10.3 Fundamental Theorems for D-Dimension 305
f Finally um havg t0 prnve the nrpquene SS [heor em cfw_0:- [he D=Canonical
bered variety. We begin by showing the uniqueness theorem for y.
Lemma 10.4. Let f : V! W, be a bered variety. Suppose that V is norm
324 11 Logarithmic Kodaira Dimension of Varieties
Then since f(V) E W, one has @(A1(V) g ,u1(W), and so (V:
7;,(p(W)> EdaXV is induced for every M. Now, dene f* to be 972*.
Thus, one has the linear map f*: TM(W)+ TM(V). By Proposition 11.2 and
330 ll Logarithmic Kodaira Dimension of Varieties
' _ _ l
DU_1]13 not smooth at pup. Then ,letting m,- i 4pm, Bil-4]), 151- A y,- (
and W = #j+1 O "' o ,u, we have the formula:
K(Si) + thi_ (P0(K(S) + D) 11: (m 2)(Pik(Ej)-
(Dm) (D) Z (mj 1) )j(m 2
11.5 Etale Endomorphisms 337
Letting Z u1(V) and A: u1(D) one has the logarithmic rami-
cation formula for(p= (pIZ: Z= V+ V; ie.
K(Z) + A ~ *(K(I7) + D) + Rw.
S_ince ,u* K(Z)~K(I7) by Lemma 11.9 and ,u*(A):D, letting Rf be
,u* R, , one has the formula:
11.4 Logarithmic Ramication Formula 335
Here I denotes Fm(V) 1.
Putting F = (1),(D), F = (Dm W(B), and F = (DmZ(A), where Z = Z\A,
one has B C @(A) and so F = (Dm W(B) ; (Dm W o @(A) = pm(p) o
(1)," Z(A ): p,(p( )(F) Furthermore, since (A) = D, one h
326 ll Logarithmic Kodaira Dimension of Varieties
when V iS a SITIUULII C011c1p1 16ti011
boundary D, the logarithmi mgenu
a nonsirgular var1ety V wit11 smooth
13m(V) = l,(m K07) + D) for each m.
Denition. The logarithmic Kodaira dimension of
10.7 Subvarieties of an Abelian Variety 315
the hypothesis ofthe following lemma; hence tI/ is birational and soh = q lg r
(p1 has the required property, where (1: Y x W+ W is the projection
I emma 10.8. Let 1h: UH Y be a dominating mo
314 10 D-Dimension and Kodaira Dimension of Varieties
0 implies pg( V) = 1. Thus, we shall prove that
-o be (I 1 A A dun.1 A (112,-, which
is an element of T,(V ). Since pg(V) = l by hypothesis, letting (I), be
dv1 A A dv", which is not zero, one has T,(V
n_ . 1
Chupl 16-1 11
Logarithmic Kodaira Dimension
11.1 Loga i thmic Forms
3. In this chapter we also x an algebraically ClosesI eld k with character-
istic zero ansl assume that all varieties are over k.
Denition. If D is a _divisor with sim
11.1 Logarithmic Forms 323
d. With the notation n as above, we shall prove the proper birational i1-
variance of TM( V, D).
Lemma 11.2. Let fzflyz V> W. Then f is proper if and only if
PROOF. Let V1 2 I7\f(B), which contains V. Since (p =17|V1 i
11.2 Logarithmic Genera 325
Theorem 11.2. Let f: V> W be a strict.
(i) Pullback offorms induces a k-linear map f*: TM(W)!> TM(V) for every
(ii) Iffis dominating, thenf* is one-toone.
(iii) Ifh: Wv U is a strictly rational map such that the composition
1 1.3 Reduced Divisor as a Boundary 329
/ / v
Then, since LO + L1 + L2 + L3 has only simple normal crossings, iE(S) can
be computed as K(K(Pk2) + 4L0, ,2), which is 2.
Lemma 11.3. Afamily oflines is oftype I, or type II, or type 11% or
10.7 Subvarieties of an Abelian Variety 317
also an Abelian variety (cf. Corollary to Theorem 10.12). Let Z be g( W) and
so n(V) = Z. Hence, letting (p = IV! one has a surjective morphism
(p: V>Zsuch that (p n ,u = g of
f l w
334 ll Logarithmic Kodaira Dimension of Varieties
b. Example ll. 11. Let W be a s111ooth 00111pletion of W with a smooth
boundary B, and C be a nonsingular closed subvariety of W contained in
F1, . S, where the F are the irreducible components of B We let
IO.7 Subvarieties of an Abelian Variety 313
(ii) 'T'lno fnllnuiing nnndihnns are enulvalo I:
(a) V is a translation of an Aeb lian subvariety (i.,e a connected closed
algebraic subgroup) of.
(1?) KW): .
(c) pg(V) = 1.
(d) q(V) = -
PROOF. (i) Sin
338 11 Logarithmic Kodaira Dimension of Varieties
3. Given a variety V with K(V) 2 0, we let I7 be a completion of V and let
(V_#, ,u) be a nonsingular model of I7 such that D# = y(V_\V) has only
simple normal crossings. Applying Theorem 10.3 to
11.2 Logarithmic Genera 327
some polynomial x]; and F can be written as a polynomial in 1/1. In par-
ticular, k(X1, . . . X,)/k(l/) is an algebraically closed extension of elds.
This can be easily shown as follows: F determines a dominating mor-
phism f f
304 [0 DDimension and Kodaira Dimension of Varieties
where m* and mx are eonstari [56! deep end in on ' A. Thus
ax m"* s m" for allm 2*,max(m m ).
On the other hand (f0 (:1: ) _ 1c(*); hence (f0 (11) )< 1c(x). [j
e. Lemma 10.3. Let f: V> W be anite surjec
10.5 Kodaira Dimension 309
10.5 Kodaira Dimension
a. Denition. Let V be a nonsingular complete variety. The Kodaira dimen-
sion K(V) of V is dened to be K(K(V), V).
For any m > 0, Pm(V) = ly(mK(V); hence by Theorem 10.2, there exist a,
B > 0 mph that
10.4 D-Dimensions of 21 K3 Surface and an Abelian Variety 307
., D, e divi
(i) For any m1 > 0, .,m, >0,ifrc(D1, V) 2 0, ., K(D, V) Z 0, then
K(Z Bi, V) = K(Z mJ-Dj, V).
(ii) IfK(D1, V) 2 0, then K(D, V) s K(D + D1, V).
cfw_$10.6 Types of Varieties 311
may accnmp V ic nnncingular pomp/data By Th6 pox-g.
- - o . . .e m
10.4a nd Lemma 10.6 on e obtains the result. 1:]
Corollary. Let f: Va W be abered variety.
(i) If ic(f"(x) = oofor general points x of W, then rc(V) = 00.
312 10 D-Dimension and Kodaira Dimension of Varieties
Finally, V is said to be ofber type or type 11%, if 0 < .C( V) < n. Then, by
Theorem 10.7 there is a canonical bered variety f: V! + W such that V!
is birationally equivalent to V, dim W = x( V), and g
332 11 Logarithmic Kodaira Dimension of Varieties
Propesmen 11.8. If q(S) = 0 and D is connected, t.e.
13., (S\D) PAS) = 71(0) - 2("11- 1)(mj - 2V2-
In particular, ifS = Pf and D is a reduced divisor of degree 6, then letting
S0 = PAD, one has
306 10 D-Dimension and Kodaira Dimension of Varieties
Composing these homomorphisms, one obtains a homomorphism
r: QA(V\D)> Rat(Z). By hypothesis (2), one has dim Z = K(D, V), i.e.,
tr. degk QA(V\D). Hence, I is nite, i.e., Rat(Z)/QA(V\D) is nite, and so
(3) If k( V) = m > 0, then V is isomorphic to a product W x G',:,", where W
is a closed subvariety of some sz with 12(W) = m = dim W.
Let D be a reduced divisor on 0"," and let V be Gn\D. Show t
318 10 D-Dimension and Kodaira Dimension of Varieties
generalbers are isomorphic to some Abelian subvariety B. Moreover, there
exists an tale cover V1 of V such that V, = B X Z], where Z1 is a closed
subvariety ofsome Abelian variety with dim Z1 2 K(Zl).
322 ll Logarithmic Kodaira Dimension of Varieties
['7 4% 7 ha a mnrn hism S"Ch that] fcfw_I/I C W. Fnr a lnanrithmic 1-fnrm 11w) nf
uu u nu. PIA u. u nub.
W along B, the pullbackf*w 1s a rational 1- form (cf. 5. 4. b). We claim that
f*w is a logarithmic l
11.1 Logarithmic Forms 321
PROOF. Let 23:1 1",- be the irreducible decomposition of D. For all subsets J
of cfw_1, ., r, D, nf1(x) is nonsingular for a general point x of W by
Theorem 7.17. Since (D nf"(x), = D, nf"(x), D nf"(x) has only
simple normal cro
316 10 D-Dimension and Kodaira Dimension of Varieties
Theorem 1.3 to U, one has a nite surjective morphism U+ A1,. Then U
can be replaced by i and so there exists a hyperplane H of U such that
HgUj for all j. Thus spm(H)U:1 (H n Uj) and dimH n Uj<
310 10 D-Dimension and Kodaira Dimension of Varieties
D TLnAanmn In) 1AA in: a 1 K '
u. my memems 1U.), um, um, and 10, one obtains the fundamental
theorems for Kodaira dimension.
Theorem 10.7 (Fibering Theorem). Given a variety V with K(V) 2 0, there
11.6 Logarithmic Canonical Fibered Varieties 339
p and p1 =f(p), respectively. Choose local coordinate systems (2, ., 21)
and (wj, ., w;) around p and p1, respectively, in such a way that
Uln V(z,1)= Ulnl" and Wan V(w;) = WanA where U,I and
WE are the coo
11.8 Some Applications 341
aboye re_sult. Therefore, G = cfw_I and so L< Ym> is a nite group. Since
f: V Ym is birational, Ker ,6," is trivial and hence SBir(V) g L< Ym>. [3
Corollary. If r2(V) = dim V, then SBir( V) coincides with PBir( V).
PROOF. Let f:
342 ll Logarithmic Kodaira Dimension of Varieties
thereader. r H H r 7 ~ 7 7 E]
b. The following theorem has been proved by Kawamata.
Theorem 11.15. Let f: V+ W be abered variety such that a generalber
f 1(x) is a curve. Ifx is a general point, then i?( V
11.4 Logarithmic Ramication Formula 333
Theref re, ic(V\A) 2 K( A
V E V_#\A, one has
K.(V) + D, V). However, by (i) and the inclusion
i2(i7#\A) 3 am 3 K(K(I7) + D, 17).
Thus the equality has been established. [3
EXAMPLE 11.5. Let ,u: S 30 be a birationai