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European Journal of Mathematics

, Volume 4, Issue 1, pp 137–161 | Cite as

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

  • Jacob Cable
  • Hendrik SüßEmail author
Open Access
Research Article

Abstract

We give a classification of all pairs \((X,\xi )\) of Gorenstein del Pezzo surfaces X and vector fields \(\xi \) which are K-stable in the sense of Berman–Witt–Nyström and therefore are expected to admit a Kähler–Ricci solition. Moreover, we provide some new examples of Fano threefolds admitting a Kähler–Ricci soliton.

Keywords

K-Stability Kähler–Ricci solitons T-Variety Torus action Fano variety 

Mathematics Subject Classification

32Q20 14L30 14J45 

1 Introduction

By a Fano orbifold we mean a complex normal variety with only orbifold singularities and an ample anti-canonical class. It is called Gorenstein if the anti-canonical class is Cartier. In dimension 2 those varieties are usually called Gorenstein del Pezzo surfaces. Let X be a Fano orbifold and \(\omega _g\) be the Kähler form of a Kähler metric g on X. The form \(\omega _g\) is called a Kähler–Ricci soliton if there exists an orbifold holomorphic vector field \(\xi \), such thatholds, where Open image in new window denotes the Lie derivative of a form \(\eta \) with respect to \(\xi \). This implies that Open image in new window is real-valued and, hence, Open image in new window , where \(\mathfrak {I}\xi \) denotes the imaginary part of \(\xi \). Therefore, \(\mathfrak {I}\xi \) generates a one-dimensional Hamiltonian torus action on X. In the following we will identify \(\xi \) with this action. If \(\xi =0\) the metric g is called Kähler–Einstein, else we speak of a non-trivial Kähler–Ricci soliton.

By [19] we know that such a soliton metric is unique if it exists. On the other hand, by [3] together with [6] the existence of such a metric (at least in the smooth case) corresponds to an algebro-geometric stability condition, known as K-stability. The key objects involved in defining K-stability are test configurations:

Definition 1.1

Let (XL) be a polarized projective variety. A test configuration for (XL) is a \(\mathbb {C}^*\)-equivariant flat family Open image in new window over Open image in new window equipped with a relatively ample equivariant \(\mathbb {Q}\)-line bundle Open image in new window such that
A test configuration with Open image in new window is called a trivial or a product configuration. A test configuration with normal special fiber Open image in new window is called special. Given an algebraic group G with action on X, a test configuration is called G-equivariant if the action extends to Open image in new window and commutes with the \(\mathbb {C}^*\)-action of the test configuration.

Remark 1.2

By Hironaka’s Lemma [7, III.9.12] the normality of the fibers induces the normality of the total space of the family. Hence, a special test configuration Open image in new window has normal total space Open image in new window .

We will primarily be concentrating on the situation where X is Fano and Open image in new window . It follows that the special fiber Open image in new window is \(\mathbb {Q}\)-Fano. We proceed to define the modified Donaldson–Futaki invariants as they appeared in [3].

Let Open image in new window be any \(\mathbb {Q}\)-Fano variety equipped with the action of an algebraic torus \(T'\), and let \(\ell \in \mathbb {N}\) be the smallest natural number such that Open image in new window is Cartier. Let \(M'\) and \(N'\) be the lattices of characters and one-parameter subgroups of \(T'\), respectively. We denote the associated \(\mathbb {R}\)-vector spaces by \(M'_\mathbb {R}\) and \(N'_\mathbb {R}\), respectively. Fix an element \(\xi \in N'_\mathbb {R}\). Consider the canonical linearisation for Open image in new window coming from the identification Open image in new window . Then we set Open image in new window and for every \(v \in N'\) Now,defines a linear form on Open image in new window , i.e. an element of Open image in new window . It is called the (modified) Futaki character.

Consider now a Fano variety X with a (possibly trivial) torus T acting on it. Fix Open image in new window , where N is the lattice of one-parameter subgroups of T. Let Open image in new window be the central fiber of a special T-equivariant test configuration Open image in new window for Open image in new window . Then Open image in new window comes equipped with a Open image in new window -action induced by the test configuration. Let \(v\in N'\) denote the one-parameter subgroup corresponding to the \(\mathbb {C}^*\)-action of the test configuration. Note that the inclusion \(T\hookrightarrow T'\) induces an inclusion of one-parameter subgroups \(N\hookrightarrow N'\).

Definition 1.3

Given a special test configuration Open image in new window of Open image in new window as above, its modified Donaldson–Futaki invariant is defined as

Definition 1.4

Consider a Fano variety X with action by a reductive group G containing a maximal torus \(T \subset G\) and Open image in new window . The pair \((X,\xi )\) is called equivariantly K-stable if Open image in new window for every G-equivariant special test configuration Open image in new window as above and we have equality exactly in the case of product test configurations. If \(\xi =0\) we say X itself is equivariantly K-stable.

The following result by Datar and Székelyhidi motivates the study of equivariant K-stability:

Theorem 1.5

([6]) For a smooth Fano G-variety X, the variety X admits a Kähler–Ricci soliton with respect to \(\xi \) if and only if the pair \((X,\xi )\) is equivariantly K-stable.

By [13, 6.1.2] the Gorenstein del Pezzo surfaces of degree \(\leqslant 4\) admitting a Kähler–Einstein metric are known to be exactly those which are either smooth or fit into one of the following combinations of degree and singularity type:
Degree 1:

\(2D_4\) or a combination of \(A_k\)-singularities with \(k \leqslant 7\),

Degree 2:

\(2A_3\) or a combination of \(A_1\) and \(A_2\) singularities,

Degree 3:

\(3A_2\) or \(\ell A_1\) with \(\ell \geqslant 1\),

Degree 4:

\(2A_1\) or \(4A_1\).

Hence, it is natural to ask which of the remaining ones admit at least a Kähler–Ricci soliton. In this paper we approach this question by giving a complete classification of pairs \((X,\xi )\) of Gorenstein del Pezzo surfaces X and vector fields \(\xi \) as above which are equivariantly K-stable with respect to the torus action generated by \(\xi \).

Theorem 1.6

Among the Gorenstein del Pezzo surfaces with non-trivial holomorphic vector fields. Exactly the following combinations of degree/singularity type admit non-trivial equivariantly K-stable pairs \((X,\xi )\):
Degree 1:

\(E_8\), \(E_7A_1\), \(E_6A_2\),

Degree 2:

\(A_5A_2\), \(D_6A_1\), \(E_7\), \(E_6\), \(D_5A_1\), \(D_43A_1\),

Degree 3:

\(E_6\), \(A_4A_1\), \(D_5\), \(D_4\), \(A_32A_1\),

Degree 4:

\(D_5\), \(D_4\), \(A_4\), \(A_3\),

toric:

\(A_32A_1\), \(A_22A_1\),

Degree 5:

\(A_4\), \(A_3\), \(A_1\),

toric:

\(2A_1\), \(A_2A_1\)

Degree 6:

\(A_1\),

toric:

\(A_1\), \(2A_1\), \(A_2A_1\),

Degree 7:

toric: smooth, \(A_1\),

Degree 8:

toric: smooth, \(A_1\).

For the toric cases the existence of a Kähler–Ricci soliton is known by [16]. Note, that Theorem 1.5 only considers the smooth case. We hope and expect that the methods from [6] will work in our setting as well. However, at the moment we do not have the corresponding statement in the case of orbifolds. This prevents us from actually proving the existence of Kähler–Ricci solitons in the cases considered in Theorem 1.6.

On the other hand, the implication of K-stability by the existence of a Kähler–Ricci soliton holds also in the singular case by [3, Theorem 1.5]. This allows us to rule out the existence of Kähler–Ricci solitons for the remaining Gorenstein del Pezzo surfaces with non-trivial vector fields. Hence, by using a classification of such surfaces from [9] we obtain the following corollary.

Corollary 1.7

The following combinations of degree/singularity type do not admit a Kähler–Ricci soliton metric (neither non-trivial nor Kähler–Einstein):
Degree 3:

\(A_5A_1\), \(2A_2A_1\), \(2A_2\),

Degree 4:

\(A_3A_1\), \(3A_1\),

Degree 5:

\(A_24\),

Degree 6:

\(A_2\).

In [18] certain smooth Fano threefolds with 2-torus action were considered and the paper [11] determined which of them admit a Kähler–Einstein metric via equivariant K-stability. Moreover, for some of the remaining ones the existence of a Kähler–Ricci soliton could be proven by the simple observation that in these cases there are no equivariant special test configurations beside the product ones. In this paper we consider some of the remaining cases and obtain the following theorem.

Theorem 1.8

The Fano threefolds 2.30, 2.31, 3.18, 3.22, 3.23, 3.24, 4.8 from Mori and Mukai’s classification [12] admit a non-trivial Kähler–Ricci soliton.

The common feature of the surfaces and threefolds considered in Theorems 1.6 and 1.8, respectively, is the presence of an effective action of an algebraic torus of one dimension less than the variety it acts on. In the following we will call these varieties T-varieties of complexity 1. Note, that surfaces with non-trivial Kähler–Ricci soliton automatically fall into this class due to the torus action generated by vector field. However, for threefolds this is indeed an additional condition.

In Sect. 2 we review the combinatorial description of Fano T-varieties of complexity 1 and their equivariant test configurations as it was developed in [11].

In Sect. 3 we state the classification of Gorenstein del Pezzo surfaces from [9] in terms of their combinatorial data and describe the computational methods, that we used to determine which of these surfaces can be complemented to a K-stable pair in the sense of Definition 1.4.

Finally, in Sect. 4 we apply the same methods to the threefolds from [11, 18] to obtain new examples of Kähler–Ricci solitons on Fano threefolds.

In an appendix we provide examples of the computer assisted calculations, which we used to obtain our results. The complete computations are available in the ancillary files [4].

2 Combinatorial description of T-varieties of complexity 1

We fix an algebraic torus \(T \cong (\mathbb {C}^*)^n\). We denote its character lattice by M and the dual lattice of co-characters or one-parameter subgroups by N. The corresponding vector spaces are denoted by Open image in new window and Open image in new window , respectively.

We will describe Gorenstein Fano T-varieties of complexity 1 by the following set of data. A lattice polytope \(\Box \) in Open image in new window , which contains the origin, together with a concave functionsuch that
  1. (i)

    \(\Phi \) is piecewise affine linear, i.e. given by affine linear functions on a finite polyhedral subdivision of \(\Box \).

     
  2. (ii)

    For \(y\in \mathbb {P}^1\), the graph of \(\Phi _y\) has integral vertices.

     
  3. (iii)

    For every u in the interior of \(\Box \), Open image in new window .

     
  4. (iv)

    The affine linear pieces of \(\Phi _y\) have the form Open image in new window , where \(v\in N\) is a primitive lattice element.

     
  5. (v)

    Every facet F of \(\Box \) with Open image in new window has lattice distance 1 from the origin.

     
This gives rise to a polarised T-variety of complexity one in the following way. Consider Open image in new window given by \(\overline{\Phi }(u) = \Phi (u)+D\), where D is some integral divisor of degree 2. Thendefines a polarised variety (XL). It is easy to see that a different divisor \(D'\) of degree 2 will give rise to the same polarised variety, since \(D-D'\) is principal. Moreover, associating Open image in new window the weight \(ku \in M\) induces an M-grading on the section ring of L and, hence, a T-action on X.

Indeed, the function \(\overline{\Phi }\) was called a Fano divisorial polytope in [11, 18] and it was shown there that (XL) is a Gorenstein canonical variety polarised by its ample anticanonical line bundle.

Example 2.1

(Cubic surface) Consider the map Open image in new window given bywith Open image in new window , Open image in new window and Open image in new window . This fulfils the conditions (i)–(v) above. Note, that we have Open image in new window and part (v) of the conditions is void in this case.
Now, we are interested in a description of the associated section ring. Choose Open image in new window then
After rounding down the corresponding divisors have degree 0 with the exception of \(\lfloor \overline{\Phi }(2)\rfloor \), which has degree \(-1\). Hence, in degree \(k=1\) with respect to the usual \(\mathbb {Z}\)-grading of the section ring we find the following four generators, where \(z^k\) keeps track of this degree and \(\chi ^u\) keeps track of the weight \(u \in M \cong \mathbb {Z}\) as in (2).One can check that these elements generate the section ring. The relations are generated by \(x_0x_2x_3 + x_1^2x_3 + x_2^3\). Hence, we obtain a cubic surface.

As in the toric case it is possible to read off many properties and invariants of the variety directly from Open image in new window , see e.g. [18]. Here, we are mainly interested in the Fano degree and in the Cox ring.

The Fano degree is the top self-intersection number of the anti-canonical divisor. It can be calculated from the combinatorial data by the following

Theorem 2.2

([14, Proposition 3.31]) For the Fano degree one obtains
For a normal complete variety X the Cox ring is defined (as a graded vector space) byIt can be equipped with a natural multiplication map making it into a Open image in new window -graded \(\mathbb {C}\)-algebra, see [8] for details.

We sketch how to obtain generators and relations of the Cox ring of X from the corresponding divisorial polytope. For this we need to introduce some notation.

Let Open image in new window be the set of facets of the graph of \(\Phi _y\) andRecall from condition (iv) above, that for Open image in new window the facet F is the graph of an affine linear function of the following form:defined on a subset of \(\Box \). By Open image in new window we denote be the set of facets G of \(\Box \) such that Open image in new window .

Theorem 2.3

Example 2.4

(Cubic surface – continued) Consider the divisorial polytope from Example 2.1. An elementary calculation shows that Open image in new window . Hence, we verify by Theorem 2.2 that X has Fano degree 3.

Now, consider the Cox ring. For \(\Phi _0\) the graph has two facets \(F_1\) and \(F_2\) corresponding to the affine linear pieces Open image in new window and \(u \mapsto 0\). This gives Open image in new window and Open image in new window . For \(\Phi _\infty \) there is a unique facet \(F_3\) with associated affine linear form Open image in new window . Hence, Open image in new window and Open image in new window and similarly Open image in new window .

We obtain the following equation for the Cox ring:
By [11, Theorem 4.3.] equivariant non-trivial special test configurations Open image in new window of Gorenstein Fano T-varieties of complexity 1 are given by the choice of \(m \in \mathbb {N}\), \(v \in N\) and \(y \in \mathbb {P}^1\), such that \(\Phi _z(0)\) is non-integral for at most one \(z \ne y\). We call such a choice of y admissible. It easy to describe the toric special fibre Open image in new window of such a test configuration. It corresponds to the polytope Open image in new window given byand the induced \(\mathbb {C}^*\)-action on the special fibre Open image in new window is given by the one-parameter subgroup of Open image in new window corresponding to Open image in new window . We denote the corresponding test configuration by Open image in new window with special fibre Open image in new window .
As observed in [11] for the Futaki character of Open image in new window one obtainswith Open image in new window . On the other hand, for Open image in new window one obtainsTo see this consider the projection Open image in new window giving a coarsened grading on the sections of Open image in new window . With this grading for \(u \in M\) one has
Since the definition of \(F_{X, \xi }\) in (1) does only depend on these dimension counts and the values Open image in new window the first equality in (5) follows. For the second equality note, that for Open image in new window and Open image in new window the integrand in (4) does not depend on the second factor of Open image in new window . Hence, integrating along this factor first gives just the height of \({\Delta }_y\) at u, which is exactly \(\deg \overline{\Phi }(u)\). Hence, one obtains the integral in (5).
Now, for a pair \((X,\xi )\) to be equivariantly K-stable the Futaki invariant \(F_{X, \xi }(v)\) has to vanish for every choice v by the condition for product test configurations. With exactly the same arguments as in [5, Section 3.1] we may see that there always exist a unique choice Open image in new window for which \(F_{X, \xi }\) is trivial. We call this \(\xi \) a soliton canditate. To see whether with this candidate \((X,\xi )\) is indeed equivariantly K-stable it remains to check positivity offor every admissible choice of y. Note, that \({\Delta }_y = {\Delta }_{y'}\) for Open image in new window . This leaves us with a finite number of integrals to check. However, in general we cannot hope to find an exact solution for \(\xi \). We have to deal with sufficiently good approximations, instead.

Remark 2.5

As a consequence of the above classification of non-product T-equivariant special test configurations we see that if there are at least three points \(y \in \mathbb {P}^1\) with \(\Phi _y(0)\) being non-integral then such a test configuration does not exists due to the lack of an admissible choice for \(y \in \mathbb {P}^1\). Hence, we obtain equivariant K-stability for the soliton candidate for free.

Example 2.6

(Cubic surface – continued) Recall the piecewise linear function Open image in new window from Example 2.1 given byNow, with the choice of \(y=\infty \) the construction from [11] gives a test configuration with special fibre corresponding to the polytope Open image in new window (Fig. 1) with induced \(\mathbb {C}^*\)-action given by the one-parameter subgroup Open image in new window .
Fig. 1

The polytope \({\Delta }_\infty \)

This degeneration can be realised in Open image in new window by the equationwhere \(\alpha \in \mathbb {C}\) is the parameter of the degeneration. Note, that for \(\alpha \ne 0\) we may eliminate \(x_4\) and get the equation from Example 2.1 (up to scaling of variables). This degeneration is induced by the action of \(\mathbb {C}^*\) with weights \((0,0,0,-1,1)\) onTo check for the existence of a vector field Open image in new window , such that \((X,\xi )\) is K-stable, we first have to determine the unique candidate with sufficient precision. Hence, we have to approximate a solution ofSolving the integral leads toWhen evaluating the exponential functions occurring in (6) with a guaranteed precision of 11 binary digits at the values \(-1.247\) and \(-1.246\), it can be shown by elementary estimations (e.g. interval arithmetic) that \(f(-1.247)<0\) and \(f(-1.246)>0\). The intermediate value theorem then implies \(-1.247< \xi < -1.246\) for the solution \(\xi \) of (6).
Now it remains to check the sign of the integralUsing the same approximations as above for the exponential functions evaluated at the lower and upper bounds for \(\xi \) gives an estimateIn any case this shows that the Donaldson–Futaki invariant is negative and the pair \((X,\xi )\) gets destabilised.

Remark 2.7

Although it is in principle possible to do the calculations/estimates in Example 2.6 by hand it becomes quite tedious. Hence, we used interval arithmetic library MPFI [15] via the SageMath [17] computer algebra system to verify K-stability for our example, see Appendix 5.1.

Remark 2.8

As for Example 2.6 the standard integrals appearing in (4) and (5) can be solved analytically in general, either by elementary methods or by using Stoke’s theorem to reduce to similar integrals along the boundary facets and by iterating this process eventually obtaining a formula which involves evaluations of exponential functions in the vertices of the polytope and rational functions, see [2, Lemma 1].

3 Classification

In this section we are considering all Gorenstein del Pezzo surfaces, which admit a non-trivial \(\mathbb {C}^*\)-action. We give a list of the corresponding combinatorial data in Table 1. For every del Pezzo surface we state the closed interval \(\Box \), the functions \(\Phi _y:\Box \rightarrow \mathbb {R}\) with \(y \in \mathrm{supp}\,\Phi \). One can check that these data fulfil the conditions (i)–(iv) from above and, hence, define Gorenstein del Pezzo surfaces with \(\mathbb {C}^*\)-action. Note, that for all considered surfaces the support of \(\Phi \) contains either three or four points. Hence, with the appropriate choice of coordinates on \(\mathbb {P}^1\) we may assume that the support consists of the points \(0,\infty ,1\) and possibly a fourth point c. If the support consists of four points the combinatorial data gives rise to a 1-parameter family of del Pezzo surfaces parametrised by Open image in new window .

A classification of Gorenstein del Pezzo surfaces with \(\mathbb {C}^*\)-action in terms of their Cox-rings is available from [9, Section 5.3]. One the other hand, Theorems 2.3 and 2.2 allow us to compare the Cox rings and Fano degrees of the del Pezzo surfaces in Table 1 with those in [9]. Doing so we see that the list below is complete and hence provides the combinatorial description of all Gorenstein del Pezzo surfaces with \(\mathbb {C}^*\)-action. Recently, the same classification was obtained in [10] directly in terms of divisorial polytopes.
Table 1

Gorenstein del Pezzo surfaces with \(\mathbb {C}^*\)-action

No.

K-stab

\(\xi \)

\(\Box \)

\(\Phi _{0},\Phi _{\infty },\Phi _1, (\Phi _{c})\)

\((K_X)^2\)

Sing.

\(\rho \)

1

\({\varvec{\surd }}\)

0

\({[-1,1]}\)

Open image in new window

1

\({2}{D}_{{4}}\)

1

2

\(\surd \)

\(-1.99761\)

\([-1,5]\)

Open image in new window

1

\(E_8\)

1

3

\(\surd \)

\( -1.94024\)

\([-1,3]\)

Open image in new window

1

\(E_7A_1\)

1

4

\(\surd \)

\( -1.69131\)

\([-1,2]\)

Open image in new window

1

\(E_6A_2\)

1

5

\({\varvec{\surd }}\)

0

\({[-1,1]}\)

Open image in new window

2

\({2}{A}_{3}{A}_{1}\)

1

6

\(\surd \)

\( -0.97052\)

\([-1,2]\)

Open image in new window

2

\(A_5A_2\)

1

7

\(\surd \)

\( -1.79675\)

\([-1,3]\)

Open image in new window

2

\(D_6A_1\)

1

8

\(\surd \)

\( -1.99186\)

\([-1,5]\)

Open image in new window

2

\(E_7\)

1

9

\(\surd \)

\( -1.94024\)

\([-1,3]\)

Open image in new window

2

\(E_6\)

2

10

\({\varvec{\surd }}\)

0

\({[-1,1]}\)

Open image in new window

2

\({2}{A}_{3}\)

2

11

\(\surd \)

\( -1.69131\)

\([-1,2]\)

Open image in new window

2

\(D_5A_1\)

2

12

\(\surd \)

\( -1.34399\)

\([-1,1]\)

Open image in new window

2

\(D_43A_1\)

1

13

\({\times }\)

\( -1.24607\)

\([-1,3]\)

Open image in new window

3

\(A_5A_1\)

1

14

\(\surd \)

\( -1.96766\)

\([-1,5]\)

Open image in new window

3

\(E_6\)

1

15

\(\surd \)

\( -1.19618\)

\([-1,2]\)

Open image in new window

3

\(A_4A_1\)

2

16

\({\varvec{{\times }}}\)

0

\({[-1,1]}\)

Open image in new window

3

\({2}{A}_{2}{A}_{1}\)

2

17

\(\surd \)

\( -1.83879\)

\([-1,3]\)

Open image in new window

3

\(D_5\)

2

18

\({\varvec{{\times }}}\)

0

\({[-1,1]}\)

Open image in new window

3

\({2}{A}_{2}\)

3

19

\(\surd \)

\( -1.69131\)

\([-1,2]\)

Open image in new window

3

\(D_4\)

3

20

\(\surd \)

\( -0.94468\)

\([-1,1]\)

Open image in new window

3

\(A_32A_1\)

2

21

\(\surd \)

\( -1.85969\)

\([-1,5]\)

Open image in new window

4

\(D_5\)

1

22

\({\times }\)

\( -0.97052\)

\([-1,2]\)

Open image in new window

4

\(A_3A_1\)

2

23

\(\surd \)

\( -1.79675\)

\([-1,3]\)

Open image in new window

4

\(D_4\)

2

24

\(\surd \)

\( -1.38176\)

\([-1,3]\)

Open image in new window

4

\(A_4\)

2

25

\({\varvec{{\times }}}\)

0

\({[-1,1]}\)

Open image in new window

4

\({3}{A}_{1}\)

3

26

\(\surd \)

\( -1.31047\)

\([-1,2]\)

Open image in new window

4

\(A_3\)

3

27

\({\varvec{\surd }}\)

0

\({[-1,1]}\)

Open image in new window

4

\({2}{A}_{1}\)

4

28

\({\times }\)

\( -0.74373\)

\([-1,1]\)

Open image in new window

4

\(A_2A_1\)

3

29

\(\surd \)

\( -1.42059\)

\([-1,5]\)

Open image in new window

5

\(A_4\)

1

30

\(\surd \)

\( -1.43886\)

\([-1,3]\)

Open image in new window

5

\(A_3\)

2

31

\({\times }\)

\( -1.10613\)

\([-1,2]\)

Open image in new window

5

\(A_2\)

3

32

\(\surd \)

\( -0.61790\)

\([-1,1]\)

Open image in new window

5

\(A_1\)

4

33

\({\times }\)

\( -1.24607\)

\([-1,3]\)

Open image in new window

6

\(A_2\)

2

34

\(\surd \)

\( -0.97052\)

\([-1,2]\)

Open image in new window

6

\(A_1\)

3

Example 3.1

(Cubic surface – continued) Consider once again the cubic surface from the Examples 2.1 and 2.4. From Example 2.4 we know that the Fano degree equals 3 and the Cox ring is isomorphic to Open image in new window . Comparing this with the data from [9, Theorem 5.25] we find that the corresponding surface has Picard rank \(\rho =1\) and singularity type \(A_5A_1\).

Proof of Theorem 1.6

We are running through the classification given in Table 1. First note, that the cases 1, 5, 10 and 27 are known to admit Kähler–Einstein metrics, hence are K-stable, see [13, Section 6.1.2]. For the cases 16, 18 and 25 one also calculates Open image in new window . Hence, we have \(\xi =0\) for the soliton candidate. On the other hand, these surfaces are known to be not Kähler–Einstein, see loc. cit.

For the remaining cases we follow the approach outlined in Example 2.6. Hence, we apply the following steps:
  1. (i)

    Find a closed form for \(F_{X,\xi }(1)\) in terms of exponential functions in \(\xi \). This can be done by solving the integral (here over an interval) appearing in (5) analytically using standard methods.

     
  2. (ii)

    Find sufficiently good bounds \(\xi _-\) and \(\xi _+\) with \(\xi _-< \xi < \xi _+\) for a solution \(\xi \) of \(F_{X,\xi }(1)=0\). To show that the interval \((\xi _-, \xi _+)\) contains a solution we calculate \(F_{X,\xi _-}(1)\) and \(F_{X,\xi _+}(1)\) with sufficient precision (i.e. we need to guarantee error bounds for the approximation of the exponential function) and then use the intermediate value theorem.

     
  3. (iii)
    For every admissible choice of \(y \in \mathbb {P}^1\) find a closed form for Open image in new window in terms of exponential functions. For this we have to analytically solve the integral in (4) for Open image in new window and Open image in new window . This comes down to solving
    Here, the right hand side just involves standard integrals in one variable and can be solved by elementary methods.
     
  4. (iv)

    Ultimatively we have to plug in the value of \(\xi \) into the found closed form and check positivity. However, we have only estimates for \(\xi \) and also for the evaluations of the exponential functions appearing in the closed form obtained in (iii). Hence, we need to use elementary estimations to ensure positivity for all values within the known error bounds.

     
The complete calculations are done using SageMath and can be found in the ancillary files [4] and as an online worksheet.1 For an example which can be adopted to the other cases see also Appendix 5.1.
Table 2

Destabilising test configurations

No.

Destabilising degeneration

Amb. space

Weights

Special fiber

13

Open image in new window

\(\mathbb {P}(1,1,1,1,2)\)

\((0,0,0,-1,1)\)

22

Open image in new window

\(\mathbb {P}^4\)

\((0,0,0,-1,1)\)

28

Open image in new window

\(\mathbb {P}^4\)

\((1,-1,0,-1,0)\)

31

Open image in new window

\(\mathbb {P}^5\)

\((0,1,1,0,0,-1)\)

33

Open image in new window

\(\mathbb {P}^6\)

\((1,0,-1,0,-1,-1,-1)\)

Note, that as in Remark 2.5 for the cases no. 1, 2, 3, 4, 7, 8 , 9, 11, 12, 14, 17, 18, 19, 21 and 23 we obtain the existence of a K-stable pair \((X,\xi )\) without any calculation, since in these cases there is no admissible choice of y. However, to obtain an approximation for \(\xi \) we still have to do the calculations in (i) and (ii). In particular, in all cases with non-trivial candidate vector fields \(\xi \) we obtain a K-stable pair with the exception of nos. 13, 22, 28, 31 and 33. \(\square \)

The cases no. 13, 22, 28, 31, 33 do not admit a K-stable pair and, hence, no Kähler–Ricci soliton. We provide a description of the destabilising test configurations in Table 2. Indeed, from the calculations one obtains as destabilising test configurations Open image in new window for no. 13 and Open image in new window for all other cases. The description of the special fibre from (3) immediately provides the last column of the table. To obtain the equations and the ambient space we refer to the explicit construction of the test configuration given in [11, Section 4.1]. The third column states the weights for the \(\mathbb {C}^*\)-action on the ambient space, which induces the \(\mathbb {C}^*\)-action on the total space of the test configuration.

Remark 3.2

Note, that some of the K-unstable examples seem to be closely related to each other. Indeed, nos. 18, 28, 31 are (weighted) blowups of no. 33 and no. 13 is a quotient of 33 by \(\mathbb {Z}/2\mathbb {Z}\). Surface no. 16 lies at the boundary of the family of del Pezzo surfaces of type 18.

4 New Kähler–Ricci solitons on Fano threefolds

In this section we consider Fano threefolds admitting an effective 2-torus action within the classification of [12]. 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. We determine a small region in Open image in new window which has to contain the soliton candidate \(\xi \). Then we use these bounds for \(\xi \) to show positivity of the Donaldson–Futaki invariants Open image in new window for every admissible choice of y. However, now \(\xi \) is two-dimensional and we cannot use the intermediate value theorem directly to bound the solution for \(\xi \). In some cases we can make use of additional symmetries to reduce to a one-dimensional problem. Here, the key observation is that given an automorphism Open image in new window permuting the vertices of Open image in new window such that Open image in new window , by (5) we have Open image in new window . Since Open image in new window is the unique solution to \(F_{X,\xi } = 0\), this gives Open image in new window . Now we show how to utilise this observation.

Example 4.1

(2.30—Blowup of quadric threefold in a point) Consider the threefold 2.30. The combinatorial data for this threefold was given in [18], although the piecewise affine \({\Psi }\) discussed there is \(\overline{\Phi }\) as denoted in Sect. 2, with Open image in new window . The function \(\Phi \) is given in Fig. 2.
Fig. 2

The combinatorial data for threefold 2.30

We now find the unique candidate vector field Open image in new window for a K-stable pair \((X,\xi )\). We see that \(\deg \Phi \) is symmetric with respect to reflection \(\sigma \) along the vertical axis. Hence, we have \(\xi = \xi _2e_2\) for some \(\xi _2 \in \mathbb {R}\) and must find a solution \(\xi _2\) to Open image in new window , which is equivalent to \(F_{X,\,\xi _2e_2}(e_2)=0\). Indeed, we haveHence, \(F_{X,\,\xi _2e_2}(e_1)=0\) and the claim follows by linearity.
By (4) the vanishing of \(F_{X,\,\xi _2e_2}(e_2)\) is equivalent to that ofwhere the integral on the right hand side can be solved analytically. We obtainEvaluating the exponential functions with a precision of 16 binary digits and using elementary estimations it can be shown that \(g(0.514) <0\) and \(g(0.515)>0\). By the intermediate value theorem then \(0.514< \xi _2 < 0.515\). It remains to check the positivity of the Donaldson–Futaki invariant for each degeneration. The degenerations of this threefold correspond to the polytopes:In each case we have induced \(\mathbb {C}^*\)-action given by Open image in new window . DenoteClearly positivity of \(h_y\) implies the positivity of Open image in new window . Once more solving the integrals appearing in (4) analytically with Open image in new window we obtain:
Using the same precision as above for the evaluations of the exponential functions at the lower and upper bounds for \(\xi _2\) gives estimates:We can therefore conclude that the threefold 2.30 is K-stable, and must admit a non-trivial Kähler–Ricci soliton.

Proof of Theorem 1.8

Looking at the combinatorial description for the threefolds given in [18] one sees that for the cases no. 2.31, 3.18, 3.22, 3.24 and 4.8, \(\deg \Phi \) admits a (non-trivial) involution Open image in new window . Then \(\xi \) has to be contained in the line Open image in new window and after choosing a basis \(e_1, e_2\) of Open image in new window with Open image in new window and \(\sigma ^*(e_2)=e_2\) we may proceed as in Example 4.1 with the following steps:
  1. (i)

    Find a closed form for \(F_{X,\xi }(e_2)\).

     
  2. (ii)

    Find sufficiently good bounds for \(\xi _2\) with \(F_{X,\,\xi _2e_2}(e_2)=0\) via the intermediate value theorem.

     
  3. (iii)

    For every admissible choice of \(y \in \mathbb {P}^1\) find a closed form for Open image in new window .

     
  4. (iv)

    Use elementary estimations to ensure positivity of Open image in new window for all values of \(\xi _2\) within the error bounds.

     
For the case of threefold no. 3.23 there is no involution fixing \(\deg \Phi \). In this case we take a more general approach to bound the value of the candidate \(\xi \). Here we make use of some elementary calculus. Note, that \(\xi \) is the unique solution to the equation \(\nabla G = 0 \), whereNow we identify a small closed rectangular region \(D \subset \mathbb {R}^2 \) such that \(\nabla _n G > 0 \) holds along \(\partial D\), for n being an outer normal of the rectangle D. This guarantees a solution to \(\nabla G = 0 \) in the interior of D. Indeed, due to compactness, D has to contain a minimum of G, which by our condition cannot lie on the boundary. Hence, the minimum is located in the interior and has to coincide with \(\xi \), since \(\nabla G\) necessarily vanishes. After bounding the value of \(\xi \) we proceed with step (iii).

However, for showing positivity of \(\nabla _n G\) along \(\partial D\) we have to use computer assistance. The approach is simple but computationally intensive. First we again determine a closed form for \(\nabla G_n(\xi )\) which coincides with \(F_{X,\xi }(n)\) up to a positive constant. Then we subdivide the faces of the boundary in sufficiently small segments, where one of coordinates is fixed and the other varies in a small interval. Using interval arithmetic when evaluating the closed form for \(\nabla _n G(\xi )\) provides the positivity result. See also Example 4.2 for details of the computation and Appendix 5.3 for the implementation in SageMath.

The complete calculations are done using SageMath and can be found in the ancillary files [4] and as an online worksheet.2 \(\square \)

Example 4.2

(3.23—Blowup of the quadric in a point and a line passing through) We follow the calculations outlined in the above proof of Theorem 1.8. As before we first have to find a closed form for Open image in new window or \(\nabla _n G(\xi )\), respectively. Then numerically we can find an approximation to \(\xi \) as the point:Setting \(\epsilon = 10^{-5}\), consider the square containing our approximation, given by:Subdividing each edge of the boundary \(\partial D\) into line segments of length \(\epsilon /1500\), we use interval arithmetic to verify that the gradient of h is positive in the outer normal direction for each of these segments, in fact Open image in new window along \(\partial D\). Once again it remains to check the positivity of the Donaldson–Futaki invariant for each degeneration. The degenerations of this threefold correspond to the polytopes:for Open image in new window . Interval arithmetic gives the following lower bounds on the Donaldson–Futaki invariants:
Table 3

Fano threefolds and their soliton vector fields in the canonical coordinates coming with the representation of the combinatorial data in [18]

Threefold

\(\xi \)

Q

(0, 0)

\({2}.{24}^{\star }\)

(0, 0)

2.29

(0, 0)

2.30

(0, 0.51489)

2.31

(0.28550, 0.28550)

2.32

(0, 0)

\(3.8^{\star }\)

\((0,-0.76905)\)

\({3}.{10}^{\star }\)

(0, 0)

3.18

(0, 0.37970)

3.19

(0, 0)

3.20

(0, 0)

3.21

\((-0.69622,-0.69622)\)

3.22

(0, 0.91479)

3.23

(0.26618, 0.67164)

3.24

(0, 0.43475)

4.4

(0, 0)

\(4.5^{\star }\)

\((-0.31043,-0.31043)\)

4.7

(0, 0)

4.8

(0, 0.62431)

\(^{\star }\)This refers only to a particular element of the family admitting a 2-torus action

We can therefore conclude that the threefold 3.23 is K-stable, and must admit a non-trivial Kähler–Ricci soliton. See also Appendix 5.3 for the SageMath code of the calculations.

Remark 4.3

Note, that by Theorem 1.8 and [11, Theorems 6.1, 6.2] all known smooth Fano threefolds with complexity-one torus action admit a Kähler–Ricci soliton.

In the Table 3 we give the estimates found for the vector field \(\xi \) for each threefold in the list of [18]. The threefolds \(3.8^{\star }\), 3.21, \(4.5^{\star }\) were shown to admit a non-trivial Kähler–Ricci soliton in [11]. Applying steps (i)–(ii) from the proof of Theorem 1.8 provides also an approximation for the vector field \(\xi \) for these threefolds. These are included in the table, together with those threefolds shown in [11] to be Kähler–Einstein, to show the complete picture for the Fano threefolds described in [18]. We can show that our approximations are correct to the nearest \(10^{-5}\).

5 Appendix: SageMath code for examples

5.1 No. 13: Degree 3/singularity type \(A_5A_1\)

The combinatorial data is given by \(\Box =[-1,3]\) and Open image in new window , Open image in new window , Open image in new window .

Step (i)—obtain a closed form for \(F_{X,\xi }(1)\)

For this we have to analytically solve the integral Open image in new window .

Step (ii)—find an estimate for the soliton candidate vector field \(\xi \)

Steps (iii) and (iv)—obtain closed forms for  Open image in new window and plug in \(\xi \)

First have to symbolically solve the integrals(which equal Open image in new window up to scaling by a positive constant) for every admissible choice of \(y \in \mathbb P^1\) and then plug in the estimate for \(\xi \) into the resulting expression.

5.2 Example 4.1 (No. 2.30)

The combinatorial data is given by Open image in new window and Open image in new window , Open image in new window , Open image in new window .

Step (i)—obtain a closed form for \(F_{X,\xi }\)

For this we have to analytically solve the integral Open image in new window for v varying over a basis of Open image in new window .

Step (ii)—find an estimate for the soliton candidate vector field \(\xi \)

Steps (iii) and (iv)—obtain closed forms for  Open image in new window  and plug in \(\xi \)

For this we have to symbolically solve the integrals Open image in new window for every (admissible) choice of \(y \in \mathbb P^1\) and then plug in the estimate for \(\xi \) into the resulting expression.

5.3 Example 4.2 (No. 3.23)

The combinatorial data is given by Open image in new window Open image in new window and Open image in new window , Open image in new window .

Step (i)—obtain a closed form for \(F_{X,\xi }\)

For this we have to analytically solve the integral Open image in new window for v varying over a basis of Open image in new window .

Step (ii)—find an estimate for the soliton candidate vector field \(\xi \)

We identify a small closed rectangle containing our estimate such that \( \nabla _n G > 0 \) for any outer normal of this rectangle, where Open image in new window . This and uniqueness guarantee our candidate lies within the rectangle.

Steps (iii) and (iv)—obtain closed forms for  Open image in new window  and plug in \(\xi \)

For this we have to symbolically solve the integrals Open image in new window for every choice of \(y \in \mathbb P^1\) and then plug in the estimate for \(\xi \) into the resulting expression.

Footnotes

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Authors and Affiliations

  1. 1.School of Mathematics, Faculty of Science and EngineeringThe University of ManchesterManchesterUK

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