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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
Outline
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
◮ We work over an algebraically closed field k (of char 0). Let
Q = (Q0, Q1) be a finite quiver without oriented cycles. The space of representations of a fixed dimension vector α ∈ Nn is Repα(Q) :=
Hom(kα(ta), kα(ha)). The group GLα :=
v∈Q0 GLα(v) acts on Repα(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
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◮ Fix a weight σ ∈ HomZ(Zn, 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 Repσ·ss
α
(Q) (resp. Repσ·st
α
(Q)) the set of all σ-semi-stable (resp. σ-stable) α-dim representations.
Jiarui Fei Wall Crossings in Moduli of Quiver Representations
Outline
◮ Fix a weight σ ∈ HomZ(Zn, 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 Repσ·ss
α
(Q) (resp. Repσ·st
α
(Q)) the set of all σ-semi-stable (resp. σ-stable) α-dim representations.
◮ Facts: There is a good categorical quotient (by GIT)
q : Repσ·ss
α
(Q) → Modσ
α(Q), and its restriction to the stable
representations is a geometric quotient.
Jiarui Fei Wall Crossings in Moduli of Quiver Representations
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◮ The weights σ for which Repσ·ss α
(Q) is nonempty form a polyhedral cone Σα(Q). It is known that such a σ is of form −−, βQ (or equivalently β, −Q), where −, −Q is the usual Euler form of Q.
Jiarui Fei Wall Crossings in Moduli of Quiver Representations
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◮ The weights σ for which Repσ·ss α
(Q) is nonempty form a polyhedral cone Σα(Q). It is known that such a σ is of form −−, βQ (or equivalently β, −Q), where −, −Q is the usual Euler form of Q.
◮ Definition The walls of Σα(Q) consists of all σ such that
Repσ·ss
α
(Q) containing a strictly semi-stable points. The walls “divides” Σα(Q) into several chambers where the moduli space is constant.
Jiarui Fei Wall Crossings in Moduli of Quiver Representations
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◮ A representation is called exceptional if HomQ(E, E) = k and
ExtQ(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
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◮ A representation is called exceptional if HomQ(E, E) = k and
ExtQ(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
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◮ For M, N ∈ Rep(Q), N is said to be right orthogonal to M
denoted by M ⊥ N if HomQ(M, N) = ExtQ(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
◮ For M, N ∈ Rep(Q), N is said to be right orthogonal to M
denoted by M ⊥ N if HomQ(M, N) = ExtQ(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 QE having no oriented cycles with |Q0| − 1 vertices. The inverse functor ιE : Mod(QE) → E ⊥ is a full exact embedding into Mod(Q).
Jiarui Fei Wall Crossings in Moduli of Quiver Representations
Outline
◮ For M, N ∈ Rep(Q), N is said to be right orthogonal to M
denoted by M ⊥ N if HomQ(M, N) = ExtQ(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 QE having no oriented cycles with |Q0| − 1 vertices. The inverse functor ιE : Mod(QE) → E ⊥ is a full exact embedding into Mod(Q).
◮ Lemma [Schofield] Repα(E ⊥) := E ⊥ ∩ Repα(Q) is isomorphic
to the homogeneous fibre space GLα ×GLαǫ Repαǫ(QE).
Jiarui Fei Wall Crossings in Moduli of Quiver Representations
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◮ Lemma [Geigel and Lenzing] There is a functor
˜ πE : Mod(Q) → E ⊥ left adjoint to the inclusion functor E ⊥ → Mod(Q).
Jiarui Fei Wall Crossings in Moduli of Quiver Representations
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◮ Lemma [Geigel and Lenzing] There is a functor
˜ πE : Mod(Q) → E ⊥ left adjoint to the inclusion functor E ⊥ → Mod(Q).
◮ By AR-duality, there is a representation τE such that
E ⊥ =⊥ τE. So we obtain a dual projection π∨
E to E ⊥ through
τE.
Jiarui Fei Wall Crossings in Moduli of Quiver Representations
Outline
◮ Lemma [Geigel and Lenzing] There is a functor
˜ πE : Mod(Q) → E ⊥ left adjoint to the inclusion functor E ⊥ → Mod(Q).
◮ By AR-duality, there is a representation τE such that
E ⊥ =⊥ τE. So we obtain a dual projection π∨
E to E ⊥ through
τE.
◮ More notation on the dimension vectors:
α
˜ πE
− → ˜ αǫ
iso
− → αǫ, β
˜ π∨
E
− − → ˜ β∨
ǫ iso
− → β∨
ǫ .
Jiarui Fei Wall Crossings in Moduli of Quiver Representations
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◮ Theorem
Assume some mild conditions. If a wall S with supporting hyperplane ǫ, ·Q is the only wall intersecting β ˜ β∨
ǫ , then
ϕE : Modσβ
α (Q) → Mod σ ˜
β∨ ǫ
α
(Q) is the blow-up of Mod
σ ˜
β∨ ǫ
α
(Q) along the irreducible subvariety q(Rep
˜ β∨
ǫ ·ss
α։ǫ (Q)).
If the blow-up loci is non-empty, then it has dimension −α − ǫ, ǫQ and its exceptional loci is q(Repβ·ss
ǫ֒ →α(Q)).
Jiarui Fei Wall Crossings in Moduli of Quiver Representations
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◮ Proposition
Mod
σ ˜
β∨ ǫ
α
(Q) ∼ = Mod
σβ∨
ǫ
αǫ (QE) and under this identification
q(Rep
˜ β∨
ǫ ·ss
α։ǫ (Q)) goes to qEπE(Rep ˜ β∨
ǫ ·ss
α։ǫ (Q)).
Jiarui Fei Wall Crossings in Moduli of Quiver Representations
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◮ Consider the quiver B4
1
a4
e4
b4
4
d4
c4
exceptional representations left orthogonal to α. They are 0 → P6 → Pu ⊕ Pv → Euv → 0, where {u, v} ⊂ {1, 2, 3, 4, 5}.
Jiarui Fei Wall Crossings in Moduli of Quiver Representations
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◮ Consider the quiver B4
1
a4
e4
b4
4
d4
c4
exceptional representations left orthogonal to α. They are 0 → P6 → Pu ⊕ Pv → Euv → 0, where {u, v} ⊂ {1, 2, 3, 4, 5}.
◮ Choose the canonical weight: σ = −−, α + ταQ, then it
turns out the moduli is the blow-up of P2 along four general
Jiarui Fei Wall Crossings in Moduli of Quiver Representations
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◮ Let E = E45. By calculation, the projected quiver is
4 1
b3
d3
e3
The projected dimension vector is αǫ = (1, 1, 1, 1, 1). The projected stability is still canonical. The projected moduli can be shown as the blow-up of P2 along three general points. The blow-up loci is another point in general position.
Jiarui Fei Wall Crossings in Moduli of Quiver Representations
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◮ By varying the stability, the quiver B4 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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◮ By varying the stability, the quiver B4 contains all
2-dimensional moduli spaces of quiver representations. They are Fano surfaces of degree from 9 to 5.
◮ In the meantime, over one hundred of different moduli in
dimension three have been found. They can be singular, weakly Fano.
Jiarui Fei Wall Crossings in Moduli of Quiver Representations
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◮ The GIT construction also gives us an ample divisor on the
quotient for free, namely the one induced from the G-linearization. We denote by Dβ the one from the stability −−, βQ.
Theorem
Under some mild assumption, Dβ = ϕ∗
E(Dβ∨
ǫ ) − ǫ, βQEβ.
Here, roughly speaking Eβ is the exceptional divisor of the blow-up.
Jiarui Fei Wall Crossings in Moduli of Quiver Representations
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◮ Consider quiver:
1
a
2
b
3
with dimension vector α = (1,4,3) and weight σβ = (3,3,−5). Take ǫ = (0, 3, 2), then the projected quiver with dimension vector is the four-arrow Kronecker quiver with αǫ = (1, 1) and σβ∨
ǫ = (3, −3). Clearly the projected moduli is P3 embedded
by O(3). By inspection, the blow-up loci is a twisted cubic curve.
Jiarui Fei Wall Crossings in Moduli of Quiver Representations
Outline
◮ Consider quiver:
1
a
2
b
3
with dimension vector α = (1,4,3) and weight σβ = (3,3,−5). Take ǫ = (0, 3, 2), then the projected quiver with dimension vector is the four-arrow Kronecker quiver with αǫ = (1, 1) and σβ∨
ǫ = (3, −3). Clearly the projected moduli is P3 embedded
by O(3). By inspection, the blow-up loci is a twisted cubic curve.
◮ The induced ample divisor
Dβ = ϕ∗
E(Dβ∨
ǫ ) − ǫ, βQEβ = ϕ∗
E(O(3)) − Eβ. So its linear
series corresponds to |O(3)| with assigned base points a twisted cubic. Then one can easily compute that h0(Modσβ
α (Q), Dβ) = 10.
Jiarui Fei Wall Crossings in Moduli of Quiver Representations
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Wall Crossings in Moduli of Quiver Representations