Continuity of the Yang-Mills flow on the set of semistable bundles
Key words and phrases:Yang-Mills flow, semistable bundles, Donaldson–Uhlenbeck compactification
2010 Mathematics Subject Classification:14D20, 14J60, 32G13, 53C07
Let be a compact Kähler manifold of dimension and a hermitian vector bundle on . The celebrated theorem of Donaldson-Uhlenbeck-Yau states that if is an integrable unitary connection on that induces an -slope stable holomorphic structure on , then there is a complex gauge transformation such that satisfies the Hermitian-Yang-Mills (HYM) equations. The proof in [UhlenbeckYau:86] uses the continuity method applied to a deformation of the Hermitian-Einstein equations for the metric . The approach in [Donaldson:85, Donaldson:87] deforms the metric using a nonlinear parabolic equation, the Donaldson flow. Deforming the metric is equivalent to acting by a complex gauge transformation modulo unitary ones, and in this context the Donaldson flow is equivalent (up to unitary gauge transformations) to the Yang-Mills flow on the space of integrable unitary connections. The proof in [Donaldson:87] assumes that is a projective algebraic manifold (more precisely, that is a Hodge metric) whereas the argument in [UhlenbeckYau:86] does not. The methods of Uhlenbeck-Yau and Donaldson were combined by Simpson [Simpson:88] to prove convergence of the Yang-Mills flow for stable bundles on all compact Kähler manifolds. The Yang-Mills flow thus defines a map from the space of smooth integrable connections on inducing stable holomorphic structures to the moduli space of irreducible HYM connections.111The notion of (semi)stability depends on the choice of Kähler class ; however, the class will remain fixed throughout, and we shall suppress this dependency from the notation. Continuity of this map follows by a comparison of Kuranishi slices (see [FujikiSchumacher:87, Miyajima:89]).
When the holomorphic bundle is strictly semistable, then the Donaldson flow fails to converge unless splits holomorphically into a sum of stable bundles (i.e. it is polystable). If it is still true, however, that the Yang-Mills flow converges to a smooth HYM connection on for any semistable initial condition. This was proven by Daskalopoulos and Råde [Daskal:92, Rade:92]. Moreover, the holomorphic structure of the limiting connection is isomorphic to the polystable holomorphic bundle obtained from the associated gradation of the Jordan-Hölder filtration of . For , there is an obstruction to a smooth splitting into an associated graded bundle, and may not be locally free. The new phenomenon of bubbling occurs, and one must talk of convergence in the sense of Uhlenbeck, that is, away from a singular set of complex codimension at least (see Theorem 2.2 below). In [DaskalWentworth:04] (see also [DaskalWentworth:07b]) it was shown for that the Yang-Mills flow converges in the sense of Uhlenbeck to the reflexification , which is a polystable bundle. The bubbling locus, which in this case is a collection of points with multiplicities, is precisely the set where fails to be locally free [DaskalWentworth:07a]. The extension of these results in higher dimensions was achieved in [Sibley:15, SibleyWentworth:15]. Here, even the reflexified associated graded sheaf may fail to be locally free, and one must use the notion of an admissible HYM connection introduced by Bando and Siu [BandoSiu:94]. Convergence of the flow to the associated graded sheaf for semistable bundles in higher dimensions was independently proven by Jacob [Jacob:15].
In a different direction, a compactification of was proposed by Tian in [Tian:00] and further studied in [TianYang:02]. This may be viewed as a higher dimensional version of the Donaldson-Uhlenbeck compactification of ASD connections on a smooth manifold of real dimension (cf. [FreedUhlenbeck:84, DonaldsonKronheimer:90]). It is based on a finer analysis of the bubbling locus for limits of HYM connections that is similar to the one carried out for harmonic maps by Fang-Hua Lin [Lin:99]. More precisely, Tian proves that the top dimensional stratum is rectifiable and calibrated by with integer multiplicities, and as a consequence of results of King [King:71] and Harvey-Shiffman [HarveyShiffman:74], it represents an analytic cycle. The compactification is then defined by adding ideal points containing in addition to an admissible HYM connection the data of a codimension cycle in an appropriate cohomology class (see Section 2). At least when is projective, the space of ideal HYM connections is a compact topological space (Hausdorff), and the compactification of is obtained by taking its closure . Under this assumption, we recently showed, in collaboration with Daniel Greb and Matei Toma, that admits the structure of a seminormal complex algebraic space [GSTW:18].
The purpose of this note is to point out the compatibility of this construction with the Yang-Mills flow. For example, in the case of a Riemann surface, the flow defines a continuous deformation retraction of the entire semistable stratum onto the moduli space of semistable bundles. This is precisely what is to be expected from Morse theory (see [AtiyahBott:82]). In higher dimensions, as mentioned above, bubbling along the flow needs to be accounted for. The result is the following.
Let be a hermitian vector bundle over a compact Kähler manifold with . Let be the set of semistable integrable unitary connections on with the smooth topology (see Section 2). Then the Yang-Mills flow defines a continuous map
In particular, the restriction of gives a continuous map , where is the closure of in the smooth topology.
The proof of the Main Theorem is a consequence of the work in [GSTW:18], with small modifications. For the case of Kähler surfaces, this result was claimed in [DaskalWentworth:07a, Thm. 2]. Unfortunately, there is an error in the proof of Lemma 8 of that paper, and hence also in the proof of Theorem 2. The Main Theorem above validates the statement in [DaskalWentworth:07a, Thm. 2], at least in the projective case. We do not know if the result holds when is only Kähler. The advantage of projectivity is that a twist of the bundle is generated by global holomorphic sections. These behave well with respect to Uhlenbeck limits and provide a link between the algebraic geometry of geometric invariant theory quotients and the analytic compactification. We review this in Section 3 below.
2. Uhlenbeck limits and admissible HYM connections
In this section we briefly review the compactification of by ideal HYM connections. As in the introduction, let be a hermitian vector bundle on a compact Kähler manifold of dimension , and let denote the bundle of skew-hermitian endomorphisms of . The space of unitary connections on is an affine space over , and we endow it with the smooth topology. A connection is called integrable if its curvature form is of type (1,1). Let denote the set of integrable unitary connections on . Then inherits a topology as a closed subset. The locus of stable holomorphic structures is open in (cf. [LubkeTeleman:95, Thm. 5.1.1]). Under the assumption that is a Hodge metric we shall prove below that the subset of semistable holomorphic structures is also open in (see Corollary LABEL:cor:open).
We call the contraction of with the Kähler metric the Hermitian-Einstein tensor. It is a hermitian endomorphism of . The key definition is the following (cf. [BandoSiu:94] and [Tian:00, Sect. 2.3]).
An admissible connection is a pair where
is a closed subset of finite Hausdorff -measure;
is a smooth integrable unitary connection on ;
An admissible connection is called admissible HYM if there is a constant such that on .
The fundamental weak compactness result is the following.
Theorem 2.2 (Uhlenbeck [UhlenbeckPreprint]).
Let be a sequence of smooth integrable connections on with uniformly bounded Hermitian-Einstein tensors. Then for any there is
a subsequence still denoted ,
a closed subset of finite -Hausdorff measure,
a connection on a hermitian bundle , and
local isometries on compact subsets of
such that with respect to the local isometries, and modulo unitary gauge equivalence, weakly in .
We call the limiting connection an Uhlenbeck limit. The set
where and are universal constants depending only on the geometry of , is called the (analytic) singular set.
For the definition of a gauge theoretic compactification more structure is needed. This is provided by the following, which is a consequence of work of Tian [Tian:00] and Hong-Tian [HongTian:04].
The Uhlenbeck limit of a sequence of smooth HYM connections on is an admissible HYM connection. Moreover, the corresponding singular set is a holomorphic subvariety of codimension at least . The same is true for Uhlenbeck limits of sequences along the Yang-Mills flow.
To be more precise, there is a decomposition , where
has codimension , and is the support of a codimension 2 cycle . The cycle appears as the limiting current of the Yang-Mills energy densities, just as in the classical approach of Donaldson-Uhlenbeck in real dimension . This structure motivates the following
Definition 2.4 ([Gstw:18, Def. 3.15]).
An ideal HYM connection is a triple satisfying the following conditions:
is an -cycle on ;
the pair is an admissible HYM connection on the hermitian vector bundle , where is given as in eq. (2.1);
, in ;
Here we have denoted by the -current given by
for smooth -forms . This is well defined by Definition 2.1 (3), and in [Tian:00, Prop. 2.3.1] it is shown to be a closed current. It thus defines a cohomology class as above. By [BandoSiu:94], there is a polystable reflexive sheaf extending the holomorphic bundle . The singular set of , that is, the locus where fails to be locally free, coincides with (see [TianYang:02, Thm. 1.4]). By the proof of [SibleyWentworth:15, Prop. 3.3], represents the class . Thus we may alternatively regard an ideal connection as a pair , where is a reflexive sheaf, is a codimension cycle with , and where the underlying smooth bundle of on the complement of is isomorphic to . See [GSTW:18, Sec. 3.3] for more details.
There is an obvious notion of gauge equivalence of ideal HYM connections. The main result is the following.
Assume is a Hodge metric. Let . Then there is a subsequence also denoted by , and an ideal HYM connection such that converges to a subcycle of , and (up to gauge transformations) in on . Moreover,
in the mass norm; in particular, also in the sense of currents.
For more details we refer to [Tian:00, TianYang:02, GSTW:18].
3. The method of holomorphic sections
Admissibility of a connection is precisely the correct analytic notion to make contact with complex analysis. Bando [Bando:91] and Bando-Siu [BandoSiu:94] show that bundles with admissible connections admit sufficiently many local holomorphic sections to prove coherence of the sheaf of -holomorphic sections. This local statement only requires the Kähler condition. The key difference between the projective vs. Kähler case is, of course, the abundance of global holomorphic sections. These provide a link between the algebraic and analytic moduli. They are also well-behaved with respect to limits. The technique described here mimics that introduced by Jun Li in [Li:93].
We henceforth assume . Let be a complex line bundle with . Define the numerical invariant:
Since is a class, may be endowed with a holomorphic structure making it the ample line bundle defining the polarization of . We also fix a hermitian metric on with respect to which the Chern connection of has curvature . Use the following notation: . The key property we exploit is the following, which is a consequence of Maruyama’s boundedness result [Maruyama:81], as well as the Hirzebruch-Riemann-Roch theorem.
There is such that for all and all , if then the bundle is globally generated and all higher cohomology groups vanish. In particular, for .
In the following, we shall assume has been fixed sufficiently large (possibly larger than in the previous proposition). Fix a vector space of dimension , and let
The Grothendieck Quot scheme is a projective scheme parametrizing isomorphism classes of quotients , where is a coherent sheaf with Hilbert polynomial [Grothendieck:61, AltmanKleiman:80]. Proposition 3.1 states that there is a uniform such that for every there is a quotient in with . The next result begins the comparison between Uhlenbeck limits and limits in .
Let , and suppose in the sense of Uhlenbeck (Theorem 2.2), and assume uniform bounds on the Hermitian-Einstein tensors. Then there is a quotient in and an inclusion such that .
The proof of this result for sequences of HYM connections is in [GSTW:18, Sec. 4.2], but the proof there works as well under the weaker assumption of bounded Hermitian-Einstein tensor. Indeed, the first key point is the application of the Bochner formula to obtain uniform bounds on -holomorphic sections. The precise statement is that if then there is a constant depending only on the geometry of , , and the uniform bound on the Hermitian-Einstein tensor, such that . In this way one can extract a convergent subsequence of orthonormal sections to obtain a map . The limiting sections may no longer form a basis of , nor necessarily do they generate the fiber of . Remarkably, though, it is still the case that the rank of the image sheaf of agrees with and has Hilbert polynomial (for this one may have to twist with a further power of ). In fact, the quotient sheaf turns out to be supported in complex codimension (the first Chern class is preserved under Uhlenbeck limits). Hence, in particular, . See [GSTW:18, proof of Lemma 4.3] for more details.
The second ingredient in the proof is the fact that is compact in the analytic topology. Hence, after passing to a subsequence, we may assume the converge. Convergence in means the following: there is a convergent sequence of quotients and isomorphisms making the following diagram commute