Aspects of Algebraic Geometry

We would like to discuss the main open parts of the Kawaguchi–Silverman Conjecture, asserting that for a birational self-map \(f\) of a smooth projective variety \(X\) defined over \(\overline{\mathbf{Q}}\), the arithmetic degree \(\alpha_f(x)\) exists and coincides with the first dynamical degree \(\delta_f\) for any closed point \(x \in X\) with a Zariski dense orbit. Among other results, we show that this holds when \(X\) has Kodaira dimension zero and irregularity \(q(X) \geq \dim X -1\) or \(X\) is an irregular threefold (modulo one possible exception). We discuss then the existence of Zariski dense orbits, with explicit examples. This talk is based on a joint work with Professors Jungkai Chen and Huesh-Yung Lin.

For a curve of genus at least four which is either very general or very general hyperelliptic, we classify all ways in which a power of its Jacobian can be isogenous to a product of Jacobians of curves. We use this to show that for a very general principally polarized abelian variety of dimension at least four, or the intermediate Jacobian of a very general cubic threefold, no power is isogenous to a Jacobian of a curve. This confirms various cases of the Coleman–Oort conjecture and has some relation to the question whether cubic threefolds are stably irrational. Joint work with Olivier de Gaay Fortman.

I will report on joint work with E. Colombo and G.P. Pirola, where we study asymptotic directions in the tangent bundle of the moduli space of curves of genus \(g\), namely those tangent directions that are annihilated by the second fundamental form of the Torelli map. I will exhibit examples of asymptotic directions for any \(g \geq 4\). I will show that if the rank \(d\) of a tangent direction at \([C]\) (with respect to the infinitesimal deformation map) is less than the Clifford index of the curve \(C\), then the tangent direction is not asymptotic. Finally I will determine all asymptotic directions of rank \(1\) and give an almost complete description of asymptotic directions of rank \(2\).

Given homeomorphic compact complex manifolds \(X\) and \(Y\), mostly projective, I will discuss in various cases when \(X\) and \(Y\) are actually biholomorphic. An application to a compactification problem will be given.