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Branched Coverings of Surfaces with Ample Cotangent Bundle

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Branched Coverings of Surfaces with Ample Cotangent Bundle Pacific Journal of Mathematics BRANCHED COVERINGS OF SURFACES WITH AMPLE COTANGENT BUNDLE MICHAEL JEROME SPURR Volume 164 No. 1 May 1994 PACIFIC JOURNAL OF MATHEMATICS Vol. 164, No. 1, 1994 BRANCHED COVERINGS OF SURFACES WITH AMPLE COTANGENT BUNDLE MICHAEL J. SPURR Let /: X —> Y be a branched covering of compact complex sur- faces, where the ramification set in X consists of smooth curves meet- ing with at most normal crossings and Y has ample cotangent bun- dle. We further assume that / is locally of form (u,υ) —> (un ,vm). We characterize ampleness of T*X . A class of examples of such X, which are branched covers of degree two, is provided. 1. Introduction. An interesting problem in surface theory is the con- struction and characterization of surfaces with ample cotangent bun- dle. They are necessarily algebraic surfaces of general type. Natural examples occur among the complete intersection surfaces of abelian varieties. More subtle examples are those constructed by Hirzebruch [6] using line-arrangements in the plane. The characterization of those of Hirzebruch's line-arrangement surfaces with ample cotangent bun- dle is due to Sommese [8]. In this article, we will give a characteriza- tion of ampleness of the cotangent bundle of a class of surfaces which branch cover another surface with ample cotangent bundle. We will also construct certain branched coverings of explicit line-arrangement surfaces; these constructions will again have ample cotangent bundle. For any vector bundle E over a base manifold M, the projectiviza- tion P(E) is a fiber bundle over M, with fiber Pq(E) over q e M given by T?g(E) « (2?*\0)/C*. There is a tautological linebundle ξβ over P(E) satisfying (i) ζE\F ^ « 0(1)P ^ V<? e M, and (ii) the pro- jection pE: P(E) -» M gives /?^ (ξβ) ~ E. In the case that E = T*X we will denote p£ = Pτ*x simply by p. DEFINITION. The vector bundle E is ample if ζβ over ~P(E) is ample. In §2 we prove preliminary results along with: THEOREM 1.1. Let X and Y be compact complex surfaces, with Y having ample cotangent bundle. Let f: X -> Y be a branched covering which can be locally represented with coordinate charts of form f: (u, υ) —• (un, vm). Let f have ramification set \JBj in X 129 130 MICHAEL J. SPURR consisting of smooth curves meeting in normal crossings. Let |J Ca be the branch locus in Y. Then: T*X is ample & Bj Bj< 0 Vj <= Cα . Ca < 0 Vα. In §3 we give explicit examples of Theorem 1.1 which lie in a class of degree 2 branched covers. In what follows e{C) will denote the euler number of a curve C. 2. Ample cotangent bundles. Let f: X -+Y be a branched cover- ing of compact complex surfaces, with T*Y ample and with rami- fication set U Bj in X consisting of smooth curves meeting in nor- mal crossings. Note that T*Y ample gives that the canonical bundle Ky is ample [4], Hence Y is projective algebraic, which gives that X is also projective algebraic [1], and in turn that P(T*X) is also projective algebraic. Let / be locally represented with coordinate charts of form /: (u,v)-> {un, υm). The differential /*: TX -> TY induces a meromorphic mapping F: P(T*X) -» P(T*Y) given by F(x, [w]) = (f(x), lf*(w)]), where [w] denotes the line in the tan- gent bundle containing the tangent vector w . The indeterminacy set I of F corresponds to [w] such that f*(w) = 0. Blowing up / to resolve the indeterminacy of / (see [5], [10]) one gets Π(T*X) and obtains b: Π(Γ*Z)^P(Γ*X) and Φ: Π(Γ*X) -> P(Γ*7), holomor- phic, with Fob = Φ on Π(Γ*Z)\&"1(/). Let E = b~\l) be the exceptional set over / in Π(T*X). We need to precisely describe the indeterminacy set /. Before proceeding, we mention that, in the case that / is locally of form (u, υ) —• (un , vm), over each curve Bj in the ramification set in X there is a splitting of TX, due to Sommese [8], [9], namely TX\B. « TBj © NB . In particular NB., the normal bundle to Bj, is a subbundle of TX\B., and the pair if,, JV# gives a curve Bj in P(Γ*X). PROPOSITION 2.1. The indeterminacy set IofF: w of form where the Fa are fibers of p and the Bj are the curves in P(Γ*Z) corresponding to the pairs Bj and NB for Bj in the ramification set of X. Proof. In local coordinates let /: X —> Y be given by f(u9υ) = (un, vm). Then note that the indeterminacy set in P(T*X) locally BRANCHED COVERINGS 131 corresponds to the (non-zero) annihilators in ann(d(un), d(vm)) := {w e TX\d(un)(w) = 0 and d(vm)(w) = 0}. By analyzing the Ja- cobian /* one sees that ann(d(un), d(vm)) = {w e TX\f*{w) = 0}. Therefore locally / = {ann{d(un), d(vm))\0}/C*. If both n, m > 1 ι then / includes the whole fiber p~(0, 0) giving an Fa. If only one of n, m is greater than 1, say n > 1 with m = 1, then the curve B\ cor- responding to u = 0 is in the ramification set and ann(d(un), d(vm)) corresponds to the normal bundle of B\ via Sommese's splitting lem- ma [8], [9], Thus B\ and Nβ determine B\. Similarly one obtains the remaining Bj . D Let ζ\ be the tautological bundle over P(Γ*X) and let ζι be the tautological bundle over P(T*Y), as in the second paragraph of §1. Then b*(ξχ) on Π(Γ*X) relates to Φ*(ξ2) in a key manner via the following: 1 1 PROPOSITION 2.2. b*(ξ^ ) + D = Φ*^ ) where D = ΣnaDa is an effective divisor on T1(T*X) supported on the exceptional set E of U(T*X). Proof. /*: TX -> TY given by (x, w) -+ (f(x), f*(w)) induces F: P(Γ*X) ^P(Γ*Γ) which is given by (x, [w]) -> (/(JC) , [/*(w)]). Here x E X, K; is a tangent vector at x, and [tί;] denotes the line in the tangent bundle containing w . F has indeterminacy set / as described in Proposition 2.1. Over P(T*X)\I, f induces the map- ping /*: ζ~ι -• ξϊι given by (x, [w], ty) -• (f(x), [/*(^)], /*(n;)). This in turn yields the globally defined holomorphic mapping over Π(Γ*X) β: b*(ξϊι) - ξ;1 given by (p, tι;) -+ (f(p(b(p))),Φ(p), f(w)) where p e Π(T*X),b(p) = [w], and ρ(b(p)) = x. Fur- thermore Φ(p) = [/*(^)] if /*(ty) ^ 0 (i.e. off δ"1^))- In turn, β gives the mapping γ over Π(Γ*X) y: b*(ξ^1) -> Φ*^1) given by (p,w) —• (/?, Φ(p), f*{w)). There is vanishing of Λ(^) over E = b~ι(I), giving Z>. D We will prove Theorem 1.1 using the Nakai Criterion for ample- ness [7]: the holomorphic line bundle ξ\ on the projective algebraic manifold P(Γ*X) is ample if and only if for every subvariety Vn of dimension n < dimP(Γ*X) one has that Jv c^(ξ\) > 0. For brevity we define if. ^^/^ Proof of Theorem 1.1. Assume that Ca Ca < 0 for each Ca in the branch locus in Y. We show that Bj i?7 < 0 for all Bj in the 132 MICHAEL J. SPURR ramification set. Let π*(Cα) = Σk nakBak . Then Baj π*(Ca) = Baj 2^ nakBak k = najBaj Baj + Baj ]Γ Λαfc5αA: = deg(π|*JCα Ca kφj and hence 5αJ Baj = n-} deg(π\BJCa Cα - n-}Baj £ nα^ < 0 giving the implication. Assume next that Γ*X is ample. We show that Bj Bj < 0 for any Bj in the ramification set in X. Now for any Bj in the ramification set in X the splitting lemma of Sommese [8], [9] gives that TX\B. « TBj Θ Λfe . For Bj the curve in P(Γ*X) determined by 5y along ι with NB , we have 0 > ζ~ 5,- = N5 .57 = Bj . J?7. Conversely, assume that 57 J?7 < 0 for each Bj in (J^/ We show that T*X is ample. First note that since T*Y is ample we have that e(f(Bj)) is negative; hence by Riemann-Hurwitz e(Bj) < e(f(Bj)) < 0. To prove ampleness of Γ*X (i.e. of ζ\) we show that ξ\ Vn > 0 for all subvarieties Vn in P(Γ*X) where ζ\ is the tautological bundle over P(Γ*X). We handle the three cases n = 1, 2, 3 separately. (1) /ί = 1. Let Vn = C be an effective irreducible curve in P(Γ*X). We show that C fj"1 < 0. This is accomplished in three sub-cases: Case (li). Suppose that p'ι(\JBj) ~fi C. Let C7 be the proper transform of C in Π(T*X). By Proposition 2.2, Φ*^1) = b*(ξ^x)+ D where D = Σa naDa is an effective divisor. So C ξ~ι = C; ^(ίf1) - C (Φ*^1) - D) λ = deg(Φ|cOΦ(C) ξj - C D< 0. The last inequality follows since: Φ(C') is a curve in P(Γ*7), ^2 is 1 ample (which gives that Φ(C) -ί^" < 0), and D|c/ is effective on C as C7 is not contained in D.
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