# Alpha-invariants and purely log terminal blow-ups

## Abstract

We prove that the sum of the \(\alpha \)-invariants of two different Kollár components of a Kawamata log terminal singularity is less than 1.

## Keywords

Alpha-invariant of Tian Log Fano variety Purely log terminal blow-up Log terminal singularity## Mathematics Subject Classification

14E05 14J45 32Q26*V*be a normal irreducible projective variety of dimension \(n\geqslant 1\), and let \({\Delta }_V\) be an effective \({\mathbb {Q}}\)-divisor on

*V*. Writewhere each \({\Delta }_i\) is a prime divisor, and each \(a_i\) is a positive rational number. Suppose that the log pair \((V,{\Delta }_{V})\) has at most Kawamata log terminal singularities. Then, in particular, each \(a_i\) is less than 1. Suppose also that the divisor Open image in new window is ample, so that \((V,{\Delta }_{V})\) is a log Fano variety. Finally, suppose that

*V*is faithfully acted on by a finite group

*G*such that the divisor \({\Delta }_{V}\) is

*G*-invariant. Let \(\alpha _G(V,{\Delta }_V)\) be the real number defined byThis number is known as the \(\alpha \)-invariant of the log Fano variety \((V,{\Delta }_V)\), or its global log canonical threshold (see [12, Definition 3.1]). If

*G*is trivial, we put \(\alpha (V,{\Delta }_{V})=\alpha _{G}(V,{\Delta }_{V})\).

### Example 1

We put \(\alpha _G(V)=\alpha _{G}(V,{\Delta }_{V})\) if \({\Delta }_{V}=0\).

### Example 2

*G*acting faithfully on \({\mathbb {P}}^1\) is one of the following finite groups: the alternating group \({\mathfrak {A}}_5\), the symmetric group \({\mathfrak {S}}_4\), the alternating group \({\mathfrak {A}}_4\), the dihedral group \({\mathfrak {D}}_{2m}\) of order 2

*m*, or the cyclic group \(\varvec{\mu }_{m}\) of order

*m*(including the case \(m=1\), that is, the trivial group). The number Open image in new window is equal to the length of the smallest

*G*-orbit in \({\mathbb {P}}^1\), which gives

If both Open image in new window and *G* is trivial, we put \(\alpha (V)=\alpha _{G}(V,{\Delta }_{V})\). This is the most classical case. Namely, if *V* is a smooth Fano variety, then by [11, Theorem A.3] the number \(\alpha (V)\) coincides with the \(\alpha \)-invariant of *V* defined by Tian in [45]. Its values were found or estimated in many cases. For example, in the toric case the explicit formula for \(\alpha (V)\) is given by Cheltsov and Shramov in [11, Lemma 5.1]. It gives Open image in new window , which can also be verified by an easy explicit computation.

The \(\alpha \)-invariants of all del Pezzo surfaces with at worst Du Val singularities were computed in [2, 4, 7, 37, 38, 43]. Furthermore, the \(\alpha \)-invariants of many non-Gorenstein singular del Pezzo surfaces that are quasi-smooth well-formed complete intersections in weighted projective spaces were computed in [9, 15, 24]. The \(\alpha \)-invariants of many smooth and singular Fano threefolds were computed or estimated in [3, 5, 6, 11, 23, 25]. The \(\alpha \)-invariants of smooth Fano hypersurfaces were estimated in [1, 8, 10, 40].

*V*is a smooth Fano variety, then

*V*admits a

*G*-invariant Kähler–Einstein metric provided thatThis was proved by Tian in [45]. In [21], this result was improved by Fujita. He proved that

*V*admits a Kähler–Einstein metric if it is smooth andIn particular, all smooth hypersurfaces in Open image in new window of degree

*d*are Kähler–Einstein, because their \(\alpha \)-invariants are at least \(({d-1})/{d}\) by [1, 8].

The \(\alpha \)-invariant also plays an important role in birational geometry. It was first observed by Park in [35], where he proved a theorem that evolved into the following:

### Theorem 3

*X*be a variety with at most terminal \({{\mathbb {Q}}}\)-factorial singularities. Suppose that there is a proper morphism \(\phi :X\rightarrow Z\) such that

*Z*is a smooth curve, and \(-K_X\) is \(\phi \)-ample. Let

*P*be a point in

*Z*, and let \(E_X\) be the scheme fiber of \(\phi \) over

*P*. Suppose that \(E_X\) is irreducible, reduced, normal, and has at most Kawamata log terminal singularities, so that \(E_X\) is a Fano variety by the adjunction formula. Suppose also that there is a commutative diagramsuch that

*Y*is a variety with at most terminal \({\mathbb {Q}}\)-factorial singularities, \(\psi \) is a proper morphism, the divisor \(-K_{Y}\) is \(\psi \)-ample, and \(\rho \) is a birational map that induces an isomorphismwhere \(E_Y\) is the scheme fiber of \(\psi \) over

*P*. Suppose, in addition, that \(E_Y\) is irreducible. Then \(\rho \) is an isomorphism provided that \(\alpha (E_X)\geqslant 1\). Moreover, if \(E_Y\) is reduced, normal and has at most Kawamata log terminal singularities, then \(\rho \) is an isomorphism provided that \(\alpha (E_X)+\alpha (E_Y)>1\).

Theorem 3 gives a necessary condition in terms of \(\alpha \)-invariants for the existence of a non-biregular fiberwise birational transformation of a Mori fibre space over a curve. It follows from [29, Theorem 1.1] that this condition is not a sufficient condition. Nevertheless, the bound is sharp (one can find many examples in [35, 36]).

### Example 4

Let *S* be a \({\mathbb {P}}^1\)-bundle over a curve. Then we have an elementary transformation to another \({\mathbb {P}}^1\)-bundle over the same curve. Note that the Open image in new window by Example 2.

### Example 5

*S*be a smooth cubic surface in \({\mathbb {P}}^3\) with an Eckardt point

*O*. Denote by \(L_{1},L_{2},L_{3}\) the lines in

*S*passing through

*O*. Put Open image in new window , and let \(\phi \) be the natural projection Open image in new window . Let us identify

*S*with a fiber of \(\phi \). Then there is a commutative diagramwhere \(\alpha \) is the blow-up of the point

*O*, the map \(\psi \) is the anti-flip along the proper transforms of the curves \(L_{1},L_{2},L_{3}\), and \(\beta \) is the contraction of the proper transform of the surface

*S*. The scheme fiber of \(\psi \) over the point \(\phi (S)\) is a cubic surface in \({\mathbb {P}}^3\) that has one singular point of type \({\mathbb {D}}_{4}\). Its \(\alpha \)-invariant is 1 / 3 (see [4, Theorem 1.4]). On the other hand, we have \(\alpha (S)={2}/{3}\) (see [2, Theorem 1.7]).

### Example 6

*X*and

*Y*be subvarieties in Open image in new window given by equationsrespectively, where

*t*is a coordinate on Open image in new window , and Open image in new window are homogeneous coordinates on \({\mathbb {P}}^{3}\). Then the projections Open image in new window and Open image in new window are fibrations into cubic surfaces, and the mapgives a non-biregular birational fiberwise map \(\rho :X\dasharrow Y\). The fiber of \(\phi \) over the point \(t=0\) is a cubic surface that has one Du Val singular point of type \({\mathbb {E}}_{6}\), so that its \(\alpha \)-invariant is 1 / 6 (see [4, Theorem 1.4]), and the fiber of \(\psi \) over the point \(t=0\) is a smooth cubic surface with an Eckardt point, so that its \(\alpha \)-invariant is 2 / 3 (see [2, Theorem 1.7]).

*purely log terminal blow-up*of the singularity \(U\ni P\).

*different*of the pair \((X,E_X)\). One hasFurthermore, the singularities of the log pair \((E_X,\mathrm{Diff}_{E_X}(0))\) are Kawamata log terminal by Adjunction (see [44, 3.2] or [26, 17.6]). This means that \((E_X,\mathrm{Diff}_{E_X}(0))\) is a log Fano variety with Kawamata log terminal singularities, because \(-E_X\) is \(\phi \)-ample.

### Definition 7

(cf. [31, Definition 1.1]) The log Fano variety \((E_X,\mathrm{Diff}_{E_X}(0))\) is a *Kollár component* of \(U\ni P\).

Let us show how to compute \(\alpha (E_X,\mathrm{Diff}_{E_X}(0))\) in three simple cases.

### Example 8

*f*are smooth rational curves whose self-intersections are Open image in new window , and their dual graph is of type \({\mathbb {A}}_m,{\mathbb {D}}_m,{\mathbb {E}}_6,{\mathbb {E}}_7\), or \({\mathbb {E}}_8\). Let \(E_W\) be one of the exceptional curves that is chosen as follows. If \(U\ni P\) is not a singularity of type \({\mathbb {A}}_m\), let \(E_W\) be the only

*f*-exceptional curve that intersects three other

*f*-exceptional curves, i.e., \(E_W\) is the “fork” of the dual graph. If \(U\ni P\) is a singularity of type \({\mathbb {A}}_m\), choose \(E_W\) to be the

*k*-th curve in the dual graph. In this case, we may assume that \(k\leqslant ({m+1})/{2}\). In all the cases, there exists a commutative diagramwhere

*h*is the contraction of all

*f*-exceptional curves except \(E_W\), and

*g*is the contraction of the proper transform of \(E_W\) on the surface

*Y*. Denote the

*g*-exceptional curve by \(E_Y\). Then

*Y*has at most Du Val singularities of type \({\mathbb {A}}\), the curve \(E_Y\) is smooth, and it contains all the singular points of the surface

*Y*, if any. One can check that the log pair \((Y,E_Y)\) has purely log terminal singularities (see [28, Theorem 4.15 (3)]). Also, the divisor Open image in new window is

*g*-ample. Thus, the curve \(E_Y\) is a Kollár component of the singularity \(U\ni P\). Moreover,

*Y*that lie on \(E_Y\). The singular point \(P_i\) (resp. Open image in new window and \(R_\ell \)) is a Du Val singular point of type Open image in new window (resp. \({\mathbb {A}}_{j}\) and \({\mathbb {A}}_{\ell }\)). Since Open image in new window , it follows from Example 1 that

### Example 9

Let \(U\ni P\) be a germ of a Du Val singularity of type \({\mathbb {A}}_m\), and let \(f:W\rightarrow U\) be the minimal resolution of this singularity.

Let *Q* be a point on one of the *f*-exceptional curves. We consider two cases, one is the case where the point *Q* belongs to one of the two exceptional curves that correspond to “tails” of the dual graph but it is not contained in any other exceptional curve, the other is the case where *Q* is the intersection point of the *k*-th and Open image in new window -th *f*-exceptional curves, \(1\leqslant k\leqslant {m}/{2}\).

*Q*, and \(\zeta \) be the contraction of the proper transforms of all the

*f*-exceptional curves. Thus, there exists a commutative diagramDenote the

*g*-exceptional curve by \(E_Y\). It is a smooth rational curve. The dual graphs of the exceptional curves of the minimal resolution of singularities \(\zeta :\widehat{W}\rightarrow Y\) are chains such that the self-intersection numbers of the exceptional curves are \(-3, -2\), \(\ldots , -2\), and the proper transform of \(E_Y\) intersects only the “tail” components of these chains. In the former case,

*Y*has a unique singular point

*O*, which is a quotient of \({\mathbb {C}}^2\) by the cyclic group \(\varvec{\mu }_{2m+1}\). In the latter case, it contains two singular points \(P_1\) and \(P_2\), which are quotients of \({\mathbb {C}}^2\) by the cyclic groups \(\varvec{\mu }_{2k+1}\) and \(\varvec{\mu }_{2(m-k)+1}\), respectively.

*g*-ample. Thus, the curve \(E_Y\) is a Kollár component of the singularity \(U\ni P\). Moreover,Therefore,

*m*is even, and

*Q*is the “central point” of the configuration of the

*f*-exceptional curves.

It is easy to see from [28, Theorem 4.15] that if \(U\ni P\) is a Du Val singularity of type \({\mathbb {D}}\) or \({\mathbb {E}}\), and the exceptional curve \(E_W\) in Example 8 is not the one corresponding to the “fork” of the dual graph, then the curve \(E_Y\) is not a Kollár component (see [39, Example 4.7]). We will see later that in these cases the singularity \(U\ni P\) has a unique Kollár component, which is described in Example 8. This is not true in general, i.e., a Kollár component of a singularity \(U\ni P\) may not be unique, as one can see from Examples 8 and 9. Nevertheless, Li and Xu established in [31, Theorem B] the following:

### Theorem 10

A K-semistable Kollár component of \(U\ni P\) is unique if it exists.

The K-semistable Kollár components of two-dimensional Du Val singularities are described in our Examples 8 and 9. They are precisely the Kollár components whose \(\alpha \)-invariants are at least 1 / 2 (cf. [32, Example 4.7]).

Note that Du Val singularities are two-dimensional rational quasi-homogeneous isolated hypersurface singularities. The K-semistable Kollár components of many three-dimensional rational quasi-homogeneous isolated hypersurface singularities have been described in [9, 15]. Similarly, the K-semistable Kollár components of many four-dimensional rational quasi-homogeneous isolated hypersurface singularities have been described in [23].

The purpose of this paper is to prove the following analogue of Theorem 3.

### Theorem 11

Theorem 11 also implies

### Corollary 12

If \(\alpha (E_X,\mathrm{Diff}_{E_X}(0))\geqslant 1\), then the Kollár component of \(U\ni P\) is unique.

This corollary is well known: it follows from [39, Theorem 4.3] and [30, Theorem 2.1]. Recall from [39, Definition 4.1] that the singularity \(U\ni P\) is said to be *weakly exceptional* if it has a unique purely log terminal blow-up. This is equivalent to the condition that there is a Kollár component \(E_X\) of \(U\ni P\) such that \(\alpha (E_X,\mathrm{Diff}_{E_X}(0))\geqslant 1\) (see [39, Theorem 4.3], [30, Theorem 2.1], and [12]). It follows from Example 8 that Du Val singularities of types \({\mathbb {D}}\) and \({\mathbb {E}}\) are weakly exceptional. On the other hand, Du Val singularities of type \({\mathbb {A}}\) are not weakly exceptional, since each of them admits several Kollár components (see Examples 8 and 9), and thus has several purely log terminal blow ups.

### Remark 13

*W*, and \(\psi \) is a birational morphism. Denote by \({\widetilde{E}}\) the exceptional divisor of \(\pi \), and denote by \(E_Y\) the exceptional divisor of \(\psi \). Then Open image in new window , and \(E_Y\) is naturally isomorphic to the quotient Open image in new window , where

*G*is the image of the group \({\widehat{G}}\) in \(\mathrm{PGL}_{n+1}({\mathbb {C}})\). Moreover, the log pair \((Y,E_Y)\) has purely log terminal singularities, and the divisor Open image in new window is \(\psi \)-ample. Thus, the log Fano variety \((E_Y,\mathrm{Diff}_{E_Y}(0))\) is a Kollár component of the singularity \(U\ni P\). Also, it follows from [31, Example 7.1 (1)] and [31, Theorem 1.2] that \((E_Y,\mathrm{Diff}_{E_Y}(0))\) is K-semistable. Furthermore, one has(see [12, Proof of Theorem 3.16]). Thus, if Open image in new window , then this Kollár component is unique by Corollary 12. One can find many subgroups Open image in new window in [12, 13, 14, 16, 33, 41, 42]. Note also that one always has \(\alpha _G({\mathbb {P}}^{n})\leqslant 1184036\) by [46].

In the remaining part of the paper, we prove Theorem 11. Let us use its assumptions and notations. We have to show that \(\rho \) is an isomorphism. Suppose that this is not the case. Let us seek for a contradiction.

*U*is affine. There exists a commutative diagramsuch that

*W*is a smooth variety, and

*f*and

*g*are birational morphisms. Denote by \(E_X^W\) and \(E_Y^W\) the proper transforms of the divisors \(E_X\) and \(E_Y\) on the variety

*W*, respectively. Then \(E_X^W\) is

*g*-exceptional, and \(E_Y^W\) is

*f*-exceptional. We may assume that \(E_X^W,E_Y^W\) and the remaining exceptional divisors of

*f*and

*g*form a divisor with simple normal crossings.

*W*that are contracted by both

*f*and

*g*. Thenfor some rational numbers \(a,a_1,\ldots ,a_m\). Since the log pair \((X,E_X)\) has purely log terminal singularities, all numbers \(a,a_1,\ldots ,a_m\) are strictly less than 1. Also, we havewhere \(b,b_1,\ldots ,b_m\) are non-negative rational numbers. Then \(b>0\), since \(f(E_Y^W)\subset E_X\).

*Y*and

*W*by \({\mathscr {M}}_X^Y\) and \({\mathscr {M}}_X^W\), respectively. Thenwhich implies that Open image in new window . On the other hand, we have Open image in new window for some positive rational number \(\delta _Y\). Put \(\epsilon _X={\delta _Y}/({nb})\). Then Open image in new window , so thatfor some rational numbers \(\alpha ,\alpha _1,\ldots ,\alpha _m\). Similarly, let \({\mathscr {M}}_Y\) be the base point free linear system Open image in new window . Denote by \({\mathscr {M}}_Y^X\) and \({\mathscr {M}}_Y^W\) its proper transforms on

*X*and

*W*, respectively. Then there is a positive rational number \(\epsilon _Y\) such that Open image in new window , so thatfor some rational numbers \(\beta ,\beta _1,\ldots ,\beta _m\).

### Lemma 14

One has \(\alpha >1\) and \(\beta >1\). In particular, the singularities of the log pairs Open image in new window and Open image in new window are not log canonical.

### Proof

*f*-nef. Then

*B*is effective if and only if Open image in new window is effective by Negativity Lemma (see [28, Lemma 3.39]). Since \(a<1\), the divisor

*B*is not effective, which implies that \(\alpha >1\). \(\square \)

### Lemma 15

At least one of the log pairs Open image in new window and Open image in new window is not log canonical.

### Proof

*W*is smooth, the linear systems \({\mathscr {M}}_Y^W\) and \({\mathscr {M}}_X^W\) are free from base points, and the divisors \(E_X^W,E_Y^W,F_1,\ldots ,F_m\) form a simple normal crossing divisor. Since \(K_X+E_X+\lambda \epsilon _Y{\mathscr {M}}_Y^X+\mu \epsilon _X{\mathscr {M}}_X\) is \(\phi \)-ample, it follows from [28, Corollary 3.53] that the log pair \(\bigl (X,E_X+\lambda \epsilon _Y{\mathscr {M}}_Y^X+\mu \epsilon _X{\mathscr {M}}_X\bigr )\) is the canonical model of the log pair \((W,D_W)\). Similarly, the log pair \(\bigl (Y,E_Y+\lambda \epsilon _Y{\mathscr {M}}_Y+\mu \epsilon _X{\mathscr {M}}_X^Y\bigr )\) is also the canonical model of the log pair \((W,D_W)\), because \(K_Y+E_Y+\lambda \epsilon _Y{\mathscr {M}}_Y+\mu \epsilon _X{\mathscr {M}}_X^Y\) is \(\psi \)-ample. Since the canonical model is unique by [28, Theorem 3.52], we see that \(\rho \) is an isomorphism. Since \(\rho \) is not an isomorphism by assumption, we obtain a contradiction. \(\square \)

## Notes

### Acknowledgements

We are grateful to the referees for helpful comments on the earlier version of the paper.

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