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

GIT characterizations of Harder-Narasimhan filtrations Alfonso Zamora Instituto Superior Técnico Lisboa, Portugal AMS-EMS-SPM Meeting Porto, June 2015

Introduction Correspondence for sheaves Correspondence for other problems Further directions Index 1 Introduction Correspondence for sheaves 2 Harder-Narasimhan filtration Gieseker construction of a moduli space Kempf theorem Correspondence for other problems 3 Holomorphic pairs Higgs sheaves Rank 2 tensors Quiver representations ( G , h ) -constellations Further directions 4 2/ 17

Introduction Correspondence for sheaves Correspondence for other problems Further directions Classification problems in geometry 3/ 17

Introduction Correspondence for sheaves Correspondence for other problems Further directions Classification problems in geometry Want to classify geometric objects ( A ) up to equivalence relation ( ∼ ) 3/ 17

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 3/ 17

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 : Sch k → Sets, F ( S ) set of equivalence classes of families parametrized by S . Triple ( A , ∼ , F ) is a moduli problem. 3/ 17

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 : Sch k → 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 3/ 17

Introduction Correspondence for sheaves Correspondence for other problems Further directions GIT constructions of moduli spaces 4/ 17

Introduction Correspondence for sheaves Correspondence for other problems Further directions GIT constructions of moduli spaces Different Aut ( A ) , A ∈ A ⇒ Notion of stability 4/ 17

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) 4/ 17

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) 4/ 17

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 , d 1 ) G ∼ ( A , d 2 ) 4/ 17

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 , d 1 ) G ∼ ( A , d 2 ) Want to take the quotient Q / G . Have to remove some orbits (GIT-unstables) and identify others ( S -equivalence) ⇒ Geometric Invariant Theory (GIT) [Mumford] 4/ 17

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 , d 1 ) G ∼ ( A , d 2 ) Want to take the quotient Q / G . Have to remove some orbits (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 Q ss / / G . 4/ 17

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 , d 1 ) G ∼ ( A , d 2 ) Want to take the quotient Q / G . Have to remove some orbits (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 Q ss / / G . For unstable objects ⇒ Harder-Narasimhan filtration 4/ 17

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 , d 1 ) G ∼ ( A , d 2 ) Want to take the quotient Q / G . Have to remove some orbits (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 Q ss / / G . For unstable objects ⇒ Harder-Narasimhan filtration For GIT-unstable orbits ⇒ Maximal 1-parameter subgroups 4/ 17

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 , d 1 ) G ∼ ( A , d 2 ) Want to take the quotient Q / G . Have to remove some orbits (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 Q ss / / 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 4/ 17

Introduction Correspondence for sheaves Correspondence for other problems Further directions Stability for sheaves 5/ 17

Introduction Correspondence for sheaves Correspondence for other problems Further directions Stability for sheaves ( X , O X ( 1 )) polarized smooth complex projective variety of dim n . Fix P , degree n polynomial and consider E → X torsion free coherent sheaves with P E = P 5/ 17

Introduction Correspondence for sheaves Correspondence for other problems Further directions Stability for sheaves ( X , O X ( 1 )) polarized smooth complex projective variety of dim n . Fix P , degree n polynomial and consider E → X torsion free coherent sheaves with P E = P Observation: If X is Riemann surface, holomorphic vector bundles E of rank r and degree d 5/ 17

Introduction Correspondence for sheaves Correspondence for other problems Further directions Stability for sheaves ( X , O X ( 1 )) polarized smooth complex projective variety of dim n . Fix P , degree n polynomial and consider E → X torsion free coherent sheaves with P E = P Observation: If X is Riemann surface, holomorphic vector bundles E of rank r and degree d E ( m ) = E ⊗ O X ( m ) , P E ( m ) = χ ( E ( m )) = � n i = 0 ( − 1 ) i h i ( E ( m )) 5/ 17

Introduction Correspondence for sheaves Correspondence for other problems Further directions Stability for sheaves ( X , O X ( 1 )) polarized smooth complex projective variety of dim n . Fix P , degree n polynomial and consider E → X torsion free coherent sheaves with P E = P Observation: If X is Riemann surface, holomorphic vector bundles E of rank r and degree d E ( m ) = E ⊗ O X ( m ) , P E ( m ) = χ ( E ( m )) = � n i = 0 ( − 1 ) i h i ( E ( m )) Definition [Gieseker] E is semistable if ∀ F � E , P F rk F ≤ P E rk E . If not E is unstable 5/ 17

Introduction Correspondence for sheaves Correspondence for other problems Further directions Stability for sheaves ( X , O X ( 1 )) polarized smooth complex projective variety of dim n . Fix P , degree n polynomial and consider E → X torsion free coherent sheaves with P E = P Observation: If X is Riemann surface, holomorphic vector bundles E of rank r and degree d E ( m ) = E ⊗ O X ( m ) , P E ( m ) = χ ( E ( m )) = � n i = 0 ( − 1 ) i h i ( E ( m )) Definition [Gieseker] E is semistable if ∀ F � E , P F rk F ≤ P E rk E . If not E is unstable If dim C X = 1, P E ( m ) = rm + d + r ( 1 − g ) , µ ( E ) = deg E rk E = d r 5/ 17