Stable Higgs Bundles with trivial Chern Classes — Several Examples —

Yoichi Miyaoka

1 Introduction

The Bogomolov inequality for semistable vector bundes on smooth complex projective n-folds X reads

c_{2}(E)A^{n−2} ≥ \frac{r − 1}{2r}c_{1}(E)^{2}A^{n−2},

where A is an ample divisor and E is an A-semistable vector bundle of rank r on X. In case E is A-stable with vanishing c_{1}(E), the lower bound of this inequality c_{2}(E) ≥ 0 is attained if and only if E admits the structure of a flat hermitian bundle associated with an irreducible unitary representation of the fundamental group π_{1}(X), thereby establishing the one-to-one Kobayashi-Hitchin correspondence between the stable bundles with vanishing Chern classes and the irreducible unitary representation of π_{1}(X) [2]. The Bogomolov inequality is natural enough to have several proofs by completely different approaches (geometric invariant theory [1]; characteristic p method [3]; the theory of effective cones on ruled surfaces [8]; Yang-Mills theory of connections [2]). Because of this naturality, the Bogomolov inequality generalizes to certain classes of generalized vector bundles, including parabolic bundles and orbibundles.

Another important class of generalized vector bundles is that of Higgs bundles (see [9]), and the Bogomolov inequality was as well extended to this class by Simpson [9] through a modified version of Yang-Mills theory. ^{1} One of the implications of Simpson’s result is that, if a stable Higgs bundle has trivial Chern classes, then it comes from an irreducible representation of π_{1}(X) to the special linear group SL(r, C). However, we do not have many such examples except for some standard ones. In this note, we give a sequence of examples of stable Higgs bundles with trivial Chern classes or, equivalently, of stable flat Higgs bundles.

1In contrast to the aforementioned classes, an algebro-geometric proof of the Bogomolov inequality is so far not available for Higgs bundles except for very special cases: when several standard examples listed in Section 1 as Examples 0-1, 1 and 2, and the bundles of small ranks 2, 3 [7]. For instance, Simpson’s theorem implies that, if X is a ball quotient surface and n is a positive integer, then Sym^{3n}Ω_{X}^{1}(−nK_{X}) has no non-zero global section (see Proposition 2.1 below); this vanishing does not seem to follow from the algebrogeometric method we have at hand.

2 Higgs bundles: definition and basic examples

Let E be a vector bundle on a comlex manifold X and θ : E → Ω_{X}^{1} ⊗ E an O_{X}-linear mapping. The pair (E, θ) is said to be a Higgs bundle if the natural composite map θ ∧ θ : E → Ω_{X}^{2} ⊗ E identically vanishes. Alternatively, E is a Higgs bundle if an O_{X}-linear action of the sheaf of the local vector fields Θ_{X} on E is given in such a way such that ξ_{1}(ξ_{2}(e)) = ξ_{2}(ξ_{1}(e)) for arbitrary ξ_{i} ∈ Θ_{X} and e ∈ E. In other words, a Higgs bundle is a vector bundle with a Sym Θ_{X}-module structure, where

Sym Θ_{X} = ⨁_{i=0}^{∞} Sym^{i} Θ_{X}

is the symmetric tensor algebra generated by Θ_{X}. Higgs subsheaves are, by definition, Sym Θ_{X}-submodules.

Given an ample divisor A on X, the notion of A-(semi)stable Higgs bundles is naturally defined. Namely, a Hiiggs bundle E is A-stable if

\frac{c_{1}(S)A^{n−1}}{rank S} < \frac{c_{1}(E)A^{n−1}}{rank E}

for any nontrivial saturated Higgs subsheaf S ⊂ E, S ≠ 0, E, where n = dim X.

Historically Higgs structure was introduced in the study of moduli of integrable connections [5]. Let E be a vector bundle with an integrabale connection ∇_{0} : E → Ω_{X}^{1} ⊗ E. Given another integrable connection ∇, the difference θ = ∇ − ∇_{0} is a Higgs bundle structure and this correspondence translates the moduli of the integrable connections on a flat vector bundle E into the moduli of the Higgs bundle structures.

With the above definition in mind, we give below several standard examples of Higgs bundles.

Example 0-1. An ordinary vector bundle is viewed as a Higgs bundle with zero action of Θ_{X} (trivial Higgs structure). In this case, the Higgs stability is nothing but the usual stability. Usually a line bundle is thought of as a Higgs bundle with trivial Higgs structure.

Example 0-2. Starting from given Higgs bundles, we can construct new Hiiggs bundles by taking tensor products, duals and pull-backs.

Given two Higgs bundles E_{1}, E_{2}, the tensor bundle E_{1} ⊗ E_{2} is a Higgs bundle by defining ξ(e_{1} ⊗ e_{2}) = ξ(e_{1}) ⊗ e_{2} + e_{1} ⊗ ξ(e_{2}) for ξ ∈ Θ_{X}.

The dual bundle E^{∨} of a Higgs bundle is again a Higgs bundle by ⟨e|ξ(e^{∨})⟩ = −⟨ξ(e)|e^{∨}⟩, where e ∈ E, e^{∨} ∈ E^{∨}, ξ ∈ Θ_{X}. The canonical coupling E ⊗ E^{∨} → O_{X} is a Higgs bundle homomorphism.

If g : X → Y is a morphism between complex manifolds and E is a Higgs bundle on Y, then the pull-back g^{∗}E is naturally a Sym g^{∗}Θ_{Y} -module. Then the natural O_{X}-algebra homomorphism Sym Θ_{X} → Sym g^{∗}Θ_{Y} defines a canonical Higgs bundle structure of g^{∗}E in an obvious manner.

Thanks to Simpson’s theorem, the tensor product of two stable Higgs bundle is semi-stable and a direct sum of stable Higgs bundles.

Example 1. Let X be a complex manifold. The symmetric tensor algebra E_{0}^{∞}(X) = Sym Θ_{X} is naturally a Higgs bundle of infinite rank and so are its

ideals. In particular, the graded ideal E_{l+1}^{∞}(X) = ⨁_{i=l+1}^{∞} Sym^{i} Θ_{X} is a Higgs

subbundle of infinite rank. Given l ≥ k ≥ 0, the subquotient E_{k}^{l}(X) =

E_{k}^{∞}(X)/E_{l+1}^{∞}(X) is a coherent Higgs bundle isomorphic to ⨁_{i=k}^{l} Sym^{i} Θ_{X}.

The action of Θ_{X} on E_{k}^{l}(X) is given by zero on Sym^{l} Θ_{X} and by the standard multiplication Θ ⊗ Sym^{i} Θ_{X} → Sym^{i+1} Θ_{X} on the other components.

If K_{X}A^{n−1} > 0 and Θ_{X} is A-semistable as an ordinary vector bundle [resp. If K_{X}A^{n−1} ≥ 0 and Θ_{X} is A-semistable], then E_{0}^{1}(X) = O_{X} ⊕ Θ_{X} is A-stable [resp. A-semistable]. After some easy arguments, this implies that E_{0}^{l}(X) = Sym^{l}(E_{0}^{1}) is also an A-stable [resp. A-semistable] Higgs bundle. If K_{X} is ample and A = K_{X}, then the Yau inequality [10]

c_{2}(X)K_{X}^{n−2} ≥ \frac{dim X − 1}{2 dim X}K_{X}^{n}

yields the Bogomolov inequality for E_{k}^{l}(X).

When X is a two-dimensional compact ball quotient, then the Hirzebruch proportionality theorem yields c_{1}^{2}(X) = 3c_{2}(X), so that E_{0}^{1}(X) has

c_{1} = −K_{X}, c_{2} = \frac{1}{3}K_{X}^{2}; namely E_{0}^{1}(X) attains the equality of the Bogomolov

inequality. In particular, E_{0}^{3l}(lK_{X}) has trivial Chern classes (O_{X}(lK_{X}) is viewed as a trivial Higgs bundle) and, by Simpson’s theorem, it carries an integrable connection ∇. More precisely, there is a C-vector space V

of dimension ( \begin{matrix}3l + 2 \\ 2 \end{matrix} ) such that V = V ⊗_{C} O_{X} has a filtration 0 =

V^{3l+1} ⊂ V^{3l} ⊂ ⋯ ⊂ V^{0} = V with the p-th graded quotient Gr^{p} is isomorphic to Sym^{3l−p}Θ_{X}(lK_{X}). Furthermore, the Higgs action of ξ ∈ Θ_{X} on Sym^{3l−p}Θ_{X}(lK_{X}) is identified with the O_{X}-linear map (Kodaira-Spencer map) Gr^{p} → Gr^{p−1} induced by the natural derivation ∇_{ξ} of V ⊗ O_{X} by ξ. Thus E_{0}^{3l} looks very like a variation of Hodge structure (VHS) of weight 3l. However, it does not have the Hodge symmetry (the ranks of the successive quotients are 3l +1, 3l, 3l −1, …, 1), and the underlying vector space V may not have a Q-vector space structure.

Example 2. Given integers l ≥ k ≥ 0, we define the Higgs bundle F_{l}^{k}(X)

as the vector bundle ⨁_{i=k}^{l} Sym^{i} Ω_{X}^{1} with the Θ_{X}-action defined by 0 on

Sym^{k} Ω_{X}^{1} and by (−1) times the standard contraction map Θ_{X} ⊗Sym^{i} Ω_{X}^{1} → Sym^{i−1} Ω_{X}. F_{l}^{k}(X) is the dual E_{0}^{∞}(X)-module Hom_{E_{0}^{∞}(X)}(E_{k}^{l}(X), O_{X}) of E_{k}^{l}(X), where O_{X} is viewed as a Higgs bundle with trivial Θ_{X}-action. For l ≥ m ≥ k, F_{m}^{k}(X) is naturally a Higgs subbundle of F_{l}^{k}(X) with the quotient F_{l}^{k}(X)/F_{m}^{k}(X) isomorphic to F_{l}^{m+1}(X). The stability condition and the Bogomolov inequality for F_{l}^{k} are similar as for E_{k}^{l}.

Let p : X → Y be a surjective morphism between smooth projective surfaces. Assume that K_{Y} is ample and that p^{∗}K_{Y} is divisible by 3 in Pic(X). Then the normalized Higgs bundle

F̃_{1}^{0}(Y) = (O_{X} ⊕ p^{∗}Ω_{Y}^{1} ) (−\frac{π^{∗}K_{Y}}{3} )

is a Higgs bundle ⊂ F_{1}^{0}(X)(−p^{∗}K_{Y} /3) on X with trivial first Chern class. Its l-th symmetric power F̃_{l}^{0}(Y) is also a Higgs bundle ⊂ F_{l}^{0}(X)(−lp^{∗}K_{Y} /3). By easy computation, we get

c_{2}(F̃_{1}^{0}(Y)) = p^{∗} (c_{2}(Y) − \frac{K_{Y}^{2}}{3} ) .

For simplicity, we consider only two dimensional cases from now on; a complex manifold X will be a smooth projective surface unless otherwise mentioned.

Example 3. Let Y be a smooth projective surface and p : X → Y a surjective morphism such that p^{∗}K_{Y} is divisible by 3. If L be a line bundle contained in (Sym^{l+1}π^{∗}Ω_{Y}^{1} ) (−\frac{lπ^{∗}K_{Y}}{3} ), then L⊕F̃_{l}^{0}(Y) is a Higgs bundle

contained in

(Sym^{l+1}π^{∗}Ω_{Y}^{1} ) (−\frac{lπ^{∗}K_{Y}}{3} ) ⊕ F̃_{l}^{0}(Y) = F̃_{l+1}^{0}(Y) ⊗ O_{X} (\frac{π ∗ K_{Y}}{3} ) .

Its Chern classes are: c_{1} = c_{1}(L), c_{2} = c_{2}(F̃_{l}^{0}(Y)). This bundle is Asemistable if and only if 3c_{1}(L)A ≥ 0, and under this semistability condition we get c_{1}(L)^{2} ≤ c_{2}(F̃_{l}^{0}(A)).

When Y is a ball quotient and A = π^{∗}K_{Y}, we have c_{1}(L)^{2} ≤ 0 whenever c_{1}(L)π^{∗}K_{Y} ≥ 0. In particular, a line bundle which is numerically equivalent to O_{X}((l + 1)π^{∗}K_{Y} /3) cannot be contained in π^{∗}Sym^{l+1}Ω_{Y}^{1} . Hence

Proposition 2.1. Let Y be a compact ball quotient surface and 1 be the tautological divisor on the projective bundle π : P(Ω_{Y}^{1} ) → Y. Then the numerical equivalence class of 1 − (1/3)π^{∗}K_{Y} is not effective in the rational Néron-Severi group NS(P(Ω_{Y}) ⊗ Q. (On the other hand, 1 − (1/3)π^{∗}K_{Y} is known to be pseudo-effective, i.e. a limit of effective classes.)

An algebro-geometric proof of the proposition above is not known.

Example 4. Let m ≥ l ≥ 0 be integers. We define a Higgs bundle structure on

(⨁_{i=0}^{l} Sym^{i} Ω_{X}^{1}) ⊕ ( ⨁_{i=m−l}^{0} Sym^{i} Θ_{X} ⊗ Sym^{m+1} Ω_{X}^{1})

by defining the action of ξ ∈ Θ_{X} as follows:

— For α ∈ Sym^{i} Ω_{X}^{1}, l ≥ i ≥ 0, ξ(α) is the (−1)× the natural contraction ∈ Sym^{i−1} Ω_{X}^{1}.

— For α ∈ Sym^{l−m} Θ_{X} ⊗ Sym^{m+1} Ω_{X}^{1}, ξ(α) is defined by the composition of the natural product ξα ∈ Sym^{m−l+1} Θ_{X} ⊗ Sym^{m+1} Ω_{X} and (−1)^{m−l+1} times the contraction map Sym^{m−l+1} Θ_{X} ⊗ Sym^{m+1} Ω_{X} → Sym^{l} Ω_{X}^{1}.

— For α ∈ Sym^{i} Θ_{X} ⊗Sym^{m+1} Ω_{X}^{1}, i < m−l, ξ(α) is the natural product ξα ∈ Sym^{i+1} Θ_{X} ⊗ Sym^{m+1}.

This Higgs bundle is an extension of E_{0}^{m−l}(X) ⊗ Sym^{m+1} Ω_{X}^{1} by F_{l}^{0}(X).

Example 5. Let L be an invertible subsheaf of Ω_{X}^{1}. Then O_{X} ⊕L is a Higgs subsheaf of F_{1}^{0}. In this case, the A-stability condition is c_{1}(L)A > 0 and the Bogomolov inequality is equivalent to the classical de Franchis lemma c_{1}(L)^{2} ≤ 0. Hence we get a stable flat Higgs bundle L^{−1/2} ⊕ L^{1/2} once we have L ⊂ Ω_{X}^{1} with c_{1}(L)A > 0, L^{2} = 0. If X has a fibration over a curve C of genus ≥ 2, the pull back of ω_{C} gives an example of such L. In this case, our flat Higgs bundle is the pull back of the theta Higgs bundle

ω_{C}^{−1/2} ⊕ ω_{C}^{1/2}

on C.

If X has several fibrations over curves of genus ≥ 2, then the tensor product of the pull back of the theta Higgs bundles is again a flat Higgs bundle.

In the five examples above, flat Higgs bundles on surfaces were produced via tensor products, duals and pull backs from the following three basic examples:

(1) Stable vector bundle (of arbitrary rank) with trivial Higgs structure.

(2) Pull back of the theta Higgs bundle of rank two on curves of genus ≥ 2 (which is uniformaized by the upper half plane and has projectively flat connection).

(3) The 3-bundle F̃_{1}^{0}(Y) = F_{1}^{0}(Y)⊗O (−\frac{K_{Y}}{3} ), where Y is a two-dimensional ball quotient (again a projectively flat manifold);

In Section 3 and 4, we construct new examples which do not directly derive from the above basic ones.

3 Hirzebruch’s Kummer covers X^{(n)} attached to the complete qradrilateral on ℙ^{2} and construction of stable Higgs bundles with vanishing Chern classes

We briefly review Hirzebruch’s construction of Kummer covers of projective plane branching along a complete quadrilateral [4].

Take general four points P_{1}, …, P_{4} on ℙ^{2}, and let L_{ij} = L_{ji} denote the line connecting P_{i} and P_{j} (i ≠ j). The reduced divisor D = ⋃ L_{ij} is the so called complete quadrilateral consisting of six lines, the P_{i} being the triple points of D. D has extra three double points of the form L_{i_{1},i_{2}} ∩ L_{j_{1},j_{2}}, where {i_{1}, i_{2}, j_{1}, j_{2}} = {1, 2, 3, 4}. Exactly three singlular points of D lies on each L_{ij}, two of which are the triple points P_{i}, P_{j} and one a double point. Thus the Euler number of the non-singular locus of D is 6 × (2 − 3) = −6, while that of D is −6 + 4 + 3 = 1. Therefore the Euler number of of the complement of D is given by e(ℙ^{2} \ D) = 3 − 1 = 2.

Let μ : X → ℙ^{2} be the blowing up at the four triple points P_{1}, …, P_{4} and let E_{i} ⊂ X denote the exceptional divisor over P_{i}. X is a Del Pezzo surface of degree five with very ample anticanonical divisor −K_{X} ∼ 3H − ∑ E_{i}, where H stands for the pullback of the hyperplane of ℙ^{2}. The effective divisor μ^{∗}D is supported by a reduced effective divisor

D̃ ∼ μ^{∗} ∑ L_{ij} − 2 ∑ E_{i} ∼ 6μ^{∗}H − 2 ∑ E_{i} ∼ −2K_{X}.

D̃ has only simple normal crossings as singularities and consists of ten irreducible components: four exceptional curve E_{i} and six strict transforms L̃_{ij}. The L̃_{ij} meet each other at the three points lying over the double points of D, while each E_{i} contains three singular points of D̃. Hence D̃ has exactly 3+4×3 = 15 double points, so that e(D̃) = 4×(2−3)+6×(2−3)+15 = 5.

Given a positive integer n, there exists a finite Kummer covering π^{(n)} : X^{(n)} → X of degree n^{5} branching along D̃ [4]. The function field of X^{(n)} is simply obtained by adjoining the n-th roots \sqrt[n]{l_{ij}/l_{12}} (\{i, j\} ≠ \{1, 2\} ∈ {1, 2, 3, 4}) to ℂ(ℙ^{2}), where l_{ij} is a linear differential equation of the line L_{ij}.

X^{(n)} is a smooth projective surface and the local description of X^{(n)} is quite simple: if D̃ is locally defined by the equation x = 0 or xy = 0, then π^{(n)∗} : O_{X} → O_{X^{(n)}} is given by (x, y) ↦ (t^{n}, u) or (x, y) ↦ (t^{n}, u^{n}), where (x, y) and (t, u) are local coordinates of X and X^{(n)}. In particular, the inverse image (π^{(n)})^{−1} (p) ⊂ X^{(n)} of a closed point p ∈ X consists of n^{5} [resp. n^{4}, n^{3}] points when p ∈ X \ D̃ [resp. p ∈ D̃ \ Sing(D̃), p ∈ Sing(D̃)]. The topological Euler number of X̃^{(n)} of X^{(n)} is thus given by

\frac{c_{2}(X^{(n)})}{n^{5}} = \frac{e(X^{(n)})}{n^{5}} = e(X \ D̃) + \frac{e(D̃ \ Sing(D̃)}{n} + \frac{e(Sing(D̃)}{n^{2}}

= 2 − \frac{10}{n} + \frac{15}{n^{2}}.

On the other hand we calculate K_{X^{(n)}} by

K_{X^{(n)}} ∼ π^{(n)∗} (K_{X} + (1 − \frac{1}{n}) D̃) ∼ (1 − \frac{2}{n}) π^{(n)∗}(−K_{X}),

and hence

\frac{c_{1}(X^{(n)})^{2}}{n^{5}} = 5 (1 − \frac{2}{n})^{2} .

X^{(n)} has ample canonical divisor if n ≥ 3 (X^{(2)} is a K3 surface). When n = 5, we have c_{1}(X^{(5)})^{2} = 5^{4} × 9, c_{2}(X^{(5)}) = 5^{4} × 3, meaning that X^{(5)} is a surface of general type which attains the upper bound of the Miyaoka-Yau inequality K ≤ 3c_{2}.

The Del Pezzo surface X carries five linear pencils |2H −∑ E_{i}|, |H −E_{1}|, …, |H − E_{4}|, defining five surjective morphisms f_{0}, f_{1}, … f_{4} from X onto ℙ^{1}. Each f_{i} of these morphisms has exactly three fibres contained in D̃, which are the singular fibres of f_{i}. For f_{1} associated with |H −E_{1}, L̃_{1j} +E_{j}, j = 2, 3, 4 are such fibres, and so are the three curves L̃_{12} + L̃_{34}, L̃_{13} + L̃_{24}, L̃_{14} + L̃_{23} for f_{0} associated with |2H − ∑ E_{i}|.

Upstairs on X^{(n)}, there are thus five morphisms f_{0}^{(n)}, f_{1}^{(n)}, …, f_{4}^{(n)} onto the curve C^{(n)}, an n^{2}-sheeted Kummer cover of ℙ^{1} branching at three points, 0, 1, ∞, say. The pullback line bundle L_{i}^{(n)} = f_{i}^{(n)∗}ω_{C^{(n)}} is an invertible subshef of Ω_{X^{(n)}}^{1}. We easily check that L_{i}^{(n)} is saturated in Ω_{X^{(n)}}^{1} and that

\begin{aligned}L_{0}^{(n)} & ∼ (1 − \frac{3}{n}) π^{(n)∗} (2H − ∑_{i=1}^{4} E_{i}) \\ L_{i}^{(n)} & ∼ (1 − \frac{3}{n}) π^{(n)∗}(H − E_{i}), i = 1, 2, 3, 4. \end{aligned}

Ishida [6] showed that the natural map

⨁_{j=0}^{4} f_{j}^{(n)∗}H^{0}(C^{(n)}, Ω_{C^{(n)}}^{1}) → H^{0}(X^{(n)}, Ω_{X^{(n)}}^{1})

is an isomorphism. In particular, the irregularity of X^{(n)} is given by

q(X^{(n)}) = \frac{5(n − 2)(n − 1)}{2}.

Now let us construct a stable Higgs bundle of rank 12 on X^{(n)} (n ≥ 6).

The tensor product L^{(n)} = ⨂_{i=0}^{4} f_{i}^{∗}Ω_{C^{(n)}}^{1} is an invertible subsheaf of

Sym^{5} Ω_{X^{(n)}}^{1}. Easy calculation shows that

c_{1}(L^{(n)}) = \frac{n − 3}{n}π^{(n)∗}D̃,

while

K_{X^{(n)}} = \frac{n − 2}{2n}π^{(n)∗}D̃.

Hence

L^{(n)} ≡ \frac{2(n − 3)}{n − 2}K_{X^{(n)}}.

This subsheaf ⊂ Sym^{5}Ω induces a rank 12 Higgs subsheaf

E^{(n)} = (⨁_{i=0}^{2} Sym^{i} Ω_{X^{(n)}}) ⊕ (⨁_{i=2}^{0} Sym^{i} Θ_{X} ⊗ L^{(n)})

⊂ (⨁_{i=0}^{2} Sym^{i} Ω_{X^{(n)}}) ⊕ (⨁_{i=2}^{0} Sym^{i} Θ_{X} ⊗ Sym^{5} Ω_{X^{(n)}}^{1})

(see Section 2, Examples 4 and 5). We claim:

Proposition 3.1. (1) c_{2}(E^{(n)}) = \frac{11}{24}c_{1}(E^{(n)})^{2}.

(2) If n ≥ 5, then E^{(n)} is K_{X^{(n)}}-semistable. If n ≥ 6, then E^{(n)} is K_{X^{(n)}}-stable.

Proof. The first statement follows from direct computation, which is a little messy. An alternative proof is the following: c_{2}(E^{(n)})/n^{5} and c_{1}(E^{(n)})^{2}/n^{5} are both quadratic polynomials in 1/n. Hence the equality holds for every n if we check it for three special values of 1/n (1, 1/3, 1/5, for example).

The Higgs subbundle ⨁_{i=0}^{2} Sym^{i} Ω_{X^{(n)}} and the quotient Higgs bundle ⨁_{i=2}^{0} Sym^{i} Θ_{X}⊗L^{(n)} are both stable Higgs bundles of rank 6 with first Chern class 4K_{X} and 6c_{1}(L^{(n)}) − 4K_{X}, respectively. Hence the semistability of the bundle is equivalent to the inequality

\frac{n − 3}{n − 2} ≥ \frac{2}{3}

or, equivalently, n ≥ 5.

Corollary 3.2. If n ≥ 6, then E^{(n)} is projectively flat. The flat bundle

E^{(n)} (−\frac{n − 3}{2n}π^{(n)∗}D̃) of rank 12 looks like a variation of Hodge structure

of weight five with Hodge numbers (1, 2, 3, 3, 2, 1).

4 Example of a flat Higgs bundle over a family

In the previous section, we constructed countably many examples of stable Higgs bundles of rank 12 defined over a dicrete series of surfaces. In this section, we construct a stable Higgs bundle of rank 6 defined over a oneparameter family of surfaces.

Let C be an elliptic curve and Δ ⊂ C × C the diagonal. Let μ : C → C be the multiplication by two and define three effective divisors D, F, G on C × C by

D = (μ, μ)^{∗}Δ
F = (μ, μ)^{∗} ({0} × C)
G = (μ, μ)^{∗} (C × {0}) .

These three divisors are divisible by two (actually by four) in Pic(C × C) so that there are double coverings h_{i} : X_{i} → C × C which branch along D, F and G, respectively. By forming the fibre product over C × C, we get a Galois (Z/2Z)^{⊕3}-covering X′ → C × C. The resulting surface X′ has 16 ordinary double points over the 2-torsion points _{2}C ×_{2}C ⊂ C × C.

Proposition 4.1. There exists a double covering g : X → X′ ramifying exactly at the 16 double points of X′ so that X is smooth.

Proof. Let X̃′ be the minimal resolution of X′ and E ⊂ X̃′ the sum of the sixteen (−2)-curves over the double points. We show that E ∈ Pic(X̃′) is divisible by 2.

By construction, h^{∗}G is a non-reduced Weil divisor with multiplicity two. This means g^{∗}h^{∗}G is of the form 2G̃_{0} + E, G̃_{0} denoting the reduced part of the strict transform of h^{∗}G. On the other hand G is algebraically equivalent to 4× (a fibre of the first projection C × C → C) and is divisible by 4. Hence E ≡ 2G̃_{0} + E = g^{∗}h^{∗}G ≡ 0 mod 2. □

Proposition 4.2. The Chern numbers of X is given by c_{1}^{2}(X) = 2^{7} × 3, c_{2}(X) = 2^{4} × 3^{2}.

Proof. The projection f = (μ, μ) ∘ h ∘ g : X → C × C has degree 2^{8} with branch locus B = Δ ∪ ({0} × C) ∪ (C × {0}). Ecxept at the origin (0, 0), the branch index of f along B is exactly two, while it is 2^{4} at the origin. The

canonical divisor K_{X} is thus \frac{1}{2}f^{∗}B so that K_{X}^{2} = 2^{6}B^{2} = 2^{6} × 6.

The Euler numbers of B and B\(0, 0) are −2 and −3, so that e (C × C \ B) = 2. Hence

c_{2}(X) = e(X) = 2^{8} × 2 + 2^{7} × (−3) + 2^{4} × 1 = 2^{4} × 9,

concluding the proof. □

Our surface X has three independent pencils. The two canonical projections C × C → C induce two fibrations p_{i} : X → C̃, where C̃ is the double cover of C branching at the four points _{2}C. The group homomorphism C × C → (C × C)/Δ ≃ C yields a third fibration p_{3}; X → C̃. The pull-backs of ωC̃ via the three projections p_{1}, p_{2}, p_{3} are linearly equivalent to

\frac{1}{2}f^{∗}(\{0\} × C), \frac{1}{2}f^{∗}(C × \{0\}) and \frac{1}{2}f^{∗}Δ, respectively, so that their product

gives an invertible subsheaf L ≃ O_{X}(K_{X}) ⊂ Sym^{3}Ω_{X}^{1}.

Proposition 4.3. The Higgs bundle

O_{X} ⊕ Ω_{X}^{1} ⊕ (Θ_{X} ⊗ L) ⊕ L

is stable and, by tensoring −\frac{1}{2}K_{X}, we get a stable flat Higgs bundle

O_{X}(−\frac{K_{X}}{2} ) ⊕ Ω_{X}^{1}(−\frac{K_{X}}{2} ) ⊕ Θ_{X}(\frac{K_{X}}{2} ) ⊕ O_{X}(\frac{K_{X}}{2} ).

The proof is easy.

Since an elliptic curves C has one-dimensional moduli, so does our surface X. We have thus constructed a stable flat Higgs bundle on a one-parameter family of surfaces.

References

[1] F. Bogomolov, Holomorphic tensors and vecor bundles on projective varieties, Math. USSR Izv. 13(1979), 499 – 555.

[2] S.K. Donaldson, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable bundles, Proc. London Math. Soc. 50 (1985), 1 – 26.

[3] D. Gieseker, On a theorem of Bogomolov on Chern classes of stable bundles, Amer. J. Math. 101 (1979), 77 – 85.

[4] F. Hirzebruch, Arrangements of lines and algebraic surfaces, Progr. Math. 36 (1983), 113 – 140.

[5] N.J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3) 55 (1987), 59 – 126.

[6] M.-N. Ishida, The irregularities of Hirzebruch’s examples of surfaces with c_{1}^{2} = 3c_{2}, Springer Lecture Notes in Math. 1016 (1983), 412 – 431.

[7] A. Langer, A note on Bogomolov’s instability and Higgs sheaves, In: Algebraic Geometry, Göttingen, de Gruyter, Berlin, 2002, pp. 234 – 256.

[8] Y. Miyaoka, The Chern classes and Kodaira dimension of a minimal variety, Adv. Stud. in Pure Math. 10 (1987), 449 – 476.

[9] C. Simpson, Higgs bundles and local systems, Publ. Math. I.H.E.S. 75 (1992), 5 – 95.

[10] S.-T. Yau, On Calabi’s conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), 1798 – 1799.