Einstein-Kähler metrics on symmetric toric Fano manifolds

Abstract

Let X be a complex toric Fano n-fold and N (T ) the normalizer of a maximal torus T in the group of biholomorphic authomorphisms Aut(X). We call X symmetric if the trivial character is a single N (T )-invariant algebraic character of T . Using an invariant α G (X) introduced by Tian, we show that all symmetric toric Fano n-folds admit an Einstein-Kähler metric. We remark that so far one doesn't know any example of a toric Fano n-fold X such that Aut(X) is reductive, the Futaki character of X vanishes, but X is not symmetric.

The α-invariant on toric Fano manifolds

“…This gives a generalization of the result by V. Batyrev and E. Selivanova [2] and also this formula confirms the earlier result [12] on the estimates of α invariants on CP 2 #1CP 2 and CP 2 #2CP 2 .…”

Multiplier ideal sheaves and integral invariants on toric Fano manifolds

“…In fact, it is easy to check that sup N Rψ ti ≤ 0 and (See Lemma 4.3 in [19] and [2].) Since there is a constant C ǫ depending only on ǫ such that In the above inequalities, we usedũ(0, s) ≥ −s for all s ≤ 0.…”

On homothetic balanced metrics

“…6 We do not know if in the previous theorem B g B is empty. For example, in [13] it is shown that there exist cscK polarizations L on the blow up M of CP 2 at four points (all but one aligned) constructed from Theorem 5.2 such that (M, L m ) is not asymptotically Chow polystable for m large enough, so that B c (L) is finite.…”