MODULI SPACES AND INVARIANT THEORY 81 §6. M2 (and A2) - Part II (GIT quotients and stability). Let’s summarize where we stand. We want to construct M2 as an orbit space for SL2 acting on P(V6) D. \ We use our standard approach using invariants. The classical invariant SL2 theory tells us that (V6) is generated by I2, I4, I6, I10 = D, and I15 with O 2 a single quadratic relation I15 = g(I2, I4.I6, I10). So our natural candidate for the quotient is Proj (V )SL2 , and the quo- O 6 tient map is f [I2(f) : . : I15(f)] P(2, 4, 6, 10, 15). !→ ∈ Here we got lucky: since Proj R = Proj R(2), we can also write the quotient map as f [I2(f) : . : I10(f)] P(2, 4, 6, 10) = P(1, 2, 3, 5). !→ ∈ Since there are no relations between I2, . , I10 we actually expect the quo- tient to be P(1, 2, 3, 5). If we throw away the vanishing locus of the discriminant, we get the affine chart D = 0 P(1, 2, 3, 5). { $ } ⊂ So our hope is that 3 M2 = A /µ5, where µ5 acts with weights 1, 2, 3. We’ve seen that if we want to embed this cyclic quotient singularity in the affine space, we need at least A8. Of course this construction alone does not guarantee that each point of 3 A /µ5 corresponds to a genus 2 curve and that different points correspond to different curves: this is something we are trying to work out in general. §6.1. Algebraic representations of reductive groups. Consider any repre- sentation G GL(V ). → We want to define the quotient of P(V ) by G, or even more generally a quotient of any projective variety by some action of G. 6.1.1. DEFINITION. Suppose that G is a group and an affine algebraic variety such that the multiplication map G G G and the inverse map G G × → → are regular. Then G is called a linear algebraic group Examples: GL = D(det) Mat , • n ⊂ n SL = V (det 1) Mat , • n − ⊂ n “the maximal torus” (diagonal matrices in GL ), • n “the Borel subgroup” (upper-triangular matrices in GL ), • n SO , Sp . • n n finite groups. • 82 JENIA TEVELEV 6.1.2. REMARK. The terminology “linear algebraic group” can be explained by a theorem of Chevalley: any linear algebraic group is isomorphic to a (Zariski closed) subgroup of GLn for some n. And vice versa, it is clear that any such subgroup is linear algebraic. If we remove “affine” from the definition of an algebraic group, then there are other possibilities, for example an elliptic curve (or an Abelian surface or more generally an Abelian variety) is a projective algebraic group. Non-examples: SL2(Z) and other non-finite discrete groups. • SUn SLn(C) and other non-finite compact linear Lie groups. • ⊂ In fact, we can show that 6.1.3. LEMMA. SUn is Zariski dense in SLn(C). Proof. Indeed, let f be any regular function on SLn(C) that vanishes on SUn. We have to show that it vanishes on SLn(C). It is equally easy to show this for any function holomorphic in the neighborhood of Id SLn(C). ∈ Consider the exponential map A2 exp : Matn(C) GLn(C), A exp(A) = Id + A + + . → !→ 2 This map is biholomorphic in the neighborhood of the origin (the inverse map is given by log) and (locally) identifies SLn(C) with a complex vector subspace sln of complex matrices with trace 0 and SUn with a real sub- t space sun of skew-Hermitian matrices (i.e. matrices such that A + A¯ = 0). So g = f(exp(A)) is a function holomorphic near the origin which vanishes on sun. But since sun + isun = sln, this function vanishes on sln as well: indeed, the kernel of its differential at any point of sun contains sun, and therefore contains sln (being a complex subspace). So all partial derivatives of g vanish along sun, and continuing by induction all higher-order par- tial derivatives of g vanish along sun. So g is identically zero by Taylor’s formula. ! 6.1.4. DEFINITION. A finite-dimensional representation of a linear algebraic group is called algebraic (or rational) if the corresponding homomorphism G GL(V ) is a regular morphism. In other words, an algebraic represen- → tation is given by a homomoprhism a11(g) . a1n(g) G GL(V ), g . .. , → !→ . a (g) . a (g) n1 nn where a (G). ij ∈ O We can generalize this definition to non-linear actions of G on any alge- braic variety X: the action is called algebraic if the “action” map G X X × → is a regular map. Why is this definition the same as above for the linear action? (explain). Finally, we need a notion of a linearly reductive group. MODULI SPACES AND INVARIANT THEORY 83 6.1.5. THEOREM. An algebraic group is called linearly reductive if it satisfies any of the following equivalent conditions: (1) Any finite-dimensional algebraic representation V of G is completely re- ducible, i.e. is a direct sum of irreducible representations. (2) For any finite-dimensional algebraic representation V of G, there exists a G-equivariant projector π : V V G (which is then automatically V → unique). (3) For any surjective linear map A : V W of algebraic G-representations, → the induced map V G W G is also surjective. → Proof. (1) (2). Decompose V = V . V . Suppose G acts trivially ⇒ 1 ⊕ ⊕ k on the first r sub-representations and only on them. Let U V be an ⊂ irreducible subrepresentation. By Schur’s lemma, its projection on any Vi is either an isomorphism or a zero map. It follows that if U is trivial, it is contained in V . V and if it is not trivial, it is contained in V 1 ⊕ ⊕ r r+1 ⊕ . V . So in fact we have a unique decomposition ⊕ k V = V G V , ⊕ 0 where V0 is the sum of all non-trivial irreducible subWe will representa- tions. The projector V V is the projector along V . → G 0 (2) (3). Suppose that the induced map V G W G is not onto. Choose ⇒ → w W G not in the image of V G and choose any projector W G w that ∈ → * + annihilates the image of V G. Then the composition π V A W W W G w −→ −→ → * + is a surjective G-invariant linear map f : V C that annihilates V G. After → dualizing, we have a G-invariant vector f V which is annihilated by all ∈ ∗ G-invariant linear functions on V ∗. However, this is nonsense: we can eas- ily construct a G-invariant linear function on V ∗ which does not annihilate f by composing a G-invariant projector V (V )G (which exists by (2)) ∗ → ∗ with any projector (V )G f . ∗ → * + (3) (1). It is enough to show that any sub-representation W V ⇒ ⊂ has an invariant complement. Here we get sneaky and apply (2) to the restriction map of G-representations Hom(V, W ) Hom(W, W ). → The G-invariant lift of Id Hom(W, W ) gives a G-invariant projector V ∈ → W and its kernel is a G-invariant complement of W . ! Next we study finite generation of the algebra of invariants. Suppose we have a finite-dimensional representation G GL(V ). → We are looking for criteria that imply that (V )G is a finitely generated O algebra. In fact, we already know (Lemma 3.4.4) that it is enough to show existence of a Reynolds operator (V ) (V )G. O → O 6.1.6. LEMMA. The Reynolds operator exists for any algebraic finite-dimensional representation G GL(V ) of a linearly reductive group. In particular, (V )G → O in this case is finitely generated. 84 JENIA TEVELEV Proof. (V ) is an algebra graded by degree and each graded piece (V ) O O n has an induced representation of G. By linear reductivity, there exists a unique G-invariant linear projector R : (V ) (V )G for each n. We n O n → O n claim that this gives a Reynolds operator R. The only thing to check is that R(fg) = fR(g) for any f (V )G. Without loss of generality we can assume that f ∈ O ∈ (V )G and g (V ) . Then f (V ) is a G-sub-representation in (V ) , O n ∈ O m O m O m+n so its G-invariant projector fR(g) should agree with a G-invariant projector R(fg) on (V ) . O m+n ! §6.2. Finite Generation Theorem via the unitary trick. 6.2.1. LEMMA. Any algebraic representation of SLn is completely reducible, i.e. SLn is a linearly reductive group. Proof. We will use a unitary trick introduced by Weyl (and Hurwitz). An algebraic representation of SLn induces a continuous representation of SUn, and any sub-representation of SUn is in fact a sub-representation for SLn by Lemma 6.1.3 (explain). So it is enough to show that any continuous complex representation SU GL(V ) is completely reducible. There are two ways to prove this. n → One is to use the basic lemma above and to construct an equivariant pro- jector V V G for any finite-dimensional continuous representation. Just → like in the case of finite groups, one can take any projector p : V V G and → then take it average p(gv) dµ π(v) = SUn . dµ ' SUn Here µ should be an equivariant measur' e on SUn (a so-called Haar measure), and then of course we would have to prove its existence.