Event description
Abstract: The canonical bundle of a complex manifold of complex dimension n is defined as the n-th exterior power of its holomorphic tangent bundle and turns out to be a holomorphic line bundle over the manifold. Complex manifolds with holomorphically trivial canonical bundle play a significant role in differential geometry, as well as in theoretical physics. For example, Calabi-Yau manifolds are compact Kähler manifolds with holomorphically trivial canonical bundle.
Cavalcanti-Gualtieri [2004] and Barberis-Dotti-Verbitsky [2009] independently showed that any nilmanifold endowed with an invariant complex structure has holomorphically trivial canonical bundle, due to the existence of a trivializing holomorphic section that is invariant under the action of the nilpotent Lie group. For complex solvmanifolds, such a section may or may not exist.
In this talk, we will present examples of complex solvmanifolds that admit a holomorphic trivializing section of their canonical bundle which is not invariant under the action of the solvable Lie group. This phenomenon leads to a two-stage study of the existence of trivializing holomorphic sections. In the invariant case, we characterize this existence in terms of the Koszul 1-form, defined in the Lie algebra of the solvable Lie group. In the non-invariant case, we provide an algebraic obstruction (also in terms of the Koszul form) for a solvmanifold to have holomorphically trivial (or more generally, torsion) canonical bundle, and we show how to explicitly construct a non-invariant trivializing section of the canonical bundle in certain examples. Finally, we will present some results in real dimension 6, including a new example of a solvable Lie algebra that admits complex structures with a non-zero holomorphic (3,0)-form, thereby providing new examples of complex solvmanifolds in dimension 6 with holomorphically trivial canonical bundle.
This talk is based on joint work with Adrián Andrada (https://link.springer.com/article/10.1007/s00031-024-09866-z) and more recent work (https://arxiv.org/abs/2412.02325).