Journal Article On the limit of spectral measures associated to a test configuration of a polarized Kähler manifold

Hisamoto, Tomoyuki

We apply our integral formula of volumes to the family of graded linear series constructed from any test configuration. This solves the conjecture raised by Witt Nyström to the effect that the sequence of spectral measures for the induced ℂ*-action on the central fiber converges to the canonical measure defined by the associated weak geodesic ray in the space of Kähler metrics. This limit measure coincides with the classical Duistermaat–Heckmann measure if the test configuration is product. As a consequence, we show that the algebraic p-norm of the test configuration is equal to the Lp-norm of tangent vectors on the geodesic ray. Using this result, we give a natural energy theoretic explanation for the lower bound estimate on the Calabi functional by Donaldson, extending the statement to any p-norm (p ≥ 1), and prove an analogous result for Kähler–Einstein metrics.
Published Online: 2014-04-11

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