On a Conjecture About Higgs Bundles and Some Inequalities

. We brieﬂy review an open conjecture about Higgs bundles that are semistable after pulling back to any curve, and prove it in the rank 2 case. We also prove some results in higher rank under suitable additional assumptions. Moreover, we establish a set of inequalities holding for H-nef Higgs bundles that generalize some of the Fulton–Lazarsfeld inequalities for numerically eﬀective vector bundles.


Introduction
The progenitor of the results discussed in this paper may be traced back to a theorem by Miyaoka [20], which characterizes the semistability of a vector bundle E on a smooth projective curve X in terms of the nefness of a numerical class in the projectivized bundle PE: where π 1 : PE → X is the projection, and r = rk E, then E is semistable if and only if λ(E) is nef (note that rλ(E) is the relative anticanonical class of PE over X).

Curve semistable (Higgs) bundles
The following theorem was proved in [21] and rediscovered in [6] in a slightly different and seemingly stronger, albeit equivalent form.It may be regarded as a higher dimensional generalization of Miyaoka's theorem.Let X be an n-dimensional smooth connected complex projective variety.For any coherent O X -module F of positive rank define its discriminant as Moreover, if E is a vector bundle on X, the class λ(E) is defined as in equation (1).
Theorem 1.1.Let E be a vector bundle on X.The following conditions are equivalent: (i) E is semistable with respect to some polarization H, and ∆(E) = 0; (ii) for any morphism f : C → X, where C is a smooth projective curve, the vector bundle f * E is semistable; (iii) the class λ(E) is nef.
(In Nakayama the condition on the discriminant was ∆(E) • H n−2 = 0, but via Theorem 2 in [24] this is readily shown to be equivalent to ∆(E) = 0 whenever E is semistable with respect to H.) We shall call curve semistable the vector bundles satisfying condition (ii).It may be natural to wonder if Theorem 1.1 also holds true for Higgs bundles.We recall that a Higgs sheaf is a pair F = (F, φ), where F is a coherent O X -module, and φ : is zero.A Higgs bundle is a Higgs sheaf with F locally free.Semistability and stability are defined as for vector bundles but only with reference to φ-invariant subsheaves.Curve semistability is defined as for vector bundles.So the Higgs bundle version of Theorem 1.1 is the following conjecture: Conjecture 1.2.Let E = (E, φ) be a Higgs bundle on X.The following conditions are equivalent: (i) E is semistable with respect to some polarization H, and ∆(E) = 0; (ii) for any morphism f : C → X, where C is a smooth projective curve, the Higgs bundle f * E is semistable.
(We shall state the condition generalizing the nefness of the class λ(E) later on.)The fact that condition (i) implies condition (ii) was proved in [6].A motivation for expecting that the opposite implication may hold true is Bogomolov inequality [14]: if E is a vector bundle on an n-dimensional smooth projective variety, semistable with respect to a polarization H, then ∆(E) • H n−2 ≥ 0. The underlying vector bundle E of a semistable Higgs bundle E = (E, φ) satisfies the same inequality, even when E itself is not semistable [23]; i.e., semistability is a sufficient but non-necessary condition for the non-negativity of the quantity ∆(E)•H n−2 , and one can imagine the same happens for the vanishing of ∆(E) for curve semistable bundles.
We conjecture that the reverse implication holds true for any smooth projective variety.In this paper we prove this when E has rank two; actually, we prove the implication in any rank when the Grassmannian of Higgs quotients of some rank (to be defined later) has a component which is a divisor in the full Grassmannian and surjects onto X.Then we prove that such a component always exists in rank two.

Higgs varieties
One easily shows that a curve semistable Higgs bundle is semistable with respect to any polarization.So the nontrivial content of the conjecture is the following statement: A curve semistable Higgs bundle has vanishing discriminant.
Here curve semistability for Higgs bundles is defined as in condition (ii) of Conjecture 1.2.Waiting for the conjecture to be eventually settled in the positive or negative, it makes sense to prove it for specific classes of varieties.The authors of [8] defined a Higgs variety X as one on which the conjecture holds.The easiest case is that of varieties with slope-semistable cotangent bundle of nonnegative degree, simply because in this situation the underlying vector bundle E of a curve semistable Higgs bundle is itself curve semistable.Starting from this one can identify other Higgs varieties, such as: • rationally connected varieties; • abelian varieties; • fibrations over a Higgs variety whose fibers are rationally connected; • bases of finite étale covers whose total space is a Higgs variety; • varieties of dimension ≥ 3 containing an effective ample divisor which is a Higgs variety; • varieties with nef tangent bundle (in dimension 2 and 3 these were classified in [10]); • varieties birational to a Higgs variety.Moreover, in [7] it was shown that algebraic K3 surfaces are Higgs varieties, and this was extended, using different techniques, to simply connected Calabi-Yau varieties in [3].Some results in the case of elliptic surfaces are proved in [9].A review of this problem updated to 2017 can be found in [18].

Contents
The main tool we use in this paper is the Higgs Grassmannian of a Higgs bundle E = (E, φ), a notion that some of us introduced in [6].This object is defined in Section 2, where some of its basic properties are studied.It seems quite difficult to find general results about the Higgs Grassmannian, but its structure is quite clear in the case rk E = 2, and this is indeed the key to the proof of the conjecture in the rank 2 case that we give in Section 4.2.Actually in Section 3 we prove the conjecture assuming that the rank d Higgs Grassmannian Gr d (E) has a component that is a divisor in the full Grassmann bundle Gr d (E) which surjects onto X.Such a divisor always exists in the rank 2 case, due to the fact that the Higgs Grassmiannian of a rank 2 Higgs bundle over a curve is never empty, thus providing a full proof of the conjecture in the rank 2 case.
The Higgs Grassmannian allows one to introduce a notion of numerical effectiveness for Higgs bundles, a notion that "feels" the Higgs field.This was studied in [5,4].In the final Section 5 of this paper we show that Higgs bundles that are numerically effective in this sense satisfy some inequalities which generalize some of the Fulton-Lazarsfeld inequalities for numerically effective vector bundles ( [13], see also [10]).
Notation and conventions.All varieties and schemes are over the complex numbers, and, unless otherwise stated, all varieties are supposed to be connected.A "sheaf" on a scheme X will be a coherent O X -module.

The Higgs Grassmannian
The Higgs Grassmannian is an object that parameterizes locally free Higgs quotients of a Higgs bundle exactly as the usual Grassmann bundle parameterizes locally free quotients of a vector bundle.This was introduced in [6].We recall here its definition and some of its properties.

Definition of the Higgs Grassmannian
Let X be a smooth variety over C. For a given rank r vector bundle E on X, and for every d in the range 0 < d < r, we denote the Grassmann bundle of rank d locally free quotients of E as Gr d (E).Since Gr 1 (E) = PE we shall use the latter notation.One has the universal exact sequence 0 of vector bundles on Gr d (E), where Q d is the rank d universal quotient bundle, S d is the corresponding kernel, and The The scheme Gr d (E) may be singular, reducible, nonreduced, non-equidimensional.On the positive side it enjoys the analogous universal property of the usual Grassmann bundles: if f : Y → X is a scheme morphism, and G is a rank d locally free Higgs quotient of f * E, there is a morphism g : Now assume that X is projective.Given a rank r Higgs bundle E = (E, φ) on X, for every 0 < d < r we define the following classes in N 1 (Gr It was proved in [6] (see also [5]) that E is curve semistable if and only if all classes θ d (E) are nef.Note that θ 1 (E) is the restriction of the class λ(E) ∈ N 1 (PE) ⊗ Q to Gr 1 (E).Here one can note a different behavior of Higgs bundles as opposed to vector bundles: while in the latter case the condition that the class λ(E) is nef is equivalent to curve semistability, in the Higgs case one needs the nefness of all classes θ d (E); see [6] for an example of a rank 3 Higgs bundle on a curve with θ 1 (E) nef, θ 2 (E) not nef, which is not semistable.

Higgs numerical effectiveness
In [5] by means of the Higgs Grassmannians a notion of numerical effectiveness for Higgs bundles was introduced.It is a definition based on recursion on the rank of the successive universal quotient bundles.Since we are going to use this definition later on, we recall it here.
Definition 2.1.A Higgs bundle E = (E, φ) of rank one on a smooth projective variety is said to be Higgs-numerically effective (for short, H-nef ) if the underyling vector bundle E is numerically effective in the usual sense.If rk E ≥ 2 we require that: (i) all bundles Q k are Higgs numerically effective; (ii) the line bundle det(E) is nef.
If both E and E * are Higgs-numerically effective, E is said to be Higgs-numerically flat (Hnflat).
3 The conjecture in any rank

A push-forward formula
We recall from [16] a push-forward formula for the Segre classes of the universal quotient bundle over Grassmann bundles π d : Gr d (E) → X.Here X will be a smooth projective variety of dimension n and E a rank r > 1 vector bundle.Greek letters such as λ, µ will denote a partition, i.e., a finite nonincreasing sequence of natural numbers.We let where λ = (λ 1 , . . ., λ q ), while π * will denote the push-forward of Chow groups Moreover, we define the Segre classes of the vector bundle F on a variety X by the formula where c(F ) is the total Chern class of F (we follow the normalization of [12], hence the minus signs).Lemma 3.1.[15,16] Let Q be the rank d universal quotient bundle of a rank r vector bundle E over X.The following push-forward formula holds: Here (i) ε is the partition of length d whose elements are all r − d; (ii) for every c ∈ A • (X), ∆ λ (c) is the Schur polynomial associated with λ computed on the components of c in A • (X), that is, where f λ is the number of standard Young tableaux of shape λ.1

Some notation and facts:
• A natural number k is regarded as a partition of length 1.For every k, ∆ k (c) = c k , i.e., the degree k term of c.In particular, • Conjugate partitions: given a partition λ, let λ be the conjugate partition, i.e., the partition which describes the conjugate Young tableau of λ (the one obtained by flipping it with respect to its diagonal.)Then ∆λ(c(E)) = ∆ λ (s(E)).
• We shall denote by p(k) the partition made up by k 1's.Then k and p(k) are conjugate partitions, so that ∆ p(k) (s(E)) = c k (E).
• An explicit formula for f λ is the following [11,19].Let λ be a partition of length q > 1.Then We note that

The result
In this Section we prove the main result of this paper.
Theorem 3.3.Given a curve semistable Higgs bundle E = (E, φ) on a surface X, if, for some d in the range 0 < d < r = rk E, the Higgs Grassmannian Gr d (E) has an irreducible component Z which is a divisor in Gr d (E) and surjects onto X, then ∆(E) = 0.
Proof.Using the Leray-Hirsch Theorem we define the classes where [Z] is the class of Z in A 1 (Gr d (E)) (for a version of the Leray-Hirsch Theorem for Chow groups which applies to the present case see [17]).Recalling equation ( 3), a rather lengthy computation yields2 In the last line, by integrating over X, we think of β 0 and ∆(E) as integers.β 0 is positive by the following argument.Denote by π Z the restriction of π d to Z, and by Z x its fiber at a point x ∈ X.We also denote F x = π −1 d (x).We assume that π Z is surjective so that by [1, Lemma 29.28.2]every irreducible component of its fibers has either dimension d(r − d) or d(r − d) − 1 (to apply that result we need Z to be integral but we can achieve that by replacing it with its reduced subscheme if needed).On the other hand, since π Z is proper, by the semicontinuity of the fiber dimension (see e.g.[1, Lemma 37.29.5]) the locus in X where the fiber then Gr d (E) would coincide with Gr d (E), a situation which we may exclude.So the generic fiber of π Z has dimension d(r − d) − 1, the "expected dimension".Hence for generic x, Z x determines a class in A 1 (F x ).Since the restriction χ x of χ to F x is ample, Now by the Bogomolov inequality ∆(E) ≥ 0 we obtain ∆(E) = 0.
Note the "miraculous disappearance" of β 1 ! 4 The conjecture in rank two

A non emptiness result
The following result is a key to our proof of the Conjecture in rank two.
Theorem 4.1.Let X be a smooth curve, which may be projective or affine, and let E = (E, φ) be a rank 2 Higgs bundle on X.The Higgs Grassmannian Gr 1 (E) of rank one Higgs quotients of E is not empty.
Lemma 4.2.Let X be a smooth curve (projective or affine), and let E = (E, φ) be a rank 2 Higgs bundle on X. Assume that the Higgs Grassmannian Gr 1 (E) of rank one Higgs quotients of E is empty.Then the Higgs field φ induces a splitting of the exact sequence Note that since Gr 1 (E) is assumed to be empty, the Higgs field φ is necessarily nonzero.
Proof.We refer to diagram (2).Note that PE/X is a line bundle, and Ω 1 PE is locally free, the morphism s is either zero or is injective; but if it were zero, since the Higgs Grassmannian Gr 1 (E) is the zero locus of the composition b the Higgs Grassmannian Gr 1 (E) would be the entire PE, and therefore would not be empty.So we have an exact sequence where R is by definition the quotient, which has rank one.
As the Higgs Grassmannian is empty, s has no zeroes, so that R is locally free.We form the diagram 0 We show that the morphism g = r • i : π * 1 Ω 1 X → R cannot be zero.Indeed if it were zero we would have a morphism h : π * 1 Ω 1 X → Ω 1 PE/X which is not zero as i = s • h.However since the fiber degree of Ω 1 PE/X is −2, the restriction of h to each fiber of π is zero, i.e., h = 0, which is a contradiction.Thus, g is nonzero, hence is injective.We prove it is an isomorphism.We have an exact sequence 0 where N has rank zero.For any fiber F of π 1 , by a standard argument, we have an exact sequence 0 Since R has fiber degree 0, R |F is isomorphic to O F , so that N |F = 0.As this holds for every fiber, N = 0, hence g is an isomorphism.Now we have a diagram / / 0 which shows that the sequence (6) splits.
Proof of Theorem 4.1.Note that the first line in diagram ( 7) splits as i • g −1 is a section of r.Let t ′ be a retraction of the morphism s.
= id S 1 so that the first line in diagram (2) splits.But this is impossible as on each fiber of π 1 that sequence reduces to the Euler exact sequence.
Corollary 4.3.Let E = (E, φ) be a rank two Higgs bundle on a smooth n-dimensional projective variety X.The Higgs Grassmannian Gr 1 (E) has a component of dimension at least n which surjects onto X.
Proof.If Gr 1 (E) does not have such a component, let Y be its image in X (actually taking its reduced subscheme if it happens to be nonreduced), let C be a curve in X not contained in Y , and let C ′ be C minus its intersection points with Y , and minus its possible singular points.Then E |C ′ has an empty Higgs Grassmannian, a contradiction to Corollary 4.1.
Remark 4.4.The splitting of the exact sequence (6) means that E is projectively flat, i.e., PE comes from a projective representation π 1 (X) → PGL 2 (C) of the fundamental group of X.This agrees with a result in [22], whose authors, as a particular case of their equivalence of categories, prove that semistable Higgs bundles on a curve are projectively flat.Note indeed that if the Higgs Grassmannian is empty, the Higgs bundle is stable.

The proof
We start with the case dim X = 2, i.e., X is smooth projective surface.From Corollary 4.3 we get that Gr 1 (E) has a component Z of dimension 2, and we are in the hypotheses of Section 3.2, so that ∆(E) = 0.This can be extended to the higher dimensional case dim X = n.Let H be the class of an ample line bundle L = O X (D) and let Y be the intersection of n − 2 generic divisors in the linear system |mD| for m ≫ 0. The result for dimension 2 implies that For k < n we have Since 1 ≤ N ≤ r − 1, the last summation in both equations may contain terms with i > N but these are zero as β i = 0 in that range.
Remark 5.2.In the non-Higgs case we have β 0 = 1, β i = 0 for i > 0 and we recover the identities of [13,10] when the Schur polynomial is a Segre class.
d-th Higgs Grassmannian of E, denoted Gr d (E), is the subscheme of Gr d (E) defined by the zero locus of the composition b d • π * d φ • a d .By construction, the restrictions of the bundles S d and Q d to Gr d (E) carry Higgs fields induced by π * d φ, so that we have an exact sequence of Higgs bundles on Gr d