On a conjecture of Tian
Abstract
We study Tian’s \(\alpha \)invariant in comparison with the \(\alpha _1\)invariant for pairs \((S_d,H)\) consisting of a smooth surface \(S_d\) of degree d in the projective threedimensional space and a hyperplane section H. A conjecture of Tian asserts that \(\alpha (S_d,H)=\alpha _1(S_d,H)\). We show that this is indeed true for \(d=4\) (the result is well known for \(d\leqslant 3\)), and we show that \(\alpha (S_d,H)<\alpha _1(S_d,H)\) for \(d\geqslant 8\) provided that \(S_d\) is general enough. We also construct examples of \(S_d\), for \(d=6\) and \(d=7\), for which Tian’s conjecture fails. We provide a candidate counterexample for \(S_5\).
Keywords
Log canonical threshold \(\alpha \)Invariant of Tian Smooth surfaceMathematics Subject Classification
Primary 14J25 14J70 Secondary 32Q201 Introduction
In order to prove the existence of a Kähler–Einstein metric, known as the Calabi problem, on a smooth Fano variety, in [12] Gang Tian introduced a quantity, known as the \(\alpha \)invariant, that measures how singular plurianticanonical divisors on the Fano variety can be. There, he proved that a smooth Fano variety of dimension m admits a Kähler–Einstein metric provided that its \(\alpha \)invariant is bigger that \(\frac{m}{m+1}\).
Despite the fact that the Calabi problem for smooth Fano varieties has been solved (see [7, 9, 11, 14]) this result of Tian is often the only way to prove the existence of the Kähler–Einstein metric for a given Fano.
Conjecture 1.1
Theorem 1.2
Let \(S_4\) be a smooth quartic surface in \({\mathbb {P}}^3\). Then \(\alpha (S_4,H)=\alpha _1(S_4,H)\).
Hence, Conjecture 1.1 holds for the pair \((S_d,H)\) provided that \(d\leqslant 4\). In particular, this gives an easy way to compute all possible values of \(\alpha (S_d,H)\) for \(d=4\), because the number \(\alpha _1(S_d,H)\) is easy to compute. However, Conjecture 1.1 fails for general surfaces of large degree in \({\mathbb {P}}^3\). This follows from
Theorem 1.3
Let \(S_d\) be a general surface in \({\mathbb {P}}^3\) of degree \(d\geqslant 8\). Then \(\alpha (S_d,H)<\alpha _1(S_d,H)\).
This result shows that it is hard to compute \(\alpha (S_d,H)\) for \(d\gg 0\). In fact, we do not know what the exact value of \(\alpha (S_d,H)\) is when \(d\geqslant 5\) and the surface \(S_d\) is general. One the other hand, we prove that \(\alpha _1(S_d,H)=\frac{3}{4}\) for these hypersurfaces (see Lemmas 3.1 and 3.2).
We prove Theorem 1.3 in Sect. 5. In Sect. 6, we show that Conjecture 1.1 also fails for some smooth sextic and septic surfaces in \({\mathbb {P}}^3\). We believe that it fails for some smooth quintic surfaces as well. Unfortunately, we are unable to verify this claim at this stage, due to enormous computations required in our method (see Remark 6.4).
By [2, Theorem 1.7], Conjecture 1.1 holds for all smooth del Pezzo surfaces, i.e. smooth Fano varieties of dimension two, polarized by their anticanonical divisors. Surfaces considered in Theorems 1.2 and 1.3 have nonnegative Kodaira dimension, so that, in particular, they are not del Pezzo surfaces. Unfortunately, we do not know whether Conjecture 1.1 holds for smooth del Pezzo surfaces polarised by arbitrary ample divisors. Thus, we conclude by posing
Question 1.4
All varieties are assumed to be algebraic, projective and defined over \({\mathbb {C}}\).
2 Singularities of pairs
In this section we present local results about effective \({\mathbb {Q}}\)divisors on smooth surfaces. Almost all these results can be found in [10, § 6] in much more general forms.
Let S be a smooth surface, let D be an effective nonzero \({\mathbb {Q}}\)divisor on the surface S, and let P be a point in the surface S. Put \(D=\sum _{i=1}^{r}a_iC_i\), where each \(C_i\) is an irreducible curve on S, and each \(a_i\) is a nonnegative rational number. We assume here that all curves \(C_1,\ldots ,C_r\) are different. We call (S, D) a log pair.
Definition 2.1

\(a_i\leqslant 1\) for every \(C_i\) such that \(P\in C_i\),

\(b_j\leqslant 1\) for every \(F_j\) such that \(\pi (F_j)=P\).
This definition is independent on the choice of birational morphism \(\pi :\widetilde{S}\rightarrow S\) provided that the surface \(\widetilde{S}\) is smooth and \(\sum _{i=1}^r\widetilde{C}_i+\sum _{j=1}^n F_j\) is a divisor with simple normal crossings. The log pair (S, D) is said to be log canonical if it is log canonical at every point of S.
Remark 2.2
The following result is wellknown and is very easy to prove.
Lemma 2.3
([10, Exercise 6.18]) If (S, D) is not log canonical at P, then \({\mathrm {mult}}_{P}(D)>1\).
Remark 2.4
The log pair (S, D) is log canonical at P if and only if \((S_1, D^1+({\mathrm {mult}}_{P}(D)1)E_1)\) is log canonical at every point of the curve \(E_1\).
Corollary 2.5
If \({\mathrm {mult}}_{P}(D)>2\), then (S, D) is not log canonical at P.
Theorem 2.7
([10, Exercise 6.31], [3, Theorem 7]) Suppose that \(r\geqslant 2\). Put \(\Delta =\sum _{i=2}^{r}a_iC_i\). Suppose that \(C_1\) is smooth at P, \(a_1\leqslant 1\), and the log pair (S, D) is not log canonical at P. Then \({\mathrm {mult}}_{P}(C_1\cdot \Delta )>1\).
This theorem implies
Lemma 2.8
Suppose that (S, D) is not log canonical at P, and \({\mathrm {mult}}_{P}(D)\leqslant 2\). Then there exists a unique point in \(E_1\) such that \((S_1, D^1+({\mathrm {mult}}_{P}(D)1)E_1)\) is not log canonical at it.
Proof
A crucial role in the proof of Theorems 1.2 is played by
Theorem 2.9
Recall that \(\pi \) is a composition of n blow ups of smooth points. We encourage the reader to prove both Theorems 2.7 and 2.9 using induction on n.
3 Smooth surfaces in \({\mathbb {P}}^3\)
In this section we collect global results about smooth surfaces in \({\mathbb {P}}^3\). These results will be used in the proof of Theorems 1.2 and 1.3.
Lemma 3.1
Suppose that \(d\geqslant 3\). Then \(\alpha _1(S_d,H)\leqslant \frac{3}{4}\).
Proof
We proved the required assertion in the case \(d=3\). Now let us prove it for \(d=4\). The proof is similar for higher degrees.
Therefore it follows that \({\mathcal {Z}}\) is irreducible and has dimension \(34+66=34\). In order to complete the proof, we need to show that the first projection is surjective. Since it is a projective map, the image \({\mathcal {W}}\subset {\mathcal {X}}\) is closed. We claim that there exists a point \(X\in W\) with finite fibre. Then the generic fibre is finite and \(\dim ({\mathcal {W}})=\dim ({\mathcal {Z}})=34\).
Arguing as in the proof of [5, Proposition 2.1], we get
Lemma 3.2
Suppose that \(S_d\) is a general surface in \({\mathbb {P}}^3\) of degree d. Then \(\alpha _1(S_d,H)\geqslant \frac{3}{4}\).
Proof
Similar as in the proof of Lemma 3.1, we define \({\mathcal {X}}\cong \mathbb {P}^{{d+3\atopwithdelims ()3}1}\), \({\mathcal {Y}}\) the variety of all complete flag varieties, and \({\mathcal {Z}}\subset {\mathcal {X}}\times {\mathcal {Y}}\) the incidence consisting of all pairs (X, Y), where \(Y=(P, L, E)\), such that \(X\cap E\) has an \({\mathbb {A}}_4\), or worse, singularity at P with tangent L. Now the fibers of the second projection have codimension 7 (defined by 6 linear and one quadratic equation). Since \(\dim ({\mathcal {Y}})=6\), it follows that \(\dim ({\mathcal {Z}})<\dim ({\mathcal {X}})\), hence the first projection cannot be surjective and the generic surface has no corresponding point in \({\mathcal {Z}}\). This shows that its hyperplane sections have only singularities of type \({\mathbb {A}}_1\), \({\mathbb {A}}_2\), and \({\mathbb {A}}_3\). \(\square \)
The following result is due to Pukhlikov.
Lemma 3.3
Let D be an effective \({\mathbb {Q}}\)divisor on \(S_d\) such that \(D\sim _{{\mathbb {Q}}} H\), and let P be a point in the surface \(S_d\). Put \(D=\sum _{i=1}^{r}a_iC_i\), where each \(C_i\) is an irreducible curve, and each \(a_i\) is a nonnegative rational number. Then each \(a_i\) does not exceed 1.
Proof
For an alternative proof of Pukhlikov’s lemma, see the proof of [10, Lemma 5.36].
4 Quartic surfaces
In this section, we prove Theorem 1.2. Let \(S_4\) be a smooth quartic surface in \({\mathbb {P}}^3\). Denote by H its hyperplane section. By definition, one has \(\alpha (S_4,H)\leqslant \alpha _1(S_4,H)\). We must show that \(\alpha (S_4,H)=\alpha _1(S_4,H)\). Suppose that \(\alpha (S_4,H)<\alpha _1(S_4,H)\). Let us seek for a contradiction.
Lemma 4.2
The curve \(T_P\) contains all lines in \(S_4\) that passes through P.
Proof
Lemma 4.4
Let L be a line in \(S_4\) that passes through P. Then L is contained in \({\mathrm {Supp}}(D)\).
Proof
Corollary 4.6
Suppose that \(m\leqslant \frac{2}{\lambda }\). Then the log pair (4.5) is log canonical at every point of the curve E that is different from Q.
Corollary 4.9
Suppose that \(m+\widetilde{m}\leqslant \frac{3}{\lambda }\). Then the log pair (4.8) is log canonical at every point of F that is different from O.
Lemma 4.11
Suppose that \(m\leqslant \frac{2}{\lambda }\), \(m+\widetilde{m}\leqslant \frac{3}{\lambda }\) and \(Q\not \in \widetilde{T}_P\). Then \(O=\overline{E}\cap F\).
Proof
Suppose \(O\ne \overline{E}\cap F\). Then the linear system \((f\circ g)^*(H)2F\overline{E}\) is a free pencil. Thus, it contains a unique curve that passes through the point O. Denote this curve by \(\overline{M}\), and denote its proper transform on \(S_4\) by M. Then M is a hyperplane section of the surface \(S_4\) and \(P\in M\). In particular, M is reduced by Lemma 3.3. Since \(Q\not \in \widetilde{T}_P\), we have \(M\ne T_P\), so that M is smooth at P. Thus, \(\overline{M}\) is the proper transform of the curve M on the surface \(\overline{S}_4\).
Since M is smooth at P, the log \(\hbox {pair}(S_4,\lambda M)\) is log canonical at P. Thus, it follows from Remark 2.2 that there exists an effective \(\mathbb {Q}\)divisor \(D^\prime \) on the surface \(S_4\) such that \(D^\prime \sim _{{\mathbb {Q}}} H\), the log pair \((S_4,\lambda D^\prime )\) is not log canonical at P, the support of the divisor \(D^\prime \) is contained in the support of the divisor D and does not contain at least one irreducible component of the curve M. Replacing D by \(D^\prime \), we may assume that D enjoys all these properties.
Lemma 4.14
If \(m\leqslant 2\), then \(m\leqslant \frac{2}{\lambda }\), \(m+\widetilde{m}\leqslant \frac{3}{\lambda }\) and \(O\ne \overline{E}\cap F\).
Proof
 (A)
\({{\mathrm{mult}}}_P(T_P)=4\), hence \(T_P\) consists of four lines that intersect at P.
 (B)\({{\mathrm{mult}}}_P(T_P)=3\) and \(T_P\)
 (B1)
consists of four lines and three of them intersect at P, or
 (B2)
it is an irreducible quartic with a singular point P of multiplicity 3, or
 (B3)
it consists of a conic and two lines, all intersecting at P, or
 (B4)
it consists of a cubic curve with a singular point P of multiplicity 2 and a line passing through P.
 (B1)
 (C)\({{\mathrm{mult}}}_P(T_P)=2\) and \(T_P\)
 (C1)
consists of four lines, two of which pass through P, or
 (C2)
it consist of a conic and two lines, and the two lines intersect at P and P does not lie on the conic, or
 (C3)
it consist of a conic and two lines and P is the intersection point of the conic with one of the lines, or
 (C4)
it consists of a cubic curve and a line and P is the intersection of the two at a smooth point of the cubic curve, or
 (C5)
it consists of a cubic curve and a line and P is singular point of the cubic curve with multiplicity 2 and does not lie on the line, or
 (C6)
it consists of two conics and they intersect at P, or
 (C7)
it is an irreducible quartic curve with a singular point P of multiplicity 2.
 (C1)
Lemma 4.15
We may assume that the support of the divisor D does not contain at least one irreducible component of the plane quartic curve \(T_P\).
Proof
Note that \((S_4,\lambda T_P)\) is log canonical at P, because \(\lambda <\alpha _1(S_4,H)\). Thus, it follows from Remark 2.2 that there exists an effective \(\mathbb {Q}\)divisor \(D^\prime \) on the surface \(S_4\) such that \(D^\prime \sim _{{\mathbb {Q}}} H\), the log pair \((S_4,\lambda D^\prime )\) is not log canonical at P, and the support of \(D^\prime \) does not contain at least one irreducible component of the curve \(T_P\). Replacing D by \(D^\prime \), we obtain the required assertion. \(\square \)
We denote by \(C_\star \) the irreducible component of the curve \(T_P\) that is not contained in the support of the divisor D. By Lemma 4.4, if \(P\in C_\star \), then \(C_\star \) is not a line. This gives
Corollary 4.16
The case (A) is impossible.
Lemma 4.18
The case (B1) is impossible.
Proof
By Lemma 4.4, the lines \(L_1\), \(L_2\), and \(L_3\) are contained in the support of D, and \(C_\star =L_4\). Hence, we put \(D=a_1L_1+a_2L_2+a_3L_3+\Omega \), where \(a_1\), \(a_2\), and \(a_3\) are positive rational numbers, and \(\Omega \) is an effective \({\mathbb {Q}}\)divisor whose support does not contain the lines \(L_1\), \(L_2\), \(L_3\), and \(L_4\). Put \(n={{\mathrm{mult}}}_P(\Omega )\). Then \(m=n+a_1+a_2+a_3\).
Lemma 4.19
The case (B2) is impossible.
Proof
Lemma 4.20
The case (B3) is impossible.
Proof
By Lemma 4.4, both lines \(L_1\) and \(L_2\) are contained in the support of the divisor D. Hence we can write \(D=a_1L_1+a_2L_2+\Omega \), where \(a_1\) and \(a_2\) are positive rational numbers, and \(\Omega \) is an effective \({\mathbb {Q}}\)divisor whose support does not contain the lines \(L_1\) and \(L_2\). Recall that the support of \(\Omega \) does not contain the curve \(C_\star \) by assumption. In our case, the curve \(C_\star \) is the conic \(C_1\).
Lemma 4.21
The case (B4) is impossible.
Proof
Lemma 4.22
The cases (C1) and (C2) are impossible.
Proof
Recall that \(m=n+a_1+a_1\). We see that \(m\leqslant \frac{5}{2}\), because \(a_1+a_2\leqslant 1\) and \(n\leqslant \frac{3}{2}\). In particular, \(\lambda m<\frac{15}{8}\), because \(\lambda <\frac{3}{4}\) by (4.1).
We see that \(Q\not \in \widetilde{L}_1\). Similarly, we see that \(Q\not \in \widetilde{L}_2\).
Lemma 4.23
The case (C3) is impossible.
Proof
Lemma 4.24
The case (C4) is impossible.
Proof
Note that \(a\leqslant 1\) by Lemma 3.3. This also follows from \(n+3a\leqslant 3\). We also know that \(a>0\). In fact, one can show that \(a>\frac{1}{6}\). Indeed, we have \(\lambda \big (1+2a\big )=\lambda \Omega \cdot L_1>1\) by Theorem 2.7. This gives \(a>\frac{1}{6}\), since \(\lambda >\frac{3}{4}\).
Denote by \(\widetilde{\Omega }\) the proper transform of the divisor \(\Omega \) on the surface \(\widetilde{\Omega }\). Similarly, denote by \(\widetilde{L}_1\) the proper transform of the line \(L_1\) on the surface \(\widetilde{\Omega }\). Then we can rewrite the log pair (4.5) as \((\widetilde{S}_4,\lambda a\widetilde{L}_1+\lambda \widetilde{\Omega }+(\lambda (a+n)1)E)\). Put \(\widetilde{n}={\mathrm {mult}}_{Q}(\widetilde{\Omega })\). Then \(\widetilde{n}\leqslant n\).
Recall that \(n2a\leqslant 1\) and \(n+3a\leqslant 3\). Adding these two inequalities together, we obtain \(m+\widetilde{m}=a+n+\widetilde{n}\leqslant a+2n\leqslant 4<\frac{3}{\lambda }\), since \(\lambda <\frac{3}{4}\). Thus, Corollary 4.9 implies that the log pair (4.8) is log canonical at every point of the curve F that is different from O. By Lemma 4.11, we have \(O=F\cap \overline{E}\), because \(m<\frac{2}{\lambda }\), \(m+\widetilde{m}<\frac{3}{\lambda }\) and \(Q\not \in \widetilde{L}_1\cup \in \widetilde{C}_1\).
Lemma 4.25
The case (C5) is impossible.
Proof
Denote by \(\widetilde{\Omega }\) the proper transform of the divisor \(\Omega \) on the surface \(\widetilde{\Omega }\). Similarly, denote by \(\widetilde{C}_1\) the proper transform of the curve \(L_1\) on the surface \(\widetilde{\Omega }\). Then we can rewrite the log pair (4.5) as \((\widetilde{S}_4,\lambda a\widetilde{C}_1+\lambda \widetilde{\Omega }+(\lambda (n+2a)1)E)\). Put \(\widetilde{n}={\mathrm {mult}}_{Q}(\widetilde{\Omega })\). Then \(\widetilde{n}\leqslant n\). If \(Q\not \in \widetilde{C}_1\), then \(\widetilde{m}=\widetilde{n}\). If \(Q\in \widetilde{C}_1\), then \(\widetilde{m}=\widetilde{n}+a\).
Lemma 4.26
The case (C6) is impossible.
Proof
Without loss of generality, we may assume that \(C_1=C_\star \). This gives \(2=C_1\cdot D\geqslant m\). Then \(m\leqslant \frac{2}{\lambda }\) and \(m+\widetilde{m}\leqslant \frac{3}{\lambda }\) by Lemma 4.14. Hence, Corollary 4.6 implies that the log pair (4.5) is log canonical at every point of the curve E that is different from Q. Moreover, Corollary 4.9 implies that the log pair (4.8) is log canonical at every point of the curve F that is different from O. Furthermore, Lemma 4.14 implies that \(O\ne \overline{E}\cap F\).
Lemma 4.27
The case (C7) is impossible.
Proof
By Corollary 4.16 and Lemmas 4.18, 4.19, B3, B4, 4.22, 4.23, 4.24, 4.25, 4.26, and 4.27, we obtain the desired contradiction. This completes the proof of Theorem 1.2.
5 General surfaces of large degree
In this section, we prove Theorem 1.3. By Lemmas 3.1 and 3.2, it follows from
Lemma 5.1
Let \(S_d\) be a smooth surface in \(\mathbb {P}^3\) of degree d, and let H be its hyperplane section. Then \(\alpha (S_d,H)\leqslant \frac{2}{\sqrt{d}}\).
Proof
Let us fix a positive integer n such that mn is an integer and \(f^*(nH)nmE\) is not empty. Pick a divisor \(\widetilde{M}\) in this linear system, so that \(\widetilde{M}\sim n\widetilde{H}nmE\). Denote by M the proper transform of the divisor \(\widetilde{M}\) on the surface \(S_d\). Put \(D=\frac{1}{n}M\). Then \({{\mathrm{mult}}}_P(D)\geqslant m\), so that \({\mathrm {lct}}_P(S_d,D)\leqslant \frac{2}{m}\) by (2.6). This gives \(\alpha (S_d, H)\leqslant \frac{2}{m}\), because \(D\sim _{{\mathbb {Q}}} H\). Since we can choose rational number \(m<\sqrt{d}\) as close to \(\sqrt{d}\) as we wish, we obtain \(\alpha (S_d,H)\leqslant \frac{2}{\sqrt{d}}\). \(\square \)
The idea of the proof of this lemma comes from [4, Example 1.26].
6 Quintic, sextic and septic
Lemma 6.1
Suppose that \(d\leqslant 7\). Then \(\alpha _1(X_d, H)>\frac{1}{2}\).
Proof
Let \(C\subset {\mathbb {P}}^3\) be the curve defined by the intersection of the surface \(S_d\) and the Hessian surface \({\mathrm {Hess}}(S_d)\) of \(S_d\). For the tangent hyperplane \(T_P\) at a point \(P\in S_d\), if the multiplicity of the curve \(T_P\cap S_d\) at the point P is at least 3, then the curve C is singular at the point P. Using the computer algebra system Magma, we checked that the curve C is smooth. Thus, the intersections of \(S_d\) with its tangent planes do not have points of multiplicity 3 or higher. The later implies that \(\alpha _1(S_d,H)>\frac{1}{2}\). Indeed, each singular hyperplane section of \(S_d\) is reduced by Lemma 3.3, so that each its singular point is of type \({\mathbb {A}}_n\). Then \(\alpha _1(S_d,H)=\frac{1}{2}+\frac{1}{m}\), where m is the greatest integer such that a hyperplane section of \(S_d\) has a singular point of type \({\mathbb {A}}_m\). \(\square \)
On the other hand, we have
Lemma 6.2
One has \(\alpha _2(S_d,H)\leqslant \frac{3}{d}\).
Proof
We may assume that \(d\geqslant 3\). Put \(P=[0:0:0:1]\). Let M be the divisor that is cut out on \(S_d\) by the equation \(xw+yz=0\). Locally at P, the divisor M is given by \((yz)^d=(yz)^d=0\), which implies that \({\mathrm {lct}}_P(S_4,M)=\frac{3}{2d}\). Since \(M\sim 2H\), we obtain \(\alpha _2(S_d,H)\leqslant \frac{3}{d}\). \(\square \)
Corollary 6.3
If \(d>5\), then \(\alpha (S_d,H)<\alpha _1(S_d,H)\).
Remark 6.4
We expect that \(\alpha (S_d,H)<\alpha _1(S_d,H)\) for \(d=5\) as well. By Lemma 6.1, this claim follows from \(\alpha _1(S_d,H)>\frac{3}{5}\). To check the latter inequality one would have to find out if the intersections of \(S_d\) with its tangent planes have a singularity of type \({\mathbb {A}}_9\) or worse. This can be expressed as a system of polynomial equations in 4 variables x, y, z, w.
Start with the equation of the quintic in variables x, y, z, w. Then intersect this with a symbolic plane \(w=ax+by+cz\), by substitution. This gives a polynomial in a, b, c, x, y, z. Now we compute the discriminant of this equation with respect to z, which results in a huge polynomial in a, b, c, x, y. Let us denote this polynomial by h. If there is an \({\mathbb {A}}_9\) singularity, or worse, then the discriminant, as a polynomial in x, y (when a, b, c are treated as as parameters), should have a zero of multiplicity 10 or higher. So the system of equations to consider consists of h and all its derivatives of order up to 10, as a system of polynomial equations in a, b, c, and x.
We used computer algebra to check whether or not this system has a solution, but the computations did not finish after 1500 CPU seconds on a Pentium Pro with 2.7 GHz. After reducing the system of equations modulo some small prime numbers (up to 293), the program finished with the answer that the reduced system has no solution. This can be interpreted as a strong evidence that \(\alpha (S_d,H)<\alpha _1(S_d,H)\) for \(d=5\).
Notes
Acknowledgements
The authors would like to thank Konstantin Shramov for fruitful discussions. A substantial part of this research was carried out when Ahmadinezhad and Cheltsov visited the Max Plank Institute for Mathematics in Bonn. We would like to thank MPIM for their hospitality and great research environment.
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