2 G. D. DASKALOPOULOS AND R. A. WENTWORTH general, is not true. Thus, unlike the case of divisors, there are situations where k?1 0 and W k?1 = ;. r;d

Transcription

1 ON THE BRILL-NOETHER PROBLEM FOR VECTOR BUNDLES GEORGIOS D. DASKALOPOULOS AND RICHARD A. WENTWORTH Abstract. On an arbitrary compact Riemann surface, necessary and sucient conditions are found for the existence of semistable vector bundles with slope between zero and one and a prescribed number of linearly independent holomorphic sections. Existence is achieved by minimizing the Yang-Mills-Higgs functional. 1. Introduction In this note we exhibit the existence of semistable vector bundles on compact Riemann surfaces with a prescribed number of linearly independent holomorphic sections. This may be regarded as a higher rank version of the classical Brill-Noether problem for line bundles. Fix a compact Riemann surface of genus g 2 and integers r and d satisfying (1.1) 0 d r ; r 2 : Then the main result may be stated as follows: Main Theorem. Let k be a positive integer and suppose that r and d satisfy (1.1). Then the necessary and sucient conditions for the existence of a semistable bundle of rank r and degree d on with at least k linearly independent holomorphic sections are k r and if d 6= 0, r d + (r? k)g. That such a criterion should hold was originally conjectured by Newstead. By analogy with the classical situation of special divisors (cf. [1, 7]) one can dene the higher rank version of the Brill-Noether number: (1.2) k?1 r;d = r2 (g? 1) + 1? k(k? d + r(g? 1)) : Then k?1 is the formal dimension of the locus W k?1 in the moduli space of r;d r;d semistable bundles of rank r and degree d. W k?1 is dened as the closure of r;d the set of stable bundles with at least k linearly independent sections. Note that the condition in the Main Theorem implies that k?1 1, except in r;d the trivial case d = 0 where W k?1 is necessarily empty. The converse, in r;d Date: September 16, 1996 (revised) Mathematics Subject Classication. Primary: 14J60; Secondary: 14D20. Supported in part by NSF grant DMS Supported in part by NSF grant DMS and a Sloan Fellowship. 1

2 2 G. D. DASKALOPOULOS AND R. A. WENTWORTH general, is not true. Thus, unlike the case of divisors, there are situations where k?1 0 and W k?1 = ;. r;d r;d By a dimension counting argument, we can also give a statement, rst proved by Brambila Paz, et. al., concerning the existence of stable bundles: Corollary. (see [5]) For 0 < d < r and r d + (r? k)g, there exists a stable bundle of rank r and degree d with at least k linearly independent holomorphic sections. Instead of the constructive approach to theorems of this type taken in references [9, 10], we use a variational method. More precisely, we study the Morse theory of the Yang-Mills-Higgs functional (cf. [3]). The idea is simply the following: Let (A i ; ~' i ) be a minimizing sequence with respect to the Yang-Mill-Higgs functional (2.1). Here, ~' i = (' i 1 ; : : : ; 'i k ) is a k- tuple of linearly independent holomorphic sections with respect to A i. If the sequence converges to a solution to the k--vortex equation, then for an appropriate choice of the limiting holomorphic structure is semistable (cf. [2]). Otherwise, we show that under the assumptions of the Main Theorem there exist \negative directions" which contradict the fact that the sequence is minimizing. The energy estimates used closely follow [6]. However, an extra combinatorial argument is needed to ensure that the bundles constructed have the correct number of holomorphic sections, and this is where the assumption r d + (r? k)g is needed. Acknowledgements. We would like to thank L. Brambila Paz for introducing us to this problem and for several useful discussions during the preparation of this note. Finally, we are grateful for the warm hospitality of UAM, Mexico and the Max-Planck Institute in Bonn, where a portion of this work was completed. 2. The Yang-Mills-Higgs Functional Let, d, and r be as in the Introduction, and let k be a positive integer. Let E be a xed hermitian vector bundle on of rank r and degree d. Let A denote the space of hermitian connections on E, 0 (E) the space of smooth sections of E, and H A 0 (E) k the subspace consisting of holomorphic k-pairs. Thus, H = (A; ~' = (' 1 ; : : : ; ' k )) : D 00 A ' i = 0; i = 1; : : : ; k : Given a real parameter, we dene the Yang-Mills-Higgs functional: (2.1) f : A 0 (E) k?! R f (A; ~') = kf A k 2 + kd A ' i k ' i '? I i? 2d

3 BRILL-NOETHER PROBLEM FOR VECTOR BUNDLES 3 In the above, the k k denotes L 2 norms. Using a Weitzenbock formula we obtain (cf. [3, Theorem 4.2]) f (A; ~') = 2 kda' 00 i k 2 +?1FA + p 1 2 ' i ' i? 2 2 I and therefore the absolute minimum of f consists of holomorphic k-pairs satisfying the k--vortex equations discussed in [2]. Proposition 2.1. (i) The L 2 -gradient of f? r(a;~') f 1 = D A F A + 1 2? r(a;~') f is given by? DA ' j '? ' j j D A ' j 2;i = A' i? 2 ' i h' i ; ' j i' j (ii) If (A; ~') 2 H, then under the usual identication 1 (; ad E) ' 0;1 (; End E), we have? r(a;~') f? r(a;~') f 1 =?D00 A p?1f A ;i = p?1f A (' i )? 2 ' i ' j ' j 1 A h' i ; ' j i' j (iii) If (A; ~') 2 H is a critical point of f, then either (I) ~' 0 and A is a direct sum of Hermitian-Yang-Mills connections (not necessarily of the same slope), or (II) A splits as A = A 0 A Q on E = E 0 E Q, where (A 0 ; ~') solves the k--vortex equations and A Q is a direct sum of Hermitian-Yang-Mills connections (not necessarily of the same slope). Proof. (i) is a standard calculation, and (ii) follows from (i) via the Kahler identities. We are going to prove (iii). If (A; ~') is critical, then since p P?1FA + 1 k 2 ' j ' j is a self-adjoint holomorphic endomorphism, it must give a splitting A = A 0 A` according to its distinct (constant) eigenvalues 0 ; : : : ; `. Write p?1fa = P k ' j ' j + 0 I C 0 1 I..... C A : 0 ` I ;

4 4 G. D. DASKALOPOULOS AND R. A. WENTWORTH Thus, 0 = p?1f A (' i )? 2 ' i h' i ; ' j i' j =? 1 h' i ; ' j i' j + 0 ' i? 2 2 ' i = 0? ' i ; 2 h' i ; ' j i' j from which we obtain either Case I or Case II, depending upon whether ~' 0. Next, recall that H is an innite dimensional complex analytic variety whose tangent space is given by the kernel of a certain dierential dened in [2, 3.15]. Moreover, H is preserved by the action of the complex gauge group G C. We have the following: Proposition 2.2. If (A; ~') 2 H, then r (A;~') f is tangent to the orbits of G C. In particular, r (A;~') f is tangent to H itself. Proof. Set u = p P?1F A k ' j '? j 2I. By Proposition 2.1 (ii) we have that r (A;~') f = d 1 (u), where d 1 is the dierential dened in [2, 3.15]. The Proposition follows. Because of Proposition 2.2, the critical points of the functional f restricted to H are characterized by Proposition 2.1 (iii). A solution (A(t); ~'(t)), t 2 [0; t 0 ) to the initial value ~' (A;~') f ; (A(0); ~'(0)) = (A 0 ; ~' 0 ) ; is called the L 2 -gradient ow of f starting at (A 0 ; ~' 0 ). Notice that (2.3) d dt f (A(t); ~'(t)) =? r(a(t);~'(t)) f 2 ; and so the energy decreases along the L 2 -gradient ow. Proposition 2.3. Given (A 0 ; ~' 0 ) 2 H, there is a t 0 > 0 such that the L 2 -gradient ow exists for 0 t < t 0. Proof. The proof is an application of the implicit function theorem as in [8]. Finally, we recall from [3, 4, 2] that a holomorphic k-pair (A; ~') 2 H is called -stable if for all holomorphic subbundles 0 6= F E, (F ) < ; and for all proper holomorphic subbundles E ' E containing each ' i, (E=E ' ) >. Here, denotes the Shatz slope = deg=rank. The version of the theorem of Bradlow, Tiwari that we shall need is the following (see [2] for more details):

5 BRILL-NOETHER PROBLEM FOR VECTOR BUNDLES 5 Proposition 2.4. For generic values of the parameter, a holomorphic k- pair (A; ~') is -stable if and only if there exists a pair ( e A; ~~'), related to (A; ~') by an element of G C, satisfying the k--vortex equations: p?1f ea ~' i ~' i? 2 I = 0 : Moreover, such a solution is unique up to real gauge equivalence. 3. Technical Lemmas In this section we collect several results needed for the proof of the Main Theorem. Throughout, E will denote a holomorphic bundle of rank r and degree d on the compact Riemann surface. Lemma 3.1. Let E be as above with 0 d r and h 0 (E) = k. If either (i) E is semistable, or (ii) E satises the k--vortex equation for some 0 < < 1 and E does not contain the trivial bundle as a split factor; then k r and if d 6= 0, r d + (r? k)g. Proof. We rst show that k r. Suppose k r. Thus, E has at least r linearly independent holomorphic sections. If the sections generate E at every point, then E ' O r ; in which case d = 0 and k = r. Suppose the sections fail to generate, so that we can nd a point p 2 and a section of E vanishing at p. Thus E contains O(p) as a subsheaf, which is a contradiction to (ii) (see the denition of -stability above). If (i) is assumed, then E is strictly semistable with d = r, and the bound k r follows from induction on the rank. Note that the second inequality is also satised in this case. Assume 0 < d < r. In both cases (i) and (ii) we obtain 0! O k?! E! F! 0, where F is locally free. By dualizing and taking the resulting long exact sequence in cohomology, we nd 0?! H 0 (F )?! H 0 (E )?! H 0 (O k )?! H 1 (F ) : We are going to show that H 0 (E ) = 0. The result then follows by the Riemann-Roch formula. For (i), H 0 (E ) = 0 by semistability. For (ii), note rst that = 0. For if not, there would be a section s : O! E with s = 6= 0. But could not have any zeros, and so O would be a split factor in E ; hence, also in E. Secondly, we show that H 0 (F ) = 0. Let L F be a subbundle. Then -stability immediately implies c 1 (L ) > > 0. Thus, in particular, F cannot contain O as a subsheaf. This completes the proof. Lemma 3.2. Let E 1, E 2 be holomorphic bundles of rank r 1 ; r 2 and degree d 1 ; d 2, satisfying 0 1 = d 1 =r 1 d 2 =r 2 = 2 1. Suppose h 0 (E 1 ) = k 1 r 1, h 0 (E 2 ) = k 2 r 2, and Furthermore, d 2 + (r 2? k 2? 1)g < r 2 d 2 + (r 2? k 2 )g :

6 6 G. D. DASKALOPOULOS AND R. A. WENTWORTH If d 1 6= 0 assume r 1 d 1 + (r 1? k 1 )g, and k 1 r 2 6= k 2 r 1. If d 1 = 0 and k 1 = r 1, assume r 2 < d 2 + (r 2? k 2 )g. Then there exists a nontrivial extension 0! E 1! E! E 2! 0 such that h 0 (E) = k 1 + k 2. Proof. If k 2 = 0, the result follows from Riemann-Roch. Suppose k 2 1. The condition that the k 2 sections of E 2 lift for some nontrivial extension is k 2 h 1 (E 1 ) < h 1 (E 1 E2 ). Notice that h 1 (E 1 ) = h 0 (E 1 )? d 1 + r 1 (g? 1) = k 1? d 1 + r 1 (g? 1) h 1 (E 1 E 2) = h 0 (E 1 E 2) + r 1 r 2 ( 2? 1 + g? 1) hence, it suces to show that (3.1) or equivalently, that (3.2) r 1 r 2 ( 2? 1 + g? 1) ; k 2 (k 1? d 1 + r 1 (g? 1)) < r 1 r 2 ( 2? 1 + g? 1) ; r 1 (d 2? r 2 + (r 2? k 2 )g)? r 2 d 1 + k 2 d 1? k 1 k 2 + k 2 r 1 > 0 : Now if k 2 = r 2 = d 2, then (3.2) is trivially satised by the hypotheses. Similarly for d 1 = 0. So assume k 2 r 2? 1, d 1 6= 0. Write d 2 = r 2? (r 2? k 2 )g + p, where 0 p < g by assumption. On the other hand, d 1 r 1 d 2 r 2 (d 1 + (r 1? k 1 )g) r 2? (r 2? k 2 )g + p r 2 where if p = 0 then either the rst or the second inequality is strict. This is equivalent to? d 1p g + k 1p + (r 1? k 1 )(r 2? k 2 )(g? 1) r 1 p? r 2 d 1 + k 2 d 1? k 1 k 2 + k 2 r 1 ; and < if p = 0. Therefore, (3.2) will follow from (3.3)? d 1p g + k 1p + (r 1? k 1 )(r 2? k 2 )(g? 1) > 0 ( if p = 0.) Now if p = 0 then (3.3) is trivially satised. Assume that 1 p g? 1. Then? d 1p g + k 1p + (r 1? k 1 )(r 2? k 2 )(g? 1) >?d 1 + r 1 p? (r 1? k 1 )p + (r 1? k 1 )(r 2? k 2 )(g? 1) (r 1? d 1 ) + (r 1? k 1 )(r 2? k 2? 1)(g? 1) 0 ; which proves (3.3), (3.2), and hence the Lemma. In order to get an upper bound on the inmum of the Yang-Mills-Higgs functional in the next section, we shall need the following construction and energy estimate:

7 BRILL-NOETHER PROBLEM FOR VECTOR BUNDLES 7 Lemma 3.3. Assume 0 < d < r, k 1, and r d + (r? k)g. Let F be a holomorphic bundle of degree d and rank r? 1 with h 0 (F ) = k? 1. Then there exists a non-split extension 0! O! E! F! 0 with h 0 (E) = k. Proof. The condition for all of the sections of F to lift is (k? 1)h 1 (O) < h 1 (F ) () g(k? 1) < d + (r? 1)(g? 1) and hence the result. () r < d + (r? k)g + 1 ; Proposition 3.4 (cf. [6, Prop. 3.5]). Let E 1 ; E 2 be hermitian bundles with slope 1 ; 2. Let A 1 ; A 2 be hermitian connections on E 1 ; E 2, and ~' 1 ; ~' 2 be k 1 and k 2 tuples of holomorphic sections. Set k = k 1 + k 2. Let 1 ; 2 and be real numbers satisfying 1 1 < 2 2, and assume that (A 1 ; ~' 1 ) and (A 2 ; ~' 2 ) satisfy the 1 and 2 vortex equations, respectively. Set E = E 1 E 2, ' i = (' 1 i ; 0) for i = 1; : : : ; k 1, and ' k1 +i = (0; ' 2 i ) for i = 1; : : : ; k 2. Then there exist constants " 1 ; " 2 ; > 0 such that for all with kk = " 1, k ~ k " 2, and A = 2 H 0;1 (; Hom(E 2 ; E 1 )) ; ~ 2 0 (E) k ; A1 ; ~' + 0 A ~ 2 H ; 2 it follows that f (A ; ~' + ~ ) < f (A 1 A 2 ; ~')?. Proof. By assumption, It follows that p?1fa1 A p?1fa` k X 1 Xk` ' 1 j ('1 j ) '` j ('` j) = 2 I` ; ` = 1; 2 : k X 2 ' 2 j ('2 j)? 2 I 1 =? 2 I ? 2 I 2 The argument of [6, pp ] shows that there is a constant " 1 such that for and A as in the statement, f? A ; ' 1 1; : : : ; ' 1 k 1 ; ' 2 1; : : : ; ' 2 k 2 < f? A1 A 2 ; ' 1 1; : : : ; ' 1 k 1 ; ' 2 1; : : : ; ' 2 k 2 : Now if we take " 2 suciently small the Proposition follows (note that which norms we use is irrelevant, since and ~' + ~ satisfy elliptic equations, and hence the L 2 norm is equivalent to any other).

8 8 G. D. DASKALOPOULOS AND R. A. WENTWORTH 4. Proof of the Main Theorem Necessity of the conditions follows from Lemma 3.1, and suciency for d = 0 or d = r is clear as well, simply by taking direct sums of trivial line bundles or eective divisors of degree 1, respectively. To prove suciency in general, we shall proceed by induction on the rank. The case r = 2, d = 1 is clear from a direct construction. Assume the Main Theorem holds for bundles of rank < r. We show that it holds for r as well. Let H H denote the subspace of k-pairs (A; ~' = (' 1 ; : : : ; ' k )) such that the sections ' 1 ; : : : ; ' k are linearly independent. Fix as in Assumption 1 of [4], i.e. (E) < = (E) + < +, where + is the smallest possible slope greater that = (E) of a subbundle of E (note that 0 < < 1 and that we also normalize the volume of to be 4). Lemma 4.1. Let m = inf H f. Then 0 m < =(r? 1). Proof. Let F be a vector bundle of degree d and rank r? 1. Then by the inductive hypothesis, Lemma 3.2, and Proposition 2.4, we may assume there exist hermitian connections A 1 and A 2 on O and F, respectively, and holomorphic sections ' 1 6= 0 in H 0 (; O), and ' 2 ; : : : ; ' k linearly independent sections in H 0 (; F ), such that (A 1 ; ' 1 ) and (A 2 ; ' 2 ; : : : ; ' k ) satisfy the 1 and 2 vortex equations, respectively, for 1 =, 2 = d=(r? 1) +. It follows from Lemma 3.3 and Proposition 3.4 that there is a nontrivial extension : 0! O! E! F! 0, an > 0, and a (smooth) ~ such that (A ; ~' + ~ ) 2 H and f (A ; ~' + ~ ) < f (A 1 A 2 ; ' 1 ; : : : ; ' k )? d r? 1? d 2 I F r = 1 2? < r? 1 : Let (A i ; ~' i ) be a minimizing sequence in H. Thus, f (A i ; ~' i )! m. By weak compactness (more precisely, see the argument in [4, Lemma 5.2]) there is a subsequence converging to (A 1 ; ~' 1 ) in the C 1 topology. By the continuity of f with respect to the C 1 topology, Propositions 2.3 and 2.2, and equation (2.3), it follows that (A 1 ; ~' 1 ) is a critical point of f. If the holomorphic structure E 1 dened by A 1 is semistable, then by semicontinuity of cohomology we are nished. We therefore assume E 1 is unstable and derive a contradiction. According to Proposition 2.1 (iii) we must consider the following cases: (I) (II) ~' 1 = 0 ; ~' 1 6= 0 ; E 1 = E 1 E` E 1 = E ' E 1 E` Set j = (E j ), and assume 1 < < `. If ` > 1, then f (A 1 ; ~' 1 ) (`? ) 2 r` (`? 1) 2 r` r` r? 1 > m ;

9 BRILL-NOETHER PROBLEM FOR VECTOR BUNDLES 9 contradicting Lemma 4.1. Similarly, if 1 < 0, then f (A 1 ; ~' 1 ) ( 1? ) 2 r 1 ( 1 ) 2 r 1 r 1 r? 1 > m ; also a contradiction. We therefore rule out these possibilities. We will consider Cases I and II separately. Case I. Let k i = h 0 (E i ). By semicontinuity of cohomology, P` k i k. If ` = 1, then we may replace E` by a Hermitian-Yang-Mills bundle b E` with exactly ^k` = r` sections. Hence, we may assume that d` + (r`? ^k`? 1)g < r` d` + (r`? ^k`)g : For 1 < i < `, the inductive hypothesis implies that we may replace E i by a Hermitian-Yang-Mills bundle Ei b with h 0 ( Ei b ) = ^k di + r i (g? 1) i = ; g the maximal number of sections allowed for d i ; r i, and g. Note that (4.1) d i + (r i? ^k i? 1)g < r i d i + (r i? ^k i )g : If 1 6= 0, then we can replace E 1 by E1 b as above. If 1 = 0, we may replace E 1 with O r 1, with ^k 1 = r 1 k 1 sections. By our choices of ^k i, P` ^k i P` k i k. Let 0 1 < < s < s+1 < < ` 1. Suppose rst that s 6= 0. By Lemma 3.2 there is a nontrivial extension 0! Es b! G! be s+1! 0, with h 0 (G) = ^k s + ^k s+1. Thus, h 0 b E1 b Es?1 G b Es+1 b E` = `X ^k i k : On the other hand, by Proposition 3.4 there is a hermitian connection on be 1 Es?1 b G Es+1 b E` b and linearly independent sections ' 1 ; : : : ; ' k such that f (A; ~') < f (A 1 ; 0) = m, contradicting the minimality of (A 1 ; 0). Now suppose s = 1 = 0, < i for 2 i `. If for any 2 i ` we have r i < d i + (r i? ^k i )g, then by Lemma 3.2 there is a nontrivial extension 0! E1 b! G! Ei b! 0, with h 0 (G) = ^k 1 + ^k i, and Proposition 3.4 yields a contradiction as before. Suppose that for all 2 i `, r i = d i + (r i? ^k i )g. We claim that P` ^k i > k. For if P` ^k i = k, then P` i=2 (r i? ^k i ) = r? k, and hence r > `X i=2 r i = `X i=2 d i + (r i? ^k i )g = d + (r? k)g ; a contradiction. Thus, we may replace b E1 by a bundle b E 0 1 having ^k 0 1 = ^k 1?1 sections. According to Lemma 3.2 there is a nontrivial extension 0! b E 0 1!

10 10 G. D. DASKALOPOULOS AND R. A. WENTWORTH G! b E2! 0, with h 0 (G) = ^k ^k 2, ^k P` i=2 ^k i k, and Proposition 3.4 yields a contradiction as before. Case II. First notice that by the invariance of the Yang-Mills-Higgs equations under the natural action by U(k), we may assume that ' 1 ; : : : ; '1 k form an L 2 -orthogonal set of sections. In particular, we may assume that there exists s k such that ' 1 ; : : : ; '1 s are linearly independent and ' 1 s+1 ; : : : ; '1 k 0. Write E ' = E 0 ' O t, where E 0 ' contains no split factor of O. Set k i = h 0 (E i ), k ' = h 0 (E ' ), k 0 ' = h 0 (E') 0 = k '? t. By semicontinuity of cohomology, k ' + P` k i k. As in Case I, we may replace each E i by a Hermitian-Yang-Mills bundle Ei b such that h 0 ( Ei b ) = ^k i k i, and (4.1) is satised for i = 1; : : : ; `. On the other hand, since E ' satises the k--vortex equation for = + as above, it follows that E 0 ' is -stable. Therefore, 0 6= (E') 0 = (E); and since < 1, we obtain from Lemma 3.1 that r 0 ' d ' + (r 0 '? k')g. 0 Finally, notice that since E 1 is unstable, ` >. We may now apply Lemma 3.2 to E 0 ' and E` b to obtain a nontrivial extension 0! E 0! G! b ' E`! 0, with h 0 (G) = k 0 ' + ^k`. It follows that h 0 G O t E1 b E`?1 b `X = k ' + ^k i k : By Proposition 3.4 there is a hermitian connection A on GO t b E1 be`?1 and linearly independent sections ' 1 ; : : : ; ' k extending ' 1 1 ; : : : ; '1 s such that f (A; ~') < f (A 1 ; ~' 1 ) = m, again contradicting the minimality of m. This completes the proof of the Main Theorem. 5. Proof of the Corollary Let B denote the set of gauge equivalence classes of solutions to the k--vortex equation for bundles of rank r and degree d, where is chosen as in the proof of the Main Theorem. Let B denote the open subset of pairs (E; ' 1 ; : : : ; ' k ) such that the ' i are linearly independent as sections of E. By the Main Theorem and Lemma 3.2 it follows that B is nonempty. One can therefore show as in [2, 4] that B is a smooth complex manifold of dimension r 2 (g? 1) + k(d? r(g? 1)) with a holomorphic map : B! M(r; d), where M(r; d) is the moduli space of semistable bundles of rank r and degree d and where the map sends a pair [E; ~'] to [E]. Let B 0 B denote the subset where the bundle E is stable. Proposition 5.1. Let W k?1 denote the closure of (B 0 r;d ) in M(r; d). If B 0 k?1 6= ;, then every irreducible component of W has dimension k?1 = r;d r;d r 2 (g? 1) + 1? k(k? d + r(g? 1)). Proof. Consider rst a pair [E; ~'] 2 B 0 where h 0 (E) = k. Notice that B 0 is smooth. Moreover, dim [E;~'] B 0 = dim ([E]) W k?1 r;d + dim?1 ([E])

11 BRILL-NOETHER PROBLEM FOR VECTOR BUNDLES 11 and the dimension formula holds, since dim?1 ([E]) = k 2?1. Since h 0 (E) = k is an open condition in W k?1, the Proposition follows from r;d Lemma 5.2. Suppose that E 0 is a semistable (resp. stable) bundle of rank r, degree d, 0 < d r, and h 0 (E 0 ) = k 1. Then there exists a sequence of semistable (resp. stable) bundles E j of the same rank and degree with h 0 (E) = k? 1 and E j! E 0 in M(r; d). Proof. By Lemma 3.1, k r. The case where d = r and E 0 is strictly semistable is trivial. In the other cases, k < r, and we may write 0 : 0! O k! E 0! F! 0 ; where by assumption the connecting homomorphism 0 : H 0 (F )! H 1 (O k ) is injective. Consider fl t : t 2 Dg a smooth local family of line bundles parametrized by the open unit disk D C and satisfying L 0 = O and H 0 (L t ) = 0, t 6= 0. Set G t = O k?1 L t. The semistability of E 0 implies that H 0 (F G t ) = 0. Hence, fh 1 (F G t ) : t 2 Dg denes a smooth vector bundle V over D. Let = f(t) : t 2 Dg be a nowhere vanishing section of V with (0) = 0. Then denes a smooth family of nonsplit extensions 0! G t! E t! F! 0 and a smooth family of connecting homomorphisms t : H 0 (F )?! H 1 (G t ) 0;1 (U) ; where U is the trivial rank k, C 1 vector bundle on. By assumption, 0 is injective; hence, t is injective for small t. It follows that h 0 (E t ) = k? 1 for small t. Furthermore, since E 0 is semistable (resp. stable) then E t is also semistable (resp. stable) for small t. Continuing with the proof of the Corollary, we rst take care of a borderline situation: Lemma 5.3. If r = d+(r?k)g, 0 < d < r, then there exists a stable bundle of rank r and degree d with k linearly independent holomorphic sections. Proof. Note that k < r. Let F be a stable bundle of rank r? k and degree d. Consider an extension : 0! O k! E! F! 0 ; obtained by choosing a basis for H 1 (F ). We claim that any such E is stable. For suppose there is a proper semistable quotient E! Q! 0 with (Q) (E). By the stability of F we also have (Q) 0. Let ` be the rank of the image of the induced map O k! Q. Note that by the choice of we cannot have Q ' O `, and so from Lemma 3.1 we obtain a locally free quotient Q 0 ' Q=O ` of F. Again applying Lemma 3.1 we nd (Q 0 ) = deg Q rk Q? ` (Q) 1? (Q) g (E) 1? (E) g = (F ) : By the stability of F we must have Q 0 ' F, which contradicts the properness of Q.

12 12 G. D. DASKALOPOULOS AND R. A. WENTWORTH The proof of the Corollary is completed by the following Lemma 5.4. Let d and r be as in the statement of the Corollary. We choose k to be the maximal integer such that r d + (r? k)g. By Lemma 5.3, we may also assume r < d + (r? k)g. Let B B be as above. Then no irreducible component of the image of B under the map can be contained in M(r 1 ; d 1 ) M(r 2 ; d 2 ) M(r; d) for any choice of integers r 1 ; r 2 ; d 1 ; d 2 satisfying r 1 + r 2 = r, d 1 + d 2 = d, and d 1 =r 1 = d 2 =r 2 = d=r. Proof. Assume the image of is contained in such a locus; we shall derive a contradiction. Let [E; ~'] 2 B, and suppose (E; ~') ' E 1 E 2. By semicontinuity of cohomology, k k 1 + k 2, where k i = h 0 (E i ). On the other hand, notice that d 1? (r 1? k 1? 1)g < r 1, and d 2? (r 2? k 2? 1)g < r 2, since otherwise by Lemma 3.1, r d + (r? (k + 1))g, contradicting the maximality of k. Now as in the proof of Lemma 3.2 (see (3.1)) we obtain (5.1) (5.2) k 2 (k 1? d 1 + r 1 (g? 1)) < r 1 r 2 (g? 1)? 1 ; k 1 (k 2? d 2 + r 2 (g? 1)) < r 1 r 2 (g? 1)? 1 : On the other hand, we may assume that E 1 and E 2 are stable and nonisomorphic (otherwise the inequalities are even sharper), and by Proposition 5.1 we may assume E 1 E 2 lies in a subvariety S M(r 1 ; d 1 ) M(r 2 ; d 2 ) of dimension at most k 1?1 r 1 ;d 1 + k 2?1 r 2 ;d 2. Finally, we have (5.3) dim [E;~'] B dim [E1 E 2 ] S + dim?1 ([E 1 E 2 ]) : Since?1 ([E 1 E 2 ]) consists of extensions E of E 2 by E 1 such that the sections of E 2 lift, or vice versa, together with k sections of E, it follows that ( (5.4) dim?1 ([E 1 E 2 ]) = max h 1 (E1 E 2)? k 1 h 1 (E 2 ) + k 2? 1 h 1 (E2 E 1)? k 2 h 1 (E 1 ) + k 2? 1 : By combining (5.3) and (5.4) we obtain either or k 1 (k 2? d 2 + r 2 (g? 1)) r 1 r 2 (g? 1)? 1 k 2 (k 1? d 1 + r 1 (g? 1)) r 1 r 2 (g? 1)? 1 ; contradicting either (5.1) or (5.2). This completes the proof of the Lemma. References [1] E. Arbarello, M. Cornalba, P. Griths, and J. Harris, \Geometry of Algebraic Curves," Springer-Verlag, New York, [2] A. Bertram, G. Daskalopoulos, and R. Wentworth, Gromov invariants for holomorphic maps from Riemann surfaces to Grassmannians, JAMS 9, [3] S. Bradlow, Special metrics and stability for holomorphic bundles with global sections, J. Di. Geom. 33 (1991),

13 BRILL-NOETHER PROBLEM FOR VECTOR BUNDLES 13 [4] S. Bradlow and G. Daskalopoulos, Moduli of stable pairs for holomorphic bundles over Riemann surfaces, Int. J. of Math. 2 (1991), [5] L. Brambila Paz, I. Grzegorczyk, and P. Newstead, Geography of Brill-Noether loci for small slope, to appear in the Journal of Algebraic Geometry. [6] G. Daskalopoulos, The topology of the space of stable bundles on a Riemann surface, J. Di. Geom. 36 (1992), [7] P. Newstead, Vector bundles on algebraic curves, problem list prepared by Europroj, [8] J. Rade, On the Yang-Mills heat equation in two and three dimensions, J. Reine. Angew. Math. 431 (1992), [9] N. Sundaram, Special divisors and vector bundles, T^ohoku Math. J. 39 (1987), [10] M. Teixidor I Bigas, Brill-Noether theory for stable vector bundles, Duke Math. J. 62 (1991), Department of Mathematics, Brown University, Providence, RI address: daskal@gauss.math.brown.edu Department of Mathematics, University of California, Irvine, CA address: raw@math.uci.edu