arXiv subjects
Publications and source records attributed to Gunnar Þór Magnússon.
We calculate the $L^2$-norm of the holomorphic sectional curvature of a Kähler metric by representation-theoretic means. This yields a new proof that the holomorphic sectional curvature determines the whole curvature tensor. We then investigate what the holomorphic sectional curvature of a Hermitian metric determines and calculate the $L^2$-norm of the holomorphic bisectional curvature.
We consider the blowup of a point of a compact Kähler manifold and a metric of the form $μ^*h + t b$ on it, where $h$ is a Kähler metric on the original manifold and $b$ is Hermitian form that looks like the Fubini--Study metric near the exceptional divisor. We calculate the curvature tensor of this metric on the exceptional divisor and show that its holomorphic sectional curvature is negative in some directions for all small enough $t$, which torpedos a natural approach to showing that blowups of manifolds of positive holomorphic sectional curvature have positive curvature.
We give a quick direct proof of Wu's theorem on the subadditivity of Hermitian metrics of negative holomorphic sectional curvature.
We calculate curvature tensors of metrics on the total spaces of holomorphic fibrations. Our main tool is a theory of Chern connections and curvature forms for possibly degenerate Hermitian forms on holomorphic vector bundles. We prove a version of the Codazzi-Griffiths equations for curvatures of sub- and quotient bundles in that setting and apply them to the study of honest Hermitian metrics on fibrations. We apply this to calculate the curvature of a metric on Grassmannian bundles and prove they have positive holomorphic sectional curvature if the base does.
The Kähler cone of a compact Kähler manifold carries a natural Riemannian metric, given by the intersection product of its cohomology ring. We give cohomological expressions for the Levi-Civita connection and curvature tensor of this metric, and determine when the metric is complete.
We give a formula for the inner product of forms on a Hermitian vector space in terms of linear combinations of iterates of the adjoint of the Lefschetz operator. As an application, we reprove the Kobayashi-Lubke inequality for Hermite-Einstein bundles.
If $f$ is an automorphism of a compact simply connected Kähler manifold with trivial canonical bundle that fixes a Kähler class, then the order of $f$ is finite. We apply this well known result to construct compact non-Kähler manifolds. These manifolds contradict the abundance and Iitaka conjectures for complex manifolds.
The Kähler cone of a compact manifold carries a natural Riemannian metric, given by the intersection product of its cohomology ring. We write down the curvature tensor of this metric by embedding the Kähler cone in the space of hermitian metrics on the underlying manifold. After discussing weak functorality and completeness properties, we give a relative version of both the Kähler cone and the metric.
Let $π: \mathcal X \to S$ be a local universal family of compact Ricci-flat Kähler manifolds over a smooth base $S$. The complexified Kähler cones of each fiber of the family form a holomorphic fiber bundle $\mathcal K \to S$. We show that there exists a natural hermitian metric $ω$ on the fiber product $\mathcal X \times_S \mathcal K$. We then discuss the example of elliptic curves in some detail.