K-stability for Fano manifolds with torus action of complexity 1

Abstract

We consider Fano manifolds admitting an algebraic torus action with general orbit of codimension one. Using a recent result of Datar and Székelyhidi, we effectively determine the existence of Kähler-Ricci solitons for those manifolds via the notion of equivariant K-stability. This allows us to give new examples of Kähler-Einstein Fano threefolds, and Fano threefolds admitting a non-trivial Kähler-Ricci soliton.

On the classification of Kähler–Ricci solitons on Gorenstein del Pezzo surfaces

“…By [11,Theorem 4.3.] equivariant non-trivial special test configurations X of Gorenstein Fano T -varieties of complexity 1 are given by the choice of m ∈ N, v ∈ N and y ∈ P 1 , such that z (0) is non-integral for at most one z = y.…”

“…Indeed, the function was called a Fano divisorial polytope in [11,18] and it was shown there that (X, L) is a Gorenstein canonical variety polarised by its ample anticanonical line bundle. Now, we are interested in a description of the associated section ring.…”

“…Now, with the choice of y = ∞ the construction from [11] gives a test configuration with special fibre corresponding to the polytope ∞ = conv((−1, 0), (0, − 1 /2), (3, 1)) ( Fig. 1) with induced C * -action given by the one-parameter subgroup (0, 1) ∈ N ×Z = Z×Z.…”

“…In [18] a not necessarily complete list of such threefolds together with their combinatorial description was given. We use the methods described above to extend the results of [11], providing new examples of threefolds admitting a non-trivial Kähler-Ricci soliton. For this we use the same approach as in the proof of Theorem 1.6.…”

“…In the Table 3 we give the estimates found for the vector field ξ for each threefold in the list of [18]. The threefolds 3.8 , 3.21, 4.5 were shown to admit a non-trivial This refers only to a particular element of the family admitting a 2-torus action Kähler-Ricci soliton in [11]. Applying steps (i)-(ii) from the proof of Theorem 1.8 provides also an approximation for the vector field ξ for these threefolds.…”