Canonical and log canonical thresholds of Fano complete intersections

It is proved that the global log canonical threshold of a Zariski general Fano complete intersection of index 1 and codimension k in PM+k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb P}^{M+k}$$\end{document} is equal to one, if M⩾2k+3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M\geqslant 2k+3$$\end{document} and the maximum of the degrees of defining equations is at least 8. This is an essential improvement of the previous results about log canonical thresholds of Fano complete intersections. As a corollary we obtain the existence of Kähler–Einstein metrics on generic Fano complete intersections described above.


Statement of the main result
The aim of the present paper is to show that the global (log) canonical threshold of a general Fano complete intersection of index 1 is at least (respectively, equal to) one, except for a sufficiently narrow class of Fano complete intersections, defined by equations of low degree. More precisely, let d = (d 1 , . . . , d k ) be an ordered integral vector, where k 1 (the value k is not fixed), 2 d 1 · · · d k , and B Aleksandr V. Pukhlikov lct(F) > M M + 1 implies the existence of the Kähler-Einstein metric on F (this fact was shown for arbitrary Fano varieties, not only for complete intersections in the projective space), in fact, the inequality above implies the K -stability of F, see the most recent paper [9] on this subject. Since the property of being canonical is stronger than that of being log canonical, the claim of Conjecture 1.1 implies the existence of the Kähler-Einstein on a general Fano complete intersection of index 1. This application alone is sufficient to justify the importance of Conjecture 1.1. For the applications to birational geometry, see Sect. 1.3. Now let us state the main result of the present paper. Let D be the set of ordered integral vectors d, such that 2 d 1 · · · d k , k 2. For an integer a 2 set D a = {d : d k a}.
Theorem 1.2 Assume that d ∈ D 8 and |d| 3k + 3. Then for a Zariski general variety F ∈ F(d) the inequality ct(F) 1 holds. Corollary 1. 3 In the assumptions of Theorem 1.2 the equality lct(F) = 1 holds, so that on the variety F there is a Kähler-Einstein metric.
Note that the inequality ct(F) 1 was shown for a general variety F ∈ F(d), d ∈ D 8 , under the assumption that M 4k + 1 (that is, |d| 5k + 1), in [16], and under the assumption that M 3k + 4 (that is, |d| 4k + 4), in [8]. For more details about the history of this problem, see Sect. 1.5. One should keep in mind that the smaller are the degrees d i of the equations defining F (respectively, the higher is the degree d = d 1 · · · d k = deg F with the dimension M = dim F fixed), the harder is to prove the inequality ct(F) 1. The case is similar with proving the birational superrigidity of Fano complete intersections of index 1 [19]: the birational superrigidity remains an open problem in arbitrary dimension only for three types of complete intersections, d ∈ {(2, . . . , 2, 2), (2, . . . , 2, 3), (2, . . . , 2, 4)}.
Note also that the canonicity of the pair (F, (1/n)D) for any divisor D ∼ −nK F is a much stronger fact, than the canonicity of this pair for a general divisor D of an arbitrary mobile linear system ⊂ |−nK F |, and for that reason it is harder to prove the inequality ct(F) 1, than the birational rigidity.

Regular complete intersections
We understand the condition that the variety F ∈ F(d) is Zariski general in the sense that at every point o ∈ F the regularity condition (R), which we will now state, is satisfied. This condition was used in [8,16].
Let F = Q 1 ∩· · ·∩Q k ∈ F(d) and o ∈ F be an arbitrary point. Fix a system of affine coordinates z * = (z 1 , . . . , z M+k ) on A M+k ⊂ P with the origin at the point o ∈ A M+k . Let f i (z * ) be the (non-homogeneous) polynomial defining the hypersurface Q i in the affine chart A M+k , deg F i = d i . Write down where q i, j (z * ) are homogeneous polynomials of degree j. On the set {q i, j : 1 i k, 1 j d i } we introduce the standard order, setting: Thus placing the polynomials q i, j in the standard order, we get a sequence of d 1 + · · · + d k = M + k homogeneous polynomials q 1,1 , q 2,1 , ..., q k,d k (1) in M + k variables z * .

The canonical threshold and birational rigidity
Theorem 1.2 has the following application in birational geometry. For an arbitrary non-singular primitive Fano variety X (that is, Pic X = ZK X ) of dimension dim X define the mobile canonical threshold mct(X ) as the supremum of such λ ∈ Q + , that the pair (X, (λ/n)D) is canonical for a general divisor D of an arbitrary mobile linear system ⊂ |−nK X |. The inequality mct(X ) 1 is "almost equivalent" to birational superrigidity of the variety X (for the definition of birational rigidity and superrigidity see [18,Chapter 2]). In [15] the following general fact was shown.

Theorem 1.7
Assume that primitive Fano varieties F 1 , . . . , F K , K 2, satisfy the conditions lct(F i ) = 1 and mct(F i ) 1. Then their direct product V = F 1 × · · · × F K is a birationally superrigid variety. In particular, (i) Every structure of a rationally connected fiber space on the variety V is given by a projection onto a direct factor. More precisely, let β : V # → S # be a rationally connected fiber space and χ : V V # a birational map. Then there exists a subset of indices I = {i 1 , . . . , i k } ⊂ {1, . . . , K } and a birational map is the natural projection onto a direct factor. (ii) Let V # be a variety with Q-factorial terminal singularities, satisfying the condition and χ : V V # a birational map. Then χ is a (biregular) isomorphism. (iii) The groups of birational and biregular self-maps of the variety V coincide: In particular, the group Bir V is finite. (iv) The variety V admits no structures of a fibration into rationally connected varieties of dimension strictly smaller than min{dim F i }. In particular, V admits no structures of a conic bundle or a fibration into rational surfaces.
Since the inequality ct(F) 1 implies that mct(F i ) 1 and lct(F i ) = 1, Theorem 1.2 implies that generic complete intersections F ∈ F(d) with d ∈ D 8 for |d| 3k + 3 satisfy the assumptions of Theorem 1.7.

The structure of the paper
In Sects. 2-3 we prove Theorem 1.6. We reproduce the proof sketched in [16, Section 3.1] in full detail, somewhat modifying the argument given in [16], adjusting it to a wider class of Fano complete intersections. In principle, the new argument is potentially applicable to proving the inequality ct(F) 1 for complete intersections F ∈ F(d) with d / ∈ D 8 . Our main tool is the technique of hypertangent linear systems. This is a procedure (described in Sect. 3), the "input" of which is an effective divisor D ∼ n H F , such that the pair (F, (1/n)D) is not canonical (under the assumption that such pairs exist), and the "output" of which is an effective 1-cycle C that has a high multiplicity at some This contradiction proves Theorem 1.6.

Historical remarks
As we pointed out above, the connection between the existence of Kähler-Einstein metrics and the global log canonical thresholds was established in [7,12,20]. The special importance of those papers is in that they connected some concepts of complex differential geometry with some objects of higher-dimensional birational geometry, which makes it possible to use the results of birational geometry to prove the existence of Kähler-Einstein metrics. That work was started in [1] and continued in [2][3][4][5][6]8,11,16,17]. Every time, a computation or estimate for the global log canonical threshold, obtained by the methods of birational geometry (the connectedness principle, inversion of adjunction, the technique of hypertangent divisors) yielded a proof of existence of Kähler-Einstein metrics for new classes of varieties. Such results are important by themselves, not speaking of their applications to birational geometry (Theorem 1.7), that is, of new classes of birationally rigid varieties.

Tangent divisors
In this section we start the proof of Theorem 1.6. We begin (Sect. 2.1) with some preparatory work: assuming that the pair is not canonical, we show the existence of a hyperplane section of the variety F, such that the multiplicity of the restriction of the divisor D onto at the point o is strictly higher than 2n. After that (Sect. 2.2) using the regularity condition (R), we construct a subvariety Y ⊂ of codimension k + 1 with a high multiplicity at the point o.

Inversion of adjunction
Assume that there exists an effective divisor D ∼n H F such that the pair (F, (1/n)D) is not canonical, that is, there is an exceptional divisor E over F, satisfying the Noether-Fano inequality By linearity of this inequality in the divisor D (the integer n ∈ Z + depends linearly on D), we may assume that D is a prime divisor. Let B ⊂ F be the centre of the exceptional divisor E. It is well known that the estimate mult B D > n holds, whence by for example [16,Proposition 3 The hyperplane section can be viewed as a complete intersection of the type d in P M+k−1 .

Intersection with tangent hyperplanes
Now assume that F satisfies the condition (R). In the notations of Sect. 1.2 the system of linear equations Let h(z * ) be the linear form, defining the hyperplane that cuts out . In particular, . . , k, be the tangent hyperplane sections of the variety . By the condition (R), the inequality dim 2k + 2 and the Lefschetz theorem (taking into account that the singularities of the variety are at most zero-dimensional and o ∈ is a non-singular point), we may conclude that for any j = 1, . . . , k, is an irreducible subvariety of codimension j in , which has multiplicity precisely 2 j at the point o. We will show that the effective divisor D ∼ n H (where H is the class of a hyperplane section of the complete intersection ⊂ P M+k−1 ), satisfying inequality (2), cannot exist. Again by the linearity of inequality (2) (we will need no other information about the divisor D ), we assume that D is a prime divisor. In particular, inequality (2) implies that D = T 1 (since mult o T 1 = 2), so that the effective cycle (D • T 1 ) = Y * 1 of the scheme-theoretic intersection of these divisors is well defined and satisfies the inequality and moreover, Y * 1 ∼ n H 2 ; in particular, Since by construction Y 1 ⊂ T 1 and we conclude that Y 1 ⊂ T 2 and the effective cycle (Y 1 • T 2 ) = Y * 2 is well defined and satisfies the inequality Let Y 2 be an irreducible component of the cycle Y * 2 with the maximal value of mult o /deg. Continuing in the same way, we construct a sequence of irreducible subvarieties The inequality M 2k + 3 is needed to justify the last step in this construction: by the Lefschetz theorem, T 1 ∩ · · · ∩ T k = (T 1 • · · · •T k ) is an irreducible subvariety of of codimension k, with the multiplicity 2 k at the point o and degree d, which makes it possible to form the effective cycle Y * k = (Y k−1 •T k ) of codimension k + 1. We have shown the following claim.
In order to complete the proof of Theorem 1.6, we now need the technique of hypertangent divisors. It is considered in the next section.

Hypertangent divisors
In this section we complete the proof of Theorem 1.6. First (Sect. 3.1) we construct hypertangent linear systems on the variety and study their properties. After that (Sect. 3.2) we select a sequence of general divisors from the hypertangent systems. Finally, intersecting the subvariety Y with the hypertangent divisors, we complete the proof of Theorem 1.6 (Sect. 3.3).

Hypertangent linear systems
For j ∈ {1, . . . , d i } let f i, j = q i,1 + · · · + q i, j be the truncated equation of the hypersurface Q i . By the symbol P a,M+k we denote the linear space of homogeneous polynomials of degree a in the coordinates z 1 , . . . , z M+k . We use this symbol for a < 0 as well, setting in that case P a,M+k = {0}.

Definition 3.1 The linear system of divisors
is the a-th hypertangent linear system on at the point o.
Note that by our convention about the negative degrees only the polynomials f i, j of degree j a are really used in the construction of the system a . Set δ = d k and for a 2 set Obviously, k a = 0 for a δ + 1. The equality d 1 + · · · + d k = M + k implies that δ M. Obviously, k = k 2 + · · · + k δ .
Obviously, one of these cases takes place: we simply listed all options.
For a 2 set It is easy to see that m a is the number of polynomials of degree a in the sequence (1).
In the next proposition we sum up the properties of hypertangent systems that we will need. The symbol codim o stands for the codimension in a neighborhood of the point o with respect to . (iv) In Case I for a = δ − 1, in Cases IIA and IIB for a = δ − 2, and in Case III for a = δ − 3 the following equality holds: dim Bs a = 1. Note that (iii) in Case IIA for a = δ − 2 and in Case III for a = δ − 3 coincides with (iv) for these cases.
Proof These are the standard facts of the technique of hypertangent divisors, following immediately from the regularity condition (Definition 1.4), see [18,Chapter 3]. Claim (i) is obvious, claim (ii) follows from the equality where the dots stand for the components of degree j + 2 and higher, and from the regularity condition. Claims (iii) and (iv) follow from equality (4) and the counting of polynomials of degree j in the sequence (1). For the details, see [18,Chapter 3].

Hypertangent divisors
The next step is constructing a sequence of hypertangent divisors D i, j ∈ i . From each hypertangent linear system i we select l i divisors, where the integer l i is defined in the following way: where the direct product is taken over all i such that l i = 0, see the definition of the integers l i above. It is easy to see that L is the direct product of For an arbitrary equidimensional effective cycle W on , dim W 2, and a divisor D i, j ∈ i , such that none of the components of W is contained in its support |D i, j |, we denote by the symbol of the scheme-theoretic intersection of W and D i, j (see [10,Chapter 2]) by removing all irreducible components, not containing the point o.

Proof of Theorem 1.6
Now everything is ready to apply the technique of hypertangent systems to the subvariety Y ⊂ , constructed in Sect. 2. The tuple D * , * ∈ L is understood as a tuple of divisors which makes it possible to apply the construction of the scheme-theoretic intersection at the point o, described above, many times.

Proposition 3.3 For a general tuple D
is well defined and satisfies the inequalities Proof The procedure of constructing the cycle C is justified by Proposition 3.2 (iii)-(iv), and the inequalities for the degree and multiplicity follow from claims (i) and (ii).
Let us prove Theorem 1.6. Assume that δ = d k 8. Combining inequality (3) with inequalities of Proposition 3.3, we obtain the estimate and after cancellations we see that the inequality mult o C > deg C holds. (For the details, see [16,Section 3].) This contradiction completes the proof of Theorem 1.6.

Regular complete intersections
In this section we prove Theorem 1.5. First (Sect. 4.1), we reduce the problem to a local problem about violation of the regularity condition at a fixed point. After that (Sect. 4.2), we estimate the codimension of the set of tuples of polynomials, vanishing simultaneously on some line. Finally (Sect. 4.3), we estimate the codimension of the set of tuples of polynomials, the set of common zeros of which has an "incorrect" dimension, but is not a line. This completes the proof of Theorem 1.5.

Reduction to the local problem
Following the standard scheme of proving the regularity conditions (see [18,Chapter 3] or any paper that makes use of the technique of hypertangent divisors, for example, [14] or [8]), we have to show that a violation of the local regularity condition at a fixed point o (that is, Definition 1.4 (i)) imposes at least M + 1 independent conditions on the coefficients of the polynomials (1). The complete intersection F is non-singular, so let us fix the linear forms q 1,1 , . . . , q k,1 and so the linear space The last two polynomials in the sequence (1) If all polynomials p i vanish on a line L ⊂ T, then, obviously, the local regularity condition is violated: it is sufficient to take any hyperplane S ⊃ L. For that reason the case when the set { p 1 = · · · = p M−2 = 0} contains a line will be considered separately.

The case of a line
Let B line ⊂ P T be a closed subset of tuples ( p 1 , . . . , p M−2 ), such that for some line L ⊂ T, Proof The proof is obtained by elementary but not quite trivial computations.

Lemma 4.2
The following inequality holds: Proof The first component in the right hand side is the codimension of the set of tuples of polynomials vanishing on a fixed line L ⊂ T. Subtracting the dimension of the Grassmanian of lines, we complete the proof.
Considering the polynomials q i, j for each i = 1, . . . , k separately, we conclude that where a i = d i for i = 1, . . . , k − 2, a k−1 = a k = d k − 1 in Cases I and IIA and a k−1 = d k−1 , a k = d k − 2 in Cases IIB and III. In any case a i 2 and a 1 + · · · + a k = M + k − 2.

Lemma 4.3 The minimum of the quadratic function
on the set of integral vectors (a 1 , . . . , a k ) such that all a i 2 and a 1 + · · · + a k = A, where A = ka + l, a ∈ Z and l ∈ {0, 1, . . . , k − 1}, is equal to Proof Without loss of generality we assume that the set (a 1 , . . . , a k ) is ordered: a i a i+1 . It is easy to check that if two positive integers u, v satisfy the inequality u v−2, then Therefore, if a i a i+1 − 2, then, replacing the vector a = (a 1 , . . . , a k ) by the vector a = (a 1 , . . . , a k ), where a j = a j for j = i, i + 1, a i = a i + 1 and a i+1 = a i+1 − 1, we decrease the value of the function ξ . Similarly, if then, replacing the vector a by the vector a with we decrease the value of the function ξ . In both cases the vector a remains ordered and satisfies the restrictions a 1 + · · · + a k = A, a i 2. Since the set of such vectors is finite, applying finitely many modifications of the two types described above, we obtain a vector with a 1 = · · · = a k−l = a and a k−l+1 = · · · = a k = a + 1, which realizes the minimum of the function ξ . Simple computations complete the proof of the lemma. which is easy to check (recall that by assumption M 2k + 3, so that ak + l 2k + 1 and, in particular, a 2). Starting from this moment, we assume that the polynomials p 1 , . . . , p M−2 do not vanish simultaneously on a line L ⊂ T.

End of the proof of Theorem 1.5
Fix a hyperplane S ⊂ T and its isomorphism S ∼ = P M−2 . Set Since the hyperplane S varies in an (M −1)-dimensional family, it is sufficient to show that the codimension of the set of tuples ( p 1 , . . . , p M−2 ) ∈ P such that the closed set has a component of positive dimension, which is not a line, is of codimension at least (M + 1) + (M − 1) = 2M in P. Let us check this fact. The check is not difficult, arguments of that type were published many times in full detail, so we will just sketch the main steps.
Let B i ⊂ P be the set of such tuples that the closed set (if i = 0, then this set is assumed to be equal to P M−2 ) is of codimension i − 1 in P M−2 , but for some irreducible component B of this set we have p i | B ≡ 0, and moreover if i = M − 2, then B is a curve of degree at least two. Obviously, Theorem 1.5 is implied by the following fact.

Proposition 4.4
The following inequality holds: Proof Using the method that was applied in [13] (see also [18, Chapter 3, Subsection 1.3]), for i = 1, . . . , k we obtain the estimate codim(B i ⊂ P) M + 1 − i 2 (recall that δ(i) = 2 for i = 1, . . . , k). The minimum of the right hand side is attained at i = k and it is easy to check that Therefore, we may assume that i k + 1, so that δ(i) 3. Now we use the technique that was developed in [14] (see also [18, Chapter 3, Section 3]). Let B i,b ⊂ P be the set of tuples such that the closed set (5)  Now the technique of good sequences and associated subvarieties, which we do not give here, see [14] or [18, Chapter 3, Section 3], gives the estimate (taking into account the dimension of the Grassmanian of linear subspaces of codimension b in The right hand side of this inequality, considered as a function on the set {0, . . . , i −1}, can decrease or increase or first increase and then decrease. In any case the minimum of the right hand side is attained either at b = 0 (and equals 3M − 5 2M), or at , when it is also not smaller than 2M.