On the classification of Kähler–Ricci solitons on Gorenstein del Pezzo surfaces
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Abstract
We give a classification of all pairs \((X,\xi )\) of Gorenstein del Pezzo surfaces X and vector fields \(\xi \) which are Kstable 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
KStability Kähler–Ricci solitons TVariety Torus action Fano varietyMathematics Subject Classification
32Q20 14L30 14J451 Introduction
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 algebrogeometric stability condition, known as Kstability. The key objects involved in defining Kstability are test configurations:
Definition 1.1

the \(\mathbb {C}^*\)action \(\lambda \) on Open image in new window lifts the standard \(\mathbb {C}^*\)action on Open image in new window ;

the general fiber is isomorphic to X, with Open image in new window restricting to L.
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].
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 oneparameter subgroups of T. Let Open image in new window be the central fiber of a special Tequivariant 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 oneparameter subgroup corresponding to the \(\mathbb {C}^*\)action of the test configuration. Note that the inclusion \(T\hookrightarrow T'\) induces an inclusion of oneparameter subgroups \(N\hookrightarrow N'\).
Definition 1.3
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 Kstable if Open image in new window for every Gequivariant 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 Kstable.
The following result by Datar and Székelyhidi motivates the study of equivariant Kstability:
Theorem 1.5
([6]) For a smooth Fano Gvariety X, the variety X admits a Kähler–Ricci soliton with respect to \(\xi \) if and only if the pair \((X,\xi )\) is equivariantly Kstable.
 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 \(4A_1\).
Theorem 1.6
 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 Kstability 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 nontrivial vector fields. Hence, by using a classification of such surfaces from [9] we obtain the following corollary.
Corollary 1.7
 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 2torus action were considered and the paper [11] determined which of them admit a Kähler–Einstein metric via equivariant Kstability. 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 nontrivial 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 Tvarieties of complexity 1. Note, that surfaces with nontrivial 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 Tvarieties 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 Kstable 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 Tvarieties 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 cocharacters or oneparameter subgroups by N. The corresponding vector spaces are denoted by Open image in new window and Open image in new window , respectively.
 (i)
\(\Phi \) is piecewise affine linear, i.e. given by affine linear functions on a finite polyhedral subdivision of \(\Box \).
 (ii)
For \(y\in \mathbb {P}^1\), the graph of \(\Phi _y\) has integral vertices.
 (iii)
For every u in the interior of \(\Box \), Open image in new window .
 (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.
 (v)
Every facet F of \(\Box \) with Open image in new window has lattice distance 1 from the origin.
Indeed, the function \(\overline{\Phi }\) 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.
Example 2.1
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 selfintersection number of the anticanonical divisor. It can be calculated from the combinatorial data by the following
Theorem 2.2
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.
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 .
Remark 2.5
As a consequence of the above classification of nonproduct Tequivariant special test configurations we see that if there are at least three points \(y \in \mathbb {P}^1\) with \(\Phi _y(0)\) being nonintegral 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 Kstability for the soliton candidate for free.
Example 2.6
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 Kstability 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 nontrivial \(\mathbb {C}^*\)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 1parameter family of del Pezzo surfaces parametrised by Open image in new window .
Gorenstein del Pezzo surfaces with \(\mathbb {C}^*\)action
No.  Kstab  \(\xi \)  \(\Box \)  \(\Phi _{0},\Phi _{\infty },\Phi _1, (\Phi _{c})\)  \((K_X)^2\)  Sing.  \(\rho \) 

1  \({\varvec{\surd }}\)  0  \({[1,1]}\)  1  \({2}{D}_{{4}}\)  1  
2  \(\surd \)  \(1.99761\)  \([1,5]\)  1  \(E_8\)  1  
3  \(\surd \)  \( 1.94024\)  \([1,3]\)  1  \(E_7A_1\)  1  
4  \(\surd \)  \( 1.69131\)  \([1,2]\)  1  \(E_6A_2\)  1  
5  \({\varvec{\surd }}\)  0  \({[1,1]}\)  2  \({2}{A}_{3}{A}_{1}\)  1  
6  \(\surd \)  \( 0.97052\)  \([1,2]\)  2  \(A_5A_2\)  1  
7  \(\surd \)  \( 1.79675\)  \([1,3]\)  2  \(D_6A_1\)  1  
8  \(\surd \)  \( 1.99186\)  \([1,5]\)  2  \(E_7\)  1  
9  \(\surd \)  \( 1.94024\)  \([1,3]\)  2  \(E_6\)  2  
10  \({\varvec{\surd }}\)  0  \({[1,1]}\)  2  \({2}{A}_{3}\)  2  
11  \(\surd \)  \( 1.69131\)  \([1,2]\)  2  \(D_5A_1\)  2  
12  \(\surd \)  \( 1.34399\)  \([1,1]\)  2  \(D_43A_1\)  1  
13  \({\times }\)  \( 1.24607\)  \([1,3]\)  3  \(A_5A_1\)  1  
14  \(\surd \)  \( 1.96766\)  \([1,5]\)  3  \(E_6\)  1  
15  \(\surd \)  \( 1.19618\)  \([1,2]\)  3  \(A_4A_1\)  2  
16  \({\varvec{{\times }}}\)  0  \({[1,1]}\)  3  \({2}{A}_{2}{A}_{1}\)  2  
17  \(\surd \)  \( 1.83879\)  \([1,3]\)  3  \(D_5\)  2  
18  \({\varvec{{\times }}}\)  0  \({[1,1]}\)  3  \({2}{A}_{2}\)  3  
19  \(\surd \)  \( 1.69131\)  \([1,2]\)  3  \(D_4\)  3  
20  \(\surd \)  \( 0.94468\)  \([1,1]\)  3  \(A_32A_1\)  2  
21  \(\surd \)  \( 1.85969\)  \([1,5]\)  4  \(D_5\)  1  
22  \({\times }\)  \( 0.97052\)  \([1,2]\)  4  \(A_3A_1\)  2  
23  \(\surd \)  \( 1.79675\)  \([1,3]\)  4  \(D_4\)  2  
24  \(\surd \)  \( 1.38176\)  \([1,3]\)  4  \(A_4\)  2  
25  \({\varvec{{\times }}}\)  0  \({[1,1]}\)  4  \({3}{A}_{1}\)  3  
26  \(\surd \)  \( 1.31047\)  \([1,2]\)  4  \(A_3\)  3  
27  \({\varvec{\surd }}\)  0  \({[1,1]}\)  4  \({2}{A}_{1}\)  4  
28  \({\times }\)  \( 0.74373\)  \([1,1]\)  4  \(A_2A_1\)  3  
29  \(\surd \)  \( 1.42059\)  \([1,5]\)  5  \(A_4\)  1  
30  \(\surd \)  \( 1.43886\)  \([1,3]\)  5  \(A_3\)  2  
31  \({\times }\)  \( 1.10613\)  \([1,2]\)  5  \(A_2\)  3  
32  \(\surd \)  \( 0.61790\)  \([1,1]\)  5  \(A_1\)  4  
33  \({\times }\)  \( 1.24607\)  \([1,3]\)  6  \(A_2\)  2  
34  \(\surd \)  \( 0.97052\)  \([1,2]\)  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 Kstable, 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.
 (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.
 (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.
 (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.
 (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.
Destabilising test configurations
No.  Destabilising degeneration  Amb. space  Weights  Special fiber 

13  \(\mathbb {P}(1,1,1,1,2)\)  \((0,0,0,1,1)\)   
22  \(\mathbb {P}^4\)  \((0,0,0,1,1)\)   
28  \(\mathbb {P}^4\)  \((1,1,0,1,0)\)   
31  \(\mathbb {P}^5\)  \((0,1,1,0,0,1)\)   
33  \(\mathbb {P}^6\)  \((1,0,1,0,1,1,1)\)  
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 Kstable 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 nontrivial candidate vector fields \(\xi \) we obtain a Kstable 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 Kstable 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 Kunstable 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 2torus 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 nontrivial 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 twodimensional 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 onedimensional 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
Proof of Theorem 1.8
 (i)
Find a closed form for \(F_{X,\xi }(e_2)\).
 (ii)
Find sufficiently good bounds for \(\xi _2\) with \(F_{X,\,\xi _2e_2}(e_2)=0\) via the intermediate value theorem.
 (iii)
For every admissible choice of \(y \in \mathbb {P}^1\) find a closed form for Open image in new window .
 (iv)
Use elementary estimations to ensure positivity of Open image in new window for all values of \(\xi _2\) within the error bounds.
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
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) 
We can therefore conclude that the threefold 3.23 is Kstable, and must admit a nontrivial 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 complexityone 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 nontrivial 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 \(\xi \)
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
References
 1.Altmann, K., Ilten, N.O., Petersen, L., Süß, H., Vollmert, R.: The geometry of \(T\)varieties. In: Pragacz, P. (ed.) Contributions to Algebraic Geometry. EMS Series of Congress Reports, pp. 17–69. European Mathematical Society, Zürich (2012)CrossRefGoogle Scholar
 2.Barvinok, A.I.: Exponential sums and integrals over convex polytopes. Funct. Anal. Appl. 26(2), 127–129 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
 3.Berman, R.J., Witt Nyström, D.: Complex optimal transport and the pluripotential theory of Kähler–Ricci solitons (2014). arXiv:1401.8264
 4.Cable, J., Süß, H.: On the classification of Kähler–Ricci solitons on Gorenstein del Pezzo surfaces: SageMath worksheets and scripts (2017). https://doi.org/10.5281/zenodo.572021
 5.Donaldson, S.K.: Kähler geometry on toric manifolds, and some other manifolds with large symmetry. In: Lizhen, J., et al. (eds.) Handbook of Geometric Analysis, No. 1. Advanced Lectures in Mathematics, vol. 7, pp. 29–75. International Press, Somerville (2008)Google Scholar
 6.Datar, V., Székelyhidi, G.: Kähler–Einstein metrics along the smooth continuity method. Geom. Funct. Anal. 26(4), 975–1010 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
 7.Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York (1977)CrossRefGoogle Scholar
 8.Hausen, J.: Cox rings and combinatorics. II. Moscow Math. J. 8(4), 711–757 (2008)MathSciNetzbMATHGoogle Scholar
 9.Huggenberger, E.: Fano varieties with torus action of complexity one. PhD thesis, Eberhard Karls Universität Tübingen (2013). http://nbnresolving.de/urn:nbn:de:bsz:21opus69570
 10.Ilten, N., Mishna, M., Trainor, C.: Classifying Fano complexityone \(T\)varieties via divisorial polytopes (2017). arXiv:1710.04146
 11.Ilten, N., Süß, H.: Kstability for Fano manifolds with torus action of complexity 1. Duke Math. J. 166(1), 177–204 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
 12.Mori, S., Mukai, S.: Classification of Fano 3folds with \(B_{2}\geqslant 2\). Manuscripta Math. 36(2), 147–162 (1981/1982)Google Scholar
 13.Odaka, Y., Spotti, C., Sun, S.: Compact moduli spaces of del Pezzo surfaces and Kähler–Einstein metrics. J. Differential Geom. 102(1), 127–172 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
 14.Petersen, L., Süß, H.: Torus invariant divisors. Israel J. Math. 182, 481–504 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
 15.Revol, N., Rouillier, F.: MPFI a multiple precision interval arithmetic library (version 1.5) (2012). https://perso.enslyon.fr/nathalie.revol/software.html
 16.Shi, Y., Zhu, X.: Kähler–Ricci solitons on toric fano orbifolds. Math. Z. 271(3–4), 1241–1251 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
 17.Stein, W.A., et al.: Sage Mathematics Software (Version 7.6). The Sage Development Team, Newcastle upon Tyne (2017). http://www.sagemath.org
 18.Süß, H.: Fano threefolds with 2torus action—a picture book. Documenta Math. 19, 905–940 (2014)MathSciNetzbMATHGoogle Scholar
 19.Tian, G., Zhu, X.: A new holomorphic invariant and uniqueness of Kähler–Ricci solitons. Comment. Math. Helv. 77(2), 297–325 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
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