Group actions on vector bundles

Abstract: The classification of vector bundles which are invariant under the action of a determined group, has been widely studied.

We can consider, for example, the canonical action of the projective linear group $PGL(n+1)$ on $\mathbb{P}^n$ which leads to the definition of homogeneous vector bundle. The choice of specific subgroups has been often determined, in literature, restricting our attention to particular families. Recall indeed that Ancona and Ottaviani proved that the Steiner bundles on $\mathbb{P}^n$ that are invariant under an action of $SL(2,\mathbb{C})$ are the so called Schwarzenberger bundles. Another example, moving into the realm of hyperplane arrangements, in which I have been particularly interested lately, is given by the reflection arrangements. They are defined as hyperplane arrangements that are invariant under the group generated by their reflections, and it is known that their associated sheaf is free (a sum of line bundles) and therefore homogeneous. In a previous work, studying Nearly-free arrangements, we proved that their configuration of jumping lines is extremely special but remarked that it did not characterize this family of arrangements. It turns out that they are characterized by the invariance under the action of the subgroup $G_p \subset \mathrm{PGL}(3)$ that fixes the point $p$ in the projective plane. Inspired by this result, we classify vector bundles which are invariant under the action of subgroups that fix linear subspaces of the projective plane.

Finally, we will focus on the relations between the geometry of the jumping locus and the invariance under the action of the group. Recall that, historically, such question has been studied in order to relate homogeneous bundles with uniform ones, i.e. bundles for which the splitting type is constant. This is the result of two collaborations: one with Jean Vallès and one with Rosa Maria Miró-Roig.

algebraic geometry

Audience: researchers in the topic

Brazilian algebraic geometry seminar

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