Delta invariants of smooth cubic surfaces
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Abstract
We prove that \(\delta \)invariants of smooth cubic surfaces are at least \(\frac{6}{5}\).
Keywords
Cubic surface Fano variety \(\delta \)Invariant Stability threshold Kstability Kähler–Einstein metricMathematics Subject Classification
14J45 14J26 32Q20All varieties are assumed to be projective and defined over \(\mathbb {C}\).
1 Introduction
The existence of Kähler–Einstein metrics on Fano manifolds is an important problem in complex geometry. By the Yau–Tian–Donaldson conjecture (confirmed in [4, 21]), we know that all Kstable Fano manifolds are Kähler–Einstein. Moreover, we also know explicit criteria that can be used to verify Kstability in many cases. One such criterion has been found by Tian in [19] and later generalized by Fujita in [10]. It is the following
Theorem 1.1
([10, 19]) Let X be a Fano manifold of dimension \(n\geqslant 2\). If \(\alpha (X)\geqslant \frac{n}{n+1}\), then X is Kstable.
Theorem 1.2
([20]) A smooth del Pezzo surface admits a Kähler–Einstein metric if and only if it is not a blowup of \(\mathbb {P}^2\) at one or two points.
Note that smooth cubic surfaces form the hardest case in Tian’s original proof of this result, which requires Cheeger–Gromov theory, Hörmander \(L^2\) estimates, partial \(C^0\) estimates and the lower semicontinuity of log canonical thresholds. In this paper, we will give another proof of Theorem 1.2 in this case using a new criterion for Kstability, which has been recently discovered by Fujita and Odaka in [12]. They stated it in terms of the socalled \(\delta \)invariant, which we describe now.
Theorem 1.3

X is Ksemistable if and only if \(\delta (X)\geqslant 1\);

X is uniformly Kstable if and only if \(\delta (X)>1\).
How to compute or at least estimate \(\delta (X)\) effectively? In general this is not very easy. In [17], Park and Won estimated the \(\delta \)invariants of all smooth del Pezzo surfaces, which gave another proof of Tian’s Theorem 1.2. But it seems unclear to us how to generalize their approach for higherdimensional Fano manifolds. Motivated by this, in our recent joint work with Yanir Rubinstein [7], we developed new geometric tools to estimate \(\delta \)invariants of (log) del Pezzo surfaces, which enabled us to partially prove a conjecture proposed in [6]. In this paper, we will use the same methods to give a sharper estimate for the \(\delta \)invaraints of smooth cubic surfaces. To be precise, we prove
Theorem 1.4
Let S be a smooth cubic surface in \(\mathbb {P}^3\). Then \(\delta (S)\geqslant \frac{6}{5}\).
Corollary 1.5
([17, 20]) All smooth cubic surfaces in \(\mathbb {P}^3\) are uniformly Kstable, so that they are Kähler–Einstein.
This paper is organized as follows. In Sect. 2, we present known results about divisors on smooth surfaces, and, as an illustration, we give a new proof of [17, Theorem 4.7]. In Sect. 3, we give various multiplicity estimates for basis type divisors on smooth cubic surfaces, which will be important to bound their \(\delta \)invariants in the proof of Theorem 1.4. These estimates also imply that \(\delta \)invariants of smooth cubic surfaces are at least \(\frac{18}{17}\). In Sect. 4, we prove Theorem 1.4.
2 Basic tools
Definition 2.1
In the following, let D be an effective \(\mathbb {R}\)divisor on S. We will investigate how to express the singularity of the log pair (S, D) at the point P in terms of Open image in new window and Open image in new window .
Lemma 2.2
([14]) If (S, D) is not log canonical at P, then \(\mathrm {mult}_P(D)>1\).
Lemma 2.3
Corollary 2.4
Corollary 2.5
The log pair (S, D) is log canonical at P if and only if the log pair \((\widetilde{S},\widetilde{D}+(\mathrm {mult}_P(D)1)E_1)\) is log canonical along the curve \(E_1\).
Thus, using Lemma 2.2 and Corollary 2.5, we obtain the following simple criterion.
Corollary 2.6
Corollary 2.7
Corollary 2.8
Theorem 2.9
Proof
This is a very special case of [12, Lemma 2.2].\(\square \)
Theorem 2.9 plays a crucial role in the proof of Theorem 1.4. As a warm up, let us show how to use Theorem 2.9 to estimate \(\delta \)invariants of smooth del Pezzo surfaces of degree 1.
Theorem 2.10
([17, Theorem 4.7]) Let S be a smooth del Pezzo surface of degree 1. Then \(\delta (S)\geqslant \frac{3}{2}\).
Proof
Remark 2.11
The following (simple) result can be very handy.
Lemma 2.12
Proof
The assertion follows from the fact that Open image in new window is a nonincreasing function on \(x\in [0,\tau (F)]\).\(\square \)
Using (2.1), this result can be improved as follows:
Lemma 2.13
Proof
The required assertion follows from the proof of [11, Proposition 2.1].\(\square \)
We will apply both Lemmas 2.12 and 2.13 to estimate the integral in Theorem 2.9 in the cases when it is not easy to compute.
3 Multiplicity estimates
Let S be a smooth cubic surface in \(\mathbb {P}^3\), and let D be a kbasis type divisor with \(k\gg 1\). The goal of this section is to bound multiplicities of the divisor D using Theorem 2.9. As in Theorem 2.9, we denote by \(\varepsilon _k\) a small number such that \(\varepsilon _k\rightarrow 0\) as \(k\rightarrow \infty \).
Lemma 3.1
Proof

\(T_P=L_1+L_2+L_3\), where \(L_1\), \(L_2\) and \(L_3\) are lines such that \(P=L_1\cap L_2\cap L_3\);

\(T_P=L_1+L_2+L_3\), where \(L_1\), \(L_2\) and \(L_3\) are lines such that \(L_3\not \ni P=L_1\cap L_2\);

\(T_P=L+C\), where L is a line and C is a conic such that \(P\in C\cap L\);

\(T_P\) is an irreducible cubic curve.
Lemma 3.2
Proof
Lemma 3.3
Proof
Lemma 3.4
Proof
Lemma 3.5
Proof
Lemma 3.6
Proof
Lemma 3.7
Proof
Now we consider the cases when Q is not contained in the proper transform of the singular curve \(T_P\) on the surface \(\widetilde{S}\). We start with
Lemma 3.8
Proof
Let Open image in new window and Open image in new window . Denote by \(T^\prime _Q\) the unique hyperplane section of the cubic surface \(S^\prime \) that is singular at \(Q^\prime \). If P is not an Eckardt point and Q is not contained in the proper transform of the curve \(T_P\), then Open image in new window . In this case, the number \(\tau (E_2)\) can be computed using \(T^\prime _Q\). This explains why the remaining cases are (slightly) more complicated.
Lemma 3.9
Proof
Recall that \(\nu :\widetilde{S}\rightarrow S^\prime \) is the contraction of the curve \(\widetilde{L}_3\). We let \(L_1^\prime =\nu (\widetilde{L}_1)\), \(L_2^\prime =\nu (\widetilde{L}_2)\) and \(E_1^\prime =\nu (E_1)\). Then \(L^\prime _1\), \(L^\prime _2\) and \(E_1^\prime \) are coplanar lines on \(S^\prime \).
Since Open image in new window , the line \(E_1^\prime \) is an irreducible component of the curve \(T^\prime _Q\). Thus, either \(T^\prime _Q\) consists of three lines, or \(T^\prime _Q\) is a union of the line \(E_1^\prime \) and an irreducible conic.
To complete the proof, we may assume that \(T^\prime _Q=E_1^\prime +M^\prime +N^\prime \), where \(M^\prime \) and \(N^\prime \) are two lines on \(S^\prime \) such that Open image in new window . Then Open image in new window , which implies that the lines \(M^\prime \) and \(N^\prime \) do not meet the lines \(L^\prime _1\) and \(L^\prime _2\). Denote by \(\widehat{M}\) and \(\widehat{N}\) the proper transforms on the surface \(\widehat{S}\) of the lines \(M^\prime \) and \(N^\prime \), respectively.
Lemma 3.10
Proof
Recall that \(\nu :\widetilde{S}\rightarrow S^\prime \) is the contraction of the curve \(\widetilde{C}\). Let Open image in new window and \(E_1^\prime =\nu (E_1)\). Then \(L^\prime \) is a line and \(E^\prime _1\) is a conic on \(S^\prime \) such that Open image in new window .
Lemma 3.11
Proof
Recall that \(\nu :\widetilde{S}\rightarrow S^\prime \) is the contraction of the curve \(\widetilde{C}\). Let Open image in new window . Then \(E_1^\prime \) is an irreducible cubic curve that is singular at \(P^\prime \). Thus, the curve \(E_1^\prime \) is smooth at the point \(Q^\prime \), so that \(T^\prime _Q\ne E_1^\prime \). One can easily check that \(T^\prime _Q\) does not contain \(P^\prime \).
Using Corollary 2.6 and Lemmas 3.2–3.11, we immediately get
Corollary 3.12
We have \(\delta (S)\geqslant \frac{18}{17}\).
4 Proof of the main result
In this section, we prove Theorem 1.4. Let S be a smooth cubic surface. We have to prove that \(\delta (S)\geqslant \frac{6}{5}\). Fix a positive rational number \(\lambda <\frac{6}{5}\). Let D be a kbasis type divisor. To prove Theorem 1.4, it is enough to show that, the log pair \((S,\lambda D)\) is log canonical for \(k\gg 1\). Suppose that this is not the case. Then there exists a point \(P\in S\) such that \((S,\lambda D)\) is not log canonical at P for \(k\gg 1\). Let us seek for a contradiction using results obtained in Sect. 3.
Let \(T_P\) be the hyperplane section of the surface S that is singular at P. Then \(T_P\) must be reducible. This follows from (4.1) and Lemmas 3.7 and 3.11.
Denote by Open image in new window the proper transform of the curve \(T_P\) on the surface \(\widetilde{S}\). Then Open image in new window . This follows from (4.1) and Lemmas 3.9 and 3.10.
 1.
\(T_P\) is a union of three lines passing through P;
 2.
\(T_P\) is a union of three lines and only two of them pass through P;
 3.
\(T_P\) is a union of a line and a conic that intersect transversally at P;
 4.
\(T_P\) is a union of a line and a conic that intersect tangentially at P.
4.1 Case 1
4.2 Case 2
4.3 Case 3
4.4 Case 4
The proof of Theorem 1.4 is complete.
Notes
Acknowledgements
The authors thank Yanir Rubinstein for many helpful discussions. This paper was finished during the authors’ visit to the Department of Mathematics at the University of Maryland, College Park. The authors appreciate its excellent environment and hospitality.
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