Page 120 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 120
TOPICS IN COMPLEX AND QUATERNIONIC GEOMETRY (MS-74)
These are by definition stable Einstein metrics. Stability can equivalently be characterised by a
spectral condition for the Lichnerowicz Laplacian on divergence- and trace-free symmetric 2-
tensors, i.e. on so-called tt-tensors: an Einstein metric is stable if twice the Einstein constant is
a lower bound for this operator. Stability is also related to Perelman’s ν entropy and dynamical
stability with respect to the Ricci flow.
In my talk I will discuss the stability condition. I will present a recent result obtained with G.
Weingart, which completes the work of Koiso on the classification of stable compact symmetric
spaces. Moreover, I will describe an interesting relation between instability and the existence
of harmonic forms. This is done in the case of nearly Kähler, Einstein-Sasaki and nearly G2
manifolds. If time permits I will also explain the instability of the Berger space SO(5)/SO(3),
which is a homology sphere. In this case instability surprisingly is related to the existence of
Killing tensors. The latter results are contained in joint work with M. Wang and C. Wang.
Analogues of the Dolbeault resolution in higher dimensions
Vladimír Soucˇek, soucek@karlin.mff.cuni.cz
Faculty of Mathematics and Physics, Charles University, Czech Republic
Higher dimensional analogues of complex function theory are based on a suitable choice of
an elliptic system of PDEs of the first order. The most developed case is theory of several
complex variables. The old one-variable example is the Fueter equation for regular functions
of a quaternionic variable, the new example of that sort is the Dirac equation for spinor-valued
fields. Higher spin generalizations includes massless fields equations in dimension 4 and its
higher dimensional analogues.
Theory of several complex variables includes the Dolbeault resolution as a proper gener-
alization of the de Rham complex. In a similar way, function theories mentioned above can
be considered in its several variables versions and the key tool here is a generalization of the
Dolbeault complex. There is the general scheme of BGG sequences of invariant differential op-
erators on manifolds with a given parabolic structure. They can be constructed for any regular
infinitesimal character. Such sequences form complexes in homogeneous situation but in non-
flat case, there is an obstruction given by nontrivial curvatures. Particularly nice examples are
complexes of invariant differential operators on quaternionic manifolds introduced and studied
by S. Salamon and R. Baston. They form complexes even in curved situation (for quaternionic
manifolds) due to the fact that they correspond to singular infinitesimal character.
An understanding of constructions and properties of such complexes advanced a lot in case
of Hermitian symmetric spaces mainly due to work of Enright and Shelton. Interest in cases
outside this setting was iniciated by development of function theory of several Clifford vari-
ables. This corresponds to the case of |2|-graded parabolic geometry in singular infinitesimal
character. Using methods of integral geometry (in particular of the Penrose transform), it was
possible to construct analogues of the Dolbeault resolution in even dimensions and in the stable
range.
The aim of the lecture is to introduce first main known results, to describe relations of resolu-
tions with different type of symmetry (Clifford analysis in dimension 4 and several quaternionic
variables) and to describe new results in stable as well as non stable range.
118
These are by definition stable Einstein metrics. Stability can equivalently be characterised by a
spectral condition for the Lichnerowicz Laplacian on divergence- and trace-free symmetric 2-
tensors, i.e. on so-called tt-tensors: an Einstein metric is stable if twice the Einstein constant is
a lower bound for this operator. Stability is also related to Perelman’s ν entropy and dynamical
stability with respect to the Ricci flow.
In my talk I will discuss the stability condition. I will present a recent result obtained with G.
Weingart, which completes the work of Koiso on the classification of stable compact symmetric
spaces. Moreover, I will describe an interesting relation between instability and the existence
of harmonic forms. This is done in the case of nearly Kähler, Einstein-Sasaki and nearly G2
manifolds. If time permits I will also explain the instability of the Berger space SO(5)/SO(3),
which is a homology sphere. In this case instability surprisingly is related to the existence of
Killing tensors. The latter results are contained in joint work with M. Wang and C. Wang.
Analogues of the Dolbeault resolution in higher dimensions
Vladimír Soucˇek, soucek@karlin.mff.cuni.cz
Faculty of Mathematics and Physics, Charles University, Czech Republic
Higher dimensional analogues of complex function theory are based on a suitable choice of
an elliptic system of PDEs of the first order. The most developed case is theory of several
complex variables. The old one-variable example is the Fueter equation for regular functions
of a quaternionic variable, the new example of that sort is the Dirac equation for spinor-valued
fields. Higher spin generalizations includes massless fields equations in dimension 4 and its
higher dimensional analogues.
Theory of several complex variables includes the Dolbeault resolution as a proper gener-
alization of the de Rham complex. In a similar way, function theories mentioned above can
be considered in its several variables versions and the key tool here is a generalization of the
Dolbeault complex. There is the general scheme of BGG sequences of invariant differential op-
erators on manifolds with a given parabolic structure. They can be constructed for any regular
infinitesimal character. Such sequences form complexes in homogeneous situation but in non-
flat case, there is an obstruction given by nontrivial curvatures. Particularly nice examples are
complexes of invariant differential operators on quaternionic manifolds introduced and studied
by S. Salamon and R. Baston. They form complexes even in curved situation (for quaternionic
manifolds) due to the fact that they correspond to singular infinitesimal character.
An understanding of constructions and properties of such complexes advanced a lot in case
of Hermitian symmetric spaces mainly due to work of Enright and Shelton. Interest in cases
outside this setting was iniciated by development of function theory of several Clifford vari-
ables. This corresponds to the case of |2|-graded parabolic geometry in singular infinitesimal
character. Using methods of integral geometry (in particular of the Penrose transform), it was
possible to construct analogues of the Dolbeault resolution in even dimensions and in the stable
range.
The aim of the lecture is to introduce first main known results, to describe relations of resolu-
tions with different type of symmetry (Clifford analysis in dimension 4 and several quaternionic
variables) and to describe new results in stable as well as non stable range.
118
