GEORGIOS D. DASKALOPOULOS AND RICHARD A. WENTWORTH. Dedicated to Professor Karen K. Uhlenbeck, on the occasion of her 60th birthday.


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1 CONVERGENCE PROPERTIES OF THE YANGMILLS FLOW ON KÄHLER SURFACES GEORGIOS D. DASKALOPOULOS AND RICHARD A. WENTWORTH Dedicated to Professor Karen K. Uhlenbeck, on the occasion of her 60th birthday. Abstract. Let E be a hermitian complex vector bundle over a compact Kähler surface with Kähler form ω, and let D be an integrable unitary connection on E defining a holomorphic structure D on E. We prove that the YangMills flow on (, ω) with initial condition D converges, in an appropriate sense which takes into account bubbling phenomena, to the double dual of the graded sheaf associated to the ωhardernarasimhanseshadri filtration of the holomorphic bundle (E, D ). This generalizes to Kähler surfaces the known result on Riemann surfaces and proves, in this case, a conecture of Bando and Siu. 1. Introduction The main concern of this paper is the relationship between the YangMills functional and stability of holomorphic vector bundles on Kähler surfaces. By analogy with finite dimensional symplectic geometry on the one hand, and geometric invariant theory on the other, minimizing solutions to the YangMills equations can be viewed as zeros of a moment map on an infinite dimensional symplectic manifold, and the appearance of the stability condition for the existence of such solutions may be regarded as a version of the KempfNess theorem. More generally, one might expect a correspondence between two stratifications: the stableunstable manifolds defined by the gradient flow of the YangMills functional, and the algebraic stratification coming from the maximally destabilizing one parameter subgroups. This point of view originated in the work of Atiyah and Bott [AB] and was developed further by Donaldson [Do1, Do2] (see [Ki] for a general treatment of symplectic geometry vs. geometric invariant theory in finite dimensions). Let us recall some of the key points. Given a holomorphic structure E on a complex vector bundle E of rank R over a Kähler manifold, one can associate a filtration by holomorphic subsheaves, called the HarderNarasimhan filtration, whose successive quotients are semistable. The topological type of the pieces in the associated graded obect is encoded into an Rtuple µ = (µ 1,..., µ R ) of rational numbers called the HarderNarasimhan type of (E, E ). Atiyah and Bott used the HarderNarasimhan type to define a stratification of the infinite dimensional space A of holomorphic structures on E. The group G C of complex automorphisms of E, or the complex gauge group, acts on A in a manner that preserves the stratification. The Date: November 1, G.D. supported in part by NSF grant DMS R.W. supported in part by NSF grant DMS
2 2 DASKALOPOULOS AND WENTWORTH main result of [AB] is that, when is a Riemann surface, the stratification obtained from the HarderNarasimhan filtration is G C equivariantly perfect, and this leads to a recursive calculation of the cohomology of the moduli spaces of stable bundles, in certain cases. On the other hand, fixing a hermitian structure H on E, one may identify A with the space A H of unitary connections on E via the map which sends a unitary connection D to its (0,1) part D = E (in higher dimensions we require the integrability condition that the curvature F D be of type (1,1)). The YangMills functional, which associates to a connection the L 2 norm of its curvature, can be used as a Morse function on A H. For a fixed holomorphic bundle (E, E ), the HarderNarasimhan type gives an absolute lower bound on the YangMills number of any connection in the isomorphism class. Up to a topological term, the YangMills number is the same as the HermitianYangMills number, which is defined as the L 2 norm of the contraction ΛF D of the curvature with the Kähler form (see (2.4)). If µ is the HarderNarasimhan type of (E, E ), then: HYM( µ) := 2π R µ 2 i HYM(D) := i=1 ΛF D 2 dvol for all D such that (E, D ) is holomorphically isomorphic to (E, E ) (see Cor for a proof of this result for Kähler surfaces). Atiyah and Bott conectured that, on a Riemann surface, the gradient flow of the YangMills functional should converge at infinity, achieving the lower bound expressed above. Moreover, the stableunstable manifold stratification should coincide with the HarderNarasimhan stratification. That this is indeed the case follows from the work of several authors. First, Donaldson proved the long time existence of the L 2 gradient flow for the YangMills functional on any Kähler manifold [Do1, DoKr]. For Riemann surfaces, the asymptotic convergence of the gradient flow and the equivalence of the two stratifications was established in [D] (see also [R] for a more analytic approach). A key fact which makes the two dimensional case more tractable is that the YangMills functional on Riemann surfaces satisfies the equivariant PalaisSmale Condition C. In higher dimensions Condition C fails. More seriously, the YangMills flow can develop singularities in finite time. Therefore, one cannot expect a Morse theory in the classical sense. For Riemannian fourmanifolds, Taubes formulated an extended Morse theory by attaching bundles with varying topologies in order to compensate for curvature concentration, or bubbling. In this approach one uses the strong gradient of the YangMills functional associated to a complete Riemannian metric on the space A H, which exists for all time by the fundamental existence theorem for ODE s. In this way Taubes established the connectivity of the moduli space of self dual connections in certain cases. Moreover, he was able to calculate the stable homotopy groups as predicted by the conecture of Atiyah and Jones [T]. In the case of Kähler surfaces it is more natural to consider the L 2 rather than the strong gradient flow. Long time existence, as mentioned above, is guaranteed. Donaldson used this flow to prove the correspondence between antiselfdual connections and stable bundles (see [Do1], and more generally
3 THE YANGMILLS FLOW ON KÄHLER SURFACES 3 [Do2] and [UY] for holomorphic bundles in higher dimensions). This is now known as the Hitchin Kobayashi Correspondence, or the DonaldsonUhlenbeckYau Theorem. In [BS], Bando and Siu extended the correspondence to coherent analytic sheaves by considering singular hermitian metrics with controlled curvature. They also conectured that the relationship between the YangMills flow and the HarderNarasimhan filtration which holds on Riemann surfaces should analogously be true in higher dimensions. We use the word analogous because even when considering vector bundles the HarderNarasimhan filtration in higher dimensions may be given only by subsheaves, and the associated graded obects may not be locally free. Indeed, it is for this reason that the generalization of BandoSiu naturally leads to a conecture on the behavior of the flow for vector bundles. The purpose of this paper is to prove the conecture of Bando and Siu for holomorphic bundles on Kähler surfaces. To state the result precisely, let Gr hns ω (E, E ) denote the HarderNarasimhan Seshadri filtration of (E, E ) with respect to the Kähler form ω, and let Gr hns ω (E, E ) be its double dual. To clarify, we note here that the HarderNarasimhanSeshadri filtration is actually a double filtration which takes into account the possibility that the successive factors in the Harder Narasimhan filtration may only be semistable as opposed to stable (see Prop. 2.6). Thus, the individual factors in the associated graded obect are all stable. Since is now assumed to be a surface, the double dual is a vector bundle and carries a YangMills connection which realizes the hermitian structure as a direct sum of HermitianEinstein metrics. Now for unstable bundles the flow may not converge in the usual sense; again, because of bubbling. However, one can always extract subsequential Uhlenbeck limits which are YangMills connections on bundles with a possibly different topology than the original E. The bundles are isometric, and the connections converge, away from a singular set of codimension four 1. In dimension four, the singular set is a finite collection of points. For the precise definition, see Prop below. Our result is that on a Kähler surface, the Uhlenbeck limits are independent of the subsequence and are determined solely by the isomorphism class of the initial holomorphic bundle (E, E ). More precisely: Theorem 1 (Main Theorem). Let be a compact Kähler surface, E a hermitian vector bundle, and D 0 an integrable unitary connection on E inducing a holomorphic structure E = D Let D denote the YangMills connection on Gr hns ω (E, E ) referred to above. Let D t be the time t solution to the YangMills flow with initial condition D 0. Then as t, D t converges in the sense of Uhlenbeck to D. We now give a sketch of the ideas involved in the proof of the Main Theorem and explain the organization of the paper. In Section 2 we lay out the definitions of the HarderNarasimhan Seshadri filtration and its associated graded obect. We review the YangMills flow and the notion of an Uhlenbeck limit. We also discuss other YangMills type functionals associated to invariant convex functions on the Lie algebra of the unitary group. These are closely related to L p norms, 0. 1 In this paper, convergence of connections will always be modulo real gauge equivalence.
4 4 DASKALOPOULOS AND WENTWORTH and they will play an important role not only in distinguishing the various strata, but also because one actually cannot expect good L 2 behavior in the constructions that follow. Much of the difficulty in proving the Main Theorem arises from the fact that the Harder Narasimhan filtration is not necessarily given by subbundles. Since we have restricted our attention in this paper to surfaces, the individual factors in the filtration are themselves locally free, but the successive quotients may have point singularities. These points are essentially the locus where one can expect bubbling to occur along the flow when the filtration is by subbundles one can show there is no bubbling and they are therefore a fundamental aspect of the problem and not a mere technical annoyance. In Section 3 we therefore analyze the degree to which the singularities in the filtration can be resolved by blowing up. To be more precise, we are interested in comparing the HarderNarasimhan filtrations of E and π (E), where π : is a sequence of monoidal transformations, and the Kähler metric ω ε on is an εperturbation by the components of the exceptional divisor of the pullback of the Kähler metric ω on. In Thm. 3.1 we prove that under the assumption that the successive quotients of the HarderNarasimhan filtration are stable, there is a resolution such that for sufficiently small ε, the HarderNarasimhan filtration of π (E) is given by subbundles and its direct image by π coincides with the HarderNarasimhan filtration of E. The situation is more complicated for semistable factors, and the resolution of the filtration by subbundles may not correspond to the HarderNarasimhan filtration for any ε > 0. Nevertheless, this analysis is sufficient for our purposes. In particular, we introduce the notion of an L p approximate critical hermitian structure. Roughly speaking, this is a smooth hermitian metric on a holomorphic bundle whose curvature in the direction of the Kähler form is close in the L p sense to a critical value determined by its HarderNarasimhan type (see Def. 3.9). We prove Thm which states that there exist L p approximate critical hermitian structures for all 1 p <. This result is an L p version of a conecture attributed to Kobayashi. Interestingly, the method does not seem to extend to p =. The first step in the proof of the Main Theorem is to determine the HarderNarasimhan type of an Uhlenbeck limit. Since the HermitianYangMills numbers are monotone along the flow one can show that for an initial condition which is a sufficiently close approximate critical hermitian structure, the Uhlenbeck limit of a sequence along the flow must have the correct HarderNarasimhan type. Then a length decreasing argument for the YangMills flow, which closely resembles Hartman s result for the harmonic map flow, implies that any initial condition must have Uhlenbeck limits of the correct type (see Thm. 4.1). The second step in the proof of the Main Theorem is to show that the holomorphic structure on the Uhlenbeck limit coincides with the double dual of the associated graded sheaf of the Harder NarasimhanSeshadri filtration. The approach here is necessarily completely different from that of [D]. The main idea is to generalize an argument of Donaldson who constructs limiting holomorphic maps from the sequence of complex gauge transformations defined by the sequence of connections
5 THE YANGMILLS FLOW ON KÄHLER SURFACES 5 along the flow. Instead of a map on the entire initial bundle, we show that maps can be formed for each of the pieces in the filtration separately. The result then follows by an inductive argument. The proof that the limiting holomorphic structure is the correct one is largely independent of the details of the flow. Indeed, we only use the fact that the YangMills numbers of a descending sequence in a complex gauge orbit are absolutely minimizing. Since this result is in some way disoint from Thm. 1, we formulate it separately (see Thm. 5.1). Recall that sequences minimize YangMills energy YM(D) if and only if they minimize HermitianYangMills energy HYM(D): Theorem 2 (Minimizing Sequences). Let be a compact Kähler surface, E a hermitian vector bundle, and D 0 an integrable unitary connection on E inducing a holomorphic structure E = D 0. Let µ 0 be the HarderNarasimhan type of (E, E ), and let D denote the YangMills connection on Gr hns ω (E, E ). Suppose D is a sequence of smooth unitary connections in the complex gauge orbit of D 0 such that HYM(D ) HYM( µ 0 ) as. Then there is a subsequence (also denoted ) and a finite set of points Z an. such that (1) E and Gr hns ω (E, E ) are L p 2,loc. isometric on \ Zan. for all p; (2) D D in L 2 loc. away from Zan.. Acknowledgement. This paper is a substantially revised version of an earlier preprint. We thank the referee for an exceptionally careful reading of that initial manuscript and for numerous and helpful comments. The references [Bu1, Bu2, Bu3] suggested by the referee, in particular, simplified some of our arguments and allowed us to remove the restriction to proective surfaces required in the original paper. 2. Preliminaries 2.1. Stability and the HarderNarasimhan Filtration. Let be a complex surface. The singular set Sing(E) of a coherent analytic torsionfree sheaf E is the closed subvariety where E fails to be locally free. Since dim C = 2, the singular set of a torsionfree sheaf is a locally finite collection of points and reflexive sheaves are locally free (cf. [Ko, Cor. V.5.15 and V.5.20]). A subsheaf S E of a reflexive sheaf E is said to be saturated if the quotient Q = E/S is torsionfree. In general, the saturation of a subsheaf S in E, denoted Sat E (S), is the kernel of the sheaf map E Q/ Tor(Q), where Tor(Q) is the torsion subsheaf of Q. Note that S is a subsheaf of Sat E (S) with a torsion quotient. A saturated subsheaf of a reflexive sheaf is reflexive (cf. [Ko, Prop. V.5.22]). We will also need the following result, whose proof is standard: Lemma 2.1. Let E be a torsionfree sheaf. Suppose S 1 S 2 E are subsheaves with S 2 /S 1 a torsion sheaf. Then Sat E (S 1 ) = Sat E (S 2 ). Now assume that is compact with a Kähler form ω. We will assume the volume of with respect to ω is normalized to be vol() = 2π. The ωslope µ(e) of a torsionfree sheaf E is
6 6 DASKALOPOULOS AND WENTWORTH defined by: (2.1) µ ω (E) = deg ω(e) = 1 c 1 (E) ω. rk(e) rk(e) We define µ max (E) to be the maximal slope of a subsheaf of E, and µ min (E) to be the minimal slope of a torsionfree quotient of E. A torsionfree sheaf E is ωstable (resp. ωsemistable) if for all subsheaves F E with 0 < rk(f ) < rk(e), µ ω (F ) < µ ω (E) (resp. µ ω (F ) µ ω (E)). When the Kähler form is understood we shall sometimes refer to E simply as stable or semistable, and we will also omit subscripts and write µ(e). Proposition 2.2 (cf. [Ko], Thm. V.7.15). Let E be a torsionfree sheaf. Then there is a filtration: 0 = E 0 E 1 E l = E, called the HarderNarasimhan filtration of E (abbr. HN filtration), such that Q i = E i /E i 1 is torsionfree and semistable. Moreover, µ(q i ) > µ(q i+1 ), and the associated graded obect Gr hn ω (E) = l i=1 Q i is uniquely determined by the isomorphism class of E It will be convenient to denote the subsheaf E i in the HN filtration by F hn i (E), or by F hn i,ω (E), when we wish to emphasize the role of the Kähler structure. The collection of slopes µ(q i ) is an important invariant of the isomorphism class of a torsionfree sheaf. For a torsionfree sheaf E of rank R construct an Rtuple of numbers µ(e) = (µ 1,..., µ R ) from the HN filtration by setting: µ i = µ(q ), for rk(e 1 ) + 1 i rk(e ). We call µ(e) the HarderNarasimhan type of E. These invariants admit a natural partial ordering which will be very relevant to this paper. For a pair µ, λ of Rtuple s satisfying µ1 µ R, λ 1 λ R, and R i=1 µ i = R i=1 λ i, we define: (2.2) µ λ λ, for all k = 1,..., R. k µ k The importance of this ordering is that it defines a stratificaton of the space of holomorphic structures on a given complex vector bundle over a Riemann surface. See [AB, 7] for more details. We will make use of the following simple fact: Lemma 2.3. Let µ = (µ 1,..., µ R ) and λ = (λ 1,..., λ R ) be nonincreasing Rtuples as above. Suppose there is a partition 0 = R 0 < R 1 < < R l = R such that µ i = µ for all pairs i, satisfying: R k i, R k, k = 1,..., l. If R k µ R k λ, for all k = 1,..., l, then µ λ. Several technical properties of the HN filtration will also play a role in this paper. We again omit the proofs. Proposition 2.4. (1) Let Ẽ be torsionfree and E Ẽ with T = Ẽ/E a torsion sheaf (E) = ker(f hn(ẽ) T ), and F hn(ẽ) = Sat E e(f hn (E)). supported at points. Then F hn i (2) Let E be a torsionfree sheaf. Let E 1 = F hn 1 (E), and Q 1 = E/E 1. Then In particular, F hn i+1 (E)/E 1 = F hn i (Q 1 ). F hn i+1(e) = ker(e Q 1 / F hn i (Q 1 )). i i i
7 THE YANGMILLS FLOW ON KÄHLER SURFACES 7 (3) Consider an exact sequence: 0 S E Q 0 of torsionfree sheaves on with µ min (S) > µ max (Q). Then the HarderNarasimhan filtration of E is given by: 0 = E 0 F hn 1 (S) F hn k (S) = S F hn k+1 (E) F hn l (E) = E, where F hn i (E) = F hn i (S), i k, and F hn k+i (E) = ker(e Q/ F hn i (Q)) for i = 0, 1,..., l k. In particular, Gr hn (E) Gr hn (S) Gr hn (Q). We point out the analogous filtrations for semistable sheaves: Proposition 2.5 (cf. [Ko], Thm. V.7.18). Let Q be a semistable torsionfree sheaf. Then there is a filtration 0 = F 0 F 1 F l = Q, called a Seshadri filtration of E, such that F i /F i 1 is stable and torsionfree. Moreover, µ(f i /F i 1 ) = µ(q) for each i. The associated graded obect Gr s ω(q) = l i=1 F i/f i 1, is uniquely determined by the isomorphism class of Q. Finally, the double filtration whose associated graded sheaf appears in the statements of the main results in the Introduction is obtained by combining Prop. s 2.2 and 2.5: Proposition 2.6. Let E be a torsionfree sheaf. Then there is a double filtration {E i, }, called a HarderNarasimhanSeshadri filtration of E (abbr. HNSfiltration), with the following properties: if {E i } l i=1 is the HN filtration of E, then E i 1 = E i,0 E i,1 E i,li = E i, and the successive quotients Q i, = E i, /E i, 1 are stable torsionfree sheaves. µ(q i, ) > µ(q i+1, ). The associated graded obect: l l i Gr hns ω (E) = i=1 =1 is uniquely determined by the isomorphism class of E. Q i, Moreover, µ(q i, ) = µ(q i,+1 ) and 2.2. YangMills Connections and Uhlenbeck Limits. Given a smooth complex vector bundle E of rank R, let Ω p,q (E) denote the space of smooth (p, q) forms with values in E. We will regard a holomorphic structure on E as given by a operator E : Ω p,q (E) Ω p,q+1 (E) satisfying the integrability condition E E = 0. We will sometimes denote the holomorphic structure on E explicitly by (E, E ). When this structure is understood, we will confuse the notation for the holomorphic bundle and the sheaf of holomorphic sections by E, as we have done in the previous section. Now suppose we are given a smooth hermitian metric H on (E, E ). Then there is a uniquely determined Hunitary connection D on E satisfying D = E, where D denotes the (0, 1) part of D (D will denote the (1, 0) part). We will sometimes denote this connection by D = ( E, H). Conversely, given a unitary connection D on E whose curvature F D = D D is of type (1, 1) (i.e. F 0,2 D = 0), then D = E defines a holomorphic structure on E, and D = (D, H). For a fixed hermitian metric, let A H denote the space of Hunitary connections D on E, and let A 1,1 0,2 H denote those satisfying FD = 0. The discussion above gives an identification of A1,1 H with the space A of integrable operators, or holomorphic structures, on E. We denote by G the space of
8 8 DASKALOPOULOS AND WENTWORTH unitary gauge transformations acting on A H by pulling back. Via the identification A 1,1 H A, it is clear that we have an action on A 1,1 H by GC, the complexification of G. We call G C the complex gauge group. Notice that G C also acts on the space of hermitian metrics on E, where g(h) is defined by: g(h)(s 1, s 2 ) = H(gs 1, gs 2 ). Since many norms will be used in this paper, let us emphasize the following: if a is a hermitian or skewhermitian endomorphism on an Rdimensional hermitian vector space with eigenvalues {λ 1,..., λ R }, we set: { R } 1/2 (2.3) a = λ i 2. i=1 For a hermitian vector bundle E, let u(e) denote the subbundle of End E consisting of skewhermitian endomorphisms. If a is a section of u(e), then a will denote the pointwise norm defined by (2.3). Given a Kähler metric ω on, the YangMills Functional (abbr. YM Functional) is defined by YM(D) = F D 2 L 2 (ω), and the HermitianYangMills Functional (abbr. HYM Functional) is defined by HYM(D) = Λ ω F D 2 L 2 (ω). Here, Λ ω denotes contraction with the Kähler form, and Λ ω F D is called the HermitianEinstein tensor associated to D. Since for any D A 1,1 H IV.3.29]): (2.4) YM(D) = HYM(D) + 4π 2 (2c 2 (E) c 2 1(E)), we have (cf. [Ko, the YM and HYM functionals have the same critical points on A 1,1 H ; namely, the YangMills Connections D F D = 0. We also note the Kähler identities: (2.5) D F D = 1 (D D )Λ ω F D. Given a holomorphic bundle E (, ω), a hermitian metric H is called a HermitianEinstein Metric if there is a constant µ such that 1 Λ ω F ( E,H) = µ I E, where I E denotes the identity endomorphism of E. If is compact, then because of the normalization vol() = 2π it is necessarily the case that µ = µ ω (E) (see (2.1)). The celebrated theorem of DonaldsonUhlenbeckYau relates stability to the existence of a HermitianEinstein metric (cf. [Do1, Do2, UY]): Theorem 2.7. A holomorphic vector bundle E on a compact Kähler manifold (, ω) admits a HermitianEinstein metric if and only if it is holomorphically split into a direct sum of ωstable bundles, all with slope = µ(e). The following is standard (cf. [AB, 4] or [Ko, IV. 3]): Proposition 2.8. Let D A 1,1 H be a YM connection on a hermitian vector bundle over a Kähler manifold. Then there is an orthogonal splitting (E, D) = l i=1 (Q i, D i ), where 1 Λ ω F Di = µ i I Qi, for constants µ i. In case is compact, µ i = µ(q i ), and the critical values of the YM functional on A 1,1 H are discrete.
9 THE YANGMILLS FLOW ON KÄHLER SURFACES 9 For the proof of the following version of Uhlenbeck compactness, see [U1] (and also, [UY, Thm. 5.2]): Proposition 2.9. Let be a compact Kähler surface and E a complex vector bundle with hermitian metric H. Assume D is a sequence of integrable unitary connections on E such that ΛF D L is bounded uniformly for all. Fix p > 4. Then there is: (1) a subsequence { k }, (2) a finite subset Z an., (3) a smooth hermitian vector bundle (E, H ) \ Z an. with a finite action connection D on E, (4) for any compact set W \ Z an., an L p 2 isometry τ W : (E, H ) W (E, H) W. such that for W W \ Z an., τ W = τ W W, and τ W (D k ) D weakly 2 in L p 1 (W ). We will call any D arising in this way an Uhlenbeck limit of the sequence D. We will often omit the isometries τ W from the notation, and simply identify (E, H ) with (E, H) on \Z an.. Also, it is useful to note here that weak L p 1,loc. convergence implies convergence of local holomorphic frames. This may be proven, for example, using Webster s proof of the NewlanderNirenberg theorem [W]. We refer to [DW] for more details on the following result: Proposition Let D D in L p 1,loc. ( \ Zan. ) for some p > 4, as in Prop Then for each x \ Z an. there is: (1) a coordinate neighborhood U \ Z an. of x, (2) a sequence {s } of D holomorphic frames on U, (3) a D holomorphic frame s on U, (4) and a subsequence { k } {}, such that s k s in C 1 (U). Next, we turn to a situation where the Uhlenbeck limits are YangMills: Proposition If in addition to the assumptions in Prop. 2.9 we assume D ΛF D L 2 0, then any Uhlenbeck limit D is YangMills. Moreover, the triple (E, D, H ) extends smoothly to, and the extension has a holomorphic orthogonal splitting as a direct sum: l (Q (i), D (i), H (i) ), where H (i) i=1 is a HermitianEinstein metric on Q (i). Proof. The last statement follows by the removable singularities theorem (cf. [U2]) and the argument cited above (cf. Prop. 2.8). To see that D is YangMills, we argue as follows: by the compactness of the embedding L p 1 C0 and the fact that D k D weakly in L p 1,loc., we may assume 2 We will denote weak convergence by and strong convergence by.
10 10 DASKALOPOULOS AND WENTWORTH C loc. D 0 k D, and ΛF Dk ΛF D in L p loc.. It follows that D ΛF Dk D ΛF D in L 2 1,loc., say, where L 2 k is the dual space to L2 k. On the other hand, D ΛF Dk = D k ΛF Dk +[D D k, ΛF Dk ], L so by the added hypothesis we also have, D ΛF 2 loc. Dk 0. This implies D ΛF D = 0. The statement that D is YangMills now follows from the Kähler identities (2.5). This completes the proof. Corollary With the assumptions as in Prop. 2.11, ΛF Dk L p ΛF D for all 1 p <. Proof. Set f k = ΛF Dk ΛF D. Then by the proof of Prop. 2.11, f k 0 in L p loc. and D f k 0 strongly in L 2 loc.. By Kato s inequality, f k is uniformly bounded in L 2 1,loc., so f k L2 loc. 0. Since f k is also uniformly bounded in L L, it follows that f k p 0 for all p. We conclude this subsection with a technical result on the boundedness of second fundamental forms. This will be important in the proof of the main result. Let Ω be an open set. Let {D }, D be integrable unitary connections on E Ω with D D in L p 1,loc. (Ω), for all p, 1 p <. Let π (resp. π ) be (smooth) proections onto D (resp. D ) holomorphic subbundles of E. Let us assume the following: (1) ΛF D is bounded in L loc. (Ω) uniformly in ; (2) π is bounded in L 2 1,loc. (Ω) uniformly in ; (3) π π L loc. (Ω) 0 as. Lemma With the assumptions above, π is bounded in L p 2,loc. (Ω) uniformly in, for all p. In particular, the second fundamental forms D π are locally uniformly bounded. Proof. Set p = π π. By [D, Lemma 3.2] we may write: (2.6) D (p ) = {D p, D p } + {D Γ, p } + {D p, Γ } + {Γ, Γ } + G, where the brackets {, } indicate a bilinear combination of the two arguments with bounded coefficients, Γ = D D, and G is uniformly bounded in L loc. (Ω). Since Γ is uniformly bounded in L p 1,loc. (Ω) and p 0 in L loc. (Ω), it follows from [GiM, Thm. 1.4] and [Gi, Thm. VI.1.5] that p is bounded in C 1,α loc. (Ω) for any 0 < α < 1, uniformly in. The Lp 2,loc. (Ω) bound then follows from the L p elliptic estimate (cf. [GT, Thm. 9.11]) applied to (2.6). For the second statement, write: D π = D π + Γ π, and note that the right hand side is locally bounded since it is in L p 1,loc. for p > The YangMills Flow. The basic obect of interest in this paper is the YangMills flow for a family of unitary connections D = D(t) = D t. This is the L 2 gradient flow of the YM functional, which may be written as follows: (2.7) D t = D F D, D(0) = D 0 A 1,1 H.
11 THE YANGMILLS FLOW ON KÄHLER SURFACES 11 By the work of Donaldson and Simpson (cf. [Do1, Si]), (2.7) has a unique solution in (A 1,1 H /G) [0, ). Furthermore, D t lies in a single G C orbit, namely G C D 0, for all t [0, ). One way to see this is to fix the operator E = D 0 on E and look at the family of hermitian metrics H = H(t) = H t satisfying the HermitianYangMills flow equations: (2.8) Ht 1 H t = 2 ( ) 1 Λ ω F Ht µi E, H(0) = H0. t In the equation above, F Ht denotes the curvature of D t = ( E, H t ), and µ = µ ω (E) (see (2.1)) depends only on the topology of E and the Kähler form ω. The two systems (2.7) and (2.8) are equivalent up to gauge. See [Do1] for more details. Also notice that it is easy to factor out the trace part of the connection and gauge transformations. Therefore, in the following we shall assume that the solutions to the above equations all preserve determinants. Also, in the following and throughout the paper, we will often omit the Kähler form ω from the notation when doing so is unambiguous. The following is an immediate consequence of (2.7) and (2.8) (see [Do1, Prop. 16] for a proof): Lemma (1) Let D t be a solution to (2.7). Then F Dt / t = t F Dt, and: d dt F D t 2 L 2 = 2 D t F Dt 2 L 2 0. Hence, t YM(D t ) and t HYM(D t ) are nonincreasing. (2) The pointwise norm ΛF Dt 2 satisifies t ΛF D t 2 + ΛF Dt 2 0. Uhlenbeck compactness applied to the flow gives the following: Proposition Let D t be a solution to (2.7) on a compact Kähler surface, and fix p > 4. For any sequence t we can find the following: (1) a subsequence {t k }, (2) a finite set of points Z an., (3) a smooth hermitian vector bundle (E, H ) with a finite action YangMills connection D on E, (4) and on any compact set W \Z an., an L p 2 isometry τ W : (E, H ) W (E 0, H 0 ) W, such that for W W \ Z an., τ W = g W W, and τ W (D tk ) D, weakly in L p 1 (W ). Moreover, the triple (E, D, H ) extends smoothly to, and the extension has a holomorphic orthogonal splitting as a direct sum: l i=1 (Q(i), D (i), H (i) ), where H (i) metric on Q (i). is a HermitianEinstein Proof. By Lemma 2.14 (2) and the maximum principle, ΛF Dt L is decreasing in t, and is therefore uniformly bounded. By [DoKr, Prop ], lim t D t ΛF Dt L 2 = 0. The weak L p 1,loc. convergence along a subsequence to a YM connection now follows from Prop As in 2.2, we will call D an Uhlenbeck limit of the flow. Note that a priori, D may depend on the choice of subsequence {t k }, however we will see shortly that this is not the case.
12 12 DASKALOPOULOS AND WENTWORTH Lemma Let D t be a sequence of connections along the YM flow with Uhlenbeck limit D. Then for t t 0 0, ΛF D L ΛF Dt L ΛF Dt0 L. Proof. As stated above, ΛF Dt L is decreasing in t. Fix t 0. Then for any 1 p < and sufficiently large we have: ΛF Dt L p (2π) 1/p ΛF Dt L (2π) 1/p ΛF Dt L (recall vol() = 2π). On the other hand, by Cor. 2.12, lim ΛF Dt L p = ΛF D L p, for all p. Hence, ΛF D L p (2π) 1/p ΛF Dt L. Letting p, we conclude ΛF D L ΛF Dt L. Lemma If D is the Uhlenbeck limit of D t, then ΛF Dt Moreover, lim t HYM(D t ) = HYM(D ). L p ΛF D for all 1 p <. Proof. The first part of the lemma follows from Cor The second part is immediate, since by Lemma 2.14, t HYM(D t ) is nonincreasing, and HYM(D t ) HYM(D ). The Uhlenbeck limits obtained from the YangMills flow are unique: Proposition Let D t be the solution to the YM flow (2.7), and suppose D is an Uhlenbeck limit for some sequence D t with singular set Z an.. Then D t D in L 2 loc. away from Z an.. In particular, the Uhlenbeck limit of the flow is uniquely defined up to gauge. Proof. By (2.4) and Lemma 2.17, lim YM(D t ) = HYM(D ) + 4π 2 (2c 2 (E) c 2 t 1(E)). From (2.7) and Lemma 2.14 (1), if t t: t D t D t 2 L D s 2 t 2 t s ds = DsF Ds 2 L ds 2 L 2 t = 1 t d 2 ds F D s 2 L ds = 1 ( 2 YM(Dt ) YM(D t ) ). 2 t L Since the limit of YM(D t ) as t exists, D t D t L 2 0 as t and. Since D 2 loc. t D, the convergence follows. For the last statement, if D and D are two Uhlenbeck limits with singular sets Z an. and Z an., then the argument above shows that D and D are gauge equivalent on \Z an. Z an.. In particular, the holomorphic bundles (E, D ) and (Ẽ, D ) are isomorphic on \Z an. Z an.. But then their reflexive extensions are isomorphic as well by Hartogs theorem. Remark In light of Prop. 2.18, we may speak of the Uhlenbeck limit of the flow D t. Note, however, that we have not established that the singular set Z an. is independent of the subsequence {t }. We next turn to the HN type of the Uhlenbeck limit: Lemma Let D = g (D 0 ) be a sequence of complex gauge equivalent integrable connections on a complex vector bundle E of rank R with hermitian metric H 0. Let S be a coherent subsheaf of (E, D 0 ) of rank r. Suppose that L 1 ΛF 1 D a, where a L 1 ( 1u(E)), and that the eigenvalues λ 1 λ R of a (counted with multiplicities) are constant almost everywhere. Then: deg(s) i r λ i.
13 THE YANGMILLS FLOW ON KÄHLER SURFACES 13 Proof. Since deg(s) deg(sat E (S)), we may assume that S is saturated. Let π denote the orthogonal proection onto g (S) with respect to the hermitian metric H 0. This is a bounded measurable hermitian endomorphism of E, smooth away from the singular set of E/S. The condition of being a weakly holomorphic proection implies π 2 = π = π, π D π = 0, where π = I E π. In fact, the π are L 2 1 sections of the smooth endomorphism bundle of E (cf. [UY, 4]), and conversely, any such π defines a unique saturated subsheaf 3. Moreover, the usual degree formula applies (see [Si, Lemma 3.2]), so: deg(s) = 1 2π = 1 (2.9) 2π ( ( ) Tr 1 ΛFD π D π 2) dvol 1 Tr ( ) 1 ΛF D π dvol 2π Tr (aπ ) dvol + 1 Tr (( 1 ΛF D a ) ) π dvol, 2π We now use the following result from linear algebra: Claim. Let V be a finite dimensional hermitian vector space of complex dimension R, L End(V ) a hermitian operator with eigenvalues λ 1 λ R (counted with multiplicities). Let π = π 2 = π denote the orthogonal proection onto a subspace of dimension r. Then Tr(Lπ) i r λ i. Proof (sketch). Let {e i } R i=1 be a unitary basis with Le i = λe i. If we set α i = πe i 2, then the claim follows by showing that the affine function F (α 1,..., α R ) = i r λ i R i=1 λ iα i, is nonnegative on the affine set 0 α i 1, R i=1 α i = r. This may be proven by considering the extreme values of the α i s and using induction on r and R. We omit the details. Given the claim, along with the fact that π L 1, and the normalization vol() = 2π, it follows from (2.9) that deg(s) i r λ i + (1/2π) 1 ΛF D a L 1. Now let in this inequality to complete the proof of the lemma. Recall the partial ordering (2.2) of HN types of holomorphic structures on E: Proposition Let D be a sequence along the YM flow on a bundle E of rank R with Uhlenbeck limit D. Let µ 0 = (µ 1,..., µ R ) be the HN type of E with the holomorphic structure D 0, and let λ = (λ 1,..., λ R ) the type of D. Then µ 0 λ. Proof. Let {E i } l i=1 be the HN filtration of D 0. Then deg(e i) = rk(e i ) µ. By Lemma 2.17, ΛF D L 1 ΛF D. The type λ corresponds to the (constant) eigenvalues of ΛF D. Lemma 2.20 applied to S = E i implies deg(e i ) rk(e i ) λ, for each i = 1,..., l. follows from Lemma 2.3. The following generalizes a result in [AB] to Kähler surfaces: The proposition now 3 We will often confuse the notation π of the proection operator with the subsheaf it defines.
14 14 DASKALOPOULOS AND WENTWORTH Corollary Let µ = (µ 1,..., µ R ) be the HarderNarasimhan type of a rank R holomorphic vector bundle (E, E ) on. Then for all unitary connections D in the G C orbit of (E, E ): R i=1 µ 2 i 1 ΛF D 2 dvol. 2π Proof. Let D t denote the YM flow with initial condition D. By Lemma 2.14 (2): (2.10) ΛF Dt 2 dvol ΛF D 2 dvol, for all t 0. Let D be the Uhlenbeck limit along a subsequence t. By Lemma 2.17: (2.11) ΛF D 2 dvol = lim ΛF Dt 2 dvol. Now 1ΛF D has constant eigenvalues λ which satisfy µ λ by Prop It follows from [AB, 12.6] (see also Prop below) that R i=1 µ2 i R i=1 λ2 i. This fact, along with (2.10) and (2.11), prove the result Other HermitianYangMills Type Functionals. One of the technical difficulties in dealing with holomorphic vector bundles of rank bigger than two is that the HermitianYangMills numbers do not distinguish the different critical levels (or equivalently, different HarderNarasimhan types) of the functionals YM and HYM. This was resolved by Atiyah and Bott in the case of vector bundles over Riemann surfaces by introducing YangMills type functionals corresponding to higher symmetric functions of the eigenvalues of 1 F D (cf. [AB, 8]). In the case of vector bundles over higher dimensional Kähler manifolds there are analytic restrictions on the type of functionals we may consider. In this subsection we will explain in some detail how to use these functionals in order to distinguish the different critical levels. Let u(r) denote the Lie algebra of the unitary group U(R). Fix a real number α 1. Then for a u(r), a skew hermitian matrix with eigenvalues 1 λ 1,..., 1 λ R, let ϕ α (a) = R =1 λ α. It is easy to see that we can find a family ϕ α,ρ, 0 < ρ 1, of smooth convex adinvariant functions such that ϕ α,ρ ϕ α uniformly on compact subsets of u(r) as ρ 0. Hence, by [AB, Prop ] it follows that ϕ α is a convex function on u(r). For a given number N, define: (2.12) HYM α,n (D) = ϕ α (ΛF D + 1N I E ) dvol, and HYM α (D) = HYM α,0 (D). Notice that HYM = HYM 2 is the ordinary HYM functional. Also, by a slight abuse of notation, we will set HYM α,n ( µ) = HYM α ( µ + N) = 2πϕ α ( 1 ( µ + N)), where µ+n = (µ 1 +N,..., µ R +N) is identified with the diagonal matrix diag(µ 1 +N,..., µ R +N). In particular: (2.13) HYM ( µ) = 2π R µ 2 i. i=1
15 Lemma The functional a equivalent to the L α norm. THE YANGMILLS FLOW ON KÄHLER SURFACES 15 ( ϕ α (a) dvol) 1/α, defines a norm on L α (u(e)) which is Proof. First, notice that there are universal constants C, C (depending on R) such that for any real numbers λ 1,..., λ R, and α 1: ( R ) α/2 ( 1 λ i 2 1 R α λ i ) C C i=1 i=1 ( R R ) α ( R ) α/2 λ i α C λ i C λ i 2. Applying this to the eigenvalues of a and by integrating over : 1 (Tr aa ) α/2 dvol ϕ α (a) dvol C (Tr aa ) α/2 dvol. C The lemma follows. We will require three important properties of the functionals HYM α,n : Proposition (1) If µ λ, then ϕ α ( 1 µ) ϕ α ( 1 λ) for all α 1. i=1 (2) Assume µ R 0 and λ R 0. If ϕ α ( 1 µ) = ϕ α ( 1 λ) for all α in some set A [1, ) possessing a limit point, then µ = λ. Proof. (1) follows from [AB, 12.6]. For (2), consider f(α) = ϕ α ( 1 λ) and g(α) = ϕ α ( 1 µ) as functions of α. As complex valued functions, f, g clearly have analytic extensions to C \ {α 0}. Suppose that f(α) = g(α) for all α A. Then by analyticity, f(α) = g(α) for all α C \ {α 0}. If µ λ, then there is some k, 1 k R, such that µ i = λ i for i < k, and µ k λ k ; say, µ k > λ k. Then for any α > 0: ( µk ) α R ( µi ) α = R ( λi i=1 ) α R. i=1 λ k λ k i=k λ k i=k Letting α, we obtain a contradiction; hence, the result. Proposition Let D t be a solution of (2.7). Then for any α 1 and any N, t HYM α,n (D t ) is nonincreasing. Proof. Because we can approximate ϕ α by smooth convex adinvariant functions ϕ α,ρ ϕ α, it suffices to show that the functional t ϕ α,ρ(λf Dt + 1N I E ) dvol, is nonincreasing along the flow for any ρ > 0. This follows from integrating the following inequality: (2.14) ( / t)ϕ α,ρ ( ΛFDt + 1N I E ) + ϕα,ρ ( ΛFDt + 1N I E ) 0. To prove (2.14), simplify the notation by setting f = ΛF Dt + 1N I E, ϕ = ϕ α,ρ. We first claim that: (2.15) (ϕ f)(x) = ϕ f(x) ( D tf, D t f) + ϕ f(x) ( D t f).
16 16 DASKALOPOULOS AND WENTWORTH Indeed, (ϕ f)(x) = d d(ϕ f)(x) = d ϕ f(x) (D tf), because ϕ is invariant under the adoint action. Then: d ϕ f(x) (D tf) = d ( ) ϕ f(x) ( D tf) = ϕ f(x) ( D tf, D t f) ϕ f(x) (D t D t f) = ϕ f(x) ( D tf, D t f) + ϕ f(x) ( D t f). The claim (2.15) follows. Since ϕ is a positive definite quadratic form: ( ϕ (ΛF Ht + 1N I E ) ) (x) ϕ ( Dt (ΛF Dt + 1N I E ) ) ( = ϕ t (ΛF D t + ) 1N I E ) = t ϕ(λf D t + 1N I E ), where the first equality follows from Lemma 2.14 (1). proof. This verifies (2.14) and completes the Proposition Let D be a subsequential Uhlenbeck limit of D t, where D t is a solution to (2.7). Then for any α 1 and any N, lim t HYM α,n (D t ) = HYM α,n (D ). L Proof. Let D be the Uhlenbeck limit of a sequence D t. By Lemma 2.17, ΛF p Dt ΛF D for all p. Hence, by Lemma 2.23, HYM α,n (D t ) HYM α,n (D ). The convergence in general follows by Prop. 2.25, since HYM α,n (D t ) is nonincreasing in t. 3. Blowup of the HarderNarasimhan Filtration and Approximate Metrics The first goal of this section is to show how to resolve the HarderNarasimhan filtration of a holomorphic bundle E by passing to a modification π : of the Kähler surface. We shall see that this procedure works well when the associated graded obect consists of stable sheaves. While this result is not directly needed for the remaining sections, we have chosen to present it here since it may be of independent interest. In the case where semistable quotients appear, the relationship between the HN filtration of E and that of π E is more complicated, and we have not attempted a complete description. As pointed out by the referee, much of this analysis has already appeared in the work of Buchdahl [Bu1, Bu2]. The second goal is to show that there is a Hermitian metric Ĥ on π E so that the Hermitian YangMills numbers of (π E, Ĥ) with respect to a natural family of Kähler metrics ω ε are arbitrarily close (in an appropriate norm) to the slopes of Gr hn ω (E) on for all ε sufficiently small. This is an important first step toward finding an approximate critical hermitian structure on itself. We formulate this latter result below in Thm The argument we give here circumvents the need for an explicit description of the HN filtration of π E Resolution of the HarderNarasimhan Filtration. By Prop. 2.2, a holomorphic bundle E admits a filtration by saturated subsheaves E i so that the successive quotients Q i = E i /E i+1 are semistable and torsionfree. It follows that the E i are locally free sheaves, i.e. vector bundles. They
17 THE YANGMILLS FLOW ON KÄHLER SURFACES 17 may, however, fail to be subbundles at finitely many points. Equivalently, the quotients Q i are not necessarily locally free. For each i, we have an exact sequence of sheaves: 0 Q i Q i T i 0, where Q i is locally free and T i is a torsion sheaf supported at finitely many points. Define the set Z i to be the support of T i, and let Z alg. = k i=1 Z i. We will refer to Z alg. as the singular set of the filtration {E i } (in this paper we ignore multiplicities). We will prove the following: Theorem 3.1. Let E be a holomorphic vector bundle over a smooth Kähler surface with Kähler form ω. Let Gr hn ω (E) = k i=1 Q i be the ωhn filtration of E. We assume that the Q i are stable. Then there is a smooth surface obtained from by a sequence of monoidal transformations π : with exceptional set e satisfying the following properties: (1) π(e) = Z alg. ; (2) There exists a smooth, closed (1, 1) form η on and a number ε 0 > 0 such that ω ε = π ω + εη is a family Kähler metrics on for all ε 0 ε > 0. (3) There { is a} number ε 1, ε 0 ε 1 > 0 such that for every 0 < ε ε 1, the ω ε HN filtration F hn i,ω (Ê) ε of Ê = π E is independent of ε and is a filtration by subbundles. (4) For 0 < ε ε 1, ε 1 as above, F hn i,ω ε (Ê) = Sat b E ( π F hn i (E) ). Moreover, if we write: Gr hn ˆk ω ε (Ê) = Q i=1 i, then for 0 < ε ε 1, ˆk = k, and (π Qi ) Q i. Remark 3.2. The assumption that Q i is stable is necessary. One can find examples where the pullback of a semistable bundle is unstable for all ε > 0. More generally, the resolution of the HarderNarasimhan filtration of E may not coincide with the HarderNarasimhan filtration of π E for any ε > 0. We begin by comparing stability of sheaves Ê with stability of their direct images E = π Ê on. To do this, we need to define Kähler metrics. First, consider the case where is the blowup of at a point and e is the exceptional divisor. Then there is a smooth, closed form η of type (1, 1) in the class of c 1 ( O b ( e) ) on such that ω ε = π ω + εη is positive for all ε > 0 sufficiently small. This can be constructed quite explicitly (cf. [GH, pp ]). In general, since is a sequence of blowups at points, we can construct a family of Kähler forms on be iterating the above argument. We state this precisely as: Lemma 3.3. Let π : be a sequence of monoidal transformations with exceptional set e, and choose a Kähler form ω on. Then there is a smooth, closed (1, 1) form η on and a number ε 0 > 0 with the following properties: (1) ω ε = π ω + εη is a Kähler form on for all ε 0 ε > 0; (2) For any closed 2form α on, π α η = 0. b Consider a family of Kähler forms in the manner of Lemma 3.3. Note that we do not normalize the volume of (, ω ε ), though we still assume the normalization on (, ω); so vol(, ω ε ) 2π as ε 0. In the following, let us agree that the slope µ(e) of a sheaf on will be taken with
18 18 DASKALOPOULOS AND WENTWORTH respect to ω. For a sheaf Ê on, we denote by µ ε (Ê) the slope of Ê with respect to the metric ω ε. Similarly, a subscript ε will indicate that the quantity in question is taken with respect to ω ε. With this understood, we have the following: Proposition 3.4. Given a holomorphic vector bundle Ê with E = π Ê, and given δ > 0, there is ε 1 > 0, depending upon Ê, such that for all 0 < ε ε 1 we have the following inequalities: (1) µ(e) δ µ ε (Ê) µ(e) + δ, (2) µ max (E) δ µ max,ε (Ê) µ max(e) + δ, (3) µ min (E) δ µ min,ε (Ê) µ min(e) + δ. Proof. Since µ min (E) = µ max (E ), part (3) will follow from part (2) applied to Ê. Parts (1) and (2) are essentially contained in [Bu1, Lemma 5]. The statement there assumed Ê is a pullback bundle, but the proof works as well for general Ê. As a consequence, we have (cf. [Bu2, Prop. 3.4 (d)]): Corollary 3.5. Let Ê and E = π Ê be as above. If E is ωstable, then there is a number ε 2 > 0, depending upon Ê, such that Ê is ω εstable for all 0 < ε ε 2. An inductive argument repeatedly using Prop. 3.4 implies convergence of the HN type: Corollary 3.6. Let Ê be a holomorphic vector bundle with E = π Ê. Let µ ε denote the HN type of Ê with respect to ω ε and µ the HN type of E with respect to ω. Then µ ε µ as ε 0. Next, we state a general result on resolution of filtrations: Proposition 3.7. Let 0 = E 0 E 1 E l 1 E l = E, be a filtration of a holomorphic vector bundle E by saturated subsheaves E i, and set Q i = E i /E i 1. Then there is a sequence of monoidal transformations π : with exceptional set e and a filtration 0 = Ê 0 Ê1 Êl 1 Êl = Ê = π E, such that each Êi = Sat be (π E i ) is a subbundle of Ê. If we let Q i = Êi/Êi 1, we also have exact sequences 0 Q i π Qi T i 0, where T i is a torsion sheaf supported at the singular set of Q i. Moreover, π(e) = Z alg., the union of the singular sets of Q i ; π Ê i = E i ; and Q i = (π Qi ). Proof. The proof is standard resolution of singularities (cf. [Bu1, 3] for the step 2 filtration; the general argument then follows by induction). The form Êi = Sat be (π E i ) follows from Lemma 2.1. The remaining statements follow easily, and we omit the details. Proposition 3.8. Let π : be a sequence of monoidal transformations with exceptional set e as above. Let Ê be a holomorphic vector bundle, and set E = π Ê. Let E i = F hn i (E) denote the HN filtration of E, and assume that the successive quotients E i /E i 1 are stable. Let ĵ : π E i Ê denote the induced map. We also assume that the sheaves Sat be (ĵ(π E i )) are subbundles of Ê. Then for ε > 0 sufficiently small, the HN filtration {F hn i,ε (Ê)} with respect to the Kähler metrics of Lemma 3.3 is independent of ε and is given by F hn i,ε (Ê) = Sat E b(ĵ(π E i )).
19 THE YANGMILLS FLOW ON KÄHLER SURFACES 19 Proof. We will proceed by induction on the length of the HN filtration of E. If E is stable, then the result follows from Cor Assume now that E is not stable, and define Êi = Sat be (ĵ(π E i )). Note that with this definition it follows as in the proof of Prop. 3.4 that π Ê i = E i. We claim that Ê1 = F hn 1,ε (Ê) for ε sufficiently small. This follows because (1) by the hypothesis and Cor. 3.5, Ê 1 is stable for ε sufficiently small, and (2) we may arrange that µ ε (Ê1) > µ max ( Q 1 ) for ε sufficiently small, where Q 1 = Ê/Ê1. The claim then follows from Prop. 2.4 (3). Let Q 1 = E/E 1. By pushing forward, we have 0 Q 1 Q 1 T 0, where Q 1 = π Q1, and T is a torsionsheaf supported at points. Hence, by Prop. 2.4 (1), the HN filtrations of Q 1 and Q 1 are related by F hn i (Q 1 ) = ker(f hn i ( Q 1 ) T ). For convenience, set F i = F hn i (Q 1 ), Fi = F hn i ( Q 1 ). Notice that Q 1 and Q 1 continue to satisfy the hypothesis of the proposition. Hence, by induction, we may assume that for ε sufficiently small the HN filtration of Q 1 is given by F hn i,ε ( Q 1 ) = {Sat bq1 (ĵ(π Fi ))} (where ĵ now is the induced map to Q 1 ). Now π F i π Fi with a torsion quotient. By Lemma 2.1, Sat bq1 (ĵ(π Fi )) = Sat bq1 (ĵ(π F i )). This, combined with Prop. 2.4 (2) gives F hn i,ε (Ê) = ker(ê / Q 1 Sat bq1 (ĵ(π F i 1 ))). Moreover, F i 1 = E i /E 1. Clearly then, F hn i,ε (Ê) contains ĵ(π E i ) with a torsion quotient. Thus, again by Lemma 2.1, F hn i,ε (Ê) = Sat E b(ĵ(π E i )). Proof of Thm By Prop. 3.7, the HN filtration {E i = F hn i,ω (E)} of E admits a resolution to a filtration {Êi} by subbundles on. By Prop. 3.8, the filtration {Êi} is the HN filtration with respect to ω ε for ε sufficiently small. The remaining assertions follow easily An Approximate Critical Hermitian Structure. For a fixed holomorphic structure on E, a critical point of the functional H HYM( E, H) is called a critical hermitian structure [Ko, p. 108]. By the Kähler identities (2.5), this occurs if and only if the connection ( E, H) is YangMills. The general form for a critical hermitian structure is therefore (see Prop. 2.8): (3.1) 1 Λω F ( E,H) = µ 1I Q1 µ l I Ql. In the above, the holomorphic structure E on E splits E = l i=1q i, and the induced metric on each factor Q i is HermitianEinstein with slope µ i. Notice that if we assume the slopes are ordered µ 1 > > µ l, then the HN filtration of E is given by: F hn i (E) = i Q. For a general holomorphic structure on E, the HN filtration will not be holomorphically split, so there can exist no smooth metric satisfying (3.1). What is more, the HN filtration may not be given by subbundles, so the right hand side of (3.1) is not even everywhere defined as a smooth endomorphism. In this subsection, we define precisely what is meant by an approximate solution to (3.1) (compare the following discussion with that in [Ko, IV. 5]). Let H be a smooth metric on E, and let F = {F i } l i=1 be a filtration of E by saturated subsheaves: F : 0 = F 0 F 1 F l = E. Associated to each F i and the metric H we have the unitary proection πi H onto F i. As mentioned previously, the πi H are bounded L 2 1 hermitian endomorphisms. For convenience, we set π0 H = 0. Next, suppose we are given a collection of real numbers µ 1,..., µ l. From the data F and (µ 1,..., µ l ) we define a bounded L 2 1 hermitian endomorphism of
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