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We prove a general form of the Second Main Theorem for algebraically nondegenerate holomorphic mappings into a smooth complex projective variety intersecting arbitrary hypersurfaces (rather than just the hypersurfaces in general position) and truncated multiplicities
A general form of the Second Main Theorem for hypersurfaces Tran Van Tan[1] and Vu Van Truong[2] Department of Mathematics, Hanoi National University of Education 136-Xuan Thuy street, Hanoi, Vietnam [1] tranvantanhn@yahoo.com, [2] vvtruong.dnb@moet.edu.vn Abstract We prove a general form of the Second Main Theorem for algebraically nondegenerate holomorphic mappings into a smooth complex projective variety intersecting arbitrary hypersurfaces (rather than just the hypersurfaces in general position) and truncated multiplicities. Mathematics Subject Classification 2000: Primary 32H30; Secondary 32H04, 32H25, 14J70. Key words: Nevanlinna theory, Second Main Theorem, Hypersurface. 1 Introduction and statements Let f be a holomorphic mapping of C into CP N , with a reduced representation f = (f0 : · · · : fN ). The characteristic function Tf (r) of f is defined by 1 Tf (r) := 2π 2π log f (reiθ ) dθ, where f := max{|f0 |, . . . , |fN |}. 0 Let M be a positive integer or +∞, and let ν be a divisor on C. Set ν [M ] (z) := min{ν(z), M }. The truncated counting function to level M of ν is defined by r Nν[M ] (r) |z| 0, q (q − n − 1 − )Tf (r) ≤ j=1 1 N (r, Dj ). dj We note that in Theorem B, the multiplicities of intersections are not truncated (all of them are taken in to the account of counting functions). Motivated by the case of hyperplanes, in this paper we generalize Theorem B to the case of arbitrary hypersurfaces, and the multiplicities of intersections are truncated (the multiplicities are taken in to the account do not exceed a common positive integer). In the case where hypersurfaces are in general position, from our below theorem we get a slight improvement of Theorem B that mulitiplicities in the counting functions are truncated by a positive integer. We will prove the following theorem. Theorem 1.1. Let V ⊂ CP N be a smooth complex projective variety of dimension n ≥ 1. Let f be an algebraically nondegenerate holomorphic mapping of C into V. Let D1 , . . . , Dq (V ⊂ Dj ) be arbitrary hypersurfaces in CP n of degree dj . Then, for every > 0, there exists a positive integer M depending on , dj , q, n, deg V such that 2π max 0 K∈K j∈K dθ 1 λDj (f (reiθ )) + dj 2π r 1 dt max t K∈K j∈K,|z| be an integer and let c = (c0 , . . . , cN ) ∈ +1 RN ≥0 . Let {i0 , . . . , in } be a subset of {0, . . . , N } such that {x = (x0 : · · · : xN ) ∈ CP N : xi0 = · · · = xin = 0} ∩ X = ∅. Then 1 mHX (m) SX (m, c) ≥ 1 (2n + 1) (ci0 + · · · + cin ) − (n + 1) m · max ci . 0≤i≤N Lemma 2.2 ([2], Lemma 3.2.13). Let f be a linearly nondegenerate holomorphic mapping of C into CP N with the reduced representation f = (f0 : · · · : fN ). Let W (f ) = W (f0 , . . . , fN ) be the Wronskian of f. Then N ν f0 ···fN ≤ W (f ) min{νfi , N }. i=0 4 3 Proof of Theorem 1.4. Let Qj , 1 ≤ j ≤ q, be homogeneous polynomials in C[x0 , . . . , xN ] of degree dj defining Dj . Denote by R the set of all subsets R ⊂ {1, . . . , q} such that #R = n + 1 and ∩j∈R Dj ∩ V = ∅. Claim: If R = ∅ and d1 = · · · = dq := d, then for every > 0, there exists a positive integer M depending on , d, q, n, deg V, such that 2π max 0 R∈R j∈R dθ 1 λDj (f (reiθ )) + d 2π r 1 dt max t R∈R j∈R,|z| 0, we choose m such that (n+1)d < 4 . Then, by (3.2) we get HY (m) 2π λDj (f (reiθ )) max 0 R∈R j∈R (2n+1)(n+1)dq m (3.2) < 4 and dθ n+1 ≤ (n + 1)d + Tf (r) − NW (F ) (r). 2π 2 mHY (m) (3.3) For each J := {j1 , . . . , jHY (m) } ∈ L, then there exists a constant cJ = 0 such that I IjH I W (F ) = cJ · W (Q1j1 1 (f ) · · · Qqj1 q (f ), . . . , Q1 1 Y (m) IjH (f ) · · · Qq q Y (m) (f )). On the other hand, by Lemma 2.2, ν Ij Ij Ij 1 Ij q H (m) 1 H (m) q Q1 1 (f )···Qq 1 (f )···Q1 Y (f )···Qq Y (f ) Ij Ij Ij 1 Ij q HY (m) 1 HY (m) q 1 1 (f )···Qq (f ),...,Q1 (f )···Qq W Q1 (f ) 6 ≤ ν 1≤i≤HY (m) [HY (m)−1] Ij 1 I i (f )···Q ji q (f ) q Q1 . Hence, for all J ∈ L, we have νW (F ) ≥ ν Ij Ij Ij q Ij 1 H (m) 1 H (m) q (f )···Qq Y (f ) Q1 1 (f )···Qq 1 (f )···Q1 Y ≥ Iij νQj (f ) − [H (m)−1] νQjY(f ) − ν 1≤i≤HY (m) [HY (m)−1] I Ij 1 i (f )···Q ji q (f ) q Q1 . (3.4) 1≤j≤q i∈J For every z ∈ C, let cz := (c1,z , . . . , cq,z ) where cj,z := νQj (f ) (z) − Then, by definition of the Hilbert weight, there exists Jz ∈ L such that [H (m)−1] νQjY(f ) (z). [H (m)−1] Ii · cz = SY (m, cz ) = Iij νQj (f ) (z) − νQjY(f ) (z) . 1≤j≤q i∈Jz i∈Jz Then, by Lemma 2.1, for very R ∈ R we have 1 [H (m)−1] mHY (m) 1≤j≤q i∈J ≥ (z) z 1 n+1 − ≥ Iij νQj (f ) (z) − νQjY(f ) [H (m)−1] νQj (f ) (z) − νQjY(f ) (2n + 1) m 1 n+1 (z) j∈R [H (m)−1] max νQj (f ) (z) − νQjY(f ) 1≤j≤q [H (m)−1] νQj (f ) (z) − νQjY(f ) (z) − j∈R (z) (2n + 1) m νQj (f ) (z). 1≤j≤q Combining with (3.4), for every R ∈ R and z ∈ C, we have 1 mHY (m) νW (F )(z) ≥ 1 n+1 [H (m)−1] νQj (f ) (z) − νQjY(f ) (z) j∈R − (2n + 1) m νQj (f ) (z). 1≤j≤q This implies that n+1 [H (m)−1] νW (F ) ≥ max νQj (f ) − νQjY(f ) R∈R mHY (m) j∈R − 7 (n + 1)(2n + 1) m νQj (f ) . 1≤j≤q Therefore, n+1 NW (F ) (r) ≥ mHY (m) r 1 − r ≥ 1 − r ≥ 1 dt max t R∈R dt max t R∈R (z) j∈R,|z| ... This completes the proof of Theorem 1.1 References [1] G Dethloff and T V Tan and D D Thai, An extension of the CartanNochka second main theorem for hypersurfaces, Internat J Math 22 (2011),... are taken in to the account of counting functions) Motivated by the case of hyperplanes, in this paper we generalize Theorem B to the case of arbitrary hypersurfaces, and the multiplicities of. .. H Fujimoto, Value didtribution theory of the Gauss map of minimal surfaces in Rm , Braunschweig, Vieweg -Sohn, 1993 [3] M Ru, On a general form of the Second Main Theorem, Trans AMS 349 (1997),