Holomorphic curves in moduli spaces of polarized Abelian varieties
aa r X i v : . [ m a t h . C V ] F e b HOLOMORPHIC CURVES IN MODULI SPACES OFPOLARIZED ABELIAN VARIETIES
XIANJING DONG
Abstract.
We first prove a version of tautological inequality (proposedby McQuillan) for an open Riemann surface by using stochastic calculus.Then we use it to establish a Second Main Theorem of Nevanlinna theoryfor a smooth logarithmic pair (
X, D ) . As a consequence, we also set upa Second Main Theorem of holomorphic curves into some moduli spacesof polarized Abelian varieties intersecting boundary divisors. Introduction
To start with, we review a conjecture of Vojta [11] in Nevanlinna theory.Given a smooth logarithmic pair (
X, D ) over C , i.e., X is a smooth complexprojective variety and D is a normal crossing divisor on X, we consider thefinite ramified covering π : B → C , where B is an open (connected) Riemannsurface. Fix an ample line bundle A over X, Vojta conjectured the followingSecond Main Theorem ([11], Conjecture 27.5)
Conjecture 1.1 (Vojta, [11]) . For any holomorphic curve f : B → X whoseimage is not contained in Supp D, we have T f,K X ( D ) ( r ) ≤ exc N [1] f ( r, D ) + N Ram( π ) ( r ) + O (cid:16) log T f,A ( r ) + log + log r (cid:17) . Here K X ( D ) = K X ⊗ [ D ] , N Ram( π ) ( r ) is the ramification counting functionof ramified covering π, and “ ≤ exc ” means that the inequality holds for r > T f,K X ( D ) ( r ) ≤ exc d + 12 N [1] f ( r, D ) + N Ram( π ) ( r ) + O (cid:16) log T f,A ( r ) + log r (cid:17) , Mathematics Subject Classification.
Key words and phrases.
Nevanlinna theory; Second Main Theorem; Holomorphic curve;Riemann surface; Brownian motion; Moduli space.
X.-J. DONG where d is the fiber dimension of a family, see Theorem B in [10]. In further,Sun used this result to obtain a Second Main Theorem of holomorphic curvesinto moduli spaces of polarized Abelian varieties, see Theorem A in [10].In this paper, we will revisit Vojta’s conjecture and Sun’s results in a verydifferent way. Instead of B, we study the value distribution of a holomorphiccurve f : S −→ X from a geometric point of view, where S is a general open Riemann surface.Instead of N Ram( π ) ( r ) , we hope to obtain a quantifiable term depending onlyon the geometry of S. The key point is the tautological inequality (Theorem4.2) for S, to prove which the Brownian motion is employed, see LogarithmicDerivative Lemma (Theorem 3.4) in Section 3 which is a preliminary lemma.We state the main theorems of the paper, some notations will be providedlater. By uniformization theorem, we can equip S with a complete Hermitianmetric ds = 2 gdzdz such that its Gauss curvature K S ≤ g, here K S is defined by K S = −
14 ∆ S log g = − g ∂ log g∂z∂z . Apparently, S is a complete K¨ahler manifold with associated K¨ahler form α = g √− π dz ∧ dz. Fix o ∈ S as a reference point. We denote by D ( r ) the geodesic ball centeredat o with radius r, and by ∂D ( r ) the boundary of D ( r ) . By Sard’s theorem, ∂D ( r ) is a submanifold of S for almost all r > . Set(1) κ ( t ) = min (cid:8) K S ( x ) : x ∈ D ( t ) (cid:9) . Then κ is a non-positive and decreasing continuous function on [0 , ∞ ) . We obtain the following main results:
Theorem A (Theorem 5.2) . For any holomorphic curve f : S → X whoseimage is not contained in Supp D, we have T f,A ( r ) ≤ exc d + 12 N [1] f ( r, D ) + O (cid:16) log T f,A ( r ) − κ ( r ) r + log + log r (cid:17) , where A is an ample divisor on X and d is a constant defined in Section . In Theorem A, we find that there is a curvature term − κ ( r ) r dependingonly on the metric of S, which is more intuitive than N Ram( π ) ( r ) . As S = C , we see that κ ( r ) ≡ OLOMORPHIC CURVES IN MODULI SPACES 3
Theorem B (Corollary 5.3) . Let f : S → X be a holomorphic curve whichramifies over D with order c > ( d + 1) / , i.e., a constant c > ( d + 1) / suchthat f ∗ D ≥ c · Supp f ∗ D. If f satisfies the growth condition lim inf r →∞ κ ( r ) r T f,A ( r ) = 0 , where A is given as in Theorem A , then f ( S ) is contained in D. Note that if S = C , such growth condition in Theorem B is automaticallysatisfied. To receive the degeneracy, a certain growth condition is necessary.However, it is hard to give a perfect estimate of Green functions in a generalRiemannian manifold, so the growth condition in Theorem B is not optimal. Theorem C (Corollary 5.4) . For any holomorphic curve f : S → X whoseimage is not contained in Supp D, we have T f,K X ( D ) ( r ) ≤ exc k ( d + 1)2 N [1] f ( r, D ) + O (cid:16) log T f,A ( r ) − κ ( r ) r + log + log r (cid:17) for an integer k such that [ A ] k ≥ K X ( D ) , where A, d are given as in Theorem
A.If X is a smooth complex projective curve, then we can replace k ( d + 1) / X, D ) as a smooth compactification of the basespace of a family, namely, U = X \ D carries a family of polarized manifolds.Assume that there is a smooth family ( ψ : V → U, L ) of polarized smoothvarieties with semi-ample canonical sheaves and fixed Hilbert polynomials h, such that the induced classifying mapping from U into the moduli scheme M h is quasi-finite. Then for any holomorphic curve f : S → X whose imageis not contained in Supp D, we have T f,K X ( D ) ( r ) ≤ exc c ψ N [1] f ( r, D ) + O (cid:16) log T f,A ( r ) − κ ( r ) r + log + log r (cid:17) for a positive constant c ψ depending only on ψ and ( X, D ) . Next we consider Siegel modular varieties. Let A [ n ] g ( n ≥
3) be the modulispace of principally polarized Abelian varieties with level- n structure. Also,let A [ n ] g denote the smooth compactification of A [ n ] g such that D := A [ n ] g \ A [ n ] g is a normal crossing (boundary) divisor. Then Theorem D (Theorem 5.5) . For any holomorphic curve f : S → A [ n ] g whoseimage is not contained in Supp D, we have T f,K A [ n ] g ( D ) ( r ) ≤ exc ( g + 1) N [1] f ( r, D ) + O (cid:16) log T f,A ( r ) − κ ( r ) r + log + log r (cid:17) , where A is given as in Theorem A. X.-J. DONG Nevanlinna’s functions
Preliminaries.
Let (
M, g ) be a Riemannian manifold with Laplace-Beltrami operator ∆ M associated to g. Fix x ∈ M, denote by B x ( r ) the geodesic ball centered at x with radius r, and by S x ( r ) the geodesic sphere centered at x with radius r. Apply Sard’s theorem, S x ( r ) is a submanifold of M for almost all r > . ABrownian motion ( X t ) t ≥ (written as X t for short) in M is a heat diffusionprocess generated by ∆ M / p ( t, x, y ) whichis the minimal positive fundamental solution of the heat equation ∂∂t u ( t, x ) −
12 ∆ M u ( t, x ) = 0 . We denote by P x the law of X t started at x ∈ M and by E x the correspondingexpectation with respect to P x . Co-area formula
Let D be a bounded domain with smooth boundary ∂D in M . Fix x ∈ D, we use dπ ∂Dx to denote the harmonic measure on ∂D with respect to x. Thismeasure is a probability measure. Set τ D := inf (cid:8) t > X t D (cid:9) which is a stopping time. Let g D ( x, y ) stand for the Green function of ∆ M / D with Dirichlet boundary condition and a pole x , namely −
12 ∆
M,y g D ( x, y ) = δ x ( y ) , y ∈ D ; g D ( x, y ) = 0 , y ∈ ∂D, where δ x is the Dirac function. For φ ∈ C ♭ ( D ) (space of bounded continuousfunctions on D ), the co-area formula [3] says that E x (cid:20)Z τ D φ ( X t ) dt (cid:21) = Z D g D ( x, y ) φ ( y ) dV ( y ) . From Proposition 2.8 in [3], we also have the relation of harmonic measuresand hitting times that(2) E x [ ψ ( X τ D )] = Z ∂D ψ ( y ) dπ ∂Dx ( y )for any ψ ∈ C ( D ). The co-area formula and (2) work in the case when φ or ψ is of a pluripolar set of singularities. Dynkin formula
Let u ∈ C ♭ ( M ) (space of bounded C -class functions on M ), we have thefamous Itˆo formula (see [1, 5, 6, 8]) u ( X t ) − u ( x ) = B (cid:18)Z t k∇ M u k ( X s ) ds (cid:19) + 12 Z t ∆ M u ( X s ) dt, P x − a.s. OLOMORPHIC CURVES IN MODULI SPACES 5 where B t is the standard Brownian motion in R and ∇ M is gradient operatoron M . Notice that B t is a martingale, take expectation on both sides of theabove formula, it follows Dynkin formula (see [1, 8]) E x [ u ( X τ D )] − u ( x ) = 12 E x (cid:20)Z τ D ∆ M u ( X t ) dt (cid:21) . Dynkin formula still works when u has a pluripolar set of singularities, par-ticularly, for a plurisubharmonic function u. Nevanlinna’s functions.
Let S be an open Riemann surface with the K¨ahler form α. Fix o ∈ S as areference point, denote by D ( r ) the geodesic ball centered at o with radius r, and by ∂D ( r ) the boundary of D ( r ) . Moreover, we use g r ( o, x ) to stand forthe Green function of ∆ S / D ( r ) with Dirichlet boundary condition anda pole o, also use dπ ro ( x ) to stand for the harmonic measure on ∂D ( r ) withrespect to o. Let X t be the Brownian motion generated by ∆ S / o ∈ S. Set the stopping time τ r = inf (cid:8) t > X t D ( r ) (cid:9) . Let f : S −→ X be a holomorphic curve into a compact complex manifold X. We introducethe generalized Nevanlinna’s functions over Riemann surface S. Let L → X be an ample holomorphic line bundle equipped with Hermitian metric h sothat the Chern form c ( L, h ) > . The height function of f with respect to L is defined by T f,L ( r ) = π Z D ( r ) g r ( o, x ) f ∗ c ( L, h )= − Z D ( r ) g r ( o, x )∆ S log h ◦ f ( x ) dV ( x ) , where dV ( x ) is the Riemannian volume measure of S. It can be easily knownthat T f,L ( r ) is independent of the choice of metrics on L, up to a boundedterm. Since a holomorphic line bundle can be represented as the differenceof two ample holomorphic line bundles, the definition of T f,L ( r ) can extendto an arbitrary holomorphic line bundle. By co-area formula, we have T f,L ( r ) = − E o (cid:20)Z τ r ∆ S log h ◦ f ( X t ) dt (cid:21) . For simplicity, we use notation T f,D ( r ) to stand for T f, [ D ] ( r ) for a divisor D on X. Similarly, a divisor can also be written as the difference of two ampledivisors. The Weil function of D is well defined by λ D ( x ) = − log k s D ( x ) k X.-J. DONG up to a bounded term, here s D is the canonical section associated to D. Wedefine the proximity function of f with respect to D by m f ( r, D ) = Z ∂D ( x ) λ D ◦ f ( x ) dπ ro ( x ) . A relation between harmonic measures and hitting times shows that m f ( r, D ) = E o (cid:2) λ D ◦ f ( X τ r ) (cid:3) . Locally, we write s D = ˜ s D e, where e is a local holomorphic frame of ([ D ] , h ) . The counting function of f with respect to D is defined by N f ( r, D ) = π X x ∈ f ∗ D ∩ D ( r ) g r ( o, x )= π Z D ( r ) g r ( o, x ) dd c (cid:2) log | ˜ s D ◦ f ( x ) | (cid:3) = 14 Z D ( r ) g r ( o, x )∆ S log | ˜ s D ◦ f ( x ) | dV ( x )in the sense of currents. Remark.
The definition of Nevanlinna’s functions is natural. As S = C , the Green function is (log r | z | ) /π and the harmonic measure is dθ/ π. So, byintegration by part, we will see that it coincides with the classical ones.Consider a holomorphic curve f : S → X such that f ( o ) Supp D, where D is a divisor on X. Apply Dynkin formula to λ D ◦ f ( x ) , it yields E o (cid:2) λ D ◦ f ( X τ r ) (cid:3) − λ D ◦ f ( o ) = 12 E o (cid:20)Z τ r ∆ S λ D ◦ f ( X t ) dt (cid:21) . The first term on the left hand side of the above equality is equal to m f ( r, D ) , and the term on the right hand side equals12 E o (cid:20)Z τ r ∆ S λ D ◦ f ( X t ) dt (cid:21) = 12 Z D ( r ) g r ( o, x )∆ S log 1 k s D ◦ f ( x ) k dV ( x )due to co-area formula. Since k s D k = h | ˜ s D | , where h is a Hermitian metricon [ D ] , then we get12 E o (cid:20)Z τ r ∆ S λ D ◦ f ( X t ) dt (cid:21) = − Z D ( r ) g r ( o, x )∆ S log h ◦ f ( x ) dV ( x ) − Z D ( r ) g r ( o, x )∆ S log | ˜ s D ◦ f ( x ) | dV ( x )= T f,D ( r ) − N f ( r, D ) . Therefore, we obtain the First Main Theorem: T f,D ( r ) = m f ( r, D ) + N f ( r, D ) + O (1) . OLOMORPHIC CURVES IN MODULI SPACES 7 Logarithmic Derivative Lemma
Let S be a complete open Riemann surface with Gauss curvature K S ≤ g. Note that every open Riemann surfaceadmits a nowhere-vanishing holomorphic vector field (see [14]). Now we fixa nowhere-vanishing holomorphic vector field X over S. In this section, S is assumed to be simply connected.3.1. Calculus Lemma.
Let κ be defined by (1). As is noted before, κ is a non-positive, decreasingcontinuous function on [0 , ∞ ) . Associate the ordinary differential equation(3) G ′′ ( t ) + κ ( t ) G ( t ) = 0; G (0) = 0 , G ′ (0) = 1 . We compare (3) with y ′′ ( t ) + κ (0) y ( t ) = 0 under the same initial conditions, G can be easily estimated as G ( t ) = t for κ ≡ G ( t ) ≥ t for κ . This implies that(4) G ( r ) ≥ r for r ≥ Z r dtG ( t ) ≤ log r for r ≥ . On the other hand, rewrite (3) as the formlog ′ G ( t ) · log ′ G ′ ( t ) = − κ ( t ) . Since G ( t ) ≥ t is increasing, then the decrease and non-positivity of κ implythat for each fixed t, G must satisfy one of the following two inequalitieslog ′ G ( t ) ≤ p − κ ( t ) for t >
0; log ′ G ′ ( t ) ≤ p − κ ( t ) for t ≥ . Since G ( t ) → t → , by integration, G is bounded from above by(5) G ( r ) ≤ r exp (cid:0) r p − κ ( r ) (cid:1) for r ≥ . Lemma 3.1 ([2]) . Let η > be a constant. Then there is a constant C > such that for r > η and x ∈ B o ( r ) \ B o ( η ) g r ( o, x ) Z rη dtG ( t ) ≥ C Z rr ( x ) dtG ( t ) holds, where G is defined by (3) . Lemma 3.2 (Borel’s lemma) . Let u be a monotone increasing function on [0 , ∞ ) such that u ( r ) > for some r ≥ . Then for any δ > , there existsa set E δ ⊆ [0 , ∞ ) of finite Lebegue measure such that u ′ ( r ) ≤ u ( r ) log δ u ( r ) holds for r > outside E δ . X.-J. DONG
Proof.
Since u is monotone increasing, then u ′ ( r ) exists for almost all r ≥ . Set S = (cid:8) r ≥ u ′ ( r ) > u ( r ) log δ u ( r ) (cid:9) . We have Z S dr ≤ Z r dr + Z S \ [0 ,r ] dr ≤ r + Z ∞ r u ′ ( r ) u ( r ) log δ u ( r ) dr < ∞ . (cid:3) Theorem 3.3 (Calculus Lemma) . Let k ≥ be a locally integrable functionon S such that it is locally bounded at o ∈ S. Then for any δ > , there isa constant C > independent of k, δ, and a subset E δ ⊆ (1 , ∞ ) of finiteLebesgue measure such that E o (cid:2) k ( X τ r ) (cid:3) ≤ exc F (ˆ k, κ, δ ) e r √ − κ ( r ) log r πC E o (cid:20)Z τ r k ( X t ) dt (cid:21) holds for r > outside E δ , where κ is defined by (1) and F is defined by F (cid:0) ˆ k, κ, δ (cid:1) = n log + ˆ k ( r ) · log + (cid:16) re r √ − κ ( r ) ˆ k ( r ) (cid:8) log + ˆ k ( r ) (cid:9) δ (cid:17)o δ with ˆ k ( r ) = log rC E o (cid:20)Z τ r k ( X t ) dt (cid:21) . Moreover, we have the estimate log F (ˆ k, κ, δ ) ≤ O (cid:16) log + log E o (cid:20)Z τ r k ( X t ) dt (cid:21) + log + r p − κ ( r ) + log + log r (cid:17) . Proof.
The argument refers to Atsuji [2]. From [4], the simple-connectednessand non-positivity of sectional curvature of S imply 2 πrdπ ro ( x ) ≤ dσ r ( x ) , where dσ r ( x ) is the induced volume measure on ∂D ( r ) . By Lemma 3.1 and(4), we have E o (cid:20)Z τ r k ( X t ) dt (cid:21) = Z D ( r ) g r ( o, x ) k ( x ) dV ( x )= Z r dt Z ∂D ( t ) g r ( o, x ) k ( x ) dσ t ( x ) ≥ C Z r R rt G − ( s ) ds R r G − ( s ) ds dt Z ∂D ( t ) k ( x ) dσ t ( x ) ≥ C log r Z r dt Z rt dsG ( s ) Z ∂D ( t ) k ( x ) dσ t ( x ) , E o [ k ( X τ r )] = Z ∂D ( r ) k ( x ) dπ ro ( x ) ≤ πr Z ∂D ( r ) k ( x ) dσ r ( x ) . OLOMORPHIC CURVES IN MODULI SPACES 9
Thus, E o (cid:20)Z τ r k ( X t ) dt (cid:21) ≥ C log r Z r dt Z rt dsG ( s ) Z ∂D ( o,t ) k ( x ) dσ t ( x ) , E o [ k ( X τ r )] ≤ πr Z ∂D ( r ) k ( x ) dσ r ( x ) . (6)Set Λ( r ) = Z r dt Z rt dsG ( s ) Z ∂D ( t ) k ( x ) dσ t ( x ) . Then Λ( r ) ≤ log rC E o (cid:20)Z τ r k ( X t ) dt (cid:21) = ˆ k ( r ) . Since Λ ′ ( r ) = 1 G ( r ) Z r dt Z ∂D ( t ) k ( x ) dσ t ( x ) , it yields from (6) that E o (cid:2) k ( X τ r ) (cid:3) ≤ πr ddr (cid:0) Λ ′ ( r ) G ( r ) (cid:1) . Using Lemma 3.2 twice and (5), then for any δ > ddr (cid:0) Λ ′ ( r ) G ( r ) (cid:1) ≤ exc G ( r ) n log + Λ( r ) · log + (cid:16) G ( r )Λ( r ) (cid:8) log + Λ( r ) (cid:9) δ (cid:17) o δ Λ( r ) ≤ re r √ − κ ( r ) n log + ˆ k ( r ) · log + (cid:16) re r √ − κ ( r ) ˆ k ( r ) (cid:8) log + ˆ k ( r ) (cid:9) δ (cid:17)o δ ˆ k ( r )= F (cid:0) ˆ k, κ, δ (cid:1) re r √ − κ ( r ) log rC E o (cid:20)Z τ r k ( X t ) dt (cid:21) holds outside a subset E δ ⊆ (1 , ∞ ) of finite Lebesgue measure. Thus, E o (cid:2) k ( X τ r ) (cid:3) ≤ F (cid:0) ˆ k, κ, δ (cid:1) e r √ − κ ( r ) log r πC E o (cid:20)Z τ r k ( X t ) dt (cid:21) . Therefore, we get the desired inequality. Indeed, for r > F (ˆ k, κ, δ ) ≤ O (cid:16) log + log + ˆ k ( r ) + log + r p − κ ( r ) + log + log r (cid:17) and log + ˆ k ( r ) ≤ log E o (cid:20)Z τ r k ( X t ) dt (cid:21) + log + log r. We have the desired estimate. The proof is competed. (cid:3)
Logarithmic Derivative Lemma.
Let ψ be a meromorphic function on ( S, g ) . The norm of the gradient of ψ is defined by k∇ S ψ k = 1 g (cid:12)(cid:12)(cid:12)(cid:12) ∂ψ∂z (cid:12)(cid:12)(cid:12)(cid:12) in a local holomorphic coordinate z. Locally, we write ψ = ψ /ψ , where ψ , ψ are local holomorphic functions without common zeros. Regard ψ asa holomorphic mapping into P ( C ) by x [ ψ ( x ) : ψ ( x )] . Define T ψ ( r ) = 14 Z D ( r ) g r ( o, x )∆ S log (cid:0) | ψ ( x ) | + | ψ ( x ) | (cid:1) dV ( x )as well as T ( r, ψ ) := m ( r, ψ ) + N ( r, ψ ) with m ( r, ψ ) = Z ∂D ( r ) log + | ψ ( x ) | dπ ro ( x ) ,N ( r, ψ ) = π X x ∈ ( ψ ) ∞ ∩ D ( r ) g r ( o, x ) . Let i : C ֒ → P ( C ) be an inclusion defined by z [1 : z ] . Via the pull-back by i, we have a (1,1)-form i ∗ ω F S = dd c log(1+ | ζ | ) on C , where ζ := w /w and[ w : w ] is the homogeneous coordinate system of P ( C ) . The characteristicfunction of ψ with respect to i ∗ ω F S is defined byˆ T ψ ( r ) = 14 Z D ( r ) g r ( o, x )∆ S log(1 + | ψ ( x ) | ) dV ( x ) . Clearly, ˆ T ψ ( r ) ≤ T ψ ( r ) . We adopt the spherical distance k· , ·k on P ( C ) , theproximity function of ψ with respect to a ∈ P ( C ) is defined byˆ m ψ ( r, a ) = Z ∂D ( r ) log 1 k ψ ( x ) , a k dπ ro ( x ) . Clearly, m ( r, ψ ) = ˆ m ψ ( r, ∞ ) + O (1) which yields T ( r, ψ ) = ˆ T ψ ( r ) + O (1) , T (cid:16) r, ψ − a (cid:17) = T ( r, ψ ) + O (1) . Hence, we arrive at(7) T ( r, ψ ) + O (1) = ˆ T ψ ( r ) ≤ T ψ ( r ) + O (1) . Theorem 3.4 (LDL) . Let ψ be a nonconstant meromorphic function on S. Then m (cid:16) r, X k ( ψ ) ψ (cid:17) ≤ exc k T ( r, ψ ) + O (cid:16) log + log T ( r, ψ ) − κ ( r ) r + log + log r (cid:17) , where X j = X ◦ X j − with X = Id, and κ is defined by (1) . OLOMORPHIC CURVES IN MODULI SPACES 11
Take a singular formΦ = 1 | ζ | (1 + log | ζ | ) √− π dζ ∧ dζ. A direct computation gives that Z P ( C ) Φ = 1 , πψ ∗ Φ = k∇ S ψ k | ψ | (1 + log | ψ | ) α. Set T ψ ( r, Φ) = 12 π Z D ( r ) g r ( o, x ) k∇ S ψ k | ψ | (1 + log | ψ | ) ( x ) dV ( x ) . Then, by Fubini’s theorem T ψ ( r, Φ) = Z D ( r ) g r ( o, x ) ψ ∗ Φ α dV ( x )= π Z ζ ∈ P ( C ) Φ X x ∈ ( ψ − ζ ) ∩ D ( r ) g r ( o, x )= Z ζ ∈ P ( C ) N ψ ( r, ζ )Φ ≤ T ( r, ψ ) + O (1) , which follows that(8) T ψ ( r, Φ) ≤ T ( r, ψ ) + O (1) . Lemma 3.5.
Assume that ψ ( x ) . Then E o (cid:20) log + k∇ S ψ k | ψ | (1 + log | ψ | ) ( X τ r ) (cid:21) ≤ exc
12 log T ( r, ψ ) + O (cid:16) log + log T ( r, ψ ) + r p − κ ( r ) + log + log r (cid:17) , where κ is defined by (1) . Proof.
By Jensen’s inequality, it is clear that E o (cid:20) log + k∇ S ψ k | ψ | (1 + log | ψ | ) ( X τ r ) (cid:21) ≤ E o (cid:20) log (cid:16) k∇ S ψ k | ψ | (1 + log | ψ | ) ( X τ r ) (cid:17)(cid:21) ≤ log + E o (cid:20) k∇ S ψ k | ψ | (1 + log | ψ | ) ( X τ r ) (cid:21) + O (1) . By Lemma 3.3 with co-area formula and (8)log + E o (cid:20) k∇ S ψ k | ψ | (1 + log | ψ | ) ( X τ r ) (cid:21) ≤ exc log + E o (cid:20)Z τ r k∇ S ψ k | ψ | (1 + log | ψ | ) ( X t ) dt (cid:21) + log F (ˆ k, κ, δ ) e r √ − κ ( r ) log r πC ≤ log T ψ ( r, Φ) + log F (ˆ k, κ, δ ) + r p − κ ( r ) + log + log r + O (1) ≤ log T ( r, ψ ) + O (cid:16) log + log + ˆ k ( r ) + r p − κ ( r ) + log + log r (cid:17) , where ˆ k ( r ) = log rC E o (cid:20)Z τ r k∇ S ψ k | ψ | (1 + log | ψ | ) ( X t ) dt (cid:21) . Indeed, we note thatˆ k ( r ) = 2 π log rC T ψ ( r, Φ) ≤ π log rC T ( r, ψ ) . Hence, we have the desired inequality. (cid:3)
Lemma 3.6.
Let ψ be a nonconstant meromorphic function on S. Then m (cid:16) r, X ( ψ ) ψ (cid:17) ≤ exc
32 log T ( r, ψ ) + O (cid:16) log + log T ( r, ψ ) − κ ( r ) r + log + log r (cid:17) , where κ is defined by (1) . Proof.
Write X = a ∂∂z , then k X k = g | a | . We have m (cid:16) r, X ( ψ ) ψ (cid:17) = Z ∂D ( r ) log + | X ( ψ ) || ψ | ( x ) dπ ro ( x ) ≤ Z ∂D ( r ) log + | X ( ψ ) | k X k | ψ | (1 + log | ψ | ) ( x ) dπ ro ( x )+ 12 Z ∂D ( r ) log(1 + log | ψ ( x ) | ) dπ ro ( x ) + 12 Z ∂D ( r ) log + k X x k dπ ro ( x ):= A + B + C. We handle
A, B, C respectively. For A, it yields from Lemma 3.5 that A = 12 Z ∂D ( r ) log + | a | (cid:12)(cid:12)(cid:12) ∂ψ∂z (cid:12)(cid:12)(cid:12) g | a | | ψ | (1 + log | ψ | ) ( x ) dπ ro ( x )= 12 Z ∂D ( r ) log + k∇ S ψ k | ψ | (1 + log | ψ | ) ( x ) dπ ro ( x ) ≤ exc
12 log T ( r, ψ ) + O (cid:16) log + log T ( r, ψ ) + r p − κ ( r ) + log + log r (cid:17) . OLOMORPHIC CURVES IN MODULI SPACES 13
For B, the Jensen’s inequality implies that B ≤ Z ∂D ( r ) log (cid:16) + | ψ ( x ) | + log + | ψ ( x ) | (cid:17) dπ ro ( x ) ≤ log Z ∂D ( r ) (cid:16) + | ψ ( x ) | + log + | ψ ( x ) | (cid:17) dπ ro ( x ) ≤ log T ( r, ψ ) + O (1) . Finally, we estimate C. By the condition, k X k > . Since S is non-positivelycurved and a is holomorphic, then log k X k is subharmonic, i.e., ∆ S log k X k ≥ . Clearly, we have ∆ S log + k X k ≤ ∆ S log k X k for x ∈ S satisfying k X x k 6 = 1 . Notice that log + k X x k = 0 for x ∈ S satisfying k X x k ≤ . Thus, by Dynkin formula we have C = 12 E o (cid:2) log + k X ( X τ r ) k (cid:3) (9) ≤ E o (cid:20)Z τ r ∆ S log k X ( X t ) k dt (cid:21) + O (1)= 14 E o (cid:20)Z τ r ∆ S log g ( X t ) dt (cid:21) + 14 E o (cid:20)Z τ r ∆ S log | a ( X t ) | dt (cid:21) + O (1)= − E o (cid:20)Z τ r K S ( X t ) dt (cid:21) + O (1) ≤ − κ ( r ) E o (cid:2) τ r (cid:3) + O (1) , where we use the fact K S = − (∆ S log g ) / . So, we deduce the conclusion byusing E o [ τ r ] ≤ r , due to Lemma 3.7 below. (cid:3) Lemma 3.7.
We have E o (cid:2) τ r (cid:3) ≤ r . Proof.
Apply Itˆo formula to r ( x )(10) r ( X t ) = B t − L t + 12 Z t ∆ S r ( X s ) ds, where B t is the standard Brownian motion in R , L t is a local time on cutlocus of o, an increasing process which increases only at cut loci of o. Since S is simply connected and non-positively curved, then∆ S r ( x ) ≥ r ( x ) , L t ≡ . By (10), we arrive at r ( X t ) ≥ B t + 12 Z t dsr ( X s ) . Since B t is a martingale, then we have E o [ B t ] = 0 . Take expectation on bothsides of the above inequality r = E o (cid:2) r ( X τ r ) (cid:3) ≥ E o (cid:20)Z τ r dtr ( X t ) (cid:21) ≥ r E o (cid:2) τ r (cid:3) , which proves the conclusion. (cid:3) Proof of Theorem . Proof.
The assertion can be confirmed by m (cid:0) r, X k ( ψ ) /ψ (cid:1) ≤ k X j =1 m (cid:0) r, X j ( ψ ) / X j − ( ψ ) (cid:1) with the following Lemma 3.8. (cid:3) Lemma 3.8.
We have m (cid:16) r, X k +1 ( ψ ) X k ( ψ ) (cid:17) ≤ exc
32 log T ( r, ψ ) + O (cid:16) log + log T ( r, ψ ) − κ ( r ) r + log + log r (cid:17) , where κ is defined by (1) . Proof.
We claim that T (cid:0) r, X k ( ψ ) (cid:1) ≤ k T ( r, ψ ) + O (cid:16) log T ( r, ψ ) − κ ( r ) r + log + log r (cid:17) . (11)By virtue of Lemma 3.6, when k = 1 T ( r, X ( ψ )) = m ( r, X ( ψ )) + N ( r, X ( ψ )) ≤ m ( r, ψ ) + 2 N ( r, ψ ) + m (cid:16) r, X ( ψ ) ψ (cid:17) ≤ T ( r, ψ ) + m (cid:16) r, X ( ψ ) ψ (cid:17) ≤ T ( r, ψ ) + O (cid:16) log T ( r, ψ ) − κ ( r ) r + log + log r (cid:17) holds for r > k ≤ n − . By induction, we only need to prove the claimin the case when k = n. By the claim for k = 1 showed above and Theorem OLOMORPHIC CURVES IN MODULI SPACES 15 T (cid:0) r, X n ( ψ ) (cid:1) ≤ T (cid:0) r, X n − ( ψ ) (cid:1) + O (cid:16) log T (cid:0) r, X n − ( ψ ) (cid:1) − κ ( r ) r + log + log r (cid:17) ≤ n T ( r, ψ ) + O (cid:16) log T ( r, ψ ) − κ ( r ) r + log + log r (cid:17) + O (cid:16) log T (cid:0) r, X n − ( ψ ) (cid:1) − κ ( r ) r + log + log r (cid:17) ≤ n T ( r, ψ ) + O (cid:16) log T ( r, ψ ) − κ ( r ) r + log + log r (cid:17) + O (cid:0) log T (cid:0) r, X n − ( ψ ) (cid:1)(cid:1) · · · · · · · · ·≤ n T ( r, ψ ) + O (cid:16) log T ( r, ψ ) − κ ( r ) r + log + log r (cid:17) . Thus claim (11) is confirmed. Using Theorem 3.6 and (11) to get m (cid:18) r, X k +1 ( ψ ) X k ( ψ ) (cid:19) ≤
32 log T (cid:0) r, X k ( ψ ) (cid:1) − κ ( r ) r G ( r ) r + O (cid:16) log + log T (cid:0) r, X k ( ψ ) (cid:1) + log + log G ( r ) (cid:17) ≤
32 log T ( r, ψ ) + O (cid:16) log + log T ( r, ψ ) − κ ( r ) r + log + log r (cid:17) . This proves the lemma. (cid:3) Tautological inequality
In this section, we provide a version of tautological inequality for a generalopen Riemann surface S from a geometric point of view by using stochasticcalculus. This inequality plays an essential role in proving the main theorem.Let ( X, D ) be a smooth logarithmic pair over C . Denote by Ω X (log D ) thelogarithmic cotangent sheaf over X which is the sheaf of germs of logarithmic1-forms with poles at most on D, namelyΩ X (log D ) = s X j =1 O X dσ j σ j + Ω X , where σ , · · · , σ s are irreducible and σ · · · σ s = 0 is a local defining equationof D. The dual of Ω X (log D ) is called the logarithmic tangent sheaf denotedby T X ( − log D ). Note that Ω X (log D ) is locally free. For a sheaf E , we havethe familiar symbols P ( E ) = Proj M d ≥ S d E , V ( E ) = Spec M d ≥ S d E , Sym • = M d ≥ S d . Associate a nonconstant holomorphic curve f : S → X whose image f ( S )is not contained in Supp D. Then f induces a curve f ′ : S −→ P (cid:0) Ω X (log D ) (cid:1) which is holomorphic outside S \ f ∗ D. Let p : B −→ P (cid:0) Ω X (log D ) ⊕ O X (cid:1) be the blow-up of P (Ω X (log D ) ⊕ O X ) along the zero section of V (Ω X (log D )) , i.e., the section corresponding to the projection Ω X (log D ) ⊕ O X → O X . Thatis to say, B is the closure of graph of the rational map P (cid:0) Ω X (log D ) ⊕ O X (cid:1) P (Ω X (log D )) . Let [0] denote the exceptional divisor on B and associate thelifted curve ∂f : S −→ P (cid:0) Ω X (log D ) ⊕ O X (cid:1) . Definition 4.1.
Let φ : S → B be the lift of ∂f. The D -modified ramificationcounting function of a holomorphic curve f : S → X is the counting functionfor φ ∗ [0] : N f, Ram( D ) ( r ) := N φ ( r, [0]) . Theorem 4.2 (Tautological inequality) . Let A be an ample line sheaf overa smooth logarithmic pair ( X, D ) . Let O (1) be the tautological line sheaf over P (Ω X (log D )) . Then T f ′ , O (1) ( r ) + N f, Ram( D ) ( r ) ≤ exc N [1] f ( r, D ) + O (cid:16) log + T f, A ( r ) − κ ( r ) r + log + log r (cid:17) , where κ is defined by (1) . We have a natural embedding V (Ω X (log D )) ֒ → P (Ω X (log D ) ⊕ O X ) thatrealizes P (Ω X (log D ) ⊕ O X ) as the projective closure on fibers of V (Ω X (log D )) . Denote by [ ∞ ] the (reduced) divisor P (Ω X (log D ) ⊕ O X ) \ V (Ω X (log D )) . To prove Theorem 4.2, we need the following lemma:
Lemma 4.3.
We have m ∂f ( r, [ ∞ ]) ≤ exc O (cid:16) log T f, A ( r ) − κ ( r ) r + log + log r (cid:17) , where κ is defined by (1) . Proof.
Without loss of generality, we may assume that S is simply connected(see Remark below). Note that there is a finite set H of rational functions on X with the properties: for each point y ∈ X, there exists a subset H y ⊆ H such that(i) H y generates Ω X (log D );(ii) dh/h is a regular section of Ω X (log D ) at y for all h ∈ H y . OLOMORPHIC CURVES IN MODULI SPACES 17
By the definition of [ ∞ ] and the compactness of X, it follows that λ [ ∞ ] ◦ ∂f ≤ log + max h ∈ H (cid:12)(cid:12)(cid:12)(cid:12) X ( h ◦ f ) h ◦ f (cid:12)(cid:12)(cid:12)(cid:12) + O (1) . Therefore, by Theorem 3.4 we arrive at m ∂f ( r, [ ∞ ]) ≤ C · max h ∈ H m (cid:16) r, X ( h ◦ f ) h ◦ f (cid:17) + O (1) ≤ O (cid:16) max h ∈ H log T ( r, h ◦ f ) − κ ( r ) r + log + log r (cid:17) ≤ O (cid:16) log T f, A ( r ) − κ ( r ) r + log + log r (cid:17) . (cid:3) Remark.
Let π : ˜ S → S be the (analytic) universal covering. We equip ˜ S with the induced metric from the metric of S. Then, ˜ S is a simply-connectedand complete open Riemann surface of non-positive Gauss curvature. Take adiffusion process ˜ X t in ˜ S satisfying that X t = π ( ˜ X t ) , then ˜ X t is a Brownianmotion with generator ∆ ˜ S / X t start at ˜ o ∈ ˜ S with o = π (˜ o ) , we have E o [ φ ( X t )] = E ˜ o (cid:2) φ ◦ π ( ˜ X t ) (cid:3) for φ ∈ C ♭ ( S ) . Set ˜ τ r = inf (cid:8) t > X t ˜ D ( r ) (cid:9) , where ˜ D ( r ) is a geodesic ball centered at ˜ o with radius r in ˜ S. If necessary,one can extend the filtration in probability space where ( X t , P o ) are definedso that ˜ τ r is a stopping time with respect to a filtration where the stochasticcalculus of X t works. With the above arguments, we can suppose S is simplyconnected by lifting f to the universal covering.Let us prove Theorem 4.2: Proof.
The proof essentially follows McQuillan [7]. Recall the blow-up p : B −→ P (cid:0) Ω X (log D ) ⊕ O X (cid:1) . The B admits a morphism q : B → P (Ω X (log D )) which extends the rationalmap P (Ω X (log D ) ⊕ O X ) P (Ω X (log D )) associated to the canonical mapΩ X (log D ) ֒ → Ω X (log D ) ⊕ O X . We use the same symbol O (1) to denote thetautological line sheaf over P (Ω X (log D )) and P (Ω X (log D ) ⊕ O X ) . Taking anonzero rational section s of Ω X (log D ) over X, then s determines a rationalsection s of O (1) over P (Ω X (log D )) . The divisor ( s ) is the sum of a generichyperplane section (on fibers over X ) and the pull-back of a divisor (on X ).Notice that ( s,
0) is also a nonzero rational section of Ω X (log D ) ⊕ O X over X, hence it determines a rational section s of O (1) over P (Ω X (log D ) ⊕ O X ) . Note ( s ) is again the sum of a generic hyperplane section and the pull-backof a divisor. Comparing q ∗ ( s ) with p ∗ ( s ) , we find that they coincide exceptthat p ∗ ( s ) contains [0] with multiplicity 1. Hence, we obtain(12) q ∗ O (1) ∼ = p ∗ O (1) ⊗ O ( − [0]) . Observe the following commutative diagram S ∂f f ′ * * f ❖❖❖❖❖❖❖ ' ' ❖❖❖❖❖❖❖❖❖ X P (Ω X (log D )) o o P (Ω X (log D ) ⊕ O X ) O O ❦❦❦ ❦❦❦❦ B, p o o q O O we see that there is a unique holomorphic lift φ : S → B satisfying f ′ = q ◦ φ and ∂f = p ◦ φ. Combining (12) with the diagram, it yields that T f ′ , O (1) ( r ) = T φ,q ∗ O (1) ( r ) + O (1)(13) = T φ,p ∗ O (1) ( r ) − T φ, O ([0]) ( r ) + O (1)= T ∂f, O (1) ( r ) − T φ, O ([0]) ( r ) + O (1) . Since the global section (0 ,
1) of Ω X (log D ) ⊕ O X over X corresponds to thedivisor [ ∞ ] on P (Ω X (log D ) ⊕ O X ) , then O ([ ∞ ]) ∼ = O (1) . This implies that T ∂f, O (1) ( r ) = m ∂f ( r, [ ∞ ]) + N ∂f ( r, [ ∞ ]) + O (1) . Indeed, notice that ∂f meets [ ∞ ] only over D, and with multiplicity at most1, hence N ∂f ( r, [ ∞ ]) ≤ N [1] f ( r, D ) . Therefore, it follows from (13) that T f ′ , O (1) ( r ) ≤ m ∂f ( r, [ ∞ ]) + N [1] f ( r, D ) − m φ ( r, [0]) − N φ ( r, [0]) + O (1) . In the above inequality, m φ ( r, [0]) is bounded from below, N φ ( r, [0]) is equalto N f, Ram( D ) ( r ) , and m ∂f ( r, [ ∞ ]) is bounded from above by O (log T f, A ( r ) − κ ( r ) r + log + log r ) due to Lemma 4.3. Thus, we conclude the proof. (cid:3) Theorem 4.4.
Let X be a smooth complex projective curve, and let D be aneffective ( reduced ) divisor on X. Then for any holomorphic curve f : S → X which is nonconstant, we have T f, K X ( D ) ( r ) ≤ exc N [1] f ( r, D ) + O (cid:16) log T f, A ( r ) − κ ( r ) r + log + log r (cid:17) , where κ is defined by (1) . Proof.
Since X is a curve, then Ω X (log D ) is isomorphic to K X ( D ) . Hence,the canonical projection π : P (Ω X (log D )) → X is an isomorphism, and thus OLOMORPHIC CURVES IN MODULI SPACES 19 O (1) ∼ = π ∗ K X ( D ) as well as f ′ = π − ◦ f. This implies that T f ′ , O (1) ( r ) = T f, K X ( D ) + O (1) . By Theorem 4.2, we have the conclusion holds. (cid:3)
Corollary 4.5.
Let X be a smooth complex projective curve with K X ample.Then there exists positive constants a, b such that T f, K X ( r ) ≤ exc a − bκ ( r ) r , where κ is defined by (1) . Proof.
By Theorem 4.4, it yields that T f, K X ( r ) ≤ exc O (cid:16) log T f, K X ( r ) − κ ( r ) r + log + log r (cid:17) . If κ ( r ) ≡ , the universal covering of S is biholomorphic to C . Since K X > , from [9] we have T f, K X ( r ) ≥ exc O (log r ) as f is nonconstant. From the aboveinequality, f must be a constant. So, this confirms the assertion. If κ ( r ) , we see log + log r ≤ exc − κ ( r ) r since − κ ( r ) ≥ T f, K X ( r ) = o ( T f, K X ( r )) , hence T f, K X ( r ) ≤ exc O ( − κ ( r ) r ) . We provethe conclusion. (cid:3) Second Main Theorem
In the section, we investigate Second Main Theorem of holomorphic curves f : S → X from an open Riemann surface S by using tautological inequalityproved in Section 4. For a further investigation, we apply it to Siegel modularvarieties.5.1. Second Main Theorem.
We first introduce an inequality of curvature currents proved by Sun [10],who obtained this inequality by Viehweg-Zuo’s construction [12, 13].Let (
X, D ) be a smooth logarithmic pair over C . Set ˆ S = S \ f ∗ D, thenthe restriction of f to ˆ S is a holomorphic curve γ : ˆ S → X \ D, which inducesanother holomorphic curve γ ′ : ˆ S −→ P ( T X ( − log D )) . Following the theory of Viehweg-Zuo (Chapter 4 in [12], Chapter 6 in [13]),one could have the following geometric objects over X : an ample line bundle A whose restriction on the smooth locus X \ D is isomorphic to the Viehwegline bundle det( ψ ∗ ω µY/X ) ν (Chapter 4 in [13]); the deformation Higgs bundle( F, τ ) associated to the family g ; a logarithmic Hodge bundle ( E, θ ) with poles along D + T, where T is some normal crossing divisor; the comparisonmap ρ which fits into the following commutative diagram F p,q τ p,q / / ρ p,q (cid:15) (cid:15) F p − ,q +1 ⊗ Ω X (log D ) ρ p − ,q +1 ⊗ ι (cid:15) (cid:15) A − ⊗ E p,q Id ⊗ θ p,q / / A − ⊗ E p − ,q − ⊗ Ω X (log( D + T )) . Denote by d the fiber dimension of g. By iterating Higgs maps τ p,q , we get τ q : F d, −→ F d − q,q ⊗ Sym q Ω X (log D ) . Note that the pull-back of (
F, τ ) by γ induces a (holomorphic) Higgs bundle( F γ , τ γ ) over ˆ S in such manner F γ := γ ∗ F, τ γ : F γ γ ∗ τ / / F γ ⊗ γ ∗ Ω X (log D ) / / F γ ⊗ Ω S . Similarly, ( E γ , θ γ ) is defined. Over ˆ S, we can have the following commutativediagram (see (2.2) in [10]) F p,qγ τ p,qγ / / ρ p,qγ (cid:15) (cid:15) F p − ,q +1 γ ⊗ Ω Sρ p − ,q +1 γ ⊗ ι (cid:15) (cid:15) A − γ ⊗ E p,qγ Id ⊗ θ p,qγ / / A − γ ⊗ E p − ,q − γ ⊗ Ω S (log γ ∗ T ) , where A γ := γ ∗ A. Also, we have the iterations of Higgs maps τ qγ : T ⊗ q ˆ S −→ F d − q,qγ , q = 0 , · · · , d. In further, we define G d − q,q as the saturation of the image of A ⊗ T ⊗ q ˆ S Id ⊗ τ qγ / / A ⊗ F d − q,qγ / / E d − q,qγ in E d − q,qγ . The Higgs subbundle G d − q,q is also the saturation of the image of A γ ⊗ γ ′∗ O ( − q ) Id ⊗ γ ′∗ ˜ τ q / / A γ ⊗ F d − q,qγ / / E d − q,qγ , where ˜ τ q : O ( − q ) → π ∗ F d − q,q is the lift of τ q , here π : P ( T X ( − log D )) → X is the projection. Then this gives maps (see (2.4) in [10]) ζ q : γ ′∗ O ( − q ) −→ A − γ ⊗ G d − q,q , q = 0 , · · · , d. Note that ρ d − , γ ◦ τ γ is nonzero, then it implies that ζ is nonzero. So, thereexists a positive integer m such that ζ m = 0 and ζ m +1 = 0 , here m is calledthe maximal length of iteration. OLOMORPHIC CURVES IN MODULI SPACES 21
Let O ( −
1) be the tautological line bundle over P ( T X ( − log D )) . For each q, we can construct a (pseudo) metric F q on O ( −
1) through the compositionmapping O ( − q ) ˜ τ q / / π ∗ F d − q,q / / π ∗ ( A − ⊗ E d − q,q ) . Here, we note that F q is a bounded pseudo metric with possible degenerationon P ( T X ( − log D )) . So, we can write F q = φ q F, where F is a smooth metricon O ( −
1) and φ q is a bounded function with at most a discrete set of zeros.Then we have c ( O ( − , F ) = c ( O ( − , F q ) + dd c log φ q . With the previous notations, we state a curvature current inequality. Thedetails can be found in [10], Proposition 2.4.
Lemma 5.1 (Curvature current inequality, [10]) . Suppose m is the maximallength of iteration, i.e., the largest integer such that ζ m = 0 . Then m + 12 f ′∗ c ( O ( − ≤ − f ∗ c ([ A ]) + 1 m m X q =1 qf ′∗ dd c log φ q . Theorem 5.2 (Second Main Theorem) . Let
S, X, A, D, d be given as before.Then for any holomorphic curve f : S → X whose image is not containedin Supp D, we have T f,A ( r ) ≤ exc d + 12 N [1] f ( r, D ) + O (cid:16) log T f,A ( r ) − κ ( r ) r + log + log r (cid:17) , where κ is defined by (1) . Proof.
By lifting f to the (analytic) universal covering, we may assume that S is simply connected. By Lemma 5.1, it follows immediately − m + 12 T f ′ , O (1) ( r ) ≤ − T f,A ( r ) + πm m X q =1 q Z D ( r ) g r ( o, x ) dd c log φ q ◦ f ′ . Since φ , · · · , φ q are bounded, by Dynkin formula π Z D ( r ) g r ( o, x ) dd c log φ q ◦ f ′ = 14 E o (cid:20)Z τ r ∆ S log φ q ◦ f ′ ( X t ) dt (cid:21) = 12 E o (cid:2) log φ q ◦ f ′ ( X τ r ) (cid:3) + O (1) ≤ O (1) . Combined with Theorem 4.2, then we conclude the proof. (cid:3)
Corollary 5.3.
Assume the same conditions as in Theorem . . Let f : S → X be a holomorphic curve ramifying over D with multiplicity c > ( d + 1) / . If f satisfies the growth condition lim inf r →∞ κ ( r ) r T f,A ( r ) = 0 , where κ is defined by (1) , then f ( S ) is contained in D. Proof.
Put b = 2 c/ ( d +1) , then b > . Otherwise, we assume that f ( S ) is notcontained in D. Combining Theorem 5.2 with condition f ∗ D ≥ c · Supp f ∗ D, we obtain bT f,A ( r ) ≤ exc cN [1] f ( r, D ) + O (cid:16) log T f,A ( r ) − κ ( r ) r + log + log r (cid:17) ≤ N f ( r, D ) + O (cid:16) log T f,A ( r ) − κ ( r ) r + log + log r (cid:17) . If κ ( r ) , from First Main Theorem, the above inequality implies b ≤ , acontradiction. If κ ( r ) ≡ , we can regard S as C by lifting f to the universalcovering. Then, T f,A ( r ) ≥ exc O (log r ) (see [9]). So, the above inequality alsocontradicts with b > . (cid:3) Corollary 5.4.
Assume the same conditions as in Theorem . . Then forany holomorphic curve f : S → X whose image is not contained in Supp D, we have T f,K X ( D ) ( r ) ≤ exc k ( d + 1)2 N [1] f ( r, D ) + O (cid:16) log T f,A ( r ) − κ ( r ) r + log + log r (cid:17) for an integer k such that [ A ] k ≥ K X ( D ) , where κ is defined by (1) . Proof.
The condition implies that T f,K X ( D ) ( r ) ≤ kT f,A ( r )+ O (1) . Therefore,the conclusion follows from Theorem 5.2. (cid:3)
Next we consider Siegel modular varieties. Let A [ n ] g ( n ≥
3) be the modulispace of principally polarized Abelian varieties with level- n structure. Also,let A [ n ] g denote the smooth compactification of A [ n ] g such that D := A [ n ] g \ A [ n ] g is a normal crossing (boundary) divisor. Then Theorem 5.5.
For any holomorphic curve f : S → A [ n ] g whose image is notcontained in Supp D, we have T f,K A [ n ] g ( D ) ( r ) ≤ exc ( g + 1) N [1] f ( r, D ) + O (cid:16) log T f,A ( r ) − κ ( r ) r + log + log r (cid:17) , where A is given as in Theorem . and κ is defined by (1) . Proof.
We just need to follow the arguments of Sun ([10], Corollary 4.3) andapply Corollary 5.4, and then the assertion can be confirmed. (cid:3)
OLOMORPHIC CURVES IN MODULI SPACES 23
Acknowledgement.
The author is very grateful to Prof. Songyan Xie andDr. Ruiran Sun for their so patient answer of my questions, and also thanksDr. Yan He for his constant discussion with me on mathematical problems.
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