Complex solvmanifolds with holomorphically trivial canonical bundle

Início: 19/04/2024 17:00

Término: 19/04/2024 18:00

Palestrante: Alejandro Tolcachier (UNCordoba)

E-mail do Palestrante: atolcachier@unc.edu.ar

Resumo: The canonical bundle of a complex manifold $(M,J)$, with $operatorname{dim}_{mathbb{C}} M=n$, is defined as the $n$-th exterior power of its holomorphic tangent bundle and it is a holomorphic line bundle over $M$. Complex manifolds with holomorphically trivial canonical bundle are important in differential, complex, and algebraic geometry and also have relations with theoretical physics. It is well known that every nilmanifold $Gammabackslash G$ equipped with an invariant complex structure has (holomorphically) trivial canonical bundle, due to the existence of an invariant (holomorphic) trivializing section. For complex solvmanifolds such a section may or may not exist. In this talk, we will see an example of a complex solvmanifold with a non-invariant trivializing holomorphic section of its canonical bundle. This new phenomenon lead us to study the existence of holomorphic trivializing sections in two stages. In the invariant case, we will characterize this existence in terms of the 1-form $psi$ naturally defined in terms of the Lie algebra of $G$ and $J$ by $psi(x)=operatorname{Tr} (Joperatorname{ad} x)-operatorname{Tr} operatorname{ad} (Jx)$. For the non-invariant case, we will provide an algebraic obstruction for a solvmanifold to have a trivial canonical bundle (or, more generally, holomorphically torsion) and we will explicitly construct, in certain examples, a trivializing section of the canonical bundle that is non-invariant. We will apply this construction to hypercomplex geometry to provide a negative answer to a question posed by M. Verbitsky. Based on joint work with Adrián Andrada.