Welcome to the website of the seminar organized by the Differential Geometry groups of Università di Torino and Politecnico di Torino.  

For any information, please send an e-mail to dgseminar.torino@gmail.com. 

The talks will take place either at the Deparment of Mathematics "G. Peano" (Università di Torino) or at the Department of Mathematical Sciences "G.L. Lagrange" (Politecnico di Torino). 

November 26, 11 am, Sala Orsi, Mathematics Department "G. Peano"

Alejandro Tolcachier (Università dell'Insubria) 

Complex solvmanifolds with non-invariant trivializing sections of their canonical bundle

Abstract:  The canonical bundle of a complex manifold (M,J), with dim_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 appear in many instances in mathematics and theoretical physics. For instance, Calabi-Yau manifolds are compact Kähler manifolds with holomorphically trivial canonical bundle. It is well-known that every nilmanifold Γ\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 give examples of complex solvmanifolds (Γ\G,J) with a trivializing holomorphic section of its canonical bundle which is not invariant under the action of G. This new phenomenon leads us to study the existence of holomorphic trivializing sections in two stages. In the invariant case, we can characterize this existence in terms of the Koszul 1-form ψ naturally defined in terms of the Lie algebra of G and J by ψ(x)=Tr (J ad x) - Tr ad (Jx). In the non-invariant case, we provide an algebraic obstruction for a solvmanifold to have holomorphically trivial canonical bundle (or, more generally, holomorphically torsion) and we show how to construct explicitly, in certain examples, a trivializing section of the canonical bundle that is non-invariant. Finally, we will present some results on 6-dimensional solvable strongly unimodular Lie algebras, including a new example of a solvable Lie algebra admitting complex structures with a holomorphic (3,0)-form, and an example of a complex solvmanifold with holomorphically trivial canonical bundle which is not biholomorphic to a complex solvmanifold with an invariant trivializing section of its canonical bundle.

This talk is based on joint work with Adrián Andrada and some recent results.

TBA