Alexander KolpakovDepartement für Mathematik, Universität FreiburgAlexander MednykhSobolev Institute of Mathematics, Novosibirsk State UniversityMarina PashkevichDepartment of Mathematics and Mechanics, Novosibirsk State University
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Alexander Kolpakov · Jun Murakami. The dual Jacobian of a generalised tetrahedron, and volumes of prisms. 2014.
The present paper considers volume formulæ, as well as trigonometric identities, that hold for a tetrahedron in 3-dimensional spherical space of constant sectional curvature +1. The tetrahedron possesses a certain symmetry: namely rotation of angle$π$in the middle points of a certain pair of its skew edges.
We show that the symmetrized bidisc may be exhausted by strongly linearly convex domains. It shows in particular the existence of a strongly linearly convex domain that cannot be exhausted by domains biholomorphic to convex ones.
We obtain an effective lower bound on the distance of the sum of co-adjoint orbits from the origin. Even when the distance is zero (thus the symplectic quotient is well defined) our result gives a nontrivial constraint on these co-adjoint orbits. In the particular case of unitary groups, we obtain the quadratic inequality for eigenvalues of Hermitian matrices satisfying
A + B = C.
This quadratic inequality can be interpreted as the Chern number inequality for semi-stable reflexive toric sheaves.
Almost toric manifolds form a class of singular Lagrangian fibered
symplectic manifolds that include both toric manifolds and the K3
surface. We classify closed almost toric four-manifolds up to diffeomorphism
and indicate precisely the structure of all almost toric fibrations
of closed symplectic four-manifolds. A key step in the proof is a geometric
classification of the singular integral affine structures that can
occur on the base of an almost toric fibration of a closed four-manifold.
As a byproduct we provide a geometric explanation for why a generic
Lagrangian fibration over the two-sphere must have 24 singular fibers.
We study the geometry of the Grassmannians of symplectic subspaces in a symplectic
vector space. We construct symplectic twistor spaces by the symplectic quotient
construction and use them to describe the symplectic geometry of the symplectic Grassmannians.