [Z.] The m′-Kempf filtrations do not depend on m′, for m′ ≥ m. Definition The Kempf filtration 0 ⊂ (L, ϕ|L) ⊂ (E, ϕ) defines, by uniqueness, the Harder-Narasimhan filtration of (E, ϕ)

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SLIDE 77

Introduction Correspondence for sheaves Correspondence for other problems Further directions Quiver representations

Quiver Q = {Q0 = {vi} vertices , Q1 = {α : vi → vj} arrows}

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SLIDE 78

Introduction Correspondence for sheaves Correspondence for other problems Further directions Quiver representations

Quiver Q = {Q0 = {vi} vertices , Q1 = {α : vi → vj} arrows} Definition [King, Reineke] M is (Θ, σ)-semistable if ∀ M′ M, µ(Θ,σ)(M′) ≤ µ(Θ,σ)(M), where µ(Θ,σ)(M) = Θ(M)

σ(M) and Θ, σ : ZQ0 → Z, σ(v) > 0

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SLIDE 79

Introduction Correspondence for sheaves Correspondence for other problems Further directions Quiver representations

Quiver Q = {Q0 = {vi} vertices , Q1 = {α : vi → vj} arrows} Definition [King, Reineke] M is (Θ, σ)-semistable if ∀ M′ M, µ(Θ,σ)(M′) ≤ µ(Θ,σ)(M), where µ(Θ,σ)(M) = Θ(M)

σ(M) and Θ, σ : ZQ0 → Z, σ(v) > 0

King ⇒ GIT moduli construction of a moduli space of (Θ, σ)-semistable representations of Q

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SLIDE 80

Introduction Correspondence for sheaves Correspondence for other problems Further directions Quiver representations

Quiver Q = {Q0 = {vi} vertices , Q1 = {α : vi → vj} arrows} Definition [King, Reineke] M is (Θ, σ)-semistable if ∀ M′ M, µ(Θ,σ)(M′) ≤ µ(Θ,σ)(M), where µ(Θ,σ)(M) = Θ(M)

σ(M) and Θ, σ : ZQ0 → Z, σ(v) > 0

King ⇒ GIT moduli construction of a moduli space of (Θ, σ)-semistable representations of Q Maximal 1-parameter subgroup produce filtration of subrepresentations ⇒ Kempf filtration

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SLIDE 81

Introduction Correspondence for sheaves Correspondence for other problems Further directions Quiver representations

Quiver Q = {Q0 = {vi} vertices , Q1 = {α : vi → vj} arrows} Definition [King, Reineke] M is (Θ, σ)-semistable if ∀ M′ M, µ(Θ,σ)(M′) ≤ µ(Θ,σ)(M), where µ(Θ,σ)(M) = Θ(M)

σ(M) and Θ, σ : ZQ0 → Z, σ(v) > 0

King ⇒ GIT moduli construction of a moduli space of (Θ, σ)-semistable representations of Q Maximal 1-parameter subgroup produce filtration of subrepresentations ⇒ Kempf filtration Theorem [Z.] The Kempf filtration is equal to the Harder-Narasimhan filtration 0 ⊂ M1 ⊂ M2 ⊂ . . . ⊂ Mt ⊂ Mt+1 = M verifying

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SLIDE 82

Introduction Correspondence for sheaves Correspondence for other problems Further directions Quiver representations

Quiver Q = {Q0 = {vi} vertices , Q1 = {α : vi → vj} arrows} Definition [King, Reineke] M is (Θ, σ)-semistable if ∀ M′ M, µ(Θ,σ)(M′) ≤ µ(Θ,σ)(M), where µ(Θ,σ)(M) = Θ(M)

σ(M) and Θ, σ : ZQ0 → Z, σ(v) > 0

King ⇒ GIT moduli construction of a moduli space of (Θ, σ)-semistable representations of Q Maximal 1-parameter subgroup produce filtration of subrepresentations ⇒ Kempf filtration Theorem [Z.] The Kempf filtration is equal to the Harder-Narasimhan filtration 0 ⊂ M1 ⊂ M2 ⊂ . . . ⊂ Mt ⊂ Mt+1 = M verifying

1

µ(Θ,σ)(M1) > µ(Θ,σ)(M2) > . . . > µ(Θ,σ)(Mt) > µ(Θ,σ)(Mt+1)

2

Mi := Mi/Mi−1 are (Θ, σ)-semistable

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SLIDE 83

Introduction Correspondence for sheaves Correspondence for other problems Further directions (G, h)-constellations

G reductive group, X affine G-scheme of finite type, Hilbert function h : Irr G → N

16/ 17

SLIDE 84

Introduction Correspondence for sheaves Correspondence for other problems Further directions (G, h)-constellations

G reductive group, X affine G-scheme of finite type, Hilbert function h : Irr G → N A (G, h)-constellation on X is an (OX, G)-module F with multiplicities given by h: H0(F) ≃

ρ∈Irr G Ch(ρ) ⊗ Vρ

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SLIDE 85

Introduction Correspondence for sheaves Correspondence for other problems Further directions (G, h)-constellations

G reductive group, X affine G-scheme of finite type, Hilbert function h : Irr G → N A (G, h)-constellation on X is an (OX, G)-module F with multiplicities given by h: H0(F) ≃

ρ∈Irr G Ch(ρ) ⊗ Vρ

Stability condition depends on θρ ∈ Q, ρ ∈ Irr G.

16/ 17

SLIDE 86

Introduction Correspondence for sheaves Correspondence for other problems Further directions (G, h)-constellations

G reductive group, X affine G-scheme of finite type, Hilbert function h : Irr G → N A (G, h)-constellation on X is an (OX, G)-module F with multiplicities given by h: H0(F) ≃

ρ∈Irr G Ch(ρ) ⊗ Vρ

Stability condition depends on θρ ∈ Q, ρ ∈ Irr G. Theorem [Becker, Terpereau] GIT construction of a moduli space of θ-stable (G, h)-constellations. Construction depends on D ⊂ Irr G and for each D there is a D-filtration GIT maximally destabilizing.

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SLIDE 87

Introduction Correspondence for sheaves Correspondence for other problems Further directions (G, h)-constellations

G reductive group, X affine G-scheme of finite type, Hilbert function h : Irr G → N A (G, h)-constellation on X is an (OX, G)-module F with multiplicities given by h: H0(F) ≃

ρ∈Irr G Ch(ρ) ⊗ Vρ

Stability condition depends on θρ ∈ Q, ρ ∈ Irr G. Theorem [Becker, Terpereau] GIT construction of a moduli space of θ-stable (G, h)-constellations. Construction depends on D ⊂ Irr G and for each D there is a D-filtration GIT maximally destabilizing. Theorem [Terpereau, Z.] Given a θ-unstable (G, h)-constellation F, the D-filtrations converge to its Harder-Narasimhan filtration when D → Irr G.

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SLIDE 88

Introduction Correspondence for sheaves Correspondence for other problems Further directions

Further directions Tensors Extend correspondence for tensors in more generality

17/ 17

SLIDE 89

Introduction Correspondence for sheaves Correspondence for other problems Further directions

Further directions Tensors Extend correspondence for tensors in more generality Principal bundles Gieseker-type canonical reduction for principal bundles?

17/ 17

SLIDE 90

Introduction Correspondence for sheaves Correspondence for other problems Further directions

Further directions Tensors Extend correspondence for tensors in more generality Principal bundles Gieseker-type canonical reduction for principal bundles? Theorem [Biswas, Z.]: Gieseker Harder-Narasimhan filtration of the underlying sheaf of an orthogonal or symplectic bundle is not the canonical reduction

17/ 17

SLIDE 91

Introduction Correspondence for sheaves Correspondence for other problems Further directions

Further directions Tensors Extend correspondence for tensors in more generality Principal bundles Gieseker-type canonical reduction for principal bundles? Theorem [Biswas, Z.]: Gieseker Harder-Narasimhan filtration of the underlying sheaf of an orthogonal or symplectic bundle is not the canonical reduction Idea: Try to define it through maximal GIT destabilizers

17/ 17

SLIDE 92

Introduction Correspondence for sheaves Correspondence for other problems Further directions

Further directions Tensors Extend correspondence for tensors in more generality Principal bundles Gieseker-type canonical reduction for principal bundles? Theorem [Biswas, Z.]: Gieseker Harder-Narasimhan filtration of the underlying sheaf of an orthogonal or symplectic bundle is not the canonical reduction Idea: Try to define it through maximal GIT destabilizers Non-abelian categories Harder-Narasimhan filtrations in non-abelian categories?

17/ 17