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

Abstract

We apply the result of [His12] to the family of graded linear series constructed from any test configuration. This solves the conjecture raised by [WN10] so that the sequence of spectral measures for the induced C * -action on the central fiber converges to the canonical Duistermatt-Heckman measure defined by the associated weak geodesic ray. As a consequence, we show that the algebraic p-norm of the test configuration equals to the L p -norm of tangent vectors. Using this result, We may give a natural energy theoretic explanation for the lower bound estimate on the Calabi functional by [Don05] and prove the analogous result for the Kähler-Einstein metric.2000 Mathematics Subject Classification. Primary 32A25, Secondary 32L10, 32W20.

K-polystability of Q-Fano varieties admitting Kähler-Einstein metrics

“…As pointed out above Lemma 4.2 is a special case of a general result of Hisamoto [38], saying that the measure ( dφ t dt ) * M A(φ t ) on R only depends on the test configuration (X , L) and moreover is equal to the limiting normalized weight measures for the C * -action, as conjectured by WittNyström [52], who settled the case of product test configurations. In particular, by [38] all the L p -norms (X , L) p of dφ t dt (integrating against M A(φ t )) only depend on (X , L) and coincide with the limits of the corresponding l p -norms of the weights {λ (k) i }.…”

“…In particular, by [38] all the L p -norms (X , L) p of dφ t dt (integrating against M A(φ t )) only depend on (X , L) and coincide with the limits of the corresponding l p -norms of the weights {λ (k) i }. In particular, letting p → ∞ gives Lemma 4.2.…”

“…In particular, thanks to Tomoyuki Hisamoto and David Witt-Nyström for discussions on norms of test configurations and the relations to their works [38] and [52] (compare Remark 4.4). Finally, thanks to the referees for many useful comments.…”

Relative K-stability for Kähler manifolds

“…The above proof should be thought of as an analogue of part of the Kempf-Ness Theorem, which gives lower bounds on the norm squared of moment maps, using the Cauchy-Schwarz inequality. Compare, for example, [15,25,30] and the survey [29]. The next step is the following result relating the various norms.…”

Uniform K-stability, Duistermaat–Heckman measures and singularities of pairs

“…Theorem 9.1] and [BWN14, Proposition 4.1]). See also[Bern09,WN12,His12] for an analytic approach to Duistermaat-Heckman measures via geodesic rays. Remark 3.8.…”