Metrics on semistable and numerically effective Higgs bundles. (English) Zbl 1138.14024
The main idea of the paper is to provide metric notions of numerical effectiveness and numerical flatness for Higgs bundles over a compact Kähler manifold \((X,\omega)\). The essential objective is to study the relations between numerical effectiveness and semistability for a Higgs bundles.
Let \((E,\phi)\) a Higgs bundle over a complex manifold equipped with a hermitian metric. Let \(D\) be the Chern connection \(D\) on \(E\) compatible with \(h\) and the complex structure. Now the operator \(\mathcal{D}=D+\phi+\bar\phi\) is called the Hitchin-Simpson connection, and one can denote \(\mathcal{R}\) its curvature. In that context, due to the work of Simpson, one has a Kobayashi-Hitchin type correspondence for Higgs bundles over a Kähler manifold which asserts that the polystability of \((E,\phi)\) is equivalent to the existence of a metric such that \(\mathcal{R}=c\text{Id}_E\).
Let \(V,W\) be two finite dimensional complex vector spaces and \(1\leq t \leq \min(\dim(V),\dim(W))\) an integer. One says for two hermitian forms \(\theta_1,\theta_2\) on \(V\otimes W\) that \(\theta_1\geq_t\theta_2\) if the hermitian form \(\theta_1-\theta_2\) is semi-positive definite on all tensors of rank \(r\leq t\) where \(r\) is the rank of the associated map \(V^*\rightarrow W\) (De Cataldo’s formalism). By analogy with the holomorphic world, the authors define a Higgs bundle to be \(t\)-H-nef if for all \(\epsilon>0\), there exists a hermitian metric \(h\) on \(E\) with \(\mathcal{R}\geq -\epsilon\omega \otimes h\). It is said to be \(t\)-H-nflat if both \(E\) and \(E^*\) are \(t\)-H-nef.
The authors show that \(1\)-H-nflat Higgs bundles are semistable. Futhermore, a Higgs bundle is \(1\)-H-nflat if and only if it has a filtration by Higgs subbundles whose quotients are hermitian flat Higgs bundles. In particular, all Chern classes of \(1\)-H-nflat Higgs bundle vanish.
In the case of a projective manifold, the authors had previously introduced a notion of H-nef Higgs bundles. They show that both definitions of 1-H-nef and H-nef Higgs bundle coincide. In the context of projective manifolds, they study semi-stable Higgs bundles for which one has Miyaoka equality \(c_2(E) - \frac{r(E)-1}{2r(E)}c_1(E)^2=0\) with \(r(E)\) the rank of \(E\). In particular, a Higgs bundle \(E\) satisfy such properties if and only if \(S^r(E) \otimes \det(E)^{-1}\) is H-nflat.
The paper is clear and well written.
Let \((E,\phi)\) a Higgs bundle over a complex manifold equipped with a hermitian metric. Let \(D\) be the Chern connection \(D\) on \(E\) compatible with \(h\) and the complex structure. Now the operator \(\mathcal{D}=D+\phi+\bar\phi\) is called the Hitchin-Simpson connection, and one can denote \(\mathcal{R}\) its curvature. In that context, due to the work of Simpson, one has a Kobayashi-Hitchin type correspondence for Higgs bundles over a Kähler manifold which asserts that the polystability of \((E,\phi)\) is equivalent to the existence of a metric such that \(\mathcal{R}=c\text{Id}_E\).
Let \(V,W\) be two finite dimensional complex vector spaces and \(1\leq t \leq \min(\dim(V),\dim(W))\) an integer. One says for two hermitian forms \(\theta_1,\theta_2\) on \(V\otimes W\) that \(\theta_1\geq_t\theta_2\) if the hermitian form \(\theta_1-\theta_2\) is semi-positive definite on all tensors of rank \(r\leq t\) where \(r\) is the rank of the associated map \(V^*\rightarrow W\) (De Cataldo’s formalism). By analogy with the holomorphic world, the authors define a Higgs bundle to be \(t\)-H-nef if for all \(\epsilon>0\), there exists a hermitian metric \(h\) on \(E\) with \(\mathcal{R}\geq -\epsilon\omega \otimes h\). It is said to be \(t\)-H-nflat if both \(E\) and \(E^*\) are \(t\)-H-nef.
The authors show that \(1\)-H-nflat Higgs bundles are semistable. Futhermore, a Higgs bundle is \(1\)-H-nflat if and only if it has a filtration by Higgs subbundles whose quotients are hermitian flat Higgs bundles. In particular, all Chern classes of \(1\)-H-nflat Higgs bundle vanish.
In the case of a projective manifold, the authors had previously introduced a notion of H-nef Higgs bundles. They show that both definitions of 1-H-nef and H-nef Higgs bundle coincide. In the context of projective manifolds, they study semi-stable Higgs bundles for which one has Miyaoka equality \(c_2(E) - \frac{r(E)-1}{2r(E)}c_1(E)^2=0\) with \(r(E)\) the rank of \(E\). In particular, a Higgs bundle \(E\) satisfy such properties if and only if \(S^r(E) \otimes \det(E)^{-1}\) is H-nflat.
The paper is clear and well written.
Reviewer: Julien Keller (Marseille)
MSC:
| 14J60 | Vector bundles on surfaces and higher-dimensional varieties, and their moduli |
| 53C55 | Global differential geometry of Hermitian and Kählerian manifolds |
| 32L05 | Holomorphic bundles and generalizations |
| 32Q20 | Kähler-Einstein manifolds |
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