A quadratic inequality for sum of co-adjoint orbits

We obtain an effective lower bound on the distance of thesum of co-adjoint orbits from the origin. Even when thedistance is zero (thus the symplectic quotient is welldefined) our result gives a nontrivial constraint on theseco-adjoint~orbits. In the particular case of unitarygroups, we obtain the quadratic inequality foreigenvalues of Hermitian matrices satisfying\begin{equation*}A+B=C.\end{equation*}This quadratic inequality can be interpreted as the Chern number inequalityfor semi-stable reflexive toric sheaves.

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Published 1 January 2009