Let X be a compact Riemann surface of genus g≥2. The moduli space M(r,d) of rank r and degree d semi-stable Higgs bundles over X admitted a stratification, called Shatz stratification, which was defined by the Harder-Narasimhan type of the Higgs bundles. There was also a C∗-action on M(r,d) given by the product on the Higgs field, which provided the Białynicki-Birula stratification by considering the Hodge limit bundles limz→0(E,z⋅φ). In this paper, these limit bundles were computed for all possible Harder-Narasimhan types when the rank of the Higgs bundles was r=5, explicit vector forms were provided for the Hodge limit bundles, and necessary and sufficient conditions were given for them to be stable. In addition, it was proved that, in rank 5, the Shatz strata traversed the Białynicki-Birula strata. Specifically, it was checked that there existed different semi-stable rank 5 Higgs bundles with the same Harder-Narasimhan type such that their associated Hodge limit bundles were not S-equivalent, and explicit constructions of those Higgs bundles were also provided.
Citation: Álvaro Antón-Sancho. The C∗-action and stratifications of the moduli space of semi-stable Higgs bundles of rank 5[J]. AIMS Mathematics, 2025, 10(2): 3428-3456. doi: 10.3934/math.2025159
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Let X be a compact Riemann surface of genus g≥2. The moduli space M(r,d) of rank r and degree d semi-stable Higgs bundles over X admitted a stratification, called Shatz stratification, which was defined by the Harder-Narasimhan type of the Higgs bundles. There was also a C∗-action on M(r,d) given by the product on the Higgs field, which provided the Białynicki-Birula stratification by considering the Hodge limit bundles limz→0(E,z⋅φ). In this paper, these limit bundles were computed for all possible Harder-Narasimhan types when the rank of the Higgs bundles was r=5, explicit vector forms were provided for the Hodge limit bundles, and necessary and sufficient conditions were given for them to be stable. In addition, it was proved that, in rank 5, the Shatz strata traversed the Białynicki-Birula strata. Specifically, it was checked that there existed different semi-stable rank 5 Higgs bundles with the same Harder-Narasimhan type such that their associated Hodge limit bundles were not S-equivalent, and explicit constructions of those Higgs bundles were also provided.
Let X be a compact Riemann surface of genus g≥2. Higgs bundles over X were first introduced by Hitchin [21] in the context of the resolution of self-duality equations. He also proved that the moduli space of Higgs bundles forms a hyperkähler manifold. After that, Nitsure [24] provided a construction of the moduli space of Higgs bundles over X by using techniques from geometric invariant theory. Simpson [26,27,28] extended the framework provided by Hitchin and linked Higgs bundles to non-Abelian Hodge theory and flat connections on varieties.
The study of the geometry and topology of the moduli space of Higgs bundles is an interesting field of study, not only for its importance in mathematics, but also for its connections with theoretical physics, particularly mirror symmetry, Langlands duality, and string theory [1,18]. The lines of research on Higgs bundles are rich and diverse, including the computation of cohomology of moduli spaces of Higgs bundles [13,19], the classification of its automorphisms [7], the study of certain subvarieties of the moduli space, such as the subvariety of fixed point of an automorphism of it [3,5,14] or automorphisms of the underlying vector bundles [4], the relation of Higgs bundles with the non-Abelian Hodge correspondence [29], Hitchin systems [10], or the relation of Higgs bundles with Donaldson invariants [12,23] or other topological properties such as Gopakumar-Vafa invariants [22].
The line in which the present work is framed is that of the study of stratifications of the moduli space of Higgs bundles over X. Specifically, there are two stratifications of interest for the purposes intended here. First, the Shatz stratification, defined from the vector of slopes (called Harder-Narasimhan type) associated with the Harder-Narasimhan filtration of the underlying vector bundle of strictly semi-stable Higgs bundles [24,25]. This is a stratification of great interest in geometry and has been intensively studied even in the field of bundles with structure groups different from the linear group, such as G2 [6]. The second is the Białynicki-Birula stratification, constructed from the limits limz→0(E,z⋅φ) of semi-stable Higgs bundles, where ⋅ denotes the action of the multiplicative group C∗ on the Higgs field [8,11,26]. The case of rank 2 Higgs bundles was studied by Hausel and Thaddeus [17,19,20]. They proved that the two mentioned stratifications coincide in rank 2. Later, Gothen and Zúñiga-Rojas proved that this is not the case for rank 3. The mentioned works were extended to rank 4 in [2], where it was proved that Shatz and Białynicki-Birula stratifications do not coincide in rank 4, but both stratifications have some common strata, defined by certain conditions of the Harder-Narasimhan types.
In the present work, the previous papers are extended to the rank 5 situation. In particular, it is proved that the Shatz and Białynicki-Birula stratifications do not coincide even in rank 5. The strategy followed is analogous to that of the previous paper [2] and the article by Gothen and Zúñiga-Rojas [15]. Specifically, a family of intermediate bundles of any strictly semi-stable Higgs bundle is constructed (Definition 1), and a set of bounds for the slopes of these Higgs bundles is provided (Proposition 1). From this, a family of cases for the Harder-Narasimhan type are distinguished that need to be differentiated to conduct the analysis. Finally, the limit bundles, which are always Hodge bundles, are explicitly computed in each of the differentiated cases (Theorems 1 to 6). Indeed, explicit vector forms of all possible Hodge limit bundles are provided, which constitutes an original contribution of the paper. Although the strategy is similar to that of previous works [2,15], the situation in rank 5 is notably more difficult, due to the much more complicated topology of the moduli space. This means that it is necessary to distinguish up to 6 different situations, as opposed to 4 in [2] and 2 in [15]. Moreover, as far as it has been possible to explore, the extension made here is not directly generalizable to any rank, so it is only possible to address the specific rank.
In addition, it is proved here that the strata of the two mentioned stratifications in rank 5 are transversal, in the sense that each Shatz stratum crosses several Białynicki-Birula strata (Theorem 7). This has been proven by analyzing the S-equivalence condition of the Hodge limit bundles in each of the differentiated situations, which could be grouped into four results (Propositions 2 to 5). This contribution is an original novelty and also an essential difference with respect to what happens in rank 4. Moreover, explicit constructions of strictly semi-stable Higgs bundles of rank 5 of the same Harder-Narasimhan type whose associated Hodge limit bundles are not S-equivalent are provided for each of the differentiated situations in the analysis made. Therefore, in addition to extending the results of [2,15] to rank 5, the present work contributes with the novelty of the explicit description of the Shatz strata in rank 5 and with the verification of the transversality of the Shatz and Białynicki-Birula stratifications in rank 5, an aspect that does not occur in all the previously studied ranks.
The structure of the article is as follows. In Section 2, the notions of stability and S-equivalence of Higgs bundles are presented. From this, the Harder-Narasimhan types, the C∗-action, and the Shatz and Białynicki-Birula stratifications are introduced. The bounds on the slopes of the intermediate vector bundles are checked in Section 3, where the main 6 situations of the study are distinguished and the explicit computation of the Hodge limit bundles for the C∗-action are provided. In Section 4 it is proved that, regardless of the situation considered, it is always possible to find semi-stable Higgs bundles with a fixed Harder-Narasimhan type whose Hodge limit bundles are not S-equivalent, from which the announced transversality of the Shatz and Białynicki-Birula stratifications follows. Finally, the main conclusions of the work are drawn.
Given a compact Riemann surface of genus g≥2, a Higgs bundle over X is a pair (E,φ) where E is a holomorphic vector bundle over X and φ:E→E⊗K is a homomorphism of vector bundles, K being the canonical line bundle over X. The rank and the degree of the Higgs bundle are the rank and the degree, respectively, of E, and the slope of the Higgs bundle is also the slope of E, μ=μ(E)=degErk E. The Higgs bundle (E,φ) is semi-stable if for every φ-invariant proper sub-bundle F of E it is satisfied that μ(F)≤μ(E). It is stable if the inequality is strict for every φ-invariant proper sub-bundle F. Finally, it is poly-stable if E is isomorphic to the direct sum of proper sub-bundles whose slopes are μ=μ(E) [21,26,27,28].
Any strictly semi-stable Higgs bundle (E,φ) over X admits a filtration 0⊂E1⊂⋯⊂En=E into φ-invariant proper sub-bundles Ek such that the pair (Ek,φk) is stable as a Higgs bundle, where φk:Ek→Ek⊗K denotes the restriction of φ. This filtration provides a graded object associated with (E,φ), which is a poly-stable Higgs bundle. Two semi-stable Higgs bundles are said to be S-equivalent if their associated graded objects are isomorphic. Then the moduli space of semi-stable Higgs bundles over X is a complex algebraic variety that parametrizes S-equivalence classes of semi-stable Higgs bundles or, equivalently, isomorphism classes of poly-stable Higgs bundles over X, since every S-equivalence class admits a unique isomorphism class of a poly-stable representative.
Given any vector bundle E over X, it admits a unique Harder-Narasimhan filtration, which is a filtration of the form 0⊂E1⊂⋯⊂Ek=E, where Ej/Ej−1 is semi-stable as a vector bundle for every j=1,…,k, being E0=0, and μ(Ej+1/Ej)<μ(Ej/Ej−1) for j=1,…,k−1. Here, E1 is the maximal destabilizing sub-bundle of E and the graded vector bundle is defined by ⊕kj=1Ej/Ej−1. Let (E,φ) be a strictly semi-stable Higgs bundle over X. Then, the underlying vector bundle E must be unstable. Consider the Harder-Narasimhan filtration 0⊂E1⊂⋯⊂Ek=E of E and let, for each j=1,…,k, μj=μ(Ej/Ej−1). Then, a vector of slopes (μ1,…,μr), where r=rk E, is defined, where each μj is repeated rk(Ej)−rk(Ej−1) times in the vector for j=1,…,k. This vector of slopes is called Harder-Narasimhan type of the underlying vector bundle of the Higgs bundle [16,23]. Notice that it must be μ1≥μ2≥⋯≥μr, with at least one of these inequalities being strict. Also, to make the presentation clearer in the following, when appealing to the Harder-Narasimhan filtration of the underlying vector bundle E of a strictly semi-stable Higgs bundle, a filtration of the form 0⊂E1⊆⋯⊆Er=E, the inclusions being non-strict, will be taken, where r=rk E and each Ej is repeated rk(Ej)−rk(Ej−1) times in the filtration. Then, μ1=μ(E1) and μk=μk+1 whenever Ek=Ek+1.
Remark 1. To fix notation, notice that the sub-bundle of the Harder-Narasimhan filtration of the underlying vector bundle E of a strictly semi-stable Higgs bundle (E,φ) of certain rank k will be denoted by Ek. Then, all sub-bundles within the mentioned Harder-Narasimhan filtration with the same rank k will be denoted as Ek. That is, if Ek=Ek+1, both of rank k, then Ek+1 will be called Ek. Of course, in this case, it will be μk=μk+1. For example, in rank 5, which is the case of interest in this work, if the Harder-Narasimhan filtration of E is of the form 0⊂E1⊂E3⊂E5 (with strict inclusions), this means that rk E1=1, rk E3=3, and the Harder-Narasimhan type of E satisfies μ1=μ2>μ3=μ4>μ5. This notation is consistent with that of [2] but rather different from that of [15]. The slope of E will be denoted by μ.
With the above notation, the moduli space M(r,d) of semi-stable Higgs bundles of given rank r and degree d admits a stratification whose strata are defined by the equality of the Harder-Narasimhan type of the underlying vector bundle of the Higgs bundles. This is called Shatz stratification of M(r,d) [24,25].
The moduli space M(r,d) also admits the Białynicki-Birula stratification, which arises from the action of the multiplicative group C∗ on Higgs bundles defined by z⋅(E,φ)↦(E,z⋅φ) [8,26]. Although this limit bundle may not be unique in the configuration space of semi-stable bundles, as the orbit space is not Hausdorff, the action is well-defined on the configuration space of poly-stable Higgs bundles, in which the orbit space is indeed Hausdorff [28, Corollary 9.20]. Two semi-stable Higgs bundles over X are in the same stratum of the Białynicki-Birula stratification if their associated limit bundles for the action of C∗ are S-equivalent. In addition, the limit bundle will be a Hodge bundle, since it is in turn fixed by the action of C∗ on the Higgs fields.
Let X be a compact Riemann surface of genus g≥2. In this section, all possible Harder-Narasimhan types of the underlying vector bundle of semi-stable rank 5 Higgs bundles with a given degree are considered. Specifically, for a given semi-stable Higgs bundle (E,φ) of rank 5, the Hodge limit bundle limz→0(E,z⋅φ) is computed, and explicit forms of these limit bundles are provided depending on the different possibilities for the Harder-Narasimhan type of the underlying vector bundle of a strictly semi-stable Higgs bundle. Notice that the case when μ1=μ2=μ3=μ4=μ5 cannot occur for strictly semi-stable Higgs bundles. This case would correspond to a stable Higgs bundle (E,φ), for which obviously limz→0(E,z⋅φ)=(E,0). To do all the above, some intermediate bundles are first introduced, for which certain bounding properties of their slopes will be proved in the following [2,15].
Given a strictly semi-stable rank 5 Higgs bundle (E,φ) with slope μ=μ(E), its Harder-Narasimhan type will be denoted by (μ1,μ2,μ3,μ4,μ5), where μ1≥μ2≥μ3≥μ4≥μ5 and 15(μ1+μ2+μ3+μ4+μ5)=μ. As in the previous section, if the Harder-Narasimhan filtration of the underlying vector bundle E of (E,φ) is 0⊂E1⊂E2⊂⋯⊂En=E, which is preserved by φ, then μ1=μ(E1) and μk=μk+1 whenever Ek=Ek+1, and the same notation convention will be followed.
Definition 1. Let (E,φ) be a strictly semi-stable rank 5 Higgs bundle over X such that the Harder-Narasimhan filtration of the underlying vector bundle E is 0⊂E1⊂E2⊂⋯⊂En=E and the Harder-Narasimhan type of E is (μ1,μ2,μ3,μ4,μ5). Then the following intermediate bundles are defined:
(1) For k=1,2,3,4, Ik is the sub-bundle of E/Ek obtained by saturating the sheaf φk+1,k(Ek)⊗K−1, where φk+1,k:Ek→E/Ek⊗K is the morphism induced by φ.
(2) For k=1,2,3,4, Nk=ker(φ5−k+1,5−k), where φ5−k+1,5−k is the morphism, E5−k→E/E5−k⊗K is induced by φ.
Remark 2. In Definition 1, the sub-bundles I4 and N4 have been included because they make sense, although these two sub-bundles will not play a role in the following description of the different Shatz strata.
Remark 3. The bundles Ik in Definition 1 are defined as the saturation of the images of certain morphisms induced by φ. Saturation guarantees that it is a vector bundle. On their part, the bundles Nk are defined as kernels of certain homomorphisms of vector bundles induced by φ, which are always homomorphisms by semi-stability of (E,φ). The ranks of these bundles depend on the rank of the morphisms φk+1,k. Specifically, the following cases can be differentiated, which are discussed in the result below:
(1) μ1>μ2.
(2) μ2>μ3 and rk(φ3,2)=1.
(3) μ2>μ3 and rk(φ3,2)=2.
(4) μ4>μ5.
(5) μ3>μ4 and rk(φ4,3)=1.
(6) μ3>μ4 and rk(φ4,3)=2.
Proposition 1. Let (E,φ) be a strictly semi-stable rank 5 Higgs bundle over X such that the Harder-Narasimhan filtration of the underlying vector bundle E is 0⊂E1⊂E2⊂⋯⊂En=E and the Harder-Narasimhan type of E is (μ1,μ2,μ3,μ4,μ5), and I1, I2, I3, N1, N2, and N3 be the vector bundles given in Definition 1. Then, the following is satisfied:
(1) If μ1>μ2, then I1 is a line bundle and μ1−(2g−2)≤μ(I1)≤μ2.
(2) If μ2>μ3 and the morphism φ3,2:E2→E/E2⊗K induced by φ has rank 1, then I2 and N3 are line bundles, μ2−(2g−2)≤μ(I2)≤μ3, and μ1+μ2−μ3−(2g−2)≤μ(N3)≤μ1.
(3) If μ2>μ3 and the rank of the morphism φ3,2:E2→E/E2⊗K induced by φ is 2, then I2 is a rank 2 vector bundle and μ2−(2g−2)≤μ(I2)≤μ3.
(4) If μ4>μ5, then N1 is a rank 2 vector bundle and 12(μ1+μ2+μ3+μ4−μ5−(2g−2))≤μ(N1)≤13(μ1+μ2+μ3).
(5) If μ3>μ4 and the morphism φ4,3:E3→E/E3⊗K induced by φ has rank 1, then I3 is a line bundle, N2 is a rank 2 vector bundle, μ3−(2g−2)≤μ(I3)≤μ4, and 12(μ1+μ2+μ3−μ4−μ5−(2g−2))≤μ(N2)≤13(μ1+μ2+μ3).
(6) If μ3>μ4 and the morphism φ4,3:E3→E/E3⊗K induced by φ has rank 2, then N2 is a rank 2 vector bundle and 12(μ1+μ2+μ3−μ4−μ5−(2g−2))≤μ(N2)≤13(μ1+μ2+μ3).
Proof. For case (1), suppose that μ1>μ2. Then, E1 is a line bundle by hypothesis and φ2,1≠0 by semi-stability of (E,φ). Then, I1 is also a line bundle and, since the morphism E1→I1⊗K induced by φ2,1 is nonzero, it follows that μ(I1)+μ(K)≥μ1, so μ(I1)≥μ1−(2g−2). In addition, since E2/E1 is a maximal destabilizer with rank 1 in E/E1, then it must be μ(I1)≤μ(E2/E1)=μ2.
The condition on I2 of case (2) is similar to case (1), by considering the morphism φ3,2. For the condition on N3, notice that the morphism φ3,2:E2→E/E2⊗K has rank 1 and, since μ2>μ3, E2 has rank 2, so there is an exact sequence of sheaves
0→N3→E2→Im(φ3,2)→0, |
where Im(φ3,2) has rank 1 (it may be considered as a vector bundle by saturating it). Then, N3 is a line bundle. Now, since (E,φ) is semi-stable, the slope of N3 must be μ(N3)≤μ1. Moreover, by the behavior of the slopes in exact sequences applied to the short exact sequence above,
μ(E2)=rk(N3)μ(N3)+rk(Im(φ3,2))μ(Im(φ3,2))rk(E2). |
Since rk(N3)=rk(Im(φ3,2))=1, rk(E2)=2, and μ(E2)=μ1+μ22, it follows that
μ1+μ22=μ(N3)+μ(Im(φ3,2))2. |
On the other hand, since Im(φ3,2) is a sub-sheaf of E/E2⊗K, it is obtained that
μ(Im(φ3,2))≤μ(E/E2⊗K)=μ(E/E2)+(2g−2)≤μ3+(2g−2). |
By substituting in the above expression,
μ1+μ22≤μ(N3)+μ3+(2g−2)2, |
thus, by clearing μ(N3), it follows that μ(N3)≥μ1+μ2−μ3−(2g−2).
For the conditions on I2 of the case (3), notice that I2 has rank 2 because the morphism φ3,2:E2→E/E2⊗K has rank 2. There is an exact sequence of the form
0→I2→E/E2→Q→0, |
where Q is the quotient bundle, which has rank 1, since E/E2 has rank 3 and I2 has rank 2. It is known that μ(E/E2)=μ3+μ4+μ53 and μ(Q)≥μ(E/E2). To check the last inequality, consider the exact sequence defined by
0→I2→E/E2→Q→0. |
If μ(Q)<μ(E/E2), then μ(I2)>μ(E/E2). This would contradict the semi-stability of (E,φ), as I2⊗K−1 would be a φ-invariant sub-bundle of E/E2 with slope higher than μ(E/E2). Thus, μ(Q)≥μ(E/E2). Therefore,
μ3+μ4+μ53≥2μ(I2)+μ3+μ4+μ533, |
and then μ(I2)≤μ3+μ4+μ53≤μ3, as stated. On the other hand, the morphism φ3,2:E2→I2⊗K is surjective, so the sequence
0→ker(φ3,2)→E2→I2⊗K→0 |
is exact. Therefore,
μ(E2)=rk(ker(φ3,2))μ(ker(φ3,2))+rk(I2)μ(I2⊗K)rk(E2). |
Since μ(I2)=μ2−(2g−2) and μ(ker(φ3,2))≤μ(E2)=μ2 (by semi-stability), it follows that
μ2≤rk(ker(φ3,2))μ2+rk(I2)(μ(I2)+(2g−2))rk(E2), |
so, finally, μ(I2)≥μ2−(2g−2).
In the case (4), consider N1=ker(φ5,4). Since μ4>μ5, E4 has rank 4, and φ5,4:E4→E/E4⊗K is nonzero (by semi-stability of (E,φ)). Thus, N1 has rank 3 or 2. However, if N1 had rank 3, then φ5,4 would factor through E4/N1, which is a line bundle. This would imply that φ4,3 has rank at most 1, contradicting the assumption. Therefore, N1 has rank 2. Moreover, it is known that μ(N1)≤μ(E3)=13(μ1+μ2+μ3), because N1⊂E4 and E3 is the maximal destabilizing sub-bundle of rank 3. For the lower bound, consider the exact sequence given by
0→N1→E4→E4/N1→0. |
It is known that E4/N1→E/E4⊗K is nonzero, so μ(E4/N1)≤μ5+(2g−2). By the properties of slopes in exact sequences, it is clear that
μ(E4)=2μ(N1)+μ(E4/N1)3 |
and
μ1+μ2+μ3+μ44=2μ(N1)+(μ5+(2g−2))3. |
Thus, μ(N1)≥12(μ1+μ2+μ3+μ4−μ5−(2g−2)), as stated.
For case (5), the condition on I3 is similar as that of I1 of case (1), by taking the morphism φ4,3. For the condition on N2, take the exact sequence
0→N2→E3→Im(φ4,3)→0, |
where the rank of Im(φ4,3) is 1, as φ4,3:E3→E/E3⊗K has rank 1, and E3 has rank 3, since μ3>μ4. This proves that the rank of N2 is 2. Moreover, as in case (2),
μ(E3)=rk(N2)μ(N2)+rk(Im(φ4,3))μ(Im(φ4,3))rk(E3), |
hence,
μ1+μ2+μ33=2μ(N2)+μ(Im(φ4,3))3, |
since μ(E3)=μ1+μ2+μ33. By semi-stability, μ(Im(φ4,3))≤μ(E/E3⊗K)=μ(E/E3)+(2g−2)=μ4+μ52+(2g−2), so, by substituting in the equation above, it is obtained that
μ1+μ2+μ33≤2μ(N2)+μ4+μ52+(2g−2)3, |
thus, μ(N2)≥12(μ1+μ2+μ3−μ4−μ5−(2g−2)).
The proof of the conditions on N2 of case (6) is similar to that of case (4), by considering E3 instead of E4 and N2 instead of N1.
Remark 4. For a better understanding of the results to be given below where the Hodge limit bundles are computed, it is convenient to briefly discuss the comparison between μ=μ(E) and the bounds of the slope of the sub-bundles Nk given in Proposition 1. To fix ideas, the conditions of the case (2) of the proposition will be assumed. The rest of the cases are analogous. Of course, under the assumption that μ2>μ3, it is satisfied that μ<μ1, so μ is always less or equal to the upper bound of μ(N3). However, it is not always true that, given a valid Harder-Narasimhan type of the underlying vector bundle of a strictly semi-stable Higgs bundle with rank 5, it must be μ≥μ1+μ2−μ3−(2g−2). This inequality is satisfied depending on the value of the genus g. For example, if the Harder-Narasimhan type is (2,2,1,1,1), then μ=μ1+μ2+μ3+μ4+μ55=75 and μ1+μ2−μ3−(2g−2)=3−(2g−2). Notice that 75≥3−(2g−2) if, and only if, g≥1+45, which is always true since, under our assumptions, g≥2. However, if the Harder-Narasimhan type is (10,2,1,1,1), then μ=3 and μ1+μ2−μ3−(2g−2)=11−(2g−2), so 3≥11−(2g−2) if, and only if, g≥5. This explains that, when μ(Nk)=μ or μ(Nk)<μ is supposed in the results below, an additional assumption is required, such as μ≥μ1+μ2−μ3−(2g−2).
In Theorems 1 to 6 below, all possible Hodge limit bundles limz→0(E,zφ) for a strictly semi-stable Higgs bundle of rank 5 (E,φ) are computed, according to the cases described in the Remark after Definition 1. This extends to rank 5 the results given in [15] for rank 3 and in [2] for rank 4. The above possibilities for the Hodge limit bundles depend on the various Harder-Narasimhan types of the underlying vector bundles (notice that μ1≥μ2≥μ3≥μ4≥μ5 is always satisfied) and ranks of the different components of the Higgs fields, and, within each case, different limit Hodge bundles arise depending on several conditions assumed on the intermediate bundles I1, I2, I3, N1, N2, and N3 given in Definition 1. In all theorems, the conditions on the rank and slope of these intermediate bundles follow from Proposition 1.
Theorem 1. Let (E,φ) be a rank 5 semi-stable Higgs bundle over X such that the Harder-Narasimhan type (μ1,μ2,μ3,μ4,μ5) of E satisfies μ1>μ2=μ3=μ4=μ5, μ=μ(E), and let I1 be the line bundle given by Definition 1. Let (E0,φ0) be the Hodge limit bundle limz→0(E,z⋅φ). Then, the explicit form of (E0,φ0) is the following:
(1) If μ(I1)<2μ−μ1, then
(E0,φ0)=(E1⊕E/E1,(00ψ2,10)), |
where ψ2,1:E1→E/E1⊗K is nonzero and induced by φ.
(2) If μ(I1)>2μ−μ1, then
(E0,φ0)=(E1⊕I1⊕(E/E1)/I1,(000ψ2,1000ψ3,20)), |
where ψ2,1:E1→I1⊗K and ψ3,2:I1→(E/E1)/I1⊗K are the nonzero morphisms induced by φ.
(3) If μ(I1)=2μ−μ1, then
(E0,φ0)=(E1⊕I1,(00ψ2,10))⊕((E/E1)/I1,0), |
where ψ2,1:E1→I1⊗K is nonzero and induced by φ.
Proof. Notice that, under the assumptions, I1 is a line bundle by Proposition 1 and the morphisms ψi,j should be nonzero by semi-stability of (E,φ). Notice also that the choice of the symbol ψ is due to that ψi,j may not coincide with the morphisms φi,j of Definition 1.
A family g(z) of gauge transformations parametrized by z∈C∗ will be found such that the limit
limz→0(g(z)⋅ˉ∂E,g(z)−1zφg(z)), |
is semi-stable, ˉ∂E being the operator ˉ∂ corresponding to the holomorphic structure of E. This is then the Hodge limit bundle.
For the first case, let E, E1, and E′1 be, respectively, the underlying C∞ vector bundles of E, E1, and E/E1. Then, E≅E1⊕E′1, and the holomorphic structure of E is given by the operator ˉ∂E defined by
ˉ∂E=(ˉ∂1β0ˉ∂′1), |
where ˉ∂1 and ˉ∂′1 are the ˉ∂-operators associated to the holomorphic structures of E1 and E/E1, respectively, and β∈A0,1(Hom(E′1,E1)) [9,21]. The Higgs field φ, expressed with respect to the decomposition above, is of the form
φ=(φ1,1φ1,2φ2,1φ2,2). |
Define the gauge transformation g(z), for z∈C∗, as
g(z)=(100z⋅I). |
Then,
g(z)⋅ˉ∂E=(ˉ∂1zβ0ˉ∂′1) |
and
g(z)−1(zφ)g(z)=(zφ1,1z2φ1,2φ2,1zφ2,2), |
so
limz→0(g(z)⋅ˉ∂E,g(z)−1zφg(z))=(E1⊕E/E1,(00ψ2,10)), |
which is the Hodge bundle (E0,φ0) of the first case. In addition, (E0,φ0) is stable. To check it, notice that the only φ0-invariant proper sub-bundles of the vector bundle E0 are E1⊕I1, E/E1, or a φ0-invariant proper vector sub-bundle F of E/E1. In the first situation, one has that μ(E1⊕I1)=12(μ1+μ(I1))<μ, since μ(I1)<2μ−μ1, by assumption. For the second possibility, note that μ(E/E1)=5μ−μ14=14(μ2+μ3+μ4+μ5)<μ, as μ1>μ2. For the third, μ(F)≤μ(E2/E1)=μ2<μ, due again to the assumption that μ1>μ2. This concludes the first case of the statement.
For the second case, suppose that μ(I1)>2μ−μ1, let E, E1, I1 and E″1 be the underlying C∞ vector bundles corresponding to E, E1, I1 y (E/E1)/I1, which satisfy E=E1⊕I1⊕E″1, ˉ∂E, ˉ∂1, ˉ∂I1; and let ˉ∂″1 be the associated ˉ∂-operators. Let βij be the forms such that
ˉ∂E=(ˉ∂1β12β130ˉ∂I1β2300ˉ∂″1). |
Then, the (3,1)-component φ3,1 of φ with respect to this decomposition is 0. Let
g(z)=(1000z000z2). |
The following is satisfied:
g(z)⋅ˉ∂E=(ˉ∂1zβ1,2z2β1,30ˉ∂I1zβ2300ˉ∂″1) |
and
g(z)−1φg(z)=(zφ1,1z2φ1,2z3φ1,3φ2,1zφ2,2z2φ2,30φ3,2zφ3,3), |
so
limz→0(g(z)⋅ˉ∂E,g(z)−1zφg(z))=(E1⊕I1⊕(E/E1)/I1,(000ψ2,1000ψ3,20)). |
To check the stability of this limit Hodge bundle, notice again that the only possibilities for a proper φ0-invariant sub-bundle of E0 are (E/E1)/I1, a φ0-invariant sub-bundle F of (E/E1)/I1, and I1⊕(E/E1)/I1. For the first, μ((E/E1)/I1)=13(μ2+μ3+μ4+μ5−μ(I1))<15(μ1+μ2+μ3+μ4+μ5)=μ, since μ(I1)>2μ−μ1; for the second, μ(F)≤μ((E/E1)/I1)<μ; and for the third,
μ(I1⊕(E/E1)/I1)=14μ(I1)+34μ((E/E1)/I1)=14μ(I1)+14(μ2+μ3+μ4+μ5−μ(I1))=14(μ2+μ3+μ4+μ5)<μ. |
This concludes the second part of the theorem.
In the case that μ(I1)=2μ−μ1, then (E/E1)/I1 is the only proper φ0-invariant sub-bundle of E0 and μ((E/E1)/I1)=μ=μ(E0), so in this case, the limit Hodge bundle is strictly semi-stable. To check this, notice that
μ((E/E1)/I1)=deg((E/E1)/I1)rk((E/E1)/I1)=deg(E)−deg(E1)−deg(I1)3=5μ−μ1−μ(I1)3=μ2+μ3+μ4+μ5−μ(I1)3, |
thus, by substituting the given condition μ(I1)=2μ−μ1=−3μ1+2μ2+2μ3+2μ4+2μ55, it follows that
μ((E/E1)/I1)=5μ−μ1−−3μ1+2μ2+2μ3+2μ4+2μ553=5μ−2μ3=3μ3=μ. |
Theorem 2. Let (E,φ) be a rank 5 semi-stable Higgs bundle over X such that the Harder-Narasimhan type (μ1,μ2,μ3,μ4,μ5) of E satisfies μ1≥μ2>μ3≥μ4≥μ5, μ=μ(E), let φ3,2 be the homomorphism E2→E/E2⊗K induced by φ, and let I2 and N3 be the vector bundles given in Definition 1. Assume that φ3,2 has rank 1. Let (E0,φ0) be the limit limz→0(E,z⋅φ). Then, (E0,φ0) has the following explicit form:
(1) If μ(I2)<3μ−μ1−μ2, μ≥μ1+μ2−μ3−(2g−2), and μ(N3)<μ, then
(E0,φ0)=(E2⊕E/E2,(00ψ2,10)), |
where ψ2,1:E2→E/E2⊗K is the nonzero homomorphism induced by φ.
(2) If μ(I2)>3μ−μ1−μ2, μ≥μ1+μ2−μ3−(2g−2), and μ(N3)<μ, then
(E0,φ0)=(E2⊕I2⊕(E/E2)/I2,(000ψ2,1000ψ3,20)), |
where ψ2,1:E2→I2⊗K and ψ3,2:I2→(E/E2)/I2⊗K are the nonzero homomorphisms induced by φ.
(3) If μ(I2)=3μ−μ1−μ2, μ≥μ1+μ2−μ3−(2g−2), and μ(N3)<μ, then
(E0,φ0)=(E2⊕I2,(00ψ2,10))⊕((E/E2)/I2,0), |
where ψ2,1:E2→I2⊗K is the nonzero homomorphism induced by φ.
(4) If μ(I2)<3μ−μ1−μ2 and μ(N3)>μ, then
(E0,φ0)=(N3⊕E2/N3⊕E/E2,(000ψ2,1000ψ3,20)), |
where ψ2,1:N3→E2/N3⊗K and ψ3,2:E2/N3→E/E2⊗K are the nonzero homomorphisms induced by φ.
(5) If μ(I2)>3μ−μ1−μ2 and μ(N3)>μ, then
(E0,φ0)=(N3⊕E2/N3⊕I2⊕(E/E2)/I2,(0000ψ2,10000ψ3,20000ψ4,30)), |
where ψ2,1:N3→E2/N3⊗K, ψ3,2:E2/N3→I2⊗K and ψ4,3:I2→(E/E2)/I2⊗K are the nonzero homomorphisms induced by φ.
(6) If μ(I2)=3μ−μ1−μ2 and μ(N3)>μ, then
(E0,φ0)=(N3⊕E2/N3⊕I2,(000ψ2,1000ψ3,20))⊕((E/E2)/I2,0), |
where ψ2,1:N3→E2/N3⊗K and ψ3,2:E2/N3→I2⊗K are the nonzero homomorphisms induced by φ.
(7) If μ(I2)<3μ−μ1−μ2, μ≥μ1+μ2−μ3−(2g−2), and μ(N3)=μ, then
(E0,φ0)=(N3,0)⊕(E2/N3⊕E/E2,(00ψ3,20)), |
where ψ3,2:E2/N3→E/E2⊗K is the nonzero homomorphism induced by φ.
(8) If μ(I2)>3μ−μ1−μ2, μ≥μ1+μ2−μ3−(2g−2), and μ(N3)=μ, then
(E0,φ0)=(N3,0)⊕(E2/N3⊕I2⊕(E/E2)/I2,(000ψ3,2000ψ4,30)), |
where ψ3,2:E2/N3→I2⊗K and ψ4,3:I2→(E/E2)/I2⊗K are the nonzero homomorphisms induced by φ.
(9) If μ(I2)=3μ−μ1−μ2, μ≥μ1+μ2−μ3−(2g−2), and μ(N3)=μ, then
(E0,φ0)=(N3,0)⊕(E2/N3⊕I2,(00ψ3,20))⊕((E/E2)/I2,0), |
where ψ3,2:E2/N3→I2⊗K is the nonzero homomorphism induced by φ.
Proof. As in the previous result, the theorem is proved by constructing appropriate gauge transformations for each case and showing how they lead to the stated limit bundles. Suppose first that μ(I2)<μ3 and μ(N3)<μ. Let E, E2, and E′2 be the underlying C∞ vector bundles of E, E2, and E/E2, respectively. Then, E≅E2⊕E′2. The holomorphic structure of the vector bundle E is given by the operator ˉ∂E defined by
ˉ∂E=(ˉ∂2β0ˉ∂′2), |
where ˉ∂2 and ˉ∂′2 are the ˉ∂-operators defining the holomorphic structures of E2 and E/E2, respectively, and β∈A0,1(Hom(E′2,E2)). The Higgs field φ is then of the form
φ=(φ1,1φ1,2φ2,1φ2,2) |
expressed concerning the above decomposition. Similar computations as those made in the case (1) of Theorem 1 show that
limz→0(g(z)⋅ˉ∂E,g(z)−1zφg(z))=(E2⊕E/E2,(00ψ2,10)), |
where g(z)=(100z⋅I) for z∈C∗; thus, g(z)⋅ˉ∂E=(ˉ∂2zβ0ˉ∂′2) and g(z)−1(zφ)g(z)=(zφ1,1z2φ1,2φ2,1zφ2,2), and ψ2,1=φ2,1 is the nonzero homomorphism induced by φ.
Notice now that the only proper φ0-invariant sub-bundles of E0 are N3, E/E2, any proper φ0-invariant subbunsle of E/E2, and E2⊕I2. First, μ(N3)<μ by hypothesis. Second, μ(E/E2)=13(μ3+μ4+μ5)<μ, since μ2>μ3. Third, μ(F)≤μ(E/E2)<μ. Finally,
μ(E2⊕I2)=23μ(E2)+13μ(I2)<23μ1+μ22+3μ−μ1−μ23=μ, |
since μ(I2)<3μ−μ1−μ2 by hypothesis. Then, the slope of every proper φ0-invariant sub-bundle of E0 is less than μ=μ(E0), so the Hodge limit bundle is stable in this case.
For the case (2), suppose that μ(I2)>μ3 and μ(N3)<μ, and let E, E2, I2, and E″2 be the C∞ vector bundles of E, E2, I2, and (E/E2)/I2, respectively, so E≅E2⊕I2⊕E″2. The holomorphic structure of E is given by:
ˉ∂E=(ˉ∂2β1,2β1,30ˉ∂I2β2,300ˉ∂″2), |
thus, by using the following induced decomposition of φ
φ=(φ1,1φ1,2φ1,3φ2,1φ2,2φ2,3φ3,1φ3,2φ3,3) |
and the gauge transformation
g(z)=(1000z000z2), |
the stated form of (E0,φ0) holds by taking the limit as z→0 in the expression of (g(z)⋅ˉ∂E,g(z)−1(zφ)g(z)), as in the case (2) of Theorem 1.
The stability of this Hodge limit bundle follows since the proper φ0-invariant sub-bundles are (E/E2)/I2, I2⊕(E/E2)/I2, and F⊕I2⊕(E/E2)/I2 for a line sub-bundle F of E2. The slopes of these bundles are the following:
μ((E/E2)/I2)=12(5μ−μ1−μ2−μ(I2))<12(μ3+μ4+μ5−(3μ−μ1−μ2))μ((E/E2)/I2)=5μ−3μ2=μ,μ(I2⊕(E/E2)/I2)=13(μ(I2)+2μ((E/E2)/I2))<13(μ3+μ4+μ5)<μ,μ(F⊕I2⊕((E/E2)/I2))=μ(F)+3⋅μ3+μ4+μ534μ(F⊕I2⊕((E/E2)/I2))<μ1+μ22+3⋅μ3+μ4+μ534μ(F⊕I2⊕((E/E2)/I2))≤μ2+μ3+μ4+μ54<μ. |
To check case (3), suppose μ(I2)=13(μ3+μ4+μ5) and μ(N3)<μ. This case is similar to case (2), but the Hodge limit bundle
(E0,φ0)=(E2⊕I2,(00ψ2,10))⊕((E/E2)/I2,0) |
is now strictly semi-stable. Indeed, notice that,
μ((E/E2)/I2)=12(5μ−μ1−μ2−μ(I2))μ((E/E2)/I2)=12(5μ−μ1−μ2−(3μ−μ1−μ2))μ((E/E2)/I2)=12⋅2μ=μ, |
where it has been used that μ(I2)=3μ−μ1−μ2. From this, it follows that μ((E/E2)/I2)=μ=μ(E0), as announced. Then, (E/E2)/I2 is a φ0-invariant destabilizing sub-bundle of E0 and the expression of the poly-stable representative of (E0,φ0) is as stated.
The remaining cases can be proved similarly to the previous cases, adjusting the gauge transformations as needed based on the slopes of I2 and N3, and choosing the gauge transformation to isolate the components that will remain nonzero in the limit. In cases (4) and (5), the Hodge limit bundles are stable (it can be checked by using similar arguments as the above cases). However, in cases (7) to (9), the limit bundles are strictly semi-stable, since N3 is a φ0-invariant sub-bundle of E0 with μ(N3)=μ=μ(E0). In case (6), the φ0-invariant sub-bundle of E0 with slope μ is I2, which is easily checked as done in case (3).
Theorem 3. Let (E,φ) be a rank 5 semi-stable Higgs bundle over X such that the Harder-Narasimhan type (μ1,μ2,μ3,μ4,μ5) of E satisfies μ1≥μ2>μ3≥μ4≥μ5, μ=μ(E), let φ3,2 be the homomorphism E2→E/E2⊗K induced by φ, and let I2 be the vector bundle given in Definition 1. Assume that φ3,2 has rank 2. Let (E0,φ0) be the limit limz→0(E,z⋅φ). Then (E0,φ0) has the following explicit form:
(1) If μ(I2)<2μ−12(μ1+μ2), then
(E0,φ0)=(E2⊕E/E2,(00ψ2,10)), |
where ψ2,1:E2→E/E2⊗K is the nonzero homomorphism induced by φ.
(2) If μ(I2)>2μ−12(μ1+μ2), then
(E0,φ0)=(E2⊕I2⊕(E/E2)/I2,(000ψ2,1000ψ3,20)), |
where ψ2,1:E2→I2⊗K and ψ3,2:I2→(E/E2)/I2⊗K are the nonzero homomorphisms induced by φ.
(3) If μ(I2)=2μ−12(μ1+μ2), then
(E0,φ0)=(E2⊕I2,(00ψ2,10))⊕((E/E2)/I2,0), |
where ψ2,1:E2→I2⊗K is the nonzero homomorphism induced by φ.
Proof. Suppose, for the first case, that μ(I2)<13(μ3+μ4+μ5), and let, as in the preceding theorems, E, E2, and E′2 be the underlying C∞ vector bundles of E, E2, and E/E2. Then, the holomorphic structure of E is given by the ˉ∂E-operator
ˉ∂E=(ˉ∂2β0ˉ∂′2), |
ˉ∂2 and ˉ∂′2 being the ˉ∂-operators of the holomorphic structures of E2 and E/E2, respectively, and β∈A0,1(Hom(E′2,E2)). Reasoning as in case (1) of Theorem 1 or Theorem 2, but with E2 instead of E1, it follows that
limz→0(g(z)⋅ˉ∂E,g(z)−1zφg(z))=(E2⊕E/E2,(00ψ2,10)), |
where ψ2,1=φ2,1 is the nonzero homomorphism induced by φ and g(z)=(100z⋅I). Then, the Hodge limit bundle is of the announced form. Moreover, (E0,φ0) is stable, as it is easily checked by reasoning as in Theorem 2 making the appropriate formal adaptations. Specifically, N2=0 since the rank of φ3,2 is 2 in this case; μ(E/E2)=μ3+μ4+μ53<μ since μ2>μ3; and
μ(E2⊕I2)=2μ(E2)+2μ(I2)4<μ1+μ22+2μ−μ1+μ222=μ, |
where it has been used that μ(I2)<2μ+μ1+μ22. This completes the analysis of the slopes of the φ0-invariant sub-bundles of E0.
For case (2) (μ(I2)>13(μ3+μ4+μ5)), the holomorphic structure of E is given by
ˉ∂E=(ˉ∂2β1,2β1,30ˉ∂I2β2,300ˉ∂″2), |
computed through the decomposition E≅E2⊕I2⊕E″2, where E, E2, I2, and E″2 are the C∞ vector bundles of E, E2, I2, and (E/E2)/I2, respectively. This case is similar to the case (2) of Theorem 2, so the same program leads to the stated form of the Hodge limit bundle (E0,φ0) follows. By reasoning again as in Theorem 2, it is easily checked that (E0,φ0) is stable. The only computation that requires some attention is the slope of the φ0-invariant sub-bundle (E/E2)/I2:
μ((E/E2)/I2)=5μ−μ1−μ2−2μ(I2)<5μ−μ1−μ2−4μ+2μ1+μ22=μ, |
where it has been used that μ(I2)>2μ−μ1+μ22.
For case (3), the same construction of case (2) works but, in this case, the Hodge limit bundle
(E0,φ0)=(E2⊕I2,(00ψ2,10))⊕((E/E2)/I2,0) |
is strictly semi-stable, since (E/E2)/I2 is a φ0-invariant sub-bundle satisfying that μ((E/E2)/I2)=μ, as it is easily checked:
μ((E/E2)/I2)=5μ−μ1−μ2−2μ(I2)<5μ−μ1−μ2−4μ+2⋅μ1+μ22=μ. |
Remark 5. Suppose that μ1≥μ2>μ3≥μ4≥μ5 and that the rank of φ3,2 is 2 (that is, the conditions of Theorem 3 are satisfied). Notice that in this case, the only possibility for N3 is N3=0.
Theorem 4. Let (E,φ) be a rank 5 semi-stable Higgs bundle over X such that the Harder-Narasimhan type (μ1,μ2,μ3,μ4,μ5) of E satisfies μ1≥μ2=μ3=μ4>μ5, μ=μ(E), let φ4,3 be the homomorphism E3→E/E3⊗K induced by φ, and let N1 be the rank 2 vector bundle given in Definition 1. Let (E0,φ0) be the limit limz→0(E,z⋅φ). Then, (E0,φ0) has the following explicit form:
(1) If μ≥12(μ1+μ2+μ3+μ4−μ5−(2g−2)) and μ(N1)<μ, then
(E0,φ0)=(E4⊕E/E4,(00ψ2,10)), |
where ψ2,1:E4→E/E4⊗K is the nonzero homomorphism induced by φ.
(2) If μ≥12(μ1+μ2+μ3+μ4−μ5−(2g−2)) and μ(N1)=μ, then
(E0,φ0)=(N1,0)⊕(E4/N1⊕E/E4,(00ψ2,10)), |
where ψ2,1:E4/N1→E/E4⊗K is the nonzero homomorphism induced by φ.
(3) If μ(N1)>μ, then
(E0,φ0)=(N1⊕E4/N1⊕E/E4,(000ψ2,1000ψ3,20)), |
where ψ2,1:N1→E4/N1⊗K and ψ3,2:E4/N1→E/E4⊗K are the nonzero homomorphisms induced by φ.
Proof. First, suppose that μ(N1)<μ, under the conditions of the statement. As in the case (1) of Theorem 1, the holomorphic structure of E is given by the operator ˉ∂E defined by
∂E=(∂4β0∂′4), |
where E, E4, and E′4 are the differentiable bundles of E, E4, and E/E4, respectively, and ˉ∂4 and ˉ∂′4 are the ˉ∂-operators associated to the holomorphic structures of E4 and E/E4 and β∈A0,1(Hom(E′4,E4)). Similar computations as those of the case (1) of Theorem 1 allow us to compute
limz→0(g(z)⋅∂E,g(z)−1zφg(z))=(E4⊕E/E4,(00ψ2,10)), |
which is the Hodge bundle (E0,φ0) of the first case, being g(z)=(100z⋅I). Notice also that the only φ0-invariant proper sub-bundles of E0 are E/E4 and F⊕(E/E4) for any φ0-invariant proper sub-bundle F of E4. For E/E4, it is clear that μ(E/E4)=μ5<μ. If F⊂E4 is φ0-invariant of rank r, then
μ(F⊕(E/E4))=rμ(F)+μ5r+1<rμ+μ5r+1<μ, |
since μ5<μ, as μ4>μ5, and μ(F)≤μ(E4)=14(μ1+μ2+μ3+μ4)<μ, since μ5<μ, by semi-stability of (E4,φ4). This proves that the Hodge limit bundle (E0,φ0) is stable.
For the second case, suppose that, under the assumptions of the statement, μ(N1)=μ, and let E, N1, E4/N1, and E″4 be the differentiable bundles corresponding to E, N1, E4/N1; let E/E4, ˉ∂E, ˉ∂N1, ˉ∂E4/N1, and ˉ∂″4 be the associated ˉ∂-operators; and let βij be the forms such that
ˉ∂E=(ˉ∂N1β12β130ˉ∂E4/N1β2300ˉ∂″4). |
Then, the (3,1)-component φ3,1 of φ with respect to the decomposition above is 0. Let
g(z)=(1000z000z2). |
Then, it is satisfied that
limz→0(g(z)⋅ˉ∂E,g(z)−1zφg(z))=(N1⊕E4/N1⊕E/E4,(000ψ2,1000ψ3,20)), |
and the announced Hodge limit bundle (E0,φ0) is obtained (this case is similar to the case (7) of Theorem 2 with N1 instead of N3 and E4 instead of E2). Moreover, it is strictly semi-stable, since N1 is a φ0-invariant sub-bundle of E0 with μ(N1)=μ=μ(E0). The S-equivalence class of this Hodge limit bundle can be represented by the poly-stable bundle
(E0,φ0)=(N1,0)⊕(E4/N1⊕E/E4,(00ψ2,10)), |
as stated.
The third case is analogous to the second one, but noticing that N1 is not a φ0-invariant sub-bundle here, which makes the Hodge limit bundle stable as a Higgs bundle.
Theorem 5. Let (E,φ) be a rank 5 semi-stable Higgs bundle over X such that the Harder-Narasimhan type (μ1,μ2,μ3,μ4,μ5) of E satisfies μ1≥μ2≥μ3>μ4≥μ5, μ=μ(E), let φ4,3 be the homomorphism E3→E/E3⊗K induced by φ, and let I3 and N2 be the vector bundles given in Definition 1. Assume that φ4,3 has rank 1. Let (E0,φ0) be the limit limz→0(E,z⋅φ). Then, (E0,φ0) has the following explicit form:
(1) If μ(I3)<μ4+μ5−μ and μ(N2)<μ, then
(E0,φ0)=(E3⊕E/E3,(00ψ2,10)), |
where ψ2,1:E3→E/E3⊗K is the nonzero homomorphism induced by φ.
(2) If μ(I3)>μ4+μ5−μ and μ(N2)<μ, then
(E0,φ0)=(E3⊕I3⊕(E/E3)/I3,(000ψ2,1000ψ3,20)), |
where ψ2,1:E3→I3⊗K and ψ3,2:I3→(E/E3)/I3⊗K are the nonzero homomorphisms induced by φ.
(3) If μ(I3)=μ4+μ5−μ and μ(N2)<μ, then
(E0,φ0)=(E3⊕I3,(00ψ2,10))⊕((E/E3)/I3,0), |
where ψ2,1:E3→I3⊗K is the nonzero homomorphism induced by φ.
(4) If μ(I3)<μ4+μ5−μ, μ≥12(μ1+μ2+μ3−μ4−μ5−(2g−2)), and μ(N2)=μ, then
(E0,φ0)=(N2,0)⊕(E3/N2⊕E/E3,(00ψ3,20)), |
where ψ3,2:E3/N2→E/E3⊗K is the nonzero homomorphism induced by φ.
(5) If μ(I3)>μ4+μ5−μ, μ≥12(μ1+μ2+μ3−μ4−μ5−(2g−2)), and μ(N2)=μ, then
(E0,φ0)=(N2,0)⊕(E3/N2⊕I3⊕(E/E3)/I3,(000ψ3,2000ψ4,30)), |
where ψ3,2:E3/N2→I3⊗K and ψ4,3:I3→(E/E3)/I3⊗K are the nonzero homomorphisms induced by φ.
(6) If μ(I3)=μ4+μ5−μ, μ≥12(μ1+μ2+μ3−μ4−μ5−(2g−2)), and μ(N2)=μ, then
(E0,φ0)=(N2,0)⊕(E3/N2⊕I3,(00ψ3,20))⊕((E/E3)/I3,0), |
where ψ3,2:E3/N2→I3⊗K is the nonzero homomorphism induced by φ.
(7) If μ(I3)<μ4+μ5−μ and μ(N2)>μ, then
(E0,φ0)=(N2⊕E3/N2⊕E/E3,(000ψ2,1000ψ3,20)), |
where ψ2,1:N2→E3/N2⊗K and ψ3,2:E3/N2→E/E3⊗K are the nonzero homomorphisms induced by φ.
(8) If μ(I3)>μ4+μ5−μ and μ(N2)>μ, then
(E0,φ0)=(N2⊕E3/N2⊕I3⊕(E/E3)/I3,(0000ψ2,10000ψ3,20000ψ4,30)), |
where ψ2,1:N2→E3/N2⊗K, ψ3,2:E3/N2→I3⊗K and ψ4,3:I3→(E/E3)/I3⊗K are the nonzero homomorphisms induced by φ.
(9) If μ(I3)=μ4+μ5−μ and μ(N2)>μ, then
(E0,φ0)=(N2⊕E3/N2⊕I3,(000ψ2,1000ψ3,20))⊕((E/E3)/I3,0), |
where ψ2,1:N2→E3/N2⊗K and ψ3,2:E3/N2→I3⊗K are the nonzero homomorphisms induced by φ.
Proof. For the first case, assume μ(I3)<12(μ4+μ5) and μ(N2)<μ. Let E=E3⊕E″3, where E″3=E/E3, and the gauge transformation
g(z)=(100z) |
for z∈C∗. Then,
g(z)⋅ˉ∂E=(ˉ∂3zβ0ˉ∂″3) and g(z)−1(zφ)g(z)=(zφ1,1z2φ1,2φ2,1zφ2,2). |
Taking the limit as z→0, the announced form for the Hodge limit bundle is obtained as in the case (1) of Theorem 1 with E3 instead of E1.
For the second case, assume μ(I3)>12(μ4+μ5) and μ(N2)<μ, let E=E3⊕I3⊕E″3, and define
g(z)=(1000z000z2). |
Applying this transformation and taking the limit provides the announced form for the Hodge limit bundle as in the case (2) of Theorem 1 or Theorem 3, but considering I3 instead of I1 or I2.
For the third case, suppose that μ(I3)=12(μ4+μ5) and μ(N2)<μ. Then, the same gauge transformations as in the preceding case work and the limit bundle is the one stated.
For the fourth case, assume μ(I3)<12(μ4+μ5) and μ(N2)=μ, and let E=N2⊕(E3/N2)⊕E″3. The stated form for the Hodge limit bundle arises from applying the gauge transformation g(z)=(1000z000z2), with respect to the decomposition above, and taking the limit as z→0.
For the fifth case, it is supposed that μ(I3)>12(μ4+μ5) and μ(N2)=μ. Take E=N2⊕(E3/N2)⊕I3⊕E″3, and define
g(z)=(10000z0000z20000z3). |
The limit bundle of the statement arises from this by computing the limit as done in the preceding theorems.
For the sixth case, the same gauge transformation as in the case above works. Similarly, the cases (7) to (9) follow by taking suitable gauge transformations.
By reasoning as in Theorems 1 and 2 with the appropriate formal adaptations, it follows that in the cases (1), (2), (7), and (8), the Hodge limit bundle is stable, and in the remaining cases it is strictly semi-stable, so the graded object gives a strictly poly-stable representative of those Hodge limit bundles. The first two cases will be computed (the remaining cases are similar). Specifically, for the first case, notice that μ(E/E3)=μ4+μ52<μ, as μ3>μ4, and that
μ(I3⊕E3)=3μ(E3)+μ(I3)4<μ1+μ2+μ3−μ+μ4+μ54=5μ−μ4=μ, |
since μ(I3)<μ4+μ5−μ in this case. For the second case, the only φ0-invariant vector sub-bundle that requires some computation is
μ((E/E3)/I3)=μ4+μ5−μ(I3)<μ4+μ5−μ4−μ5+μ=μ, |
since, in this case, μ(I3)>μ4+μ5−μ by hypothesis. In cases (4) to (6), μ(N2)=μ by hypothesis, so N4 is a φ0-invariant sub-bundle with the same slope as E0, and in cases (3), (6), and (9),
μ((E/E3)/I3)=μ4+μ5−μ(I3)=μ4+μ5−μ4−μ5+μ=μ, |
because, in these cases, μ(I3)=μ4+μ5−μ.
Theorem 6. Let (E,φ) be a rank 5 semi-stable Higgs bundle over X such that the Harder-Narasimhan type (μ1,μ2,μ3,μ4,μ5) of E satisfies μ1≥μ2=μ3>μ4≥μ5, μ=μ(E), let φ4,3 be the homomorphism E3→E/E3⊗K induced by φ, and let N2 be the vector bundle given in Definition 1. Assume that φ4,3 has rank 2. Let (E0,φ0) be the limit limz→0(E,z⋅φ). Then, (E0,φ0) has the following explicit form:
(1) If μ≥12(μ1+μ2+μ3−μ4−μ5−(2g−2)) and μ(N2)<μ, then
(E0,φ0)=(E3⊕E/E3,(00ψ2,10)), |
where ψ2,1:E3→E/E3⊗K is the nonzero homomorphism induced by φ.
(2) If μ≥12(μ1+μ2+μ3−μ4−μ5−(2g−2)) and μ(N2)=μ, then
(E0,φ0)=(N2,0)⊕(E3/N2⊕E/E3,(00ψ3,20)), |
where ψ3,2:E3/N2→E/E3⊗K is the nonzero homomorphism induced by φ.
(3) If μ(N2)>μ, then
(E0,φ0)=(N2⊕E3/N2⊕E/E3,(000ψ2,1000ψ3,20)), |
where ψ2,1:N2→E3/N2⊗K and ψ3,2:E3/N2→E/E3⊗K are the nonzero homomorphisms induced by φ.
Proof. Let E, E3, N2, I3, and E″3 be the underlying C∞ vector bundles corresponding to E, E3, N2, I3, and (E/E3)/I3, respectively. Let ˉ∂E, ˉ∂3, ˉ∂N2, ˉ∂I3, and ˉ∂″3 be their associated ˉ∂-operators.
Suppose, for the first case, that μ(I3)<12(μ4+μ5) and μ(N2)<μ. In this case, E=E3⊕E/E3. The holomorphic structure of E is given by
ˉ∂E=(ˉ∂3β0ˉ∂′3), |
where β∈A0,1(Hom(E/E3,E3)). Being the Higgs field φ of the form φ=(φ1,1φ1,2φ2,1φ2,2) with respect to the decomposition above and g(z), the gauge transformation g(z)=(100z⋅I) for z∈C∗, it follows that
g(z)⋅ˉ∂E=(ˉ∂3zβ0ˉ∂′3) and g(z)−1(zφ)g(z)=(zφ1,1z2φ1,2φ2,1zφ2,2), |
thus, by taking the limit as z→0, it follows that
limz→0(g(z)⋅ˉ∂E,g(z)−1zφg(z))=(E3⊕E/E3,(00ψ2,10)), |
similarly to the case (1) of Theorem 1 or Theorem 2. Then, the stated Hodge limit bundle is set. This limit bundle is stable because μ(N2)<μ by hypothesis and μ(E/E3)=13(μ3+μ4+μ5)<μ due to the assumption μ1≥μ2=μ3>μ4≥μ5.
For cases (2) to (9), the proofs follow a similar pattern, using suitable gauge transformations based on the decomposition of E and the conditions on μ(I3) and μ(N2) in each case.
For the cases where μ(I3)=12(μ−μ1−μ2−μ3) or μ(N2)=μ (that is, cases (3) to (6) and (9)), the resulting Hodge limit bundles are strictly semi-stable, having (E/E3)/I3 (for the cases (3), (6), and (9)) or N2 (for the cases (4) to (6)) as destabilizing sub-bundles, which is easily checked as in Theorem 2. Indeed, notice that in cases (4) to (6), μ(N2)=μ=μ(E0), so it is the destabilizing vector sub-bundle, and in the cases (3), (6), and (9), since
μ(E/E2)=2μ(I3)+μ((E/E2)/I3))3, |
it easily follows that μ((E/E2)/I3)=3μ3+μ4+μ53−2μ(I3)=μ, under the assumption that μ(I3)=12(μ−μ3−μ4−μ5).
Remark 6. Suppose that μ1≥μ2≥μ3>μ4≥μ5 and that the rank of φ4,3 is 2 (that is, the conditions of Theorem 6 are satisfied). Then, I3=E/E3, thus, μ(I3)=μ4+μ52 in this case, and it does not play a role in the description of the Hodge limit bundles given in Theorem 6.
Remark 7. It is worth noting that the bounds used for the intermediate bundles Ik and Nk in Theorems 1 to 6 (the notation is always that of Definition 1) are coherent with those proved for the same bundles in Proposition 1. To avoid being repetitive with very similar computations, this issue will be exemplified with the bounds found in Theorems 1 and 2, which will be confronted with the bounds of sections (1) and (2) of Proposition 1.
In Theorem 1, μ(I1) is compared with 2μ−μ1, and in part (1) of Proposition 1 it is proved that μ1−(2g−2)≤μ(I1)≤μ2 whenever μ1>μ2. Notice that 2μ−μ1=−3μ1+2μ2+2μ3+2μ4+2μ55, so 2μ−μ1<μ2 if, and only if, −3μ1+2μ2+2μ3+2μ4+2μ≤5μ2, that is, −3(μ1+μ2)+2(μ3+μ4+μ5)≤0. This is equivalent to stating that
μ3+μ4+μ53≤μ1+μ22, |
which is true under the assumption that μ1>μ2. This proves that, certainly, 2μ−μ1≤μ2. Moreover, 2μ−μ1≥μ1−(2g−2) if, and only if, g−1≥μ1−μ>0, and this is again true since, under our assumptions, g≥2. Therefore, μ1−(2g−2)≤2μ−μ1≤μ2.
For the case of the slopes used in Theorem 2, μ(I2) is compared with 3μ−μ1−μ2 and μ(N3) is compared with μ. Take the bounds proved in part (2) of Proposition 1 under the assumption that μ2>μ3. Notice first that, since μ1+μ2+μ33≥μ (since μ2>μ3), it follows that 3μ−μ1−μ2≤μ3. In addition, since that morphism E2/E1→E3/E2⊗K induced by the Higgs field is nonzero, it must be μ2−(2g−2)≥μ3≥3μ−μ1−μ2. It has been then proved that μ2−(2g−2)≤3μ−μ1−μ2≤μ3. For μ(N3), notice that μ≤μ1 as μ2>μ3. Now, the inequality μ≥μ1+μ2−μ3−(2g−2) can occur depending on the value of the genus g, as discussed in the Remark after Proposition 1.
In Theorems 1 to 6, not only are Hodge limit bundles computed, but it is discussed whether they are stable or strictly semi-stable. In the case of strict semi-stability, the statements of the theorems provide the expression of the strictly poly-stable representative of the respective S-equivalence class. Specifically, Hodge limit bundles are stable in the situations described in the following result, which follows immediately from the proofs of Theorems 1 to 6.
Corolary 1. Let (E,φ) be a semi-stable rank 5 Higgs bundle over X such that the Harder-Narasimhan type of E is (μ1,μ2,μ3,μ4,μ5) and μ=μ(E) be the slope of E. Then, the Hodge limit bundle for the C∗-action (E0,φ0)=limz→0(E,z⋅φ) described in Theorems 1 to 6 is stable if, and only if, one of the following conditions, expressed with the notation of Definition 1, holds:
(1) μ1>μ2=μ3=μ4=μ5 and μ(I1)≠2μ−μ1.
(2) μ1≥μ2>μ3≥μ4≥μ5, rk(φ3,2)=1, μ(I2)≠3μ−μ1−μ2, and μ(N3)≠μ.
(3) μ1≥μ2>μ3≥μ4≥μ5, rk(φ3,2)=2, and μ(I2)≠2μ−12(μ1+μ2).
(4) μ1≥μ2≥μ3=μ4>μ5, and μ(N1)≠μ.
(5) μ1≥μ2≥μ3>μ4≥μ5, rk(φ4,3)=1, μ(I3)≠μ4+μ5−μ, and μ(N2)≠μ.
(6) μ1≥μ2≥μ3>μ4≥μ5, rk(φ4,3)=2, and μ(N2)≠μ.
In this section, it will be proved that the Shatz and the Bialynycki-Birula stratifications of the moduli space of rank 5 and degree d Higgs bundles over X are transversal, in the sense that no stratum of one stratification coincides with a stratum of the other. Specifically, it will be checked that, for every possible Harder-Narasimhan type of the underlying vector bundle of a strictly semi-stable rank 5 Higgs bundle, there exist semi-stable Higgs bundles whose underlying vector bundle admits the given Harder-Narasimhan type such that the associated Hodge limit bundles are not S-equivalent. For this purpose, the following four cases will be distinguished:
(1) μ1>μ2=μ3=μ4=μ5,
(2) μ1≥μ2>μ3≥μ4≥μ5,
(3) μ1≥μ2=μ3=μ4>μ5,
(4) μ1≥μ2=μ3>μ4≥μ5,
and the descriptions of the Hodge limit bundles made in the Theorems 1 to 6 will be used. The key is that, for the same Harder-Narasimhan type, it will be possible to find different semi-stable Higgs bundles whose underlying vector bundles admit different intermediate bundles I1, I2, I3, N1, N2, or N3, introduced in Definition 1, so that, depending on the particular description of the possible Hodge limit bundles given in the mentioned theorems, the associated limits will have non-isomorphic graded objects.
Proposition 2. Let X be a compact Riemann surface of genus g≥2. For any Harder-Narasimhan type (μ1,μ2,μ3,μ4,μ5) satisfying μ1>μ2=μ3=μ4=μ5, there exist semi-stable rank 5 Higgs bundles (E,φ) and (E′,φ′) such that E admits this Harder-Narasimhan type and whose Hodge limit bundles under the C∗-action are not S-equivalent.
Proof. Suppose that a Harder-Narasimhan type (μ1,μ2,μ2,μ2,μ2) with μ1>μ2 is taken. Let E1 be a line bundle of degree d1=μ1 and F be a stable rank 4 vector bundle of degree 4μ2. Let E=E1⊕F and choose the Higgs field φ:E→E⊗K to be
φ=(00ψ0), |
for a nonzero morphism ψ:E1→F⊗K. Then, (E,φ) is semi-stable by construction, since E1 and F are stable and φ(E1)⊄E1⊗K. Let I1 be the saturation of ψ(E1)⊗K−1 in E/E1=F. Since F is stable as a vector bundle, μ(I1)≤μ(F)=μ2, and, by definition of I1, μ(I1)≥μ(E1)−(2g−2)=μ1−(2g−2). Choose ψ such that μ(I1)<2μ−μ1, which is always possible, as μ1>μ2.
Let now E′1 be a line bundle of degree d1=μ1, L be a line bundle of certain degree l such that 2μ−μ1<l≤μ2 (this is always possible, since μ<μ1+μ22, thus, 2μ−μ1<μ2), and G be a stable rank 3 vector bundle of degree 4μ2−l. Define E′=E′1⊕L⊕G and φ′:E′→E′⊗K by
φ′=(000ψ1000ψ20), |
where ψ1:E′1→L⊗K and ψ2:L→G⊗K are nonzero morphisms. Define I′1=L. By the condition on l, (E′,φ′) is semi-stable and E′ has the same Harder-Narasimhan type that E.
Since μ(I1)<2μ−μ1, the Hodge limit bundle (E0,φ0) of (E,φ) is computed in case (1) of Theorem 1:
(E0,φ0)=(E1⊕F,(00ψ2,10)). |
Similarly, since μ(I1)>2μ−μ1, the Hodge limit bundle of (E′,φ′) is given in case (2) of Theorem 1, so it is expressed by
(E′0,φ′0)=(E′1⊕L⊕G,(000ψ2,1000ψ3,20)). |
The Hodge limit bundles (E0,φ0) and (E′0,φ′0) are not S-equivalent, since they have different decompositions into stable sub-bundles, so their graded objects cannot be isomorphic. Therefore, (E,φ) and (E′,φ′) are semi-stable Higgs bundles whose underlying vector bundles have the same Harder-Narasimhan type (μ1,μ2,μ2,μ2,μ2) and whose respective Hodge limit bundles are not S-equivalent.
Proposition 3. Let X be a compact Riemann surface of genus g≥2. For any Harder-Narasimhan type (μ1,μ2,μ3,μ4,μ5) satisfying μ1≥μ2>μ3≥μ4≥μ5, there exist semi-stable rank 5 Higgs bundles (E,φ) and (E′,φ′) whose underlying vector bundles admit this Harder-Narasimhan type and whose Hodge limit bundles under the C∗-action are not S-equivalent.
Proof. Let E2 be a stable vector bundle of rank 2 and slope μ1+μ22, and F be a stable vector bundle of rank 3 and slope μ3+μ4+μ53. Let E=E2⊕F. Choose a rank 2 morphism ψ:E2→F⊗K. Define φ as
φ=(00ψ0):E2⊕F→(E2⊕F)⊗K. |
Then, (E,φ) is semi-stable and E has the Harder-Narasimhan type fixed in the statement, by construction.
Similarly, let E′2 be a stable vector bundle of rank 2 and slope μ1+μ22, chosen to be non-isomorphic to E2, and F′ be a stable vector bundle of rank 3 and slope μ3+μ4+μ53, chosen to be non-isomorphic to F. Define E′=E′2⊕F′, choose a nonzero morphism ψ′:E′2→F′⊗K, and let
φ′=(00ψ′0):E′2⊕F′→(E′2⊕F′)⊗K. |
Again, (E′,φ′) is semi-stable, E′ has the same Harder-Narasimhan type, and ψ′ can be chosen such that (E,φ) and (E′,φ′) are not S-equivalent. In both (E,φ) and (E′,φ′), the rank of ψ determines the rank of φ3,2, which is 2.
Let I2 and I′2 be the saturations of Im(ψ)⊗K−1 and Im(ψ′)⊗K−1, respectively. For the computation of the limit Hodge bundles, let I2 and I′2 be the saturations of Im(ψ)⊗K−1 and Im(ψ′)⊗K−1, respectively, as in Definition 1. Three cases can now be distinguished, depending on the slope of I2. First, if μ(I2)<2μ−12(μ1+μ2), then the limit Hodge bundles are:
(E0,φ0)=(E2⊕F,(00ψ0)) |
and
(E′0,φ′0)=(E′2⊕F′,(00ψ′0)), |
by Theorem 3. The Harder-Narasimhan filtrations of E0 and E′0 are
0⊂E2⊂E0and0⊂E′2⊂E′0, |
so the associated graded objects are gr(E0)=E2⊕F and gr(E′0)=E′2⊕F′. Since E2≇E′2 and F≇F′, these graded objects are non-isomorphic, proving that the limit Hodge bundles are not S-equivalent.
Second, if μ(I2)>2μ−12(μ1+μ2), the limit Hodge bundles are:
(E0,φ0)=(E2⊕I2⊕F/I2,(000ψ1000ψ20)) |
and
(E′0,φ′0)=(E′2⊕I′2⊕F′/I′2,(000ψ′1000ψ′20)), |
where ψ1:E2→I2⊗K, ψ2:I2→(F/I2)⊗K, and similarly for ψ′1 and ψ′2.
Third, if μ(I2)=2μ−12(μ1+μ2), then the limit Hodge bundles are
(E0,φ0)=((E2⊕I2,(00ψ10))⊕(F/I2,0)) |
and
(E′0,φ′0)=((E′2⊕I′2,(00ψ′10))⊕(F′/I′2,0)), |
so, in this case, the Harder-Narasimhan filtrations of the underlying vector bundles are
0⊂E2⊂E2⊕I2⊂E0and0⊂E′2⊂E′2⊕I′2⊂E′0. |
The associated graded objects are then gr(E0)=E2⊕I2⊕F/I2 and gr(E′0)=E′2⊕I′2⊕F′/I′2. As in the preceding case, since E2≇E′2 and I2≇I′2, these graded objects are non-isomorphic, so the limit Hodge bundles are not S-equivalent.
Therefore, in all cases, the semi-stable Higgs bundles (E,φ) and (E′,φ′) satisfy that the underlying vector bundles admit the given Harder-Narasimhan type satisfying the conditions of the statement and they admit limit Hodge bundles which are not S-equivalent.
Proposition 4. Let X be a compact Riemann surface of genus g≥2. For any Harder-Narasimhan type (μ1,μ2,μ3,μ4,μ5) satisfying μ1≥μ2=μ3=μ4>μ5, there exist semi-stable Higgs bundles (E,φ) and (E′,φ′) of rank 5 whose underlying vector bundles admit this Harder-Narasimhan type, and whose respective Hodge limit bundles (E0,φ0) and (E′0,φ′0) are not S-equivalent.
Proof. Under the assumptions of the statement, let L1 be a line bundle of degree d1=μ1(2g−1) and L5 be a line bundle of degree d5=μ5(2g−1). Take E=L1⊕F2⊕L5, where F2 is a rank 3 vector bundle fitting in the exact sequence:
0→OX→F2→O⊕2X→0, |
such that deg(F2)=3μ2(2g−1), OX being the trivial line bundle over X. Choose the Higgs field ϕ such that N1=L1⊕OX, which can be written in block form for this decomposition as
ϕ=(000ψ2,1000ψ3,20), |
where ψ2,1:L1⊕OX→F2/OX⊗K and ψ3,2:F2→L5⊗K are nonzero morphisms.
For the construction of the other Higgs bundle (E′,φ′), let L′1 be another line bundle of degree d1=μ1(2g−1) not isomorphic to L1, and define
E′=L′1⊕F2⊕L5. |
Choose ϕ′ with the same block structure as ϕ but ensuring that N′1=L′1⊕OX.
Both Higgs bundles are clearly semi-stable, the respective underlying vector bundles have the same fixed Harder-Narasimhan type and with μ(N1)>μ and μ(N′1)>μ; thus, by Theorem 4, the corresponding Hodge limit bundles are
(E0,φ0)=(N1⊕E4/N1⊕E/E4,(000ψ2,1000ψ3,20)) |
and
(E′0,φ′0)=(N′1⊕E′4/N′1⊕E′/E′4,(000ψ′2,1000ψ′3,20)), |
where ψ2,1,ψ3,2,ψ′2,1,ψ′3,2 are the nonzero morphisms induced by ϕ and ϕ′, respectively. These Hodge limit bundles are not S-equivalent, since N1 contains L1 while N′1 contains L′1 as direct summands and L1≇L′1 by construction.
Therefore, for any valid Harder-Narasimhan type under the conditions of the statement, there exist semi-stable Higgs bundles whose underlying vector bundles admit this Harder-Narasimhan type and whose Hodge limits are not S-equivalent.
Proposition 5. Let X be a compact Riemann surface of genus g≥2. For any Harder-Narasimhan type (μ1,μ2,μ3,μ4,μ5) satisfying μ1≥μ2=μ3>μ4≥μ5, there exist semi-stable rank 5 Higgs bundles (E,φ) and (E′,φ′) whose underlying vector bundles admit this Harder-Narasimhan type, and whose respective limit Hodge bundles under the C∗-action are not S-equivalent.
Proof. Fix a Harder-Narasimhan type (μ1,μ2,μ3,μ4,μ5) satisfying μ1≥μ2=μ3>μ4≥μ5. For the construction of (E,φ), take N2 to be a stable bundle of rank 2 with slope μ(N2)>μ (this is always possible), E3/N2 to be a stable line bundle such that μ(E3/N2)=3μ3−2μ(N2), and E/E3 to be a stable bundle of rank 2 with μ(E/E3)=5μ−3μ32. Choose ψ2,1:N2→E3/N2⊗K and ψ3,2:E3/N2→E/E3⊗K to be nonzero morphisms.
For (E′,φ′), choose N′2 to be a stable bundle of rank 2 with μ(N′2)=μ(N2) but not isomorphic to N2 and keep the same construction for E′3/N′2, E′/E′3, ψ′2,1, and ψ′3,2 as above.
These Higgs bundles satisfy that the underlying vector bundles admit the same fixed Harder-Narasimhan type and also verify the conditions of Theorem 6, specifically of the case (3), but the corresponding Hodge limit bundles are not S-equivalent because, by construction, N2≇N′2, being these vector sub-bundles summands of the respective graded objects of (E,φ) and (E′,φ′).
Therefore, there exist semi-stable Higgs bundles whose underlying vector bundles have the same Harder-Narasimhan type of the given form whose Hodge limits are not S-equivalent.
Theorem 7. Let X be a compact Riemann surface of genus g≥2 and M(5,d) be the moduli space of semi-stable rank 5 and degree d Higgs bundles over X. Then, the Shatz strata traverse the Białynicki-Birula strata of M(5,d), in the sense that each Shatz stratum contains semi-stable Higgs bundles with rank 5 of several Białynicki-Birula strata.
Proof. The result holds from Propositions 2 to 5 by noticing that, if (E,φ) is stable, then limz→0(E,z⋅φ)=(E,0). In this case, it suffices to take two stable vector bundles E and E′ with the same degree (hence, with the same Harder-Narasimhan type of the considered form) to ensure that the associated Hodge limit bundles are not S-equivalent, because (E,0) and (E′,0) are stable as Higgs bundles but they are not isomorphic.
Let X be a compact Riemann surface of genus g≥2 and M(5,d) be the moduli space of rank 5 semi-stable holomorphic Higgs bundles over X of a given degree d. The multiplicative group C∗ acts on M(5,d) by the product on the Higgs field. One of the main contributions of the paper is the computation of all possible limits limz→0(E,z⋅φ), where (E,φ) is a rank 5 semi-stable Higgs bundle over X. Explicit vector forms for these limit bundles are provided, which give a description of the Białynicki-Birula strata in M(5,d). The possibilities for the limit bundles are in accordance with a certain classification of the Harder-Narasimhan type associated with the underlying vector bundle of a semi-stable Higgs bundle that has been given, and with some bounds for the slope of certain intermediate vector bundles that have also been provided. In addition, necessary and sufficient conditions are given on the Harder-Narasimhan types and the slopes of the intermediate bundles for the Hodge limit bundle to be stable. The other main contribution of the paper is proving that, whatever the Harder-Narasimhan type, there exist semi-stable Higgs bundles with rank 5 whose associated limit bundles are not S-equivalent but whose underlying vector bundles have the same Harder-Narasimhan type, which is equal to the given. Indeed, in each of the situations differentiated in the paper to analyze this question, explicit constructions of these Higgs bundles are provided. As a consequence, it is proved that the Shatz strata of M(5,d), defined by the Harder-Narasimhan type of the underlying vector bundles, traverse the Białynicki-Birula strata, in the sense that every Shatz stratum contains Higgs bundles of different Białynicki-Birula strata. This behavior that has been proved in rank 5 contrasts with that of rank 4, where there are overlapping strata.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The author declare there is no conflict of interest.
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