8. Moduli of Vector Bundles on a Curve

66 VICTORIA HOSKINS

8. Moduli of vector bundles on a curve In this section, we describe the construction of the moduli space of (semi)stable vector bundles on a smooth projective curveX (always assumed to be connected) using geometric invariant theory. The outline of the construction follows the general method described in 2.6. First of all, wefix the available discrete invariants, namely the rankn and degreed. This§ gives a moduli problem (n, d), which is unbounded by Example 2.22. We can overcome this unboundedness problemM by restricting to moduli of semistable vector bundles and get a new moduli problem ss(n, d). This moduli problem has a family with the local universal property over a scheme RM. Moreover, we show there is a reductive groupG acting onR such that two points lie in the same orbits if and only if they correspond to isomorphic bundles. Then the moduli space is constructed as a GIT quotient of theG-action onR. In fact, the notion of semistability for vector bundles was introduced by David Mumford following his study of semistability in geometric invariant theory, and we will see both concepts are closely related. The construction of the moduli space of stable vector bundles on a curve was given by Seshadri [37], and later Newstead in [30, 31]. In these notes, we will essentially follow the construction due to Simpson [39] which generalises the curve case to a higher dimensional projective scheme. An in-depth treatment of the general construction following Simpson can be found in the book of Huybrechts and Lehn [16]. However, we will exploit some features of the curve case to simplify the situation; for example, we directly show that the family of semistable vector bundles with fixed invariants over a smooth projective curve is bounded, without using the Le Potier-Simpson estimates which are used to show boundedness in higher dimensions. Convention: Throughout this section,X denotes a connected smooth projective curve. By ‘sheaf’ on a schemeY , we always mean a coherent sheaf of -modules. O Y 8.1. An overview of sheaf cohomology. We briefly recall the definition of the cohomology groups of a sheaf overX. By definition, the sheaf cohomology groupsH i(X, ) are obtained by taking the rightF derived functors of the left exact global sections functorΓ(X,F ). Therefore, − H0(X, ) = Γ(X, ). F ∼ F AsX is projective,H i(X, ) arefinite dimensionalk-vector spaces and, asX has dimension 1, we haveH i(X, ) = 0F fori> 1. The cohomology groups can be calculated using Cech cohomology. ThefiFrst Cech cohomology group is the group of 1-cochains modulo the group of 1-coboundaries. More precisely, given a cover = U i ofX, we letU ij =U i U j and U =U U U denote the double and triple intersections;U { } then we define ∩ ijk i ∩ j ∩ k H1( , ) :=Z 1( , )/B 1( , ) U F U F U F where Z1( , ) := Kerδ = (f ) H0( , ): i, j, k, f f +f = 0 ( ) U F 1 { ij ∈ Uij F ∀ ij − jk ki ∈F U ijk } �i,j B1( , ) := Imageδ = (h h ) for (h ) (U ) U F 0 { i − j i ∈ F i } �i are the group of 1-cochains and 1-coboundaries respectively. If is a refinement of , then there is an induced homomorphismH 1( , ) H 1( , ) and thefirstV cohomology groupU H 1(X, ) is the direct limit of the groupsHU1(F ,→) overV allF covers ofX. In fact, these definitions ofF Cech cohomology groups make senseU forF any schemeX Uand any coherent sheaf ; however, higher dimensionalX, will in general have non-zero higher degree cohomology groups.F The above definition does not seem useful for computational purposes, but it is because of the following vanishing theorem of Serre. Theorem 8.1 ([14] III Theorem 3.7). LetY be an affine scheme and be a coherent sheaf on Y ; then for alli>0, we have F Hi(Y, )=0. F MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 67

Consequently, we can calculate cohomology of coherent sheaves on a separated scheme using an affine open cover. Theorem 8.2 ([14] III Theorem 4.5). LetY be a separated scheme and be an open affine cover ofY . Then for any coherent sheaf onY and anyi 0, the natural homomorphismU F ≥ Hi( , ) H i(Y, ) U F → F is an isomorphism. The assumption thatY is separated is used to ensure that the intersection of two open affine subsets is also affine (see [14] II Exercise 4.3). Hence, we can apply the above Serre vanishing theorem to all multi-intersections of the open affine subsets in the cover . U Exercise 8.3. Using the above theorem, calculate the sheaf cohomology groups i 1 H (P , 1 (n)) O P by taking the standard affine cover ofP 1 consisting of two open sets isomorphic toA 1. One of the main reasons for introducing sheaf cohomology is that short exact sequences of coherent sheaves give long exact sequences in cohomology. The category of coherent sheaves onX is an abelian category, where a sequence of sheaves is exact if it is exact at every stalk. Furthermore, a short exact sequence of sheaves 0 0 →E→F→G→ induces a long exact sequence in sheaf cohomology 0 H 0(X, ) H 0(X, ) H 0(X, ) H 1(X, ) H 1(X, ) H 1(X, ) 0, → E → F → G → E → F → G → which terminates at this point as dimX = 1. Definition 8.4. For a coherent sheaf onX, we leth i(X, ) = dimH i(X, ) as ak-vector space. Then we define the Euler characteristicF of by F F F χ( )=h 0(X, ) h 1(X, ). F F − F In particular, the Euler characteristic is additive on short exact sequences: χ( )=χ( )+χ( ). F E G 8.2. Line bundles and divisors on curves. Example 8.5.

(1) Forx X, we let X ( x) denote the sheaf of functions vanishing atx; that is, for U X∈, we have O − ⊂ ( x)(U)= f (U):f(x) = 0 . OX − { ∈O X } By construction, this is a subsheaf of X and, in fact, X ( x) is an invertible sheaf on X. O O − (2) Forx X, we letk x denote the skyscraper sheaf ofx whose sections overU X are given∈ by ⊂ k ifx U k (U) := x 0 else.∈ � The skyscraper sheaf is not a locally free sheaf; it is a torsion sheaf which is supported 0 1 on the pointx. SinceH (X, kx) =k x(X)=k andH (X, kx) = 0, we haveχ(k x) = 1. There is a short exact sequence of sheaves (2) 0 ( x) k 0 →O X − →O X → x → where forU X, the homomorphism X (U) k x(U) is given by evaluating a function f (U)⊂ atx ifx U. We can tensorO this exact→ sequence by an invertible sheaf to obtain ∈O X ∈ L 0 ( x) k 0 →L − →L→ x → 68 VICTORIA HOSKINS

where ( x) is also an invertible sheaf, whose sections overU X are the sections of over U whichL − vanish atx. Hence, we have the following formula ⊂ L (3) χ( ) =χ( ( x)) + 1. L L − Deﬁnition 8.6. LetX be a smooth projective curve.

(i) A Weil divisor onX is afinite formal sum of pointsD= x X mxx, form x Z. ∈ ∈ (ii) The degree ofD is degD= mx. � (iii) We sayD iseffective, denotedD 0, ifm x 0 for allx. (iv) For a rational functionf k(X�),≥ we define the≥ associated principal divisor ∈ div(f)= ordx(f)x, x X(k) ∈� where ord (f) is the order of vanishing off atx (as is a discrete valuation ring, x O X,x we have a valuation ordx :k(X) ∗ Z). (v) We say two divisors are linearly equivalent→ if their difference is a principal divisor. (vi) For a Weil divisorD, we define an invertible sheaf (D) by O X (D)(U) := 0 f k(X) ∗ : (divf+D) 0 . OX { }∪{ ∈ | U ≥ } Remark 8.7.

(1) ForD= x, this definition of X (D) coincides with the definition of X ( x) above. (2) AsX is− smooth, the notionsO of Weil and Cartier divisors coincide. TheO above− construc- tionD X (D) determines a homomorphism from the group of Weil divisors modulo linear equivalence�→O to the Picard group of isomorphism classes of line bundles, and this homomorphism is an isomorphism asX is smooth. In particular, any invertible sheaf L overX is isomorphic to an invertible sheaf X (D). For proofs of these statements, see [14] II 6. O § For an effective divisorD, the dual line bundle ( D) is isomorphic to the ideal sheaf of the (possibly non-reduced) subschemeD X given byO this− effective divisor (see [14] II Proposition 6.18) and we have a short exact sequence⊂ (4) 0 ( D) k 0, →O X − →O X → D → wherek D denotes the skyscraper sheaf supported onD; thusk D is a torsion sheaf. This short exact sequence generalises the short exact sequence (2). In particular, any effective divisor admits a non-zero section X X (D). In fact, a line bundle X (D) admits a non-zero section if and only ifD is linearlyO → equivalentO to an effective divisorDO by [14] II Proposition 7.7.

Deﬁnition 8.8. The Grothendieck group ofX, denotedK 0(X), is the free group generated by classes [ ] ,for a coherent sheaf onX, modulo the relations [ ] [ ]+[ ] = 0 for short exact sequencesE 0 E 0. E − F G →E→F→G→ We claim that there is a homomorphism

(5) (det, rk) :K 0(X) Pic(X) Z → ⊕ which sends a locally free sheaf to (det := rk , rk ). To extend this to a homomorphism E E ∧ E E E onK 0(X), we need to define the map for coherent sheaves : for this, we can take afinite resolution of by locally free sheaves, which exists becauseXF is smooth, and use the relations F definingK 0(X). This map is surjective and in fact is an isomorphism (see [14] II, Exercise 6.11). Using this homomorphism we can define the degree of any coherent sheaf onX. Definition 8.9. (The degree of a coherent sheaf). (i) IfD is a divisor, we define deg (D) := degD. O X (ii) If is a torsion sheaf, we define deg = x X length( x). (iii) IfF is a locally free sheaf, deg = deg(detF ). ∈ F (iv) IfE is a coherent sheaf, we deEfine deg :=E� deg(det ), where det is the image of inF Pic(X) under the homomorphism (5).E F F F MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 69

In fact, the degree is uniquely determined by theﬁrst two properties and the fact that the degree is additive on short exact sequences (that is, if we have a short exact sequence 0 0, then deg = deg + deg ); see [14] II, Exercise 6.12. →E→F→G→ F E G Example 8.10. The skyscraper sheafk x has degree 1. 8.3. Serre duality and the Riemann-Roch Theorem.

Proposition 8.11 (Riemann-Roch Theorem, version I). Let = X (D) be an invertible sheaf on a smooth projective curveX. Then L O χ( (D)) =χ( ) + degD OX O X Proof. We can writeD=x + +x y y and then proceed by induction on 1 ··· n − 1 −···− m n+m Z. The base case whereD = 0 is immediate. Now assume that the equality has been proved∈ forD; then we can deduce the statement forD+x (andD x) from the equality (3). − � 1 Definition 8.12. For a smooth projective curveX, the sheaf of differentialsω X :=Ω X onX 0 is called the canonical sheaf. The genus ofX isg(X) :=h (X,ω X ). The canonical bundle is a locally free sheaf of rank 1 = dimX; see [14] II Theorem 8.15. Theorem 8.13 (Serre duality for a curve). LetX be a smooth projective curve and be a locally free sheaf overX. There exists a natural perfect pairing E 0 1 H (X, ∨ ω ) H (X, ) k. E ⊗ X × E → Hence,H 0(X, ω ) = H1(X, ) andh 0(X, ω ) =h 1(X, ). E ∨ ⊗ X ∼ E ∨ E ∨ ⊗ X E Remark 8.14. 1 (1) Once one chooses an isomorphismH (X,ω X ) k, the pairing can be described as the composition � 0 1 1 1 H (X, ∨ ω ) H (X, ) H (X, ∨ ω ) H (X,ω ) k E ⊗ X × E → E ⊗E⊗ X → X � where thefirst map is a cup-product and the map ∨ X is the trace. (2) In fact, Serre duality can be generalised to any projectiveE ⊗E→ schemeO (see [14] III Theorem 7.6 for the proof) whereω X is replaced by a dualising sheaf. IfY is a smooth projective n variety of dimensionn, then the dualising sheaf is the canonical sheafω Y = ΩY , which is thenth exterior power of the sheaf of differentials, and thefirst cohomology∧ group is replaced by thenth cohomology group. An important consequence of Serre duality on curves is the Riemann–Roch Theorem. Theorem 8.15 (Riemann–Roch theorem, version II). LetX be a smooth projective curve of genusg and let be a degreed invertible sheaf onX. Then L 0 0 h (X, ) h (X, ∨ ω ) =d+1 g. L − L ⊗ X − Proof. First, we use Serre duality to calculate the Euler characteristic of the structure sheaf χ( ) :=h 0(X, ) h 1(X, ) = 1 h 0(X,ω ) = 1 g. OX O X − O X − X − Then by Serre duality and the baby version of Riemann–Roch it follows that 0 0 h (X, ) h (X, ∨ ω ) =χ( )=d+χ( (X)) =d+1 g L − L ⊗ X L O − as required. � There is a Riemann–Roch formula for locally free sheaves due to Weil. The proof is given by induction on the rank of the locally free sheaf with the above version giving the base case. To go from a given locally free sheaf to a locally free sheaf of lower rank � one uses a short exact sequence E E 0 � 0, →L→E→E → where is an invertible subsheaf of of maximal degree (this forces the quotient � to be locally free; seeL Exercise 8.23 for the existenceE of such a short exact sequence). E 70 VICTORIA HOSKINS

Corollary 8.16 (Riemann–Roch for locally free sheaves on a curve). LetX be a smooth projective curve of genusg and be a locally free sheaf of rankn and degreed overX. Then F χ( )=d+n(1 g). F − Example 8.17. On a curveX of genusg, the canonical bundle has degree 2g 2 by the Riemann–Roch Theorem: − h0(X,ω ) h 1(X, ) =g 1 = degω + 1 g. X − O X − X − 1 Therefore, onP , we haveω 1 = 1 ( 2). P O P −