Abstract

In general, the equivalence of the stability and the solvability of an equation is an important problem in geometry. In this talk, we introduce the -equation on holomorphic vector bundles over compact manifolds, as an extension of the line bundle case and the Hermitian-Einstein equation over Riemann surfaces. We investigate some fundamental properties as well as examples. In particular, we give an algebraic stability condition called the (asymptotic) -stability in terms of subbundles on compact surfaces, and a numerical criterion on vortex bundles via dimensional reduction. Also, we discuss an application for the vector bundle version of the deformed Hermitian-Yang-Mills equation in the small volume regime.

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