NOTES ON GIT AND RELATED TOPICS

FEI SI

Abstract. The reading note contains basics of GIT and some computational examples. A brief GIT construction of moduli of stable curves and stable maps will be given. At last,we discuss relations with other stability conditions in algebraic geometry and moment maps in sympletic geometry.

Contents 1. GIT Pholosophy 2 2. Basic setting of GIT under Reductive group action 2 2.1. Affine GIT 2 2.2. Projective GIT 2 2.3. Luca’s Slice Theorem 2 2.4. Hilbert-Mumford Criteria 2 2.5. Application 1:construction of coarse moduli space Mg,n 2 2.6. Application 2:construction of coarse moduli Kd of K3 2 3. Some key techniques to detect GIT Stability 3 4. moment maps and GIT 3 4.1. moment maps 3 4.2. Kempf-Ness theorem 4 5. Variation of GIT 4 6. Non reductive GIT 4 7. Derived category of GIT 4 8. stability in algebraic geometry 4 8.1. K-stability 4 8.2. Chow-stability 5 9. Appedix:Hilbrt scheme,Chow variety and Quot schemes 6 9.1. Construction of Hilbert schemes 6 9.2. Construction of Chow Variety 8 9.3. Construction of Quot schemes 9 9.4. Hilber-Chow morphism 9 9.5. Some basic Properties of Hilbert scheme and chow variety 10 References 10 1 2 Fei Si

1. GIT Pholosophy Lemma 1.1. (Key lemma)If C is a Degline-Mumford stable curve of g ≥ 2 over C.then 1 ⊗n (i) H (C, ωC ) = 0 for n ≥ 2 ⊗n (ii) ωC is very ample for n ≥ 3. Proof. (i) comes from Vanishing Theorem. (ii) can be applied by Riemann-Roch and criteror ⊗n ⊗n dim|ωC | − dim|ωC − P − Q| = 2 for ∀ P,Q ∈ C  Remark 1.2. By the the lemma,choose n = 3,universal for all stable curves,then we have the embedding 5g−6 C,→ P then we can consider the Hilbert scheme Hp,5g−6 which parametrize all i-dimensional closed subscheme in P5g−6.Then take its GIT quotien- t.This is a sample of GIT approach to construction of moduli space.The advantage of this approach is that projectivity of moduli spaces is ob- vious.

2. Basic setting of GIT under Reductive group action 2.1. Affine GIT.

2.2. Projective GIT.

2.3. Luca’s Slice Theorem. suppose G × X → X with G-linearized L ∈ P ic(X) Definition: (i) x ∈ X is semistable w.r.t L if (ii) x ∈ X is stable w.r.t L if

2.4. Hilbert-Mumford Criteria. Theorem 2.1. (Hilbert-Mumford) (i) p ∈ X is semistable ⇔ weight µ(p, λ) ≥ 0 for any 1 − P S λ (ii) p ∈ X is polystable ⇔ weight µ(p, λ) > 0 for such 1 − P S λ as lim (iii) p ∈ X is stable ⇔ weight µ(p, λ) > 0 for any 1 − P S λ

Proof. 

2.5. Application 1:construction of coarse moduli space Mg,n.

2.6. Application 2:construction of coarse moduli Kd of K3. Notes on GIT3

3. Some key techniques to detect GIT Stability In general,the advanteg of GIT’s approach to moduli is that you can show the projectivity and compatify the moduli space very easily.But the disadantege is very obvious too:very difficult to get detailed analysis the stability m.

4. moment maps and GIT

4.1. moment maps. Definition 4.1. Let (X, ω) be a symplectic manifold with a compact Lie group action G y X. the moment map ∗ ∼ dim G µ : X → g = R defined by

dhµ(x), ςi = ω(Vς , ) where ς ∈ g and Vς is the vector field generated by the flow ϕt := exp(tς) ∈ Sym(X, ω), t ∈ R,ie, dϕ t (x) = V (ϕ ) dt ς t

ϕ0 = id n i P Example 4.2. (X, ω) = (C , 2 dzidzi) and G = U(1) the natural action is given by multiplication.Then g = iR itθ itθ ϕt :(z1, ..., zn) 7→ (e z1, ...e zn), iθ ∈ g

dϕt X itθ ∂ −itθ ∂ = θe z1 − θe z1 dt ∂zl ∂zl X ∂ ∂ Vς = iθzl − iθzl ∂zl ∂zl X ω(Vς , ) = iθ (zjdzj + zjdzj) Thus, the moment map is 2 µ(z)(θ) = θ|z| θ ∈ R Remark: consider sympletic quotient,easy to see −1 n−1 µ (1)/G = P This coinside with quotient in algebraic geometry. √ n P Example 4.3. (X, ω) = (P , −1∂∂log( zizi) and G = U(n + 1) acts Pn naturally. 4 Fei Si

n i P n Example 4.4. (X, ω) = (C , 2 dzidzi) and G = U(n) acts C nat- urally.Each element A ∈ G can be diagonalized.so assume √ √ X A = diag( −1λ1, ... −1λn), λi ∈ R, λi = 0

.Then

4.2. Kempf-Ness theorem.

n Theorem 4.5. Let X ⊂ CP be projective manifold with reductive Lie group action G ⊂ GL(n + 1, Cn),then p ∈ X is poly-stable ⇔ orbit G · p T µ−1(0) 6= φ moreover if p is polystable,then #G · p T µ−1(0) = 1

Proof. Suppose G is complexication of a compact subgroup K ⊂ GL(n+ n 1, C), fixed a v0 ∈ C then consider

G → R

g 7→| g · v0 | Note that

| g · v0 |=| k · g · v0 | for each k ∈ K,then it induces

G/K → R and G/K is a homogeous space admit nonnegative curvature. We claim :the function obtain its minimum iff v0 is stable 

5. Variation of GIT 6. Non reductive GIT

7. Derived category of GIT

8. stability in algebraic geometry 8.1. K-stability. Notes on GIT5

8.1.1. Futaki invatiant:obstruction to KE metric on Fano manifold. From history,the first obstruction to KE-metric is Mas ’s Aut(X),which says it’s reductive if X admits KE. then in 1985,A.Futaki gives another one.Here we follow3. Set ∞ h := {∂f ∈ Γ(TX): f ∈ C (X, C) where ∂f := gij ∂f ∂ ∂zj ∂zi consider functional F : h → C Z f 7→ (S − Sb) · f · ωn X ij where scalar curvature of the kahler metric is S := g Rij and average R n X S·ω scalar curvature is Sb := R n It’s easy to see if X admits Csck, then X ω F ≡ 0 Note that KE is just a special Csck.

8.1.2. Donalson-Futaki invariant:Algebraic geometry enters. Definition 8.1. (Test-configuration) Let (X,L) be a polarized man- ifold with L ample.A Test-configuration for of exponent r > 0 is an Nr ⊗r ∗ embedding X,→ CP by L with 1 − P S λ : C → GL(Nr + 1, C) 0 ⊗r where Nr := dimH (X,L )

Example 8.2. (Trivial TC)

8.1.3. Tian’s analytical definition.

8.1.4. Donalson’s algebraic definition. Remark:By the work of Li-Xu,(see[LX]5,It’s enough ro only consid- er special test configurations.

8.2. Chow-stability.

Definition 8.3. A normal variety X ⊂ PN is called chow stable(semi- stable) if its chow form (chow point) is stable(semi-stable) in the sense of GIT SL(N + 1, C) y Chow Definition 8.4. A polarized variety (X,L) is called asymtopic chow Nm stable(semi-stable) if ϕm(X) ⊂ P is chow stable(semi-stable) for m  0 6 Fei Si

9. Appedix:Hilbrt scheme,Chow variety and Quot schemes In fact, Grassimian G(n, m) is a toy model.It parametrize all n− dimensional subspace of Cm,or equivalently,all (n-1)-dimensional pro- jective subspace in Pm−1 9.1. Construction of Hilbert schemes. Hilbert scheme is a moduli space parametrizing all closed subschemes with given Hilbert polyno- mial in a given projective space. Lemma 9.1. (Uniform lemma) Given polynomial P ,if Z ⊂ PN is a closed subscheme with Hilbert poly- nomial P ,Let be I its ideal sheaf.then ∃integer m = m(P ) st:for n ≥ m (i) hi(PN , I(n)) = 0 for i > 0 (ii) I(n) is generated by global sections. 0 N 0 (iii) H (P , O(n))  H (X, OX (n)) Proof. The proof follows induction on N. Recall by cohomological definition of Hilbert polynomial, 0 P (n) = χ(OX (n)) = dimH (X, OX (n)) for n  0 since OX (1) is very ample. N = 0,it’s trival. Now suppose it holds for < N Take H a hyperplane of PN st:each component of X * H. Set J := I ⊗ OH By tensoring OH with

0 → I → OPN → OX → 0 we have J ,→ OH

By induction hyperthesis,∃n0 = n0(N, J ) st:  N Let HP,N := {Z ⊂ P with PZ = P }/'.Then choose a uniform integer n in lemma9.1, HP,N is embedded into Grassimian:

N + n N + n N + n H → G(P (n), ) = G( − P (n), ) P,N n n n 0 0 via Z 7→ {H (I(n)) ⊂ H (OPN (n))} By Plucker embedding, A n m G(n, m) ,→ P = P ∧ C

span(v1, ..vn) 7→ v1 ∧ ... ∧ vn Category view: consider functor

HP,N : sch(S) → sets T 7→ {X → T flat proper fiber isomorphic to N closed subscheme in P with Hilbert polynomal P } Notes on GIT7

Theorem 9.2. HP,N is represented by HP,N with a universal family U → HP,N Proof. idea of construction: step1:Grassmannian can be represented. Given a Noetherian scheme and a vector bundle E on S,rk(E) > r,consider Gr : sch(S) → sets

T 7→ {subbundles of T ×S E with rank = r} 0 let t1, ...tn ∈ H (S,E) be global sections of E generating E. For eachs ∈ S,Es is k(s) vector space,we can associate grassmmi- an this define a scheme G(r, E) over S More precisely,using plucker embedding claim:Gr is represented by G(r, E). consider transformation T : a step2:Hilbert functor is related to Grassmannian  Remark:In general, we can consider the Hilbert scheme parametriz- ing subschemes in a general projective scheme X over S.

Example 9.3. The Hilbert scheme of hypersurface of degree d in Pn n 0 n Hilbp(P ) = P(H (P , O(6))

Example 9.4. The Hilbert scheme of points on smooth curves C of length m.

Example 9.5. The Hilbert scheme of points X[m] on K3 X of length m. Theorem 9.6. (Fogarty,Fujiki,A.Beaville,) X[m] is a compact smooth irreducible holomorphic variety of dimen- sion 2m and its betti number [m] bk(X ) Proof. The Hilbert-chow morphism is just resolution of singularities X[m] → X(m) Away from singular locus,there is a natural holomorphic symplectic (m) 2−forms on Xreg by pullback of from  8 Fei Si

2 Example 9.7. The Hilbert scheme of points Hilm(C ) on affine plane 2 of length m. Surprisingly,Hilm(C ) have very rich geometric structure to representation theory,see12. Theorem 9.8. Define n 2 n M := {(A1,A2, v) ∈ End(C ) ×C :[A1,A2] = 0; stability}//GL(n, C) n where stability means No L C st: Ai(L) ⊂ L, v(L) ⊂ L and group action is conjugate,ie −1 −1 g · (A1,A2, v) := (gA1g , gA2g , gv) Then we have ∼ 2 M = Hiln(C ) Proof. X[m] → X(m) 

Example 9.9. The Nested Hilbert scheme of points on smooth surface X of length m.

Example 9.10. Mumford’s nonreducedness example:

Local structure of Hilbert scheme:tangent space Basic tool of studying local structure of Hilbert scheme is deforma- tion theory.Here only list the main result,for deformation theory,a nice reference is [7] Theorem 9.11. For Y ⊂ X closed subset of a fixed projective scheme X, the

Theorem 9.12. (Harshone) HP,N is connected

9.2. Construction of Chow Variety. Comparing with Hilbert scheme, Chow Varieties parametrize cycles of fixed codimension in an Variety. n • Chow forms of X j P n Let X j P be an irreducible variety of dimension = k and degree = d.set B(X) := {(x, L) ∈ Gr(n − k, n + 1) : x ∈ L 6= X} Z(X) := {L ∈ Gr(n − k, n + 1) : L ∩ X 6= φ} Notes on GIT9

Then the natural projection gives two morphism

p2 B(X) / Z(X)

p1  X Theorem 9.13. Z(X) ⊆ Gr(n−k, n+1) is an irreducible hypersurface of degree = deg(X) = d

Proof. note that the general fiber of p1 : −1 p1 (x) ≈ Gr(n − k − 1, n) and by dimension counting n dim(L) + dim(X) − dim(P ) = (n − k − 1) + k − n = −1

So p1 is a degree= 1 map and thus a birational map,then run dimension counting, dim(B(X) = dim(Z(X) = dim(X) + dim(Gr(n − k − 1, n) = k + (n − k − 1)(k + 1) = dim(Gr(n − k, n + 1) − 1 Fixed a general projective subspace M,N of dim = n − k − 2, n − k respectively.Then C(M,N) := {L ∈ Gr(n − k, n + 1) : M ⊂ L ⊂ N} gives a generic line in Gr(n − k, n + 1) thus,the degree of Z(X) should be deg(Z(X)) = #C(M,N) ∩ Z(X) = deg(X) = d 

Let ΦX be defining equation of Z(X),ie, 0 ΦX ∈ H (Gr(n − k, n + 1), O(d)) .Using Plucker embedding,

Example 9.14. The chow variety parametrizing deg = 1 n Chow(P , 1, k) = Gr(k + 1, n + 1) • A Mumford’s Criterion for Chow stability of projective variety X ⊂ Pn 9.3. Construction of Quot schemes. Quot schemes is a building block for construction of moduli space of sheaves over a variety.

9.4. Hilber-Chow morphism. Given closed a closed subscheme 10 Fei Si

9.5. Some basic Properties of Hilbert scheme and chow vari- ety. Acknowledgements During writing the notes,Dr GuangSheng Yu gave me lots of help in latex.

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ShangHai Mathematical Research Center, Shanghai, 200433, People’s Republic of China E-mail address: [email protected]