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We study a variational problem whose critical point determines the Reeb vector field for a SasakiEinstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the EinsteinHilbert action, restricted to a space of Sasakian metrics on a link <i>L</i> in a CalabiYau cone <i>X</i>, is the volume functional, which in fact is a function on the space of Reeb vector fields. We relate this function both to the DuistermaatHeckman formula and also to a limit of a certain equivariant index on <i>X</i> that counts holomorphic functions. Both formulae may be evaluated by localisation. This leads to a general formula for the volume function in terms of topological fixed point data. As a result we prove that the volume of a SasakiEinstein manifold, relative to that of the round sphere, is always an algebraic number. In complex dimension <i>n</i>=3 these results provide, via AdS

We study the general problem of equidistribution of expanding translates of an analytic curve by an algebraic diagonal flow on the homogeneous space of a semisimple algebraic group G. We define two families of algebraic subvarieties of the associated partial flag variety, which give the obstructions to non-divergence and equidistribution. We apply this to prove that for Lebesgue almost every point on an analytic curve in the space of m × n real matrices whose image is not contained in any subvariety coming from these two families, Dirichlet’s theorem on simultaneous Diophantine approximation cannot be improved. The proof combines geometric invariant theory, Ratner’s theorem on measure rigidity for unipotent flows, and linearization technique.

No abstract uploaded!

No abstract uploaded!