DEFORMATIONS OF WEAK Q-FANO 3-FOLDS TARO SANO

Abstract. We prove several results on deformations of a weak Q-Fano 3-fold. First we prove that a weak Q-Fano 3-fold has unobstructed deformations. Next we give a bound of the number of singular points on a Q-Fano 3-fold with a Du Val elephant via the orbifold Euler number and the local invariant attached to a pair of a 3-fold terminal singularity and its Du Val elephant. We also treat a stacky proof of the unobstructedness of deformations of a Q-Fano 3-fold.

Contents 1. Introduction 1.1. Comments on the proof 2. Unobstructedness of deformations of a weak Q-Fano 3-fold 3. Bound of singularities on a Q-Fano 3-fold 3.1. Orbifold Riemann-Roch formula 3.2. An invariant of a pair of a 3-fold terminal singularity and its elephant 3.3. Formula for the number of singular points 4. Deformations of a Q-Fano 3-fold via its canonical covering stack Acknowledgment References

1 2 3 7 7 8 11 14 16 16

1. Introduction In this paper, we consider algebraic varieties over the complex number field C. Definition 1.1. Let X be a normal projective 3-fold. We say that X is a weak Q-Fano 3-fold (resp. Q-Fano 3-fold) if X has only terminal singularities and −KX is a nef and big (resp. ample) divisor. Weak Q-Fano 3-folds appear in the birational study of a Q-Fano 3-fold. (cf. [Tak06]) In this paper, we first study the deformation of a weak Q-Fano 3-fold. The following is a main result of this paper. Theorem 1.2. Deformations of a weak Q-Fano 3-fold are unobstructed. The author proved the unobstructedness for a Q-Fano 3-fold ([San12, Theorem 1.7]). Minagawa proved it for a weak Fano 3-fold with only terminal Gorenstein singularities([Min01, Main Theorem (1)]). Theorem 1.2 is a generalization of these results. Next we try to estimate the number of singular points on a Q-Fano 3-fold as follows. 2010 Mathematics Subject Classification. Primary 14B07, 14J30, 14J45; Secondary 14B15. Key words and phrases. Deformation theory, weak Q-Fano 3-folds, singularities. 1

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TARO SANO

Theorem 1.3. Let X be a Q-Fano 3-fold. Assume that X contains D ∈ |−KX | with only Du Val singularities and it can be deformed to a V-smooth pair (Xt , Dt ) (See Definition 3.6). Then we have an inequality (1)

N X X 1 (2ri + 1)(ri − 1) 21 − eorb (Xt ) − ≥ τ (X, D, p), 2 2ri i=1 p∈Sing X∪Sing D

where eorb (Xt ) is the topological orbifold Euler number of Xt (cf. [Bla96, (2.14)]), {(ri , ai ) | i = 1, . . . , N } is the basket of singularities of X(Definition 3.15), and τ (X, D, p) is a positive integer associated to the germ of a pair (X, D) at p (See (9) for the definition of τ (X, D, p)). This is inspired by Namikawa’s formula on the bound of singular points on a Fano 3fold with terminal Gorenstein singularities ([Nam97, Theorem 13]). The author proved the existence of a deformation as in the statement ([San12, Theorem 1.9]). Namikawa used a different invariant of a singularity in his formula. The local invariant τ (X, D, p) of the singularity p ∈ Sing X ∪ Sing D is zero if and only if the pair (X, D) is V-smooth at p (Corollary 3.12). Thus the above formula also captures the information of singularities of an elephant D. The above formula is a corollary of the proof of [San12, Theorem 1.9] and the orbifold Riemann-Roch formula (cf. [Rei87, Chapter III], [Bla96]). Remark 1.4. On a Q-Fano 3-fold X with the basket of singularities {(ri , ai )}N i=1 , we already have an inequality N X ri2 − 1 ≤ 24, ri i=1 P ri2 −1 1 since we have an equality 1 = χ(X, OX ) = 24 ( N i=1 ri + (−KX ) · c2 (X)) from the orbifold Riemann-Roch formula and the inequality −KX · c2 (X) ≥ 0 ([KMMT00, Theorem 1.2(1)]). We hope that Theorem 1.3 is useful in some situations. In section 4, we also treat the canonical covering stack associated to a 3-fold with terminal singularities or a surface with klt singularities. We shall explain the unobstructedness of deformations of a Q-Fano 3-fold and Q-Gorenstein deformations of weak log del Pezzo surface. 1.1. Comments on the proof. To prove Theorem 1.2, we apply the T 1 -lifting property to a pair (X, D) of a weak Q-Fano 3-fold X and a smooth divisor D ∈ |−mKX | for a sufficiently large m > 0. The author proved the unobstructedness for a smooth weak Fano variety by using the E1 -degeneration of the Hodge to de Rham spectral sequence associated to the log de Rham complex ([San14b, Theorem 1.1]). Since we do not have such a statement on a 3-fold with terminal singularities, we prove the necessary statements directly by using some arguments similar to those in the proof of [Nam94, Theorem 1]. Let us comment on the proof of Theorem 1.3. Let Up ⊂ X be a Stein neighborhood of 1 1 p ∈ Sing X ∪ Sing D and T(X,D) , T(U be the set of first order deformations of pairs (X, D) p ,Dp ) and (Up , Dp := D ∩ Up ) respectively. In [San12, Section 4], the author proved the existence [0] 1 1 of a good element η ∈ T(X,D) by using a homomorphism φ(Up ,Dp ) : T(U → HE2 p (U˜p , FUp ) p ,Dp ) and the composition M M ⊕ prUp ⊕φ(Up ,Dp ) [0] 1 1 T(X,D) −−−−→ T(U − − − − − − → HE2 p (U˜p , FUp ), p ,Dp ) p∈Sing X∪Sing D

p∈Sing X∪Sing D

DEFORMATIONS OF WEAK Q-FANO 3-FOLDS

3 [0]

where U˜p → Up is some partial resolution with the exceptional divisor Ep and FUp is certain coherent sheaf on U˜p . Although the restriction homomorphism ⊕ prUp is not necessarily L surjective, the composition is surjective onto Im φ(Up ,Dp ) ([San12, Claim 4.11]). By this property, we obtain an inequality X 1 dim T(X,D) ≥ τ (X, D, p), p∈Sing X∪Sing D

where we set τ (X, D, p) := dim Im φ(Up ,Dp ) . We shall show that N

1 dim T(X,D)

X (2ri + 1)(ri − 1) 1 = dim Def(X, D) = dim Def(Xt , Dt ) = 21 − eorb (Xt ) − 2 2ri i=1

by the universality and smoothness of the Kuranishi space Def(X, D) of the pair (X, D) and the orbifold Riemann-Roch formula. These imply the formula (13). Although Theorem 1.2 is stronger, in Theorem 4.4, we shall prove the unobstructedness for a Q-Fano 3-fold X by associating the canonical covering stack X → X. We shall show that the obstruction space Ext2 (Ω1X , OX ) for X vanishes. An advantage of this method is that the canonical sheaf ωX is invertible. 2. Unobstructedness of deformations of a weak Q-Fano 3-fold In this section, we shall prove Theorem 1.2. Let X be an algebraic scheme and D its closed subscheme. Let ArtC be the category of Artinian local C-algebras with residue field C. We consider the deformation functors Def X , Def (X,D) : ArtC → (Sets) of X and (X, D) respectively (cf. [San12, Definition 2.1, 2.2]). We first prepare a lemma about Ext group under restriction to a open subset over An := C[t]/(tn+1 ). Lemma 2.1. (cf. [San14a, Lemma 2.11]) Let X be a 3-fold with only terminal singularities and ξn := (Xn → Spec An ) ∈ Def X (An ). Let U ⊂ X be the smooth locus of X and Un → Spec An be a deformation of U induced by ξn . Let F, L be reflexive sheaves on Xn . Assume that R1 ι∗ L|Un = 0, where ι : U ,→ X is an open immersion. Then the restriction homomorphism (2)

r : Ext1OXn (F, L) → Ext1OUn (F|Un , L|Un )

is an isomorphism. Proof. We can construct the converse of r as follows. Given an exact sequence 0 → L|Un → GUn → F|Un → 0, its push-forward 0 → L → ι∗ GUn → F → 0 is also exact by the condition R ι∗ L|Un = 0. 1

 The following result is crucial to prove Theorem 1.2. Theorem 2.2. Let X be a weak Q-Fano 3-fold. Take a smooth member D ∈ |−mKX | for some positive integer m such that D∩Sing X = ∅ which exists by the base point free theorem. Then the deformation functor Def (X,D) is unobstructed.

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TARO SANO

Proof. We shall use the T 1 -lifting property. The proof is similar to [San14b, Theorem 2.2], but also follows some arguments in [Nam94, Section 2]. Set An := C[t]/(tn+1 ) and Bn := C[x, y]/(xn+1 , y 2 ) ' An ⊗C A1 . For [(Xn , Dn ), φ0 ] ∈ Def (X,D) (An ), let T 1 ((Xn , Dn )/An ) be the set of isomorphism classes of pairs ((Yn , En ), ψn ) consisting of deformations (Yn , En ) of (Xn , Dn ) over Bn and marking isomorphisms ψn : Yn ⊗Bn An → Xn such that ψn (En ⊗Bn An ) = Dn , where we use a ring homomorphism Bn → An given by x 7→ t and y 7→ 0. Then we see the following. Claim 2.3. We have (3)

T 1 ((Xn , Dn )/An ) ' Ext1 (Ω1Xn /An (log Dn ), OXn ).

Proof. We can prove this by a standard argument (cf. [Ser06, Proposition 3.4.17]) using Bn = An ⊗C A1 .  By the T 1 -lifting theorem ([FM99, Theorem A]), it is enough to show that the homomorphism Ext1OXn (Ω1Xn /An (log Dn ), OXn ) → Ext1OX

n−1

(Ω1Xn−1 /An−1 (log Dn−1 ), OXn−1 )

is surjective. Let U be the smooth locus of X, ι : U ,→ X be the open immersion. Note that D ⊂ U . Let (Un , Dn ) be a deformation of (U, D) over An induced by (Xn , Dn ). By Lemma 2.1, we obtain (4)

Ext1OXn (Ω1Xn /An (log Dn ), OXn ) ' Ext1OUn (Ω1Un /An (log Dn ), OUn )
' Ext1OXn (ι∗ Ω1Un /An (log Dn ) ⊗ ωUn /An , ωXn /An )

since we have R1 ι∗ ωUn /An = 0 (cf. [San12, Claim 2.12]), where ι : Un ,→ Xn is an open immersion. By the Serre duality, we obtain
(5) Ext1OXn (ι∗ Ω1Un /An (log Dn ) ⊗ ωUn /An , ωXn /An )
' HomAn (H 2 (Xn , ι∗ Ω1Un /An (log Dn ) ⊗ ωUn /An ), An ). Thus it is enough to show the homomorphism (6)


HomAn (H 2 (Xn , ι∗ Ω1Un /An (log Dn ) ⊗ ωUn /An ), An )
→ HomAn−1 (H 2 (Xn−1 , ι∗ Ω1Un−1 /An−1 (log Dn−1 ) ⊗ ωUn−1 /An−1 ), An−1 )

is surjective. Let πn : Zn → Xn be a cyclic cover branched along Dn ∈ |−mKXn /An |. We have an isomorphism Ω1Zn (log ∆n ) ' (πn∗ Ω1Xn (log Dn ))∗∗ for some divisor ∆n ∈ |−πn∗ KXn /An |, where πn∗ KXn /An is a Cartier divisor on Zn corresponding to a line bundle (πn∗ ωXn /An )∗∗ . Then we obtain the decomposition (πn )∗ Ω1Zn (log ∆n )(−∆n ) '

m−1 M

ι∗ (Ω1Un (log Dn )((i + 1)KUn /An )).

i=0

By this decomposition, the surjectivity is reduced to that of the homomorphism

DEFORMATIONS OF WEAK Q-FANO 3-FOLDS

(7)

5

HomAn (H 2 (Zn , Ω1Zn /An (log ∆n )(−∆n )), An ) → HomAn−1 (H 2 (Zn−1 , Ω1Zn−1 /An−1 (log ∆n−1 )(−∆n−1 )), An−1 ).

As in the proof of [Nam94, Section 2, Theorem 1], the above surjectivity is reduced to the following two statements; (i) H 2 (Zn , Ω1Zn /An (log ∆n )(−∆n )) is a free An -module. (ii) The reduction homomorphism H 2 (Zn , Ω1Zn /An (log ∆n )(−∆n )) → H 2 (Zn−1 , Ω1Zn−1 /An−1 (log ∆n−1 )(−∆n−1 )) is surjective. The following lemma implies (ii). Lemma 2.4. Let Z := Z0 , ∆ := ∆0 and π := π0 : Z → X be as above. Then we have H 3 (Z, Ω1Z (log ∆)(−∆)) = 0. Proof of Lemma 2.4. Since Z is smooth along ∆, there is an exact sequence 0 → Ω1Z (log ∆)(−∆) → Ω1Z → Ω1∆ → 0. This sequence induces an exact sequence H 2 (∆, Ω1∆ ) → H 3 (Z, Ω1Z (log ∆)(−∆)) → H 3 (Z, Ω1Z ). Thus it is enough to show that the both side terms are zero. We first show H 2 (∆, Ω1∆ ) = 0. By the Serre duality and Hodge symmetry, we see that 2 h (∆, Ω1∆ ) = h0 (∆, Ω1∆ ) = h1 (∆, O∆ ). There is an exact sequence H 1 (Z, OZ ) → H 1 (∆, O∆ ) → H 2 (Z, OZ (−∆)). Lm−1 1 Since we have π∗ OZ ' i=0 OX (iKX ), we see that H (Z, OZ ) = 0 by the KawamataViehweg vanishing theorem since −KX is nef and big. Since ∆ is nef and big, we also see that H 2 (Z, OZ (−∆)) = 0. Thus we obtain H 2 (∆, Ω1∆ ) = 0. Next we show H 3 (Z, Ω1Z ) = 0. By the Serre duality, we see that h3 (Z, Ω1Z ) = h0 (Z, Ω2Z ). Let µ : Z˜ → Z be a log resolution of singularities of Z. Since Z has only terminal singularities, we see that µ∗ Ω2Z˜ ' Ω2Z (cf. [Nam01, Theorem 4]). Hence we obtain h0 (Z, Ω2Z ) = ˜ Ω2˜ ) = h2 (Z, O ˜ ), where we use the Hodge symmetry on Z˜ for the latter equalh0 (Z, Z Z ˜ O ˜ ) = h2 (Z, OZ ). This is zero by the ity. Since Z has only rational singularities, h2 (Z, Z Kawamata-Viehweg vanishing theorem. Thus we obtain H 3 (Z, Ω1Z ) = 0. Concluding the proof of Lemma 2.4.  Thus we obtain (ii). In order to obtain (i), it is enough to show the surjectivity of the homomorphism Φn : H 1 (Zn , Ω1Zn /An (log ∆n )(−∆n )) → H 1 (Zn−1 , Ω1Zn−1 /An−1 (log ∆n−1 )(−∆n−1 )) by (ii) and the base change theorem (cf. [Har77, Theorem 12.11 (b)]).

6

TARO SANO ∗ ). Then we have a commutative diagram Let K(Zn ,∆n ) := Ker(OZ∗ n → O∆ n

/

0

0

/

/

H 1 (Z, K(Z,∆) ) ⊗Z C α1



/

H 1 (Z, OZ∗ ) ⊗Z C /

H 1 (Z, Ω1Z (log ∆)(−∆))



∗ H 1 (∆, O∆ ) ⊗Z C

α2

/



H 1 (Z, Ω1Z )

α3

H 1 (∆, Ω1∆ ),

∗ where the vertical homomorphisms are induced by d logZ : OZ∗ → Ω1Z , d log∆ : O∆ → Ω1∆ . 1 By [Nam94, Lemma 2.2], the homomorphism α2 is surjective. Since H (∆, O∆ ) = 0, the homomorphism α3 is injective. Thus, by the snake lemma, we see that α1 is surjective. Let K0 := Ker(K(Zn ,∆n ) → K(Zn−1 ,∆n−1 ) ). We see that K0 ' OZ (−∆) since we have a commutative diagram

0 /

0 /

0 /

0

0



/

K0





/



/

K(Zn ,∆n )

OZ

O∆

0

/





/

OZ∗ n 





OZ∗ n−1 

/

∗ O∆ n

/

∗ O∆ n−1



0

/

0 /

0

K(Zn−1 ,∆n−1 )

0.



0

0

By this and H 2 (Z, OZ (−∆)) = 0, we see that the homomorphism β1 : H 1 (Zn , K(Zn ,∆n ) ) → H 1 (Z, K(Z,∆) ) is surjective. Since we have a commutative diagram β1

H 1 (Zn , K(Zn ,∆n ) ) ⊗Z An 

H 1 (Zn , Ω1Zn /An (log ∆n )(−∆n ))

β2

/

/

H 1 (Z, K(Z,∆) ) ⊗Z C α1



H 1 (Z, Ω1Z (log ∆)(−∆)),

we see that the homomorphism β2 is surjective. We see that β2 ⊗An C is an isomorphism by the base change theorem ([Har77, Theorem 12.11 (a)]). Thus, by Nakayama’s lemma, we see that H 1 (Zn , K(Zn ,∆n ) ) ⊗ An → H 1 (Zn , Ω1Zn /An (log ∆n )(−∆n )) is surjective. By this and the commutative diagram H 1 (Zn , K(Zn ,∆n ) ) ⊗Z An 

H 1 (Zn , Ω1Zn /An (log ∆n )(−∆n ))

β10

/

Φn

/

H 1 (Zn−1 , K(Zn−1 ,∆n−1 ) ) ⊗Z An−1 

H 1 (Zn−1 , Ω1Zn−1 (log ∆n−1 )(−∆n−1 )),

we see that Φn is surjective. Hence we obtain (i). We finish the proof of Theorem 2.2.



DEFORMATIONS OF WEAK Q-FANO 3-FOLDS

7

Corollary 2.5. Let X be a weak Q-Fano 3-fold. Then its deformation functor Def X is unobstructed. Proof. Let D ∈ |−mKX | be a smooth divisor as in Theorem 2.2. Let F : Def (X,D) → Def X be the forgetful morphism. Note that H 1 (D, ND/X ) = 0 since we have an exact sequence H 1 (X, OX (D)) → H 1 (D, ND/X ) → H 2 (X, OX ) and both side terms are zero by the Kawamata-Viehweg vanishing theorem. Thus F is a smooth morphism and this implies that Def X is unobstructed since Def (X,D) is unobstructed by Theorem 2.2.  3. Bound of singularities on a Q-Fano 3-fold In this section, we give a formula on the number of singular points on a Q-Fano 3-fold with an anticanonical member with only Du Val singularities. 3.1. Orbifold Riemann-Roch formula. In this subsection, we review the orbifold RiemannRoch formula (cf. See [Rei87, Chapter III], [Bla96].). We frequently use the following notion of a V-free sheaf. Definition 3.1. Let (U, p) be a germ of an isolated quotient singularity and πU : (V, q) → (U, p) be a quotient morphism such that (V, q) is smooth and πU is ´etale outside p. We say that a reflexive coherent sheaf F on (U, p) is V-free of rank m if the reflexive sheaf (πU∗ F)∗∗ is a free sheaf on (V, q) of rank m. ([Bla96, (2,4) Definition]). Let KV0 V (U, p) be the set of isomorphism classes of V-free sheaves on (U, p). Let X be a variety with only isolated quotient singularities and F be a reflexive coherent sheaf on X. We say that F is a locally V-free sheaf on X of rank m if it induces a V-free sheaf of rank m on a germ (X, x) for all x ∈ X. We have the Chern characters and the orbifold Euler characteristics on a projective Vmanifold as follows. Definition 3.2. Let X be a n-dimensional projective variety with only isolated quotient singularities. Let F be a locally V-free sheaf of rank m on X. We have the i-th Chern class ci (F) ∈ H 2i (X, Q) of F and ci (X) := ci (ΘX ) as defined in [Bla96, (2.10)], where ΘX is the tangent sheaf of X. We have the Chern character ch(F) ∈ H 2∗ (X, Q) and the Todd class td(X) ∈ H 2∗ (X, Q) as in [Bla96, (3.3) Definition]. We define the holomorphic orbifold Euler characteristic χorb (F) ∈ Q of F as χorb (F) := (ch(F) · td(X)) · [X] by using the fundamental class [X] ∈ H2n (X, Z). Remark 3.3. If dim X = 3 in Definition 3.2, we have 1 1 ch(F) = m + c1 (F) + (c21 (F) − 2c2 (F)) + (c31 (F) − 3c1 (F) · c2 (F) + 3c3 (F)), 2 6 1 1 1 td(X) = 1 + c1 (X) + (c21 (X) + c2 (X))) + c1 (X) · c2 (X). 2 12 24 We have the following Riemann-Roch formula on a V-manifold.

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Theorem 3.4. ([Bla96, (3.5) Theorem]) Let X be a n-dimensional projective variety with only isolated quotient singularities and F be a locally V-free sheaf on X. Then there exists a unique mapping µ(X,x) : KV0 V (X, x) → Q such that we have X

χ(X, F) = χorb (X, F) +

µ(X,x) (Fx ).

x∈Sing X

Blache ([Bla96, (3.17) Theorem]) described the term µ(X,x) (Fx ). Let Cn /Zr (a1 , . . . , an ) be a cyclic quotient singularity of Cn by the Zr -action (x1 , . . . , xn ) 7→ (ζra1 x1 , . . . , ζran xn ),

√ where ζr := exp(2π −1/r). For an isolated cyclic quotient singularity, we have the following description of µ(X,x) (Fx ). Theorem 3.5. ([Bla96, (3.17) Theorem]) Let (U, p) := (Cn /Zr (a1 , . . . , an ), 0) be a germ of an isolated cyclic quotient singularity. Let Fp be a V-free sheaf on (U, p) corresponding to a representation ρ : Zr → GL(m, C) which sends the generator 1 ∈ Zr to a diagonal matrix  i1  ζr ..  ρ(1) :=  . ζrim (cf. [Bla96, (2.6) Proposition]). Then we have 1 µ(U,p) (Fp ) = · r

X

Pm

εi j , ai i=1 (1 − ε ) j=1

Qn ∈Mr \{1}

r

where Mr := {ε ∈ C | ε = 1} is the set of r-th roots of unity. 3.2. An invariant of a pair of a 3-fold terminal singularity and its elephant. In this subsection, we explain an invariant of a pair of a singularity and its elephant introduced in [San12, Section 4] to prove the existence of a simultaneous Q-smoothing of a Q-Fano 3-fold, that is, a deformation to a V-smooth pair defined as follows. Definition 3.6. Let U be a 3-fold with only terminal quotient singularities and D ⊂ U its reduced divisor. A pair (U, D) is called a V-smooth pair if, for each point p, there exists a Stein neighborhood Up such that the index one cover πp : Vp → Up satisfies that πp−1 (D ∩ Up ) ⊂ Vp is a smooth divisor. We use the following spaces to define an invariant of a singularity. Settings 3.7. Let p ∈ U be a Stein neighborhood of a 3-fold terminal singularity and D ∈ |−KU | be a member with only Du Val singularity at p. Let π : V → U be an index ˜ := ν −1 (∆) and D ˜ := µ−1 (D). Let ν : V˜ → V be one cover, q := π −1 (p), ∆ := π −1 (D), ∆ ∗ ∗ a Zr -equivariant log resolution of the pair (V, ∆) which induces a crepant resolution of ∆ and has F := ν −1 (q) as its exceptional divisor. Such ν is constructed in [San12, Proposition 4.5]. Let U˜ := V˜ /Zr with the quotient morphism π ˜ : V˜ → U˜ and a birational morphism [0] ˜ Let U 0 := U \{p}, µ : U˜ → U induced by ν. Let FU be the Zr -invariant part of π ˜∗ Ω2V˜ (log ∆).

DEFORMATIONS OF WEAK Q-FANO 3-FOLDS

9

D0 := D∩U 0 and E := µ−1 (p) ⊂ U˜ be the exceptional divisor. Then we have the coboundary homomorphism [0] φ(U,D) : H 1 (U 0 , Ω2U 0 (log D0 )) → HE2 (U˜ , FU ). 1 Let T(U,D) , TD1 be the set of first order deformations of (U, D) and D, respectively. Note that 1 ' H 1 (U 0 , Ω2U 0 (log D0 )) T(U,D)

(8)

since p ∈ U is a terminal singularity and we have KU + D ∼ 0 (cf. [San12, Lemma 4.7]). We have the following property of φ(U,D) . 1 Lemma 3.8. Under the above Settings 3.7, let prD : T(U,D) → TD1 be the forgetful homomorphism. Then we have prD (Ker φ(U,D) ) = 0.

Proof. This is [San12, Lemma 4.9].  Now let (9)

τ (U, D, p) := dim Im φ(U,D) .

We have the following proposition to describe τ (U, D, p). [0] Proposition 3.9. We consider under Settings 3.7. Let F¯U be the Zr -invariant part of ˜ + F )(−F ). π ˜∗ Ω2V˜ (log ∆ [0] (i) We have HE1 (U˜ , F¯U ) = 0. [0] (ii) We have H 2 (U˜ , F¯U ) = 0. (iii) We have [0] τ (U, D, p) = h2E (U˜ , F¯U ), As a consequence, we obtain an exact sequence [0] [0] 1 0 → H 1 (U˜ , F¯U ) → T(U,D) → HE2 (U˜ , F¯U ) → 0.

(10)

˜ + F )(−F )) = 0 . We have an exact Proof. (i) It is enough to show that HF1 (V˜ , Ω2V˜ (log ∆ sequence ˜ + F )(−F )) → H 1 (∆, ˜ Ω1˜ (log Γ)(−Γ)), H 1 (V˜ , Ω2˜ (log F )(−F )) → H 1 (V˜ , Ω2˜ (log ∆ F

F

V

Γ

V

∆

˜ We can check that the both side terms are zero by the same argument as where Γ := F ∩ ∆. the last part of the proof of [Ste97, Theorem 4] since V and ∆ has only Du Bois singularities. Thus we obtain (i). ˜ + F )(−F )) = 0. We have an exact sequence (ii) It is enough to show that H 2 (V˜ , Ω2V˜ (log ∆ ˜ + F )(−F )) → H 2 (∆, ˜ Ω1˜ (log Γ)(−Γ)). H 2 (V˜ , Ω2V˜ (log F )(−F )) → H 2 (V˜ , Ω2V˜ (log ∆ ∆ We see that the both side terms are zero by the vanishing theorem ([PS08, Proposition 7.30. (b)]). Thus we obtain (ii). ˜ is decomposed as (iii) The homomorphism φ(V,∆) : H 1 (V 0 , Ω2V 0 (log ∆0 )) → HF2 (V˜ , Ω2V˜ (log ∆)) ¯

0

φ(V,∆) φV ˜ + F )(−F )) − ˜ H 1 (V 0 , Ω2V 0 (log ∆0 )) −−−→ HF2 (V˜ , Ω2V˜ (log ∆ → HF2 (V˜ , Ω2V˜ (log ∆)).

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TARO SANO

As the Zr -invariant parts, we obtain ¯

0

φ(U,D) [0] [0] φU H 1 (U 0 , Ω2U 0 (log D0 )) −−−→ HE2 (U˜ , F¯U ) −→ HE2 (V˜ , FU ). By (ii), we see that φ¯(U,D) is surjective. Thus it is enough to show that the homomorphism 0 φU is injective. We have a diagram

HF2 (V˜ , Ω2V˜ (log F )(−F ))

φ00



˜ HF2 (V˜ , Ω2V˜ (log ∆ 

+ F )(−F ))

˜ Ω1˜ (log Γ)(−Γ)) HΓ2 (∆, ∆

φ0V

φ0∆

/

/

HF2 (V˜ , Ω2V˜ ) 

˜ HF2 (V˜ , Ω2V˜ (log ∆)) /



˜ Ω1˜ ), HΓ2 (∆, ∆

where the vertical sequences are exact. As in the proof of [NS95, Theorem (1.1)], we see that φ00 and φ0∆ are injective. By the diagram, we see that φ0V is injective, thus φ0U is also injective. Concluding the proof of Proposition 3.9.  Remark 3.10. We shall explain that τ (U, D, p) does not depend on the choice of a resolution ν : V˜ → V . By the exact sequence (10), we obtain [0] 1 τ (U, D, p) = dim T(U,D) − h1 (U˜ , F¯U ).

˜• Let (Ω ((V,∆),q) , F ) be the filtered de Rham complex associated to the pair (V, ∆) at q (cf. [Ste85, (3.1)], [PS08, p. 177]). By [Ste85, (3.4) Corollary], we have an isomorphism ˜ + F )(−F )) ' Hp+q (V, Grp (Ω ˜ •((V,∆),q) )). H q (V˜ , Ωp (log ∆ V˜

F

and its dimension does not depend on the choice of a log resolution ν : V˜ → V of (V, ∆). ˜ + F )(−F )) is independent of ν. Since h1 (U˜ , F¯ [0] ) is its Zr In particular, h1 (V˜ , Ω2V˜ (log ∆ U invariant part, we see that τ (U, D, p) is also independent of ν. Thus it makes sense to say that τ (U, D, p) is a local invariant of a pair (U, D) at p. Remark 3.11. Let (U, D) be a pair as above which is not V-smooth at p ∈ U and U is not Gorenstein. Then we see that Im prD 6= 0 as follows. 1 which induces a By the classification ([Rei87, (6.4)(B)]), there is an element η ∈ T(U,D) deformation of (U, D) to a V-smooth pair. Since it induces a smoothing of ∆ ⊂ V , we see that prD (η) 6= 0. Thus we see that Im prD 6= 0. As a corollary of Lemma 3.8, we obtain the following. Corollary 3.12. Under Settings 3.7, we have the followings. (i) We have (11)

1 τ (U, D, p) ≥ dim T(U,D) − dim Ker prD = dim Im prD .

(ii) The pair (U, D) is V-smooth at p if and only if τ (U, D, p) = 0. Proof. (i) follows from Lemma 3.8. 1 (ii) If (U, D) is V-smooth at p, then we have T(U,D) = 0 and τ (U, D, p) = 0. Suppose that (U, D) is not V-smooth. If Im prD 6= 0, then we obtain τ (U, D, p) 6= 0 by (i). If Im prD = 0,

DEFORMATIONS OF WEAK Q-FANO 3-FOLDS

11

we see that U has a Gorenstein singularity at p and p 6∈ D by Remark 3.11. Then φ(U,D) becomes the coboundary map φ : H 1 (U 0 , Ω2U 0 ) → HE2 (U˜ , Ω2˜ ) U

which is the homomorphism treated in the beginning of [NS95, Section 1]. By [NS95, Theorem (1.1)], this is nonzero since U is not rigid. Thus we obtain (ii).  Corollary 3.12 is useful for computing the invariant τ (U, D, p). Example 3.13. Let r, a be coprime positive integers such that a < r and k, m be positive integers such that m ≥ 2. Let U := (xy + z kr + um = 0)/Zr (a, −a, 1, 0) be a 3-fold terminal singularity and D := (z = 0) ⊂ U be an anticanonical divisor with a Du Val singularity at 0. Then we see that D ' (xy + um = 0)/Zr (a, −a, 0) and Im prD '

m−2 M

C · [ui ],

i=0 i

where [u ] ∈

TD1

corresponds to a deformation of D D := (xy + um + λui = 0)/Zr ⊂ C4 /Zr (a, −a, 0, 0).

Thus, by (11), we see that τ (U, D, p) ≥ m − 1 in this case. 3.3. Formula for the number of singular points. We shall need the following proposition on the universality of Def (X,D) . Proposition 3.14. Let X be a Q-Fano 3-fold with a Du Val member D ∈ |−KX |. Then we have H 0 (X, ΘX (− log D)) = 0, where ΘX (− log D) is the sheaf of tangent vectors vanishing along D. Proof. Let Sing(X, D) := Sing X ∪ Sing D, X 0 := X \ Sing(X, D) and ι : X 0 ,→ X be the ˜ 2 (log D) ˜ 2 (log D) := ι∗ Ω2 0 (log D0 ). Note that ΘX (− log D) ' Ω open immersion. Let Ω X X X since they are reflexive and they coincide outside the finite set Sing(X, D). ˜ → X be a log resolution of the pair (X, D) which is an isomorphism outside Let µ : X ˜ be the µ-exceptional ˜ := µ−1 (D) be the strict transform of D and E ⊂ X Sing(X, D). Let D ∗ 2 2 ˜ ˜ divisor. Then we see that µ∗ ΩX˜ (log D + E) ' ΩX (log D) since it is reflexive (cf.[GKKP11, Theorem 1.5]). ˜ Ω2˜ (log D ˜ + E)) = 0. We have an exact sequence Thus it is enough to show that H 0 (X, X (12)

˜ Ω2˜ (log E)) → H 0 (X, ˜ Ω2˜ (log D ˜ + E)) → H 0 (D, ˜ Ω1˜ (log Γ)), H 0 (X, X X D

˜ Let µD := µ| ˜ : D ˜ → D be the induced birational morphism. We see where Γ := E ∩ D. D 2 2 that µ∗ ΩX˜ (log E) ' µ∗ ΩX˜ and (µD )∗ Ω1D˜ (log Γ) ' (µD )∗ Ω1D˜ since µ∗ Ω2X˜ and (µD )∗ Ω1D˜ are reflexive (cf. [GKK10, Theorem 1.4]). Thus we see that ˜ Ω2˜ (log E)) = H 0 (X, ˜ Ω2˜ ) = 0, H 0 (X, X

X

˜ Ω1˜ (log Γ)) = H 0 (D, ˜ Ω1˜ ) = 0 H 0 (D, D D since X is a Q-Fano 3-fold and D is a K3 surface with only Du Val singularities. By these ˜ Ω2˜ (log D ˜ + E)) = 0. and the exact sequence (12), we see that H 0 (X, X 

12

TARO SANO

We define the basket of singularities of a 3-fold with terminal singularities as follows. Definition 3.15. Let X be a 3-fold with only terminal singularities and Sing X =: {p1 , . . . , pl }. Let Ui be a Stein neighborhood of pi for i = 1, . . . , l. We define the basket of singularities 1 {(ri , ai )}N i=1 of X as follows: each Uj has a deformation Uj → ∆ over an unit disk whose general fiber has quotient singularities of type 1/rj,1 (1, aj,1 , −aj,1 ), . . . , 1/rj,kj (1, aj,kj , −aj,kj ), P 0 −1 and, for i such that i = lj=1 kj + i0 and 0 < i0 ≤ kl0 , we set ri := rl0 ,i0 , ai := al0 ,i0 . We obtain a formula for the bound of the number of singular points of a pair (X, D) of a Q-Fano 3-fold and its elephant as follows. Theorem 3.16. Let X be a Q-Fano 3-fold. Assume that X contains D ∈ |−KX | with only Du Val singularities and it can be deformed to a V-smooth pair (Xt , Dt ). Then we have an inequality N

(13)

X (2ri + 1)(ri − 1) 1 21 − eorb (Xt ) − ≥ 2 2r i i=1

X

τ (U, D, p),

p∈Sing(X,D)

where eorb (Xt ) is the topological orbifold Euler number of Xt ([Bla96, (2.14) Definition]) and {(ri , ai ) | i = 1, . . . , N } is the basket of singularities of X. Proof. We use the same notations as in [San12, Section 4.5]. Let m > 0 be a sufficiently large integer such that |−mKX | contains a smooth member Dm . Let π : Y → X be the cyclic cover branched along Dm . Let µ : Y˜ → Y be the Zm -equivariant resolution constructed in ˜ := Y˜ /Zm be the quotient morphism and µ : X ˜ → X be [San12, Proposition 4.5], π ˜ : Y˜ → X ˜ Zm be the Zm -invariant part. the induced birational morphism. Let F [0] := (˜ π∗ Ω2Y˜ (log ∆)) Let Sing(X, D) =: {p1 , . . . , pl } and Ui be a Stein neighborhood of pi for i = 1, . . . , l. Let µi : U˜i → Ui be a partial resolution of Ui induced by µ and Ei ⊂ U˜i be its exceptional divisor. Then we have the following commutative diagram; (14)

H 1 (X 0 , Ω2X 0 (log D0 )) 

Ll

i=1

H

1

⊕ψi

/

Ll

i=1

˜ F [0] )⊕βi HE2 i (X,

⊕pUi

⊕φ(Ui ,Di ) / (Ui0 , Ω2U 0 (log Di0 )) i

Ll

O

/

˜ F [0] ) H 2 (X,

?

i=1

Im φ(Ui ,Di ) ,

where Ui0 := Ui \ {pi } and Di0 := Di ∩ Ui0 . Since we have βi ◦ φ(Ui ,Di ) = 0 by [San12, Claim 4.11], we see that βi |Im φ(Ui ,Di ) = 0 for L each i. Hence we see that ⊕ψi is surjective onto li=1 Im φ(Ui ,Di ) and, by the definition (9) of τ (U, D, p), obtain 1

dim Def(X, D) = h (X

0

, Ω2X 0 (log D0 ))

≥

l X

τ (Ui , Di , pi )

i=1 1 T(X,D)

1

0

, Ω2X 0 (log D0 ))

since Def(X, D) is smooth and ' H (X by the same reason as (8). Since Def(X, D) is smooth ([San12, Theorem 4.4]) and universal by Proposition 3.14, we see that dim Def(X, D) = dim Def(Xt , Dt ). And we can compute dim Def(Xt , Dt ) = h1 (Xt , ΘXt (− log Dt )) = −χ(Xt , ΘXt (− log Dt ))

DEFORMATIONS OF WEAK Q-FANO 3-FOLDS

13

since we have H j (Xt , ΘXt (− log Dt )) ' H j (Xt , Ω2Xt (log Dt )∗∗ ) = 0 for j = 2, 3. Indeed it is a subquotient of H j+2 (Xt \ Dt , C) = 0 by the Hodge theory on a V-manifold since j + 2 ≥ 4 and Xt \ Dt is a smooth affine 3-fold. In the following, we compute χ(Xt , ΘXt (− log Dt )). Since we have an exact sequence 0 → ΘXt (− log Dt ) → ΘXt → NDt /Xt → 0, we obtain (15) χ(Xt , ΘXt (− log Dt )) = χ(Xt , ΘXt ) − χ(Dt , NDt /Xt ) = χ(Xt , ΘXt ) − χ(Xt , OXt (Dt )) + χ(Xt , OXt ). Let ci := ci (Xt ) := ci (ΘXt ) ∈ H 2i (Xt , Q) be the Chern classes of Xt . By Theorem 3.4, we can compute (16) (17)

X 19 1 1 c1 · c2 + c31 + c3 + µXt ,x (ΘXt ), 24 2 2 x∈Sing Xt X 1 1 µXt ,x (OXt (Dt )). χ(Xt , OXt (Dt )) = c31 + c1 · c2 + 2 8 x∈Sing X

χ(Xt , ΘXt ) = −

t

By Theorem 3.5, if x ∈ Sing Xt is a quotient singularity of type 1/rx (1, ax , rx − ax ), then we obtain 1 1 (18) µXt ,x (ΘXt ) = (σ1 (x) + σax (x) + σrx −ax (x)), µXt ,x (OXt (Dt )) = σ1 (x), rx rx where, for each i ∈ Z, we define σi (x) :=

X ε∈Mrx \{1}

εi , (1 − ε)(1 − εax )(1 − ε−ax )

where Mrx is the set of rx -th roots of unity. By Theorem 3.4, we also obtain (19)

1 1 = χ(Xt , OXt ) = 24

c1 · c2 +

X x∈Sing Xt

rx2 − 1 rx

! .

By using (16), (18) and (19), we obtain (20) χ(Xt , ΘXt (− log Dt )) = −

X 1 1 11 c1 · c2 + c3 + (σa (x) + σrx −ax (x)) + 1 12 2 rx x x∈Sing X t

X 1 11 X rx2 − 1 1 + c3 + (σax (x) + σrx −ax (x)) = −21 + 12 x∈Sing X rx 2 r x x∈Sing X t

Claim 3.17. We can compute (21)

r2 − 6rx + 5 1 (σax (x) + σrx −ax (x)) = x . rx 12rx

t

14

TARO SANO

Proof of Claim. In this proof, we write r := rx , a := ax , σi := σi (x). For k ∈ Z, let k be an unique integer such that 0 ≤ k < r and k ≡ k mod r. By [Rei87, (8.10) Proposition], we have   2 i−1 r2 − 1 1 r − 1 X bj(r − bj) σ0 = , σi = −i + , 24 2 12 2 j=0 where b ∈ Z is defined by ab ≡ 1 mod r and 0 < b < r. We obtain ! r−1 X j(r − j) 1(r − 1) r2 − 1 + − . (22) σa + σr−a = (1 − r) 12 2 2 j=1 We can compute (23)

r−1 X j(r − j) j=1

2

r−1

=

r−1

rX 1X 2 j− j 2 j=1 2 j=1

r r(r − 1) 1 1 r(r − 1)(r + 1) · − · (r − 1)r(2r − 1) = . 2 2 2 6 12 Putting (22) and (23) together, we obtain the claim.  =

By putting (20) and (21) together, we obtain X (2rx + 1)(rx − 1) 1 χ(Xt , ΘXt (− log Dt )) = −21 + c3 + 2 2rx x∈Sing X and thus obtain the required inequality (13) since we have c3 = eorb (Xt ) ([Bla96, (2.14) Theorem]). Concluding the proof of Theorem 3.16.  4. Deformations of a Q-Fano 3-fold via its canonical covering stack In this section, we explain the canonical covering stack associated to a 3-fold with only terminal singularities. We use it to prove the unobstructedness of deformations of a Q-Fano 3-fold. Let X be a 3-fold with only terminal singularities. Let Sing X =: {p1 , . . . , pl }, pi ∈ Ui a small affine neighborhood of pi and πi : Vi → Ui the index one cover ` for i = 1, . . . , l. Let V0 := X \ Sing X and π0 : V0 → X the open immersion. Let V := li=0 Vi and π : V → X the morphism such that π|Vi = πi for i = 0, . . . , l. Let W := V ×X V and consider the ´etale groupoid space p1

W

p2

/ /

V.

Let X be the associated Deligne-Mumford stack (cf. [Kaw02, Definition 6.1]). Let γ : X → X be the morphism to the coarse moduli space. We can define a functor Def X : ArtC → (Sets) of deformations of the stack X over Artinian rings as in the case of schemes (cf. [San12, Definition 2.1]). Remark 4.1. Hacking ([Hac04, Section 3]) considered a canonical covering stack of a slc surface and its deformation theory. The theory is parallel in the case of a 3-fold with terminal singularities.

DEFORMATIONS OF WEAK Q-FANO 3-FOLDS

15

Lemma 4.2. (cf. [Hac04, Proposition 3.7]) There is an isomorphism of functors (24)

∼

c : Def X − → Def X

which sends a deformation of X to its coarse moduli space. Proof. We can construct a natural transformation c0 : Def X → Def X as follows; Let Xn → Spec An be a deformation of X over An . Let {Ui }ni=1 be an affine open sets of X as above and Ui,n → Spec An be a deformation of Ui induced by Xn . We see that Ui,n is induced by a Zri -equivariant deformation Vi,n → Spec An of the index one cover Vi of Ui (cf. [San12, ` Proposition 4.2]). Let V0,n be a deformation of V0 induced by Xn . Let Vn := li=0 Vi,n . We can construct an associated Deligne-Mumford stack Xn similarly as X . We see that Xn is flat over An since we can check it locally. We can check that c0 is an inverse of c.  We can construct obstructions for deformations of X as follows. Proposition 4.3. Let X be a 3-fold with terminal singularities and X its canonical covering stack. Then we can define an obstruction oξn ∈ Ext2 (Ω1X , OX ) to lift a deformation ξn ∈ Def X (An ) to An+1 . Proof. The construction is parallel to [Ser06, Proposition 2.4.8] or [San12, Proposition 2.4].  Theorem 4.4. Let X be a Q-Fano 3-fold. Then the deformation functor Def X is unobstructed. Proof. Let γ : X → X be the canonical covering stack of X constructed as above. By the isomorphism (24), it is enough to show that Def X is a smooth functor. We have isomorphisms Ext2 (Ω1X , OX ) ' Ext2 (Ω1X ⊗ ωX , ωX ) ' H 1 (X, Ω1X ⊗ ωX )∗ . The first isomorphism follows since ωX is invertible. This is a main advantage of considering the canonical covering stack. The second isomorphism follows from the Serre duality on a Deligne-Mumford stack ([Nir08, Corollary 2.10]). Moreover, we have an isomorphism H 1 (X, Ω1X ⊗ ωX ) ' H 1 (X, ι∗ (Ω1X 0 ⊗ ωX 0 )), where ι : X 0 ,→ X is an open immersion of the smooth part X 0 of X. We can check this by the construction of γ : X → X. Indeed, we have an isomorphism γ∗ (Ω1X ⊗ ωX ) ' ι∗ (Ω1X 0 ⊗ ωX 0 ) since both sheaves are reflexive. Thus it is enough to check H 1 (X, ι∗ (Ω1X 0 ⊗ ωX 0 )) = 0. This can be checked by a variant of Lefschetz hyperplane section theorem as in [San12, Theorem 2.11].  Remark 4.5. The proof of Theorem 4.4 gives a new proof of [San12, Theorem 1.7]. In the proof of [San12, Theorem 1.7], we need to compare delicately the deformations of a Q-Fano 3-fold and its smooth part. The new proof avoids this issue. By using the canonical covering stack, we can also prove the unobstructedness of QGorenstein deformations of a log del Pezzo surface.

16

TARO SANO

Theorem 4.6. (cf. [HP10, Proposition 3.1], [ACC+ 15, Lemma 6]) Let S be a log weak del Pezzo surface, that is, a normal projective surface with only klt singularities with a nef and big anticanonical divisor −KS . Then the Q-Gorenstein deformation functor Def qG S is unobstructed. (See [Tzi09, DefiniqG tion 2.2] for the definition of Def S ) Proof. Let S → S be the canonical covering stack of S induced from canonical coverings of singularities of S ([Hac04, 3.1]). Since we have Def S ' Def qG S ([Hac04, Proposition 3.7]), it is enough to show that Def S is unobstructed. We see that Ext2 (Ω1S , OS ) ' Ext2 (Ω1S ⊗ ωS , ωS ) ' H 0 (S, Ω1S ⊗ ωS ). We also see that H 0 (S, Ω1S ⊗ ωS ) ' H 0 (S, (Ω1S ⊗ ωS )∗∗ ) ' Hom(OS (−KS ), (Ω1S )∗∗ ). This vanishes by the Bogomolov-Sommese vanishing theorem on a log canonical variety. Thus we are done.  Acknowledgment The author would like to thank Professor Yoshinori Namikawa for useful comments about deformations of a weak Q-Fano 3-fold and an invariant τ (U, D, p) in Section 3.2. Theorem 1.2 is an answer to his question. The author would like to thank Professor Alessio Corti for his lecture in Udine in 2014 which leads the author to an idea of using canonical covering stacks. The author would like to thank Professors Yuri Prokhorov, Meng Chen and Doctor Chen Jiang for their suggestion to consider the bound of singularities on a Q-Fano 3-fold. He was partially supported by Max Planck Institute for Mathematics, JSPS fellowship for Young Scientists and JST tenure track program. References +

[ACC 15]

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