A note on Kawaguchi–Silverman conjecture

We collect some results on endomorphisms on projective varieties related to the Kawaguchi–Silverman conjecture. We discuss certain conditions on automorphism groups of projective varieties and positivity conditions on leading real eigendivisors of self-morphisms. We prove Kawaguchi–Silverman conjecture for endomorphisms on projective bundles on a smooth Fano variety of Picard number one. In the last section, we discuss endomorphisms and augmented base loci of their eigendivisors.

On Ulrich bundles on projective bundles

In this article, the existence of Ulrich bundles on projective bundles $${{\mathbb {P}}}({{\mathcal {E}}}) \rightarrow X$$ P ( E ) → X is discussed. In the case, that the base variety X is a curve or surface, a close relationship between Ulrich bundles on X and those on $${{\mathbb {P}}}({{\mathcal {E}}})$$ P ( E ) is established for specific polarisations. This yields the existence of Ulrich bundles on a wide range of projective bundles over curves and some surfaces.

Delta invariants of projective bundles and projective cones of Fano type

In this paper, we will give a precise formula to compute delta invariants of projective bundles and projective cones of Fano type.

Equivalences between Calabi–Yau manifolds and roofs of projective bundles

It is conjectured that many birational transformations, called K-inequalities, have a categorical counterpart in terms of an embedding of derived categories. In the special case of simple K-equivalence (or more generally K-equivalence), a derived equivalence is expected: we propose a method to prove derived equivalence for a wide class of such cases. This method is related to the construction of roofs of projective bundles introduced by Kanemitsu. Such roofs can be related to candidate pairs of derived equivalent, L-equivalent and non isomorphic Calabi–Yau varieties, we prove such claims in some examples of this construction. In the same framework, we show that a similar approach applies to prove derived equivalence of pairs of Calabi–Yau fibrations, we provide some working examples and we relate them to gauged linear sigma model phase transitions.