Page 118 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 118
TOPICS IN COMPLEX AND QUATERNIONIC GEOMETRY (MS-74)
Balanced Hermitian metrics
Anna Fino, annamaria.fino@unito.it
Università di Torino, Italy
A Hermitian metric on a complex manifold is balanced if its fundamental form is co-closed.
In the talk I will review some general results about balanced metrics and present new smooth
solutions to the Hull-Strominger system, showing that the Fu-Yau solution on torus bundles
over K3 surfaces can be generalized to torus bundles over K3 orbifolds.
Slice-by-slice and global smoothness of slice regular functions
Riccardo Ghiloni, riccardo.ghiloni@unitn.it
University of Trento, Italy
The concept of slice regular function over the real algebra H of quaternions is a generalization
of the notion of holomorphic function of a complex variable. Let Ω be an open subset of H,
which intersects R and is invariant under rotations of H around R. A function f : Ω → H is
slice regular if it is of class C1 and, for all complex planes CI spanned by 1 and a quaternionic
imaginary unit I (CI is a ‘complex slice’ of H), the restriction fI of f to ΩI = Ω ∩ CI satisfies
the Cauchy-Riemann equations associated to I, i.e., ∂I fI = 0 on ΩI , where ∂I = 1 ∂ + I ∂ .
2 ∂α ∂β
We study the continuity and the differential regularity of slice regular functions viewed as
solutions of the slice-by-slice differential equations ∂IfI = 0 on ΩI and as solutions of their
global version ϑf = 0 on Ω \ R.
Our results extend to the slice polyanalytic and monogenic cases.
On compact affine quaternionic curves and surfaces
Anna Gori, anna.gori@unimi.it
Università di Milano, Italy
The introduction of Slice Regular functions in several variables in the recent years has led to
a new definition of quaternionic manifolds. A class of manifolds appearing in this context
is that of quaternionic affine ones. This talk is devoted to the study of affine quaternionic
manifolds and to a possible classification of all compact affine quaternionic curves and surfaces.
A direct result, based on the celebrated Kodaira Theorem states that the only compact affine
quaternionic curves are the quaternionic tori. As for compact affine quaternionic surfaces, the
study of their fundamental groups, together with the inspection of all nilpotent hypercomplex
simply connected 8-dimensional Lie Groups, identifies a path towards their classification. This
talk is based on a work in collaboration with Graziano Gentili and Giulia Sarfatti.
116
Balanced Hermitian metrics
Anna Fino, annamaria.fino@unito.it
Università di Torino, Italy
A Hermitian metric on a complex manifold is balanced if its fundamental form is co-closed.
In the talk I will review some general results about balanced metrics and present new smooth
solutions to the Hull-Strominger system, showing that the Fu-Yau solution on torus bundles
over K3 surfaces can be generalized to torus bundles over K3 orbifolds.
Slice-by-slice and global smoothness of slice regular functions
Riccardo Ghiloni, riccardo.ghiloni@unitn.it
University of Trento, Italy
The concept of slice regular function over the real algebra H of quaternions is a generalization
of the notion of holomorphic function of a complex variable. Let Ω be an open subset of H,
which intersects R and is invariant under rotations of H around R. A function f : Ω → H is
slice regular if it is of class C1 and, for all complex planes CI spanned by 1 and a quaternionic
imaginary unit I (CI is a ‘complex slice’ of H), the restriction fI of f to ΩI = Ω ∩ CI satisfies
the Cauchy-Riemann equations associated to I, i.e., ∂I fI = 0 on ΩI , where ∂I = 1 ∂ + I ∂ .
2 ∂α ∂β
We study the continuity and the differential regularity of slice regular functions viewed as
solutions of the slice-by-slice differential equations ∂IfI = 0 on ΩI and as solutions of their
global version ϑf = 0 on Ω \ R.
Our results extend to the slice polyanalytic and monogenic cases.
On compact affine quaternionic curves and surfaces
Anna Gori, anna.gori@unimi.it
Università di Milano, Italy
The introduction of Slice Regular functions in several variables in the recent years has led to
a new definition of quaternionic manifolds. A class of manifolds appearing in this context
is that of quaternionic affine ones. This talk is devoted to the study of affine quaternionic
manifolds and to a possible classification of all compact affine quaternionic curves and surfaces.
A direct result, based on the celebrated Kodaira Theorem states that the only compact affine
quaternionic curves are the quaternionic tori. As for compact affine quaternionic surfaces, the
study of their fundamental groups, together with the inspection of all nilpotent hypercomplex
simply connected 8-dimensional Lie Groups, identifies a path towards their classification. This
talk is based on a work in collaboration with Graziano Gentili and Giulia Sarfatti.
116
