Outline Wall Crossings in Moduli of Quiver Representations Jiarui Fei University of California, Riverside November 18, 2012 Jiarui Fei Wall Crossings in Moduli of Quiver Representations
Outline Jiarui Fei Wall Crossings in Moduli of Quiver Representations
Outline Basics on Quiver Representations ◮ We work over an algebraically closed field k (of char 0). Let Q = ( Q 0 , Q 1 ) be a finite quiver without oriented cycles. The space of representations of a fixed dimension vector α ∈ N n is � Hom( k α ( ta ) , k α ( ha ) ) . Rep α ( Q ) := a ∈ Q 1 The group GL α := � v ∈ Q 0 GL α ( v ) acts on Rep α ( Q ) by the natural base change. Two representations are isomorphic iff they have the same orbit. Jiarui Fei Wall Crossings in Moduli of Quiver Representations
Outline Moduli of Quiver Representations ◮ Fix a weight σ ∈ Hom Z ( Z n , Z ) such that σ ( α ) = 0, an α -dim representation M is called σ -semi-stable (resp. σ -stable) if σ (dim L ) � 0 (resp. σ (dim L ) < 0) for any non-trivial subrepresentation L ⊂ M . We denote by Rep σ · ss ( Q ) (resp. α Rep σ · st ( Q )) the set of all σ -semi-stable (resp. σ -stable) α -dim α representations. Jiarui Fei Wall Crossings in Moduli of Quiver Representations
Outline Moduli of Quiver Representations ◮ Fix a weight σ ∈ Hom Z ( Z n , Z ) such that σ ( α ) = 0, an α -dim representation M is called σ -semi-stable (resp. σ -stable) if σ (dim L ) � 0 (resp. σ (dim L ) < 0) for any non-trivial subrepresentation L ⊂ M . We denote by Rep σ · ss ( Q ) (resp. α Rep σ · st ( Q )) the set of all σ -semi-stable (resp. σ -stable) α -dim α representations. ◮ Facts: There is a good categorical quotient (by GIT) q : Rep σ · ss ( Q ) → Mod σ α ( Q ), and its restriction to the stable α representations is a geometric quotient . Jiarui Fei Wall Crossings in Moduli of Quiver Representations
Outline The Chambers and Walls ◮ The weights σ for which Rep σ · ss ( Q ) is nonempty form a α polyhedral cone Σ α ( Q ). It is known that such a σ is of form −�− , β � Q (or equivalently � β, −� Q ), where �− , −� Q is the usual Euler form of Q . Jiarui Fei Wall Crossings in Moduli of Quiver Representations
Outline The Chambers and Walls ◮ The weights σ for which Rep σ · ss ( Q ) is nonempty form a α polyhedral cone Σ α ( Q ). It is known that such a σ is of form −�− , β � Q (or equivalently � β, −� Q ), where �− , −� Q is the usual Euler form of Q . ◮ Definition The walls of Σ α ( Q ) consists of all σ such that Rep σ · ss ( Q ) containing a strictly semi-stable points. α The walls “divides” Σ α ( Q ) into several chambers where the moduli space is constant. Jiarui Fei Wall Crossings in Moduli of Quiver Representations
Outline The Walls given by Real Roots ◮ A representation is called exceptional if Hom Q ( E , E ) = k and Ext Q ( E , E ) = 0. The dimension vector ǫ of E is called a real Schur root of Q . We are particularly interested in the following situation: Jiarui Fei Wall Crossings in Moduli of Quiver Representations
Outline The Walls given by Real Roots ◮ A representation is called exceptional if Hom Q ( E , E ) = k and Ext Q ( E , E ) = 0. The dimension vector ǫ of E is called a real Schur root of Q . We are particularly interested in the following situation: ◮ Let C + , C − be two adjacent chambers with W being a common wall whose supporting hyperplane is given by � ǫ, ·� Q = 0 for some real Schur root ǫ . Jiarui Fei Wall Crossings in Moduli of Quiver Representations
Outline The Orthogonal Subcategories ◮ For M , N ∈ Rep( Q ), N is said to be right orthogonal to M denoted by M ⊥ N if Hom Q ( M , N ) = Ext Q ( M , N ) = 0. The (right) orthogonal category M ⊥ is the abelian subcategory { N ∈ Mod( Q ) | M ⊥ N } . If E is exceptional, Jiarui Fei Wall Crossings in Moduli of Quiver Representations
Outline The Orthogonal Subcategories ◮ For M , N ∈ Rep( Q ), N is said to be right orthogonal to M denoted by M ⊥ N if Hom Q ( M , N ) = Ext Q ( M , N ) = 0. The (right) orthogonal category M ⊥ is the abelian subcategory { N ∈ Mod( Q ) | M ⊥ N } . If E is exceptional, ◮ Lemma [Schofield] E ⊥ is equivalent to representations of a quiver Q E having no oriented cycles with | Q 0 | − 1 vertices. The inverse functor ι E : Mod( Q E ) → E ⊥ is a full exact embedding into Mod( Q ). Jiarui Fei Wall Crossings in Moduli of Quiver Representations
Outline The Orthogonal Subcategories ◮ For M , N ∈ Rep( Q ), N is said to be right orthogonal to M denoted by M ⊥ N if Hom Q ( M , N ) = Ext Q ( M , N ) = 0. The (right) orthogonal category M ⊥ is the abelian subcategory { N ∈ Mod( Q ) | M ⊥ N } . If E is exceptional, ◮ Lemma [Schofield] E ⊥ is equivalent to representations of a quiver Q E having no oriented cycles with | Q 0 | − 1 vertices. The inverse functor ι E : Mod( Q E ) → E ⊥ is a full exact embedding into Mod( Q ). ◮ Lemma [Schofield] Rep α ( E ⊥ ) := E ⊥ ∩ Rep α ( Q ) is isomorphic to the homogeneous fibre space GL α × GL αǫ Rep α ǫ ( Q E ). Jiarui Fei Wall Crossings in Moduli of Quiver Representations
Outline The Orthogonal Projections ◮ Lemma [Geigel and Lenzing] There is a functor π E : Mod( Q ) → E ⊥ left adjoint to the inclusion functor ˜ E ⊥ → Mod( Q ). Jiarui Fei Wall Crossings in Moduli of Quiver Representations
Outline The Orthogonal Projections ◮ Lemma [Geigel and Lenzing] There is a functor π E : Mod( Q ) → E ⊥ left adjoint to the inclusion functor ˜ E ⊥ → Mod( Q ). ◮ By AR-duality, there is a representation τ E such that E ⊥ = ⊥ τ E . So we obtain a dual projection π ∨ E to E ⊥ through τ E . Jiarui Fei Wall Crossings in Moduli of Quiver Representations
Outline The Orthogonal Projections ◮ Lemma [Geigel and Lenzing] There is a functor π E : Mod( Q ) → E ⊥ left adjoint to the inclusion functor ˜ E ⊥ → Mod( Q ). ◮ By AR-duality, there is a representation τ E such that E ⊥ = ⊥ τ E . So we obtain a dual projection π ∨ E to E ⊥ through τ E . ◮ More notation on the dimension vectors: π E ˜ iso α − → ˜ α ǫ − → α ǫ , π ∨ ˜ iso → ˜ β ∨ → β ∨ E − − − β ǫ . ǫ Jiarui Fei Wall Crossings in Moduli of Quiver Representations
Outline Shell-crossing ◮ Theorem Assume some mild conditions. If a wall S with supporting hyperplane � ǫ, ·� Q is the only wall intersecting β ˜ β ∨ ǫ , then σ ˜ σ ˜ ϕ E : Mod σ β β ∨ β ∨ α ( Q ) → Mod ǫ ( Q ) is the blow-up of Mod ǫ ( Q ) α α ˜ β ∨ ǫ · ss along the irreducible subvariety q (Rep α ։ ǫ ( Q )) . If the blow-up loci is non-empty, then it has dimension −� α − ǫ, ǫ � Q and its exceptional loci is q (Rep β · ss → α ( Q )) . ǫ֒ Jiarui Fei Wall Crossings in Moduli of Quiver Representations
Outline Change to Smaller Quivers ◮ Proposition σ ˜ σ β ∨ β ∨ ( Q ) ∼ Mod ǫ = Mod α ǫ ( Q E ) and under this identification ǫ α ˜ ˜ β ∨ β ∨ ǫ · ss ǫ · ss q (Rep α ։ ǫ ( Q )) goes to q E π E (Rep α ։ ǫ ( Q )) . Jiarui Fei Wall Crossings in Moduli of Quiver Representations
� � � Outline Surface Examples ◮ Consider the quiver B 4 1 5 a 4 2 � � � � � b 4 � e 4 � � � � � � � 6 � � � � � � � c 4 � d 4 � � � � � 4 3 with dimension vector α = (1 , 1 , 1 , 1 , 1 , 2). Then there are ten exceptional representations left orthogonal to α . They are 0 → P 6 → P u ⊕ P v → E uv → 0 , where { u , v } ⊂ { 1 , 2 , 3 , 4 , 5 } . Jiarui Fei Wall Crossings in Moduli of Quiver Representations
� � � Outline Surface Examples ◮ Consider the quiver B 4 1 5 a 4 2 � � � � � b 4 � e 4 � � � � � � � 6 � � � � � � � c 4 � d 4 � � � � � 4 3 with dimension vector α = (1 , 1 , 1 , 1 , 1 , 2). Then there are ten exceptional representations left orthogonal to α . They are 0 → P 6 → P u ⊕ P v → E uv → 0 , where { u , v } ⊂ { 1 , 2 , 3 , 4 , 5 } . ◮ Choose the canonical weight: σ = −�− , α + τα � Q , then it turns out the moduli is the blow-up of P 2 along four general points. This can be seen by induction as follows. Jiarui Fei Wall Crossings in Moduli of Quiver Representations
� � � � Outline ◮ Let E = E 45 . By calculation, the projected quiver is 4 � � � � � b 3 � e 3 � � � � d 3 � � � � � � � 1 2 3 � � � � � � � � c 3 � � � a 3 � � f 3 � � � � 5 The projected dimension vector is α ǫ = (1 , 1 , 1 , 1 , 1). The projected stability is still canonical. The projected moduli can be shown as the blow-up of P 2 along three general points. The blow-up loci is another point in general position. Jiarui Fei Wall Crossings in Moduli of Quiver Representations
Outline A Conjecture ◮ By varying the stability, the quiver B 4 contains all 2-dimensional moduli spaces of quiver representations. They are Fano surfaces of degree from 9 to 5. Jiarui Fei Wall Crossings in Moduli of Quiver Representations
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