On a conjecture of Tian

Mathematische Zeitschrift, May 2017

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 three-dimensional 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 \(S_5\).

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On a conjecture of Tian

Hamid Ahmadinezhad 0 1 2 Ivan Cheltsov 0 1 2 Josef Schicho 0 1 2 0 RISC, Johannes Kepler University , Linz, Schloss Hagenberg, 4232 Hagenberg , Austria 1 Department of Mathematics, University of Edinburgh , Mayfield Rd., Edinburgh EH9 3JZ , UK 2 Department of Mathematical Sciences, Loughborough University , Loughborough LE11 3TU , UK We study Tian's α-invariant in comparison with the α1-invariant for pairs (Sd , H ) consisting of a smooth surface Sd of degree d in the projective three-dimensional space and a hyperplane section H . A conjecture of Tian asserts that α(Sd , H ) = α1(Sd , H ). We show that this is indeed true for d = 4 (the result is well known for d 3), and we show that α(Sd , H ) < α1(Sd , H ) for d 8 provided that Sd is general enough. We also construct examples of Sd , for d = 6 and d = 7, for which Tian's conjecture fails. We provide a candidate counterexample for S5. Log canonical threshold; α-Invariant of Tian; Smooth surface 1 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 α-invariant, that measures how singular pluri-anticanonical divisors on the Fano variety can be. There, he “A tragedy of mathematics is a beautiful conjecture ruined by an ugly fact.” B Ivan Cheltsov proved that a smooth Fano variety of dimension m admits a Kähler–Einstein metric provided m . that its α-invariant is bigger that 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. In fact, the α-invariant turned out to have important applications in birational geometry as well; see for example [1]. Later, Tian generalised this invariant for arbitrary polarised pairs (X, L), where X is a smooth variety and L is an ample Cartier divisor on it. For the pair (X, L), it can be defined as = sup for every effective Q-divisor D ∼Q L ∈ R>0. This definition coincides with Tian’s original definition in [12,13] by [6, Theorem A.3]. The number α(X, L) is often hard to compute but, in good situations, can be approximated by numbers that are much easier to control (see, for example, [5, Proposition 2.2]). For instance, if the linear system |n L| is not empty, Tian defined the n-th α-invariant of the pair (X, L) as is log canonical for every D ∈ |n L| ∈ Q>0. If the linear system |n L| is empty, one can simply put αn (X, L) = +∞. Then α(X, L) αn (X, L) and = ninf1 Then, Tian posed the following conjecture. Conjecture 1.1 ([13, Conjecture 5.4]) Suppose that L is very ample and defines a projectively normal embedding under its associated morphism, i.e., the graded algebra is generated by elements in H 0(X, OX (L)). Then α(X, L) = α1(X, L). Note that the very ampleness of the divisor L does not always imply that the associated morphism gives a projectively normal embedding. However, in many cases this is true, for example when X is a hypersurface and L is a hyperplane section, which includes all varieties we study in this article. Note also that [13, Conjecture 5.4] is stated in terms of the more delicate invariants αn,k (X, L), which are defined in analytic language (for their explicit definitions see [13, § 5]). Arguing as in the proof of [6, Theorem A.3], one can show that so that Conjecture 1.1 is a special case of Tian’s more general [13, Conjecture 5.4]. The purpose of this paper is to study Conjecture 1.1 for smooth surfaces in P3. Namely, let Sd be a smooth surface in P3 of degree d 1, and let H be its hyperplane section. Then the pair (Sd , H ) satisfies all hypotheses of Conjecture 1.1. Moreover, if d = 1 or d = 2, then Indeed, in these cases Sd is toric, so that the required equalities follows from [6, Lemma 5.1]. Furthermore, if d = 3, then α(Sd , H ) = α1(Sd , H ) by [2, Theorem 1.7]. In Sect. 4, we prove Hence, Conjecture 1.1 holds for the pair (Sd , H ) provided that d 4. In particular, this gives an easy way to compute all possible values of α(Sd , H ) for d = 4, because the number α1(Sd , H ) is easy to compute. However, Conjecture 1.1 fails for general surfaces of large degree in P3. This follows from This result shows that it is hard to compute α(Sd , H ) for d 0. In fact, we do not know what the exact value of α(Sd , H ) is when d 5 and the surface Sd is general. One the other hand, we prove that α1(Sd , H ) = 43 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 P3. 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 dimens (...truncated)


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Hamid Ahmadinezhad, Ivan Cheltsov, Josef Schicho. On a conjecture of Tian, Mathematische Zeitschrift, 2017, pp. 1-25, DOI: 10.1007/s00209-017-1886-z