Geometric invariant theory for non-reductive group actions and jet differentials Frances Kirwan Oxford (based on joint work with Brent Doran and Gergely Berczi) 1 Moduli spaces (or stacks) are often construct- ed as quotients of algebraic varieties by group actions. Reductive groups ; we can use Mumford’s GIT (+ techniques from symplectic geometry) Non-reductive groups? E.g. moduli spaces of hypersurfaces/complete intersections in toric varieties – automorphism group of a toric variety is not in general reduc- tive. Example: weighted projective plane P(1, 1, 2) ∼ Aut(P(1, 1, 2)) = R n U ∗ with =∼ (2) ∗ =∼ (2) reductive R GL ×C C GL ∼ + 3 U = (C ) unipotent where (x, y, z) 7→ (x, y, z + λx2 + µxy + νy2) for 3 (λ, µ, ν) ∈ C 2 Mumford’s GIT G complex reductive group X complex projective variety acted on by G We require a linearisation of the action (i.e. an ample line bundle L on X and a lift of the n action to L; think of X ⊆ P and the action given by a representation ρ : G → GL(n + 1)). X ⇒ A(X) = C[x0, . , xn]/IX L∞ 0 ⊗k | = k=0 H (X, L ) | S| ↓ X//G ⇐ A(X)G ring of invariants G reductive implies that A(X)G is a finitely generated graded complex algebra so that X//G = Proj(A(X)G) is a projective variety. 3 The rational map X − − → X//G fits into a diagram X −− → X//G cx proj variety S || onto semistable Xss −→ X//G SS open stable Xs −→ Xs/G where the morphism Xss → X//G is G-invariant and surjective. Topologically X//G = Xss/ ∼ where x ∼ y ⇔ Gx ∩ Gy ∩ Xss 6= ∅. N.B. G reductive ⇔ G is the complexification of a maximal compact subgroup (for KC K example ( ) = ( ) ), and then SL n SU n C X//G = µ−1(0)/K for a suitable moment map µ for the action of K. 4 What if G is not reductive? Problem: We can’t define a projective variety X//G = Proj(A(X)G) G where A(X) = C[x0, . , xn]/IX because A(X) is not necessarily finitely generated. [In fact G is reductive if and only if A(X)G is finitely gen- erated for all such X]. Question: Can we define a sensible ‘quotient’ variety X//G when G is not reductive? N.B. Any linear algebraic group has a unipotent normal subgroup U 6 G (its unipotent radical) such that R = G/U is reductive [for unipotent think strictly upper triangular matrices]. Moreover U has a (canonical) chain of normal subgroups {1} = U0 6 U1 6 ... 6 Us = U ∼ + + + such that each Uj/Uj−1 = C × C × · · · × C . 5 Theorem (Doran, K): Let H = R n U be a linear algebraic group over C acting linearly on n X ⊆ P . Then X has open subsets Xs (‘stable points’) and Xss (‘semistable points’) with a geometric quotient Xs → Xs/H and an ‘enveloping quo- tient’ Xss → X//H. Moreover if A(X)H is finitely generated then X//H = Proj(A(X)H). We have a similar diagram to the reductive case X −− → X//H S || semistable Xss −→ X//H SS open stable Xs −→ Xs/H BUT X//H is not necessarily projective and Xss → X//H is not necessarily onto. 6 Reductive envelopes We can choose reductive G ⊇ H and a suit- able compactification G ×H X of G×H X giving a (non-canonical) compactification G ×H X//G of X//H: s X /H ⊆ X//H ⊆ G ×H X//G However although such a compactification al- ways exists, it is not at all easy in general to decide when a compactification G ×H X of G ×H X has the properties needed. 7 + n Simple example: C acting on P We can choose coordinates in which the gener- + ator of Lie(C ) has Jordan normal form with + blocks of size k1 + 1, . , kq + 1. The linear C action therefore extends to G = SL(2) with ! 1 a + = { : a ∈ } G C 0 1 C 6 n+1 ∼ Lq ki 2 via C = i=1 Sym (C ). In fact in this case the invariants are finitely generated (Weitzenbock) so we can define n + + P //C = Proj((C[x0, . , xn])C ). + N.B. Via (g, x) 7→ (gC , gx) we have G × n =∼ (G/ +) × n =∼ ( 2 \{0}) × n C+ P C P C P 2 n 2 n ⊆ C × P ⊆ P × P and so n + ∼ 2 n P //C = (P × P )//SL(2) 8 2 n 2 n P × P −− → P × P //G S || n n n + P = {[1 : 0 : 1]} × P −− → P //C S || n ss not nec onto n + (P ) −→ P //C SS n s n s + (P ) −→ (P ) /C n ss n + Example when (P ) → P //C is not onto: 3 3 2 1 P = P(Sym (C )) = { 3 unordered points on P }. 3 ss 3 s Then (P ) = (P ) is 1 { 3 unordered points on P , at most one at ∞} and its image in 3 + 3 s + 3 P //C = (P ) /C t P //SL(2) 3 s + is the open subset (P ) /C which does not include the ‘boundary’ points coming from 2 2 0 ∈ C ⊆ P . 9 2 2 2 2 The blow-up P˜ of P at 0 ∈ C ⊆ P can be 1 identified with G ×B P where B is the Borel + subgroup of G = SL(2) containing C and the ∼ ∗ standard maximal torus T = C . 2 n n Similarly the blow-up of P × P along {0} × P 1 n can be identified with G ×B (P × P ). Let P^n//C+ be the blow-up of n + 2 n P //C = (P × P )//G n along the subvariety P //G corresponding to 2 0 ∈ P . Then the G-invariant surjection 2 n ss,G 2 n n + (P × P ) → (P × P )//G = P //C induces a B-invariant surjection 1 n ss,B n + (P × P ) → P //C 1 n ss,B from a suitable open subset (P × P ) of 1 n P × P , and thus a surjection from an open subset of the GIT quotient 1 n X = (P × P )//T n + to P //C . 10 In constructing the GIT quotient 1 n X = (P × P )//T to get a surjection from an open subset X ss of n + ∼ ∗ 1 n Xˆ to P //C , the action of T = C on P × P has to be appropriately linearised; a different choice of linearisation would give 1 n ∗ n n (P × P )//T = (C × P )/T = P . Thus the theory of variation of GIT quotients (Thaddeus, Dolgachev-Hu, Ressayre) tells us n that X and P are related by a sequence of explicit blow-ups + blow-downs (flips in the sense of Thaddeus). flips 1 n n VGIT ; X = (P × P )//T ← − → X = P flips n P ← −− → X SS n ss n + onto ss (P ) −→ P //C ←− X SSS n s n s + n s (P ) −→ (P ) /C ←− (P ) 11 Defn: Call a unipotent linear algebraic group U graded unipotent if there is a homomorphism ∗ ∗ λ : C → Aut(U) with the weights of the C action on Lie(U) all strictly positive. Then let ∗ ∗ Uˆ = U o C = {(u, t): u ∈ U, t ∈ C } with multiplication (u, t)·(u0, t0) = (u(λ(t)(u0)), tt0). Thm: Let U be graded unipotent acting lin- early on a projective variety X, and suppose ∗ that the action extends to Uˆ = U o C . Then (i) the ring A(X)U of U-invariants is finitely generated, so that X//U = Proj(A(X)U ); (ii) there is a projective variety X which is relat- ed to X via VGIT and a surjection X ss → X//U to X//U from an open subset X ss of X . flips X ← −− → X SS onto Xss −→ X//U = Proj(A(X)U ) ←− X ss SSS Xs −→ Xs/U ←− Xs 12 Example: The automorphism group of the weighted projective plane P(1, 1, 2) is ∼ Aut(P(1, 1, 2)) = R n U ∼ ∼ + 3 with R = GL(2) reductive and U = (C ) + 3 unipotent .... (λ, µ, ν) ∈ (C ) acts as (x, y, z) 7→ (x, y, z + λx2 + µxy + νy2). ∗ ∼ The central one-parameter subgroup C of R = GL(2) acts on Lie(U) with all positive weights, ∗ and the associated extension Uˆ = U oC can be identified with a subgroup of Aut(P(1, 1, 2)). Corollary When H = Aut(P(1, 1, 2)) acts lin- early on a projective variety X, the ring of in- variants A(X)H is finitely generated as a com- plex algebra, so that X//H = Proj(A(X)H), and moreover there is a projective variety X which is related to X via VGIT and a surjection X ss → X//H to X//H from an open subset X ss of X . 13 Application to jet differentials (following Demailly 1995) X complex manifold, dim X = n Jk → X bundle of k-jets of holomorphic curves f :(C, 0) → X [f and g have the same k-jet if their Taylor expansions at 0 coincide up to order k]. More generally Jk,p → X is the bundle of k-jets of p holomorphic maps f : C → X. Under composition modulo tk+1 we have a group Gk given by {k-jets of germs of biholomorphisms of (C, 0)} 2 k t 7→ φ(t) = a1t+a2t +...+akt , aj ∈ C, a1 6= 0 Gk acts on Jk fibrewise by reparametrising k- jets. Similarly we have Gk,p acting fibrewise on Jk,p.