The anticanonical system of Fano manifolds of index n − 3
M´emoire de M2 de
Enrica Floris
Directeur de stage
Andreas H¨oring
Universit´e Pierre et Marie Curie
Index
Introduction . . . . ii
1 Preliminaries 1 1.1 Positivity . . . . 2
1.2 Singularities of pairs . . . . 3
1.3 Riemann Roch Theorems . . . . 8
1.4 Stability . . . . 9
2 A theorem of Kawamata 11 3 Generalizations 18 3.1 Fano varieties of indexn−3 . . . . 18
3.2 An inductive approach . . . . 23
4 A theorem of Shokurov 26
Bibliography 29
Introduction
A Fano variety is ann-dimensionalQ-Gorenstein projective varietyX with ample anticanonical divisor−KX. Theindex of a Fano variety X is
i(X) = sup{t∈Q| −KX ∼Q tH, H ample, Cartier}.
It is well known thati(X)≤n+ 1. IfX has log terminal singularities, the Picard groupP ic(X) is torsion free. Therefore the Cartier divisor H such that −KX ∼Q i(X)H is determined up to isomorphism. It is called the fundamental divisor ofX. By the Kawamata-Viehweg vanishing theorem,
hi(X, H) = 0 ∀i >0.
It is natural to investigate the existence of global sections for the fundamen- tal divisor, that is the non vanishing of the group H0(X, H).
Once we know thatH0(X, H)&= 0, the next natural question arising is what kind of singularities a general element in |H| may have. It is said that a Fano variety has good divisors if the generic member of the fundamental divisor has at worst the same type of singularities asX.
In this work we give a detailed proof of a theorem taken from [7] that deals with the case of Fano varieties of dimension four:
Theorem 0.0.1(Kawamata). LetX be a projective variety of dimension 4 with at most Gorenstein canonical singularities. Assume that−KX ∼H is ample. Then the following hold:
• H0(X, H)&= 0;
• letY ∈ |H|be a general member, then(X, Y)is plt. HenceY has only canonical singularities.
The first point gives an answer to the problem of existence of global sections and the second to the problem of determining the singularities of the general member of the anticanonical system. The proof of the Theorem 0.0.1 is dealt with in the second Chapter.
In Chapter 3 we consider generalizations of Kawamata’s theorem to higher dimension, more precisely we study smooth Fano varieties of index
n−3. By the Kobayashi-Ochiai criterion, if i(X)≥nthenX is isomorphic to a hyperquadric or to a projective space. Smooth Fano varieties of in- dexn−1 have been classified by Fujita and smooth Fano varieties of index n−2 by Mukai. Then the varieties of indexn−3 are a class of varieties of
“big” index far from being understood. The structure of a general element Y ∈ |H|should play an important role in their classification. In this context, we propose the following theorem:
Theorem 0.0.2. Let X be a smooth Fano variety of dimensionn≥4 and indexn−3, with H the fundamental divisor.
1. If n≤5 thenh0(X, H)≥n−2.
2. If n= 6 and the Picard number is one thenh0(X, H)≥n−2.
3. Ifn≥7 and the tangent bundleTX isH-semistable, then h0(X, H)≥ n−2.
4. Suppose thath0(X, H)≥1and let Y ∈ |H|be a general member, then (X, Y) is plt.
As in the statement of Theorem 0.0.1 the first points establish the non vanishing of the global sections group and the last discuss the regularity of the general member.
The first three points of the theorem result from the following steps:
• first we prove the following formula χ(X, H) = Hn
24 (−n2+ 7n−8) +c2(X)Hn−2
12 +n−3.
• If n = 4,5 the result follows from the semipositivity of the tangent bundleTX [15] and the inequality of Miyaoka [13].
• If the tangent bundle TX is H-semistable, from the Bogomolov in- equality we obtainh0(X, H)≥n−2.
• finally in [5] Hwang proved that a Fano variety of dimension six and Picard number equal to one has semistable tangent bundle.
An important remark is that the semistability of the tangent bundle is con- jectured for all Fano varieties of Picard number equal to one.
Finally, in the last Chapter, we discuss the following famous theorem Theorem 0.0.3 (Shokurov [17]). Let X be a smooth Fano variety of di- mension three. ThenY ∈ | −KX|general is smooth.
We give a simplified proof by using an argument taken from [4].
Chapter 1
Preliminaries
We will use the standard notation from [3], everything is defined over C. In the following ≡, ∼ and ∼Q will indicate numerical, linear and Q-linear equivalence of divisors.
Definition 1.0.4. [3, p. 241] Let X be a proper scheme of dimension n.
A dualizing sheaf forX is a coherent sheaf ω◦X onX, together with a trace morphism t:Hn(X, ωX◦)→C, such that for all coherent sheaf F on X, the natural pairing
Hom(F, ωX◦)×Hn(X,F)→Hn(X, ωX◦) followed by t gives an isomorphism
Hom(F, ωX◦)→Hn(X,F).
Definition 1.0.5. Let X be a nonsingular variety and ΩX its cotangent sheaf. The canonical sheaf of X is
KX: = detΩX.
Corollary 1.0.6. IfXis a projective nonsingular variety, then the dualizing sheaf ω◦X is isomorphic to the canonical sheaf KX.
Theorem 1.0.7 (Serre duality). [3, p. 244] Let X be a scheme that is projective Cohen Macaulay of equidimensionn overC. Then for any locally free sheaf F over X there are natural isomorphisms
Hi(X,F)∼=Hn−i(X,F∨⊗ω◦X).
Remark 1.0.8. Let X be a normal variety, let KXreg be the canonical di- visor of Xreg. We can write KXreg = !
aiDi where Di is an irreducible subvariety of codimension one in Xreg. In this case we define the canonical divisor ofX as KX =!
aiD¯i where D¯i is the closure in X of Di. Since X is nonsingular in codimension one, KX is the only Weil divisor on X such thatKX|Xsing =KXsing.
Definition 1.0.9. A normal projective varietyXis said to beQ-Gorenstein if the Weil divisorKX is Q-Cartier.
Proposition 1.0.10 (Adjunction formula). [3, p. 182] LetX be a normal projective variety and let Y be an effective Cartier divisor. Let ωX be the dualizing sheaf onX. Then the dualizing sheaf of Y is given by the formula
ωY ∼(ωX⊗ OX(Y))|Y.
Theorem 1.0.11 (Bertini’s theorem). [3, p. 179] Let X be a nonsingu- lar closed subvariety of PnC. Then there exists a hyperplane H ⊆ PnC, not containing X and such that the scheme X ∩H is regular at every point.
Furthermore, the set of hyperplanes with this property forms an open dense subset of the complete linear system |H|, considered as a projective space.
Proposition 1.0.12. [2] Let X and X$ be schemes and f:X → X$ be a proper morphism. LetL be a line bundle on a scheme X, let α be a k-cycle onX$. Then
f∗(c1(f∗L)·α) =c1(L)·f∗(α).
1.1 Positivity
Definition 1.1.1. [10, p. 24] Let X be a projective variety, and L a line bundle onX.
• L is very ample if there exists a closed embedding of X into some projective space PN such that
L=OX(1) : =OPN(1)|X.
• L is ample if L⊗m is very ample for some m >0.
A Cartier divisorDis ample or very ample if the corresponding line bundle OX(D) is so.
Theorem 1.1.2. [10, p. 33] LetLbe a line bundle on a projective scheme X. Then L is ample if and only if
"
V
c1(L)dim(V)>0
for every positive dimensional irreducible subvariety V ⊆X (including the irreducible components of X).
Definition 1.1.3. [10, p. 51] Let X be a projective variety. A Cartier divisorD on X is nef if
(D·C)≥0 for all irreducible curves C⊆X.
Definition 1.1.4. [15, p. 1] A vector bundle E on a projective manifold X of dimensionnis generically nef if the following holds. Given any ample vector bundles Hj, 1 ≤ j ≤ n−1, let C be a curve cut out by general elements in|mjHj|for mj .0, then the restrictionE|C is nef.
Theorem 1.1.5. [15, p. 14] Let X be a projective manifold with −KX semiample. ThenTX is generically nef.
Theorem 1.1.6. [13, p. 468] Let X be a normal projective variety of dimension n smooth in codimension two. Let E be a torsion free sheaf on X, with first Chern class being a numerically effective Q-Cartier divisor.
Assume thatE is generically nef, then for every h1, . . . , hn−1 ample divisors onX the inequality
c2(E)h1. . . hn−1≥0 holds.
Theorem 1.1.7. [13, p. 468] Let X be a normal projective variety of dimension n and ρ: X$ → X an (arbitrary) resolution. Assume that X is smooth in codimension two and the canonical divisor is Q-Cartier and numerically effective. Then, for any numerically effective Cartier divisors H1, . . . , Hn−2, the inequality
c2(X$)(ρ∗H1). . .(ρ∗Hn−2)≥0 holds.
1.2 Singularities of pairs
Definition 1.2.1. [11, p. 143] Let D=!
Di be a divisor on a manifold X. It is said to have simple normal crossings (andDis said a SNC divisor) if eachDi is smooth, andD is defined in a neighborhood of any point by an equation in local analytic coordinates of type
z1·. . .·zk = 0 for some k ≤ n. A Q-divisor D = !
aiDi has simple normal crossing support if !
Di is SNC.
Definition 1.2.2. [9, p. 5] Let D =!
aiDi be a Q-divisor on a variety X. A log resolution of D (or of the pair (X, D)) is a projective birational morphism
µ:X$ →X,
such that X$ non singular, the exceptional locus Exc(µ) has codimension one and the divisorµ∗D+ Exc(µ)has simple normal crossing support. Here Exc(µ) denotes the sum of all the exceptional divisors of µ.
Definition 1.2.3. [11, p. 182] A pair (X,∆) consists of a normal variety X, together with a Weil Q-divisor ∆ = !
ai∆i on X with ai ∈(0,1], such that theQ-divisor KX+ ∆ is Q-Cartier onX.
Definition 1.2.4. [10, p. 183] The multiplier ideal of the divisor D for the pair (X,∆) is the sheaf
J((X,∆), D) =µ∗OX!(KX!− /µ∗(KX + ∆ +D)0).
Theorem 1.2.5 (Nadel vanishing theorem). [11, p. 191] Let (X,∆) be a pair and letD be a Q-Cartier Q-divisor on X.
• If µ:X$ →X is a log resolution of(X, D+ ∆), then RjOX!(KX!− /µ∗(KX + ∆ +D)0) = 0 ∀j >0.
• LetN be any integral Cartier divisor onXsuch thatN−(KX+∆+D) is nef and big. Then
Hi(X,OX(N)⊗ J((X,∆), D)) = 0 ∀i >0.
Definition 1.2.6. [6, p. 16] Let(X,∆)be a pair,∆ =!
ai∆i withai∈R. Suppose thatm(KX+∆)is Cartier form >0. Letf:Y →X be a birational morphism, Y normal. We can write
KY ≡f∗(KX + ∆) +#
a(Ei, X,∆)Ei.
whereEi ⊆Y are distinct prime divisors anda(Ei, X,∆)∈R. Furthermore we adopt the convention that a nonexceptional divisorE appears in the sum if and only if E=f∗−1Di for somei and then with coefficient a(E, X,∆) =
−ai.
Thea(Ei, X,∆) are called discrepancies.
Definition 1.2.7. [9, p. 52] Let (X,∆) be a pair. We set
discrep(X,∆) = inf{a(E,X,∆)|E exceptional divisor over X} and
totaldiscrep(X,∆) = inf{a(E,X,∆)|E divisor over X}.
Lemma 1.2.8. [9, p. 53] Let (X,∆)be a pair, then eitherdiscrep(X,∆) =
−∞ or −1≤totaldiscrep(X,∆)≤discrep(X,∆)≤1.
Definition 1.2.9. [9, p. 56] A pair(X,∆) is defined to be
• klt (kawamata log terminal) ifdiscrep(X,∆)>−1 and /∆0 ≤0
• plt (purely log terminal) ifdiscrep(X,∆)>−1
• lc (log canonical) if discrep(X,∆)≥ −1.
Remark 1.2.10. [11, p. 165] Since J(X,∆) =OX if and only if ordE(KX! −µ∗(KX+ ∆))>−1 ∀ E ⊆ X$,
the pair (X,∆) is klt if and only if J(X,∆) =OX and (X,∆) is lc if and only ifJ(X,(1−ε)∆) =OX for all 0< ε <1 and J(X,∆)&=OX.
Lemma 1.2.11. [9, p. 57] Let (X,∆) be a pair and ∆$ an effective Q- CartierQ-divisor. If (X,∆) is klt, then (X,∆ +ε∆$) is klt for 0≤ε21.
In the klt case the Nadel theorem gives the Kawamata-Viehweg vanishing theorem.
Corollary 1.2.12. Let (X,∆) be a klt pair. Let N be any integral Cartier divisor on X such thatN −(KX + ∆) is nef and big. Then
Hi(X,OX(N)) = 0 ∀i >0.
Remark 1.2.13. Let(X,∆)be a klt pair and letA be an ample line bundle on X. Suppose that h0(X, A) &= 0, then D ∈ |A| is connected. In fact, if we consider the exact sequence of cohomology groups associated to the exact sequence of sheaves
0→ OX(−D)→ OX → OD →0,
by the Theorem 1.2.12 he have H1(X,OX(−D)) = 0. Thus we have an exact sequence
0→H0(X,OX(−D))→H0(X,OX)→H0(D,OD)→0.
We know that H0(X,OX) is the set of the constant functions on X, and H0(X,OX(−D))⊆H0(X,OX)is the set of constant functions onXthat are zero onD and soH0(X,OX(−D)) ={0}andh0(D,OD) =h0(X,OX) = 1.
ThusD is connected.
Definition 1.2.14. [6, p. 61] A variety X is said to have rational singu- larities if, for anyµ:X$→X resolution of singularities, we have
Riµ∗OX! = 0 ∀i >0.
Theorem 1.2.15 (Elkik, Flenner). [6, p. 61] Let X be a normal variety.
Assume that(X,∆) is a klt pair. ThenX has rational singularities.
Remark 1.2.16. As a direct consequence of Theorem 1.2.15, if X is a variety with rational singularities, we can compute the cohomology groups of a Cartier divisor on X on a smooth variety. Let L be a Cartier divisor on X,µ:X$→X a resolution of singularities. By the Leray spectral sequence, we have
Hi(X$, µ∗L) =Hi(X, L) ∀i∈ {0, . . . , n}.
Definition 1.2.17. Let (X,∆) be a klt pair, D an effective Q-Cartier Q- divisor. The log canonical threshold of D for (X,∆) is
lct((X,∆),D) = sup{t∈R+|(X,∆ + tD) is LC}. Remark 1.2.18. The threshold is also
lct((X,∆),D) = sup{t|(X,∆ + tD) is klt}.
We define a pair to be properly log canonical if it is LC and not klt. Thus if c= lct((X,∆),D), the pair (X,∆ +cD) is properly log canonical.
Remark 1.2.19. [11, p. 166] The log canonical threshold is a rational number and the supremum appearing in the definition is actually a maxi- mum. In fact let (X,∆) be a pair and D ≥0 a Q-Cartier Q-divisor. Let f:X$ →X be a log resolution of Dand write
f∗(KX + ∆ +tD) =f∗(KX + ∆) +tf∗D=KX! +B∆+tBD. We writeB∆+tBD =!
i∈I(bi+tai)Ei+ ˜DwhereD˜ is sum of non exceptional divisors andEiis exceptional for alli∈I, withI a finite set. The coefficient bi refers to B∆ and ai to BD. Since (X,∆) is klt, we have bi < 1 for all i∈I. We have
bi+tai ≤1⇔t≤ 1−bi
ai and c= mini∈I{1−aibi} ∈Q.
Definition 1.2.20. Let (X,∆) be a pair that is lc,f:X$ →X a log canon- ical resolution. Let E ⊆ X$ be a divisor on X$ of discrepancy −1. Such a divisor is called a log canonical place. The closure of f(E) is called center of log canonicity of the pair(X,∆) and is denotedCenterX(E). If we write
KX! ≡µ∗(KX + ∆) +E,
we equivalently can define a place as an irreducible component of/−E0. We denote CLC(X,∆) the set of all centers.
Remark 1.2.21. If (X,∆) is log canonical,
{x|(X,∆) is not klt near x}=∪ECenterX(E) where the union runs over the places.
Proposition 1.2.22. Let (X,∆)be a klt pair and D an effectiveQ-divisor Q-Cartier divisor such that (X,∆ +D) is log canonical. Then CLC(X, D) is a finite set and if W1, W2 ∈CLC(X, D), all the irreducible components of W1∩W2 are in CLC(X, D).
Lemma 1.2.23. Let X be a normal variety and ∆ a divisor on X such that (X,∆) is klt. Let H be an ample Cartier divisor on X and Y ∈ |H| a general element. Suppose that (X,∆ +Y) is not plt and let c be the log canonical threshold. Then the union of all the centers of log canonicity of (X,∆ +cY) is contained in the base locus of |H|.
Proof. Letµ:X$ →X be a log resolution of the pair (X,∆). We have KX! ≡µ∗(KX + ∆) +#
i∈I
aiEi and ai>−1 ∀i∈I.
If h0(X, H) = 1 the statement is trivial so we can suppose h0(X, H) > 1.
If h0(X, H) > 1, then by Remark 1.2.16, we have h0(X$, µ∗H) > 1. By Bertini’s Theorem 1.0.11 there existsµ∗Y ∈ |µ∗H|that meets properly and transversally all the Ei that are not contained in the base locus. Then µ∗Y =Y$+F, whereµ(F)⊆Bs|H|. We can write
KX! ≡µ∗(KX+ ∆ +Y)−Y$+#
aiEi−F.
The divisor−Y$+!
aiEihas simple normal crossing support onµ−1(X\Bs|H|) because of the choice ofY. If we setX◦$ =µ−1(X\Bs|H|), we have
KX!
◦ ≡(µ∗(KX + ∆ +Y)−Y$+#
aiEi)|X◦!.
Thenµis a log resolution of the pair (X\Bs|H|,∆ + Y\Bs|H|), which is plt.
Finally, if (X\Bs|H|,∆+Y\Bs|H|) is plt, so is (X\Bs|H|,∆+cY\Bs|H|).
Remark 1.2.24. Let (X,∆) be a properly lc pair. Let Z the union of all centers in X. Let µ: X$ → X be a birational morphism such that X$ is smooth,
KX! =µ∗(KX + ∆)−E−F,
the coefficients of F are smaller than one and Supp(E + F) is a simple normal crossing divisor. Then
J(X,∆) = µ∗OX!(KX!− /µ∗(KX + ∆)0)
= µ∗OX!(KX!− /KX! +E+F0)
= µ∗OX!(−E− /F0)
= µ∗OX!(−E) =IZ.
Definition 1.2.25. Let (X,∆) be a log canonical pair. A centerW is said to be exceptional if there exists a log resolution µ: X$ → X for the pair (X,∆) such that:
• there exists only one place EW ⊆X$ whose image inX is W;
• for all placeE$ &=EW, we have µ(E)∩W =∅.
1.2.1 Tie break lemmas
Theorem 1.2.26. [1, p. 71] Let (X,∆) be a klt pair and D an effective Q-divisorQ-Cartier such that(X,∆+D)is properly lc. LetW be a minimal center for the pair (X,∆ +D) and H an ample Cartier divisor on X. For all rational number 0 < r 21 there exist c1, c2 ∈Q, 0< c1, c2 ≤r and an effective Q-divisor A∼Q c1H such that the pair(X,∆ + (1−c2)D+A) is log canonical and W is an exceptional center of log canonicity for it.
Theorem 1.2.27 (Kawamata, [8]). Let (X, D) be a lc pair and W an exceptional center. LetH be an ample divisor andε >0 a rational number.
ThenW is normal and there is an effectiveQ-divisor BW onW such that
• (W, BW) is a klt pair;
• KW +BW ∼Q (KX +D+εH)|W.
Theorem 1.2.28. [6, p. 263] Let X be a normal variety and S ⊆ X an irreducible Cartier divisor. Let B be an effectiveQ-divisor and assume that KX+S+B isQ-Cartier. Then
(X, S+B) is plt near S⇔(S,B|S) is klt . Assume in addition that B is Q-Cartier and S is klt. Then
(X, S+B) is LC near S⇔(S,B|S) is LC .
1.3 Riemann Roch Theorems
Theorem 1.3.1(Riemann Roch). LetX be a nonsingular projective variety of dimension 3, with Chern classes c1(X), c2(X). Then for any Cartier divisorD,
χ(X, D) = 121D(D−KX)(2D−KX) +121Dc2(X) +241c1(X)c2(X)
= 16D3−14D2KX +121D(KX2 +c2(X)) + 241c1(X)c2(X) Remark 1.3.2. We can define the intersection product (D1 ·. . .·Dn) if one of the divisors is just a Weil divisor. In fact, suppose that D1 is an irreducible hypersurface and D2. . . Dn are Cartier divisors, then
(D1·. . .·Dn) : = (D2|D1 ·. . .·Dn|D1).
