A survey to Nevanlinna-type theory based on heat diffusion
aa r X i v : . [ m a t h . C V ] J un A SURVEY TO NEVANLINNA-TYPE THEORY BASED ONHEAT DIFFUSIONS
XIANJING DONG
Abstract.
This is a survey to Nevanlinna-type theory of holomorphicmappings from a complete and stochastically complete K¨ahler manifoldinto compact complex manifolds with a positive line bundle. When someenergy and Ricci curvature conditions are imposed, the Nevanlinna-typedefect relations based on heat diffusions are obtained. Introduction and main results
In this paper, we mainly investigate the value distribution of holomorphicmappings from a complete and stochastically complete K¨ahler manifold intoa compact complex manifold using Nevanlinna-type functions introduced byAtsuji [3]. In 2010, Atsuji [3] first introduced the notions of Nevanlinna-typefunctions e T x ( t ) , e N x ( t, a ) and e m x ( t, a ) of meromorphic functions on a K¨ahlermanifold based on heat diffusions. By using a technique of Brownian motion(see [1, 2, 3, 4, 6, 7]), Atsuji proved an analogy of the Second Main Theoremin Nevanlinna theory. Theorem 1.1 (Atsuji, [3]) . Let f be a nonconstant meromorphic functionon a complete and stochastically complete K¨ahler manifold M, and a , · · · , a q be distinct in P ( C ) . Assume that e T x ( t ) < ∞ as < t < ∞ and e T x ( t ) → ∞ as t → ∞ , and | e N x ( t, Ric) | < ∞ as < t < ∞ . Then q X j =1 e m x ( t, a j ) + e N ( t, x ) ≤ e T x ( t ) + 2 e N x ( t, Ric) + O (cid:0) log e T x ( t ) (cid:1) + O (1) holds except for t in an exceptional set of finite length. Furthermore, with certain energy and Ricci curvature conditions imposed,Atsuji proved some defect relations (see [3]). In this paper, we shall develop
Mathematics Subject Classification.
Key words and phrases.
Nevanlinna theory, Second Main Theorem, K¨ahler manifold,Ricci curvature, Brownian motion.
X.J. DONG
Atsuji’s technique based on heat diffusions. In doing so, first of all, we needto extend the notions of so-called Nevanlinna-type functions, see Section 2.1in this paper. As a generalization of Theorem 1.1, we establish Theorem 1.3below. Indeed, we develop some defect relations in Section 4.3.Early in 1970s, Griffiths and his school (see [5, 11, 12]) made a significantprogress in Nevanlinna theory. Carlson-Griffiths [5] established the SecondMain Theorem and defect relations for holomorphic mappings from C m intoan algebraic variety V intersecting simple divisors under dimension assump-tion m = dim C V . The theory is referred to Griffiths’ equi-dimensional valuedistribution theory [12]. Later, Griffiths-King [11] generalized this theory toholomorphic mappings from complex affine algebraic varieties into V. We consider an analogue of Griffiths’ equi-dimensional value distributiontheory on stochastically complete K¨ahler manifolds based on heat diffusions.Our approach here is to combine the Logarithmic Derivative Lemma with aprobabilistic method. So, the first task is to establish the Logarithmic Deriv-ative Lemma for meromorphic functions on K¨ahler manifolds (see Theorem1.2 below).We state the main results of this paper.
Theorem 1.2 (Logarithmic Derivative Lemma) . Let M be a complete andstochastically complete K¨ahler manifold. Let ψ be a nonconstant meromor-phic function on M such that e T ( t, ψ ) < ∞ as < t < ∞ . Then for any δ > , there exists a set E δ ⊂ (1 , ∞ ) of finite Lebesgue measure such that e m (cid:18) t, k∇ M ψ k| ψ | (cid:19) ≤ (cid:0) δ (cid:1) log e T ( t, ψ ) + O (1) holds for t ∈ (1 , ∞ ) outside E δ . Theorem 1.3 (Second Main Theorem) . Let M be a complete and stochas-tically complete K¨ahler manifold. Let L → N be a positive line bundle overa compact complex manifold N, and D ∈ | L | such that D has only simplenormal crossings. Assume that f : M → N (dim C M ≥ dim C N ) is a differ-entiably non-degenerate holomorphic mapping such that e T f ( t, L ) < ∞ and | e T ( t, R M ) | < ∞ for < t < ∞ . Then for any δ > , we have e T f ( t, L ) + e T f ( t, K N ) + e T ( t, R M ) ≤ e N f ( t, D ) + O (cid:0) log e T f ( t, L ) (cid:1) + O (1) holds for t ∈ (1 , ∞ ) outside a set E δ ⊂ (1 , ∞ ) of finite Lebesgue measure. Theorem 1.4 (defect relation) . Let M be a complete and stochasticallycomplete K¨ahler manifold of non-negative Ricci scalar curvature. Let L → N be a positive line bundle over a compact complex manifold N, and D ∈ | L | such that D has only simple normal crossings. Assume that f : M → N EVANLINNA-TYPE THEORY BASED ON HEAT DIFFUSIONS 3 (dim C M ≥ dim C N ) is a differentiably non-degenerate holomorphic mappingsatisfying (1) Z ∞ e − ǫr dr Z B o ( r ) e f ∗ c ( L,h ) ( x ) dV ( x ) < ∞ for any ǫ > . Then e Θ f ( D ) ≤ (cid:20) c ( K ∗ N ) c ( L ) (cid:21) . First Main Theorem
Let M be a m -dimensional complete K¨ahler manifold with K¨ahler metricform α. A Brownian motion in M (as a Riemannian manifold) is a Markovprocess generated by ∆ M with transition density function p ( t, x, y ) beingthe minimal positive fundamental solution of the heat equation ∂∂t u ( t, x ) = 12 ∆ M u ( t, x ) . Let X t be the Brownian motion in M starting from o ∈ M with transitiondensity function p ( t, o, x ) , with law P o and expectation E o . The
Itˆo formula (see [1, 13, 14]) asserts that u ( X t ) − u ( o ) = B (cid:18)Z t k∇ M u k ( X s ) ds (cid:19) + 12 Z t ∆ M u ( X s ) dt, P o − a.s. for u ∈ C ♭ ( M ) , where B t is the one-dimensional standard Brownian motionin R and ∇ M is the gradient operator on M . It follows the Dynkin formula E o [ u ( X T )] − u ( o ) = 12 E o (cid:20)Z T ∆ M u ( X t ) dt (cid:21) for a stopping time T satisfying E o [ T ] < ∞ such that each term in the abovemakes sense. Due to the expectation “ E o ” , the Dynkin formula works in thecase when u is of a polar set of singularities if each term makes sense. M issaid to be stochastically complete if Z M p ( t, o, x ) dV ( x ) = 1holds for all o ∈ M. By Grigor’yan’s criterion (see [10]), M is stochasticallycomplete if R M ( x ) ≥ − cr ( x ) − c for some constant c > , where R M is thepointwise lower bound of Ricci curvatures defined by R M ( x ) = inf ξ ∈ T x M, k ξ k =1 Ric( ξ, ξ ) . X.J. DONG
Nevanlinna-type functions.
Let L → N be a holomorphic line bundle over a compact complex mani-fold N. Denoted by H ( N, L ) the vector space of holomorphic global sectionsof L over N, and by | L | the complete linear system of effective divisors D s for s ∈ H ( M, L ) . Given D ∈ | L | , let f : M → N be a holomorphic mapping into N such that f ( M ) Supp D. Endow L withHermitian metric h, which defines the Chern form c ( L, h ) = − dd c log h. Weuse the standard notations d = ∂ + ∂, d c = √− π ( ∂ − ∂ ) , dd c = √− π ∂∂. Lemma 2.1 ([8]) . ∆ M log( h ◦ f ) is globally defined on M and ∆ M log( h ◦ f ) = − m f ∗ c ( L, h ) ∧ α m − α m , where α = √− π P i,j g ij dz i ∧ dz j is the K¨ahler metric form on M. Let ( { U α } , { e α } ) be a local trivialization covering of ( L, h ) . For 0 = s ∈ H ( N, L ), write s = e se α locally on U α . Note that ∆ M log | e s ◦ f | is globallydefined on M and∆ M log k s ◦ f k = ∆ M log( h ◦ f ) + ∆ M log | e s ◦ f | . The similar argument as in the proof of Lemma 2.1 shows that∆ M log | e s ◦ f | = 4 m dd c log | e s ◦ f | ∧ α m − α m . Lemma 2.2 ([8]) . For s ∈ H ( N, L ) with D = ( s ) , we have ( i ) log k s ◦ f k is locally the difference of two plurisubharmonic functions,hence log k s ◦ f k ∈ L loc ( M ) and log k s ◦ f k ∈ L ( S o ( r )) . ( ii ) dd c log k s ◦ f k = f ∗ D − f ∗ c ( L, h ) in the sense of currents. We now introduce the notion of the so-called
Nevanlinna-type functions ofholomorphic mappings into a compact complex manifold. For a continuous(1,1)-form ω on N, we use the following convenient symbols e f ∗ ω ( x ) = 2 m f ∗ ω ∧ α m − α m , g t ( o, x ) = Z t p ( s, o, x ) ds. If ω > , we call e f ∗ ω ( x ) the energy density function of f with respect to themetrics α, ω. The characteristic function of f with respect to ω is definedby e T f ( t, ω ) = 12 E o (cid:20)Z t e f ∗ ω ( X s ) ds (cid:21) = 12 Z M g t ( o, x ) e f ∗ ω ( x ) dV ( x ) . EVANLINNA-TYPE THEORY BASED ON HEAT DIFFUSIONS 5
Apply Lemma 2.1, we have e f ∗ c ( L,h ) ( x ) = −
14 ∆ M log( h ◦ f ( x )) . It well defines e T f ( t, L ) := e T f ( t, c ( L, h ))up to a constant term, due to the compactness of N. The conditions for e T ( t, L ) < ∞ as 0 < t < ∞ and e T ( t, L ) → ∞ as t → ∞ provided L >
L > s D be the canonical section defined by D. Wemay have k s D k < N . Noticing that log k s D ◦ f ( x ) k is locally the difference of two plurisubharmonic functions from Lemma 2.2,and is thus integrable on S o ( r ) . The proximity function of f with respect to D is defined by e m f ( t, D ) = E o (cid:20) log 1 k s D ◦ f ( X t ) k (cid:21) . As for counting function , we use the expression e N f ( t, D ) = π m ( m − Z M ∩ f ∗ D g t ( o, x ) α m − = π m ( m − Z M g t ( o, x ) dd c log | e s D ◦ f ( x ) | α m − = 14 Z M g t ( o, x )∆ M log | e s D ◦ f ( x ) | dV ( x ) . We have e N f ( t, D ) = 0 if f omits D . If(2) 14 E o (cid:20)Z t ∆ M log | e s D ◦ f ( X s ) | ds (cid:21) < ∞ , < t < ∞ which is hence equal to e N f ( t, D ) by using Fubini theorem due to the absoluteconvergence of (2), then e N f ( t, D ) has an alternative expression(3) e N f ( t, D ) = lim λ →∞ λ P o (cid:18) sup ≤ s ≤ t log 1 k s D ◦ f ( X s ) k > λ (cid:19) . To see that, one can apply the arguments appeared in [9] related to the localmartingales and use Dynkin formula. It is shown that the above limit exists
X.J. DONG and equals lim λ →∞ λ P o (cid:18) sup ≤ s ≤ t log 1 k s D ◦ f ( X s ) k > λ (cid:19) = − E o (cid:20)Z t ∆ M log 1 | e s D ◦ f ( X s ) | ds (cid:21) = 14 E o (cid:20)Z t ∆ M log | e s D ◦ f ( X s ) | ds (cid:21) = 14 Z M (cid:20)Z t p ( s, o, x ) ds (cid:21) ∆ M log | e s D ◦ f ( x ) | dV ( x )= e N f ( t, D ) . Another proof of (3) will be given in Section 2.2 below. We remark that (2)can be guaranteed by e T f ( t, L ) < ∞ as 0 < t < ∞ , since Theorem 2.3 below.2.2. First Main Theorem.
We adopt the same notations as in Section 2.1.
Theorem 2.3 (FMT) . Let L → N be a positive line bundle over a compactcomplex manifold N, and D ∈ | L | . Let f : M → N be a holomorphic mappingsuch that f ( M ) Supp D . Then e T f ( t, L ) = e m f ( t, D ) + e N f ( t, D ) + O (1) . Proof.
Take a Hermitian metric h on L such that ω := c ( L, h ) > . Set T λ = inf (cid:26) t > ≤ s ≤ t log 1 k s D ◦ f ( X s ) k > λ (cid:27) . Let ( { U α } , { e α } ) be a local trivialization covering of ( L, h ). Write s D = e s D e α locally on U α . Then, we get(4) log k s D ◦ f k = log | e s D ◦ f | + log( h ◦ f ) . Note that e s D ◦ f is holomorphic and h ◦ f > k s D ◦ f k − . Consequently, E o (cid:20) log 1 k s D ◦ f ( X t ∧ T λ ) k (cid:21) = 12 E o (cid:20)Z t ∧ T λ ∆ M log 1 k s D ◦ f ( X s ) k ds (cid:21) + log 1 k s D ◦ f ( o ) k , (5)where t ∧ T λ = min { t, T λ } . Since log k s D ◦ f ( X s ) k − has no singularities as0 ≤ s ≤ T λ due to the definition of T λ , it concludes by (4) that∆ M log 1 k s D ◦ f ( X s ) k = −
12 ∆ M log( h ◦ f ( X s )) EVANLINNA-TYPE THEORY BASED ON HEAT DIFFUSIONS 7 as 0 ≤ s ≤ T λ , where we use the fact that log | e s D ◦ f | is harmonic on M \ f ∗ D. Hence, (5) turns to E o (cid:20) log 1 k s D ◦ f ( X t ∧ T λ ) k (cid:21) = − E o (cid:20)Z t ∧ T λ ∆ M log( h ◦ f ( X s )) ds (cid:21) + O (1) . Since f ∗ ω = − dd c log( h ◦ f ) , then by Lemma 2.1(6) e f ∗ ω = − m dd c log( h ◦ f ) ∧ α m − α m = −
12 ∆ M log( h ◦ f ) . By the monotone convergence theorem, it yields from (6) that − E o (cid:20)Z t ∧ T λ ∆ M log( h ◦ f ( X s )) ds (cid:21) (7) = 12 E o (cid:20)Z t ∧ T λ e f ∗ ω ( X s ) ds (cid:21) → e T f ( t, L )as λ → ∞ , where we use the fact that T λ → ∞ a.s. as λ → ∞ for that f ∗ D is polar. Write the first term appeared in (5) as two partsI + II = E o (cid:20) log 1 k s D ◦ f ( X t ) k : t < T λ (cid:21) + E o (cid:20) log 1 k s D ◦ f ( X T λ ) k : T λ ≤ t (cid:21) . Apply the monotone convergence theorem,(8) I → e m f ( r, D )as λ → ∞ . Now we look at II . By the definition of T λ , it is not hard to see(9) II = λ P o (cid:18) sup ≤ s ≤ t log 1 k s D ◦ f ( X s ) k > λ (cid:19) → e N f ( t, D )as λ → ∞ . By (7)-(9), we have the desired result. (cid:3)
Another proof of (3) . Since f ∗ D is polar, we use Dynkin formula to get E o (cid:20) log 1 k s D ◦ f ( X t ) k (cid:21) + O (1) = 12 E o (cid:20)Z t ∆ M log 1 k s D ◦ f ( X s ) k ds (cid:21) . This yields that e T f ( t, L ) + O (1) = e m f ( t, D ) + 14 E o (cid:20)Z t ∆ M log | e s D ◦ f ( X s ) | ds (cid:21) . On the other hand, the argument in the proof of Theorem 2.3 implies that e T f ( t, L ) + O (1) = e m f ( t, D ) + lim λ →∞ λ P o (cid:18) sup ≤ s ≤ t log 1 k s D ◦ f ( X s ) k > λ (cid:19) . By a comparison, we deduce (3).3.
Logarithmic Derivative Lemma
X.J. DONG
Logarithmic Derivative Lemma.
Let (
M, g ) be a complete and stochastically complete K¨ahler manifold ofcomplex dimension m , with K¨ahler metric form α and gradient operator ∇ M associated to g. Let X t be the Brownian motion in M with generator ∆ M , started at a fixed point o ∈ M with transition density function p ( t, o, x ) . Lemma 3.1 (Calculus Lemma) . Let k be a non-negative function on M sothat E o [ k ( X t )] < ∞ and E o [ R t k ( X s ) ds ] < ∞ for < t < ∞ . Then for any δ > , there exists a set E δ ⊂ [0 , ∞ ) with finite Lebesgue measure such that E o (cid:2) k ( X t ) (cid:3) ≤ (cid:18) E o h Z t k ( X s ) ds i(cid:19) δ holds for r E δ . Proof.
Set γ ( t ) := E o [ R t k ( X s ) ds ] and E δ := { t ∈ (0 , ∞ ) : γ ′ ( t ) > γ δ ( t ) } , then γ ′ ( t ) = E o (cid:2) k ( X t ) (cid:3) . The claim clearly holds for k ≡ . If k
0, supposethat γ (1) = 0 without loss of generality. Note that Z E δ dt ≤ Z ∞ γ ′ ( t ) γ δ ( t ) dt ≤ δ − γ − δ (1) < ∞ . This finishes the proof. (cid:3)
Let ψ be a meromorphic function on M. The norm of the gradient of ψ is defined by k∇ M ψ k = X i,j g ij ∂ψ∂z i ∂ψ∂z j , where ( g ij ) is the inverse of ( g ij ) . Locally, we write ψ = ψ /ψ , where ψ , ψ are holomorphic functions so that codim C ( ψ = ψ = 0) ≥ C M ≥ . Identify ψ with a meromorphic mapping into P ( C ) by x [ ψ ( x ) : ψ ( x )] . The characteristic function of ψ with respect to the Fubini-Study form ω F S on P ( C ) is defined by e T ψ ( t, ω F S ) = 14 E o (cid:20)Z t ∆ M log (cid:0) | ψ ( X s ) | + | ψ ( X s ) | (cid:1) ds (cid:21) . 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 by b T ψ ( t, ω F S ) = 14 E o (cid:20)Z t ∆ M log (cid:0) | ψ ( X s ) | (cid:1) ds (cid:21) . Clearly, b T ψ ( t, ω F S ) ≤ e T ψ ( t, ω F S ) . EVANLINNA-TYPE THEORY BASED ON HEAT DIFFUSIONS 9
We adopt the spherical distance k· , ·k on P ( C ) , the proximity function of ψ with respect to a ∈ P ( C ) = C ∪ {∞} is defined by b m ψ ( t, a ) = E o (cid:20) log 1 k ψ ( X t ) , a k (cid:21) . Again, set b N ψ ( t, a ) = lim λ →∞ λ P o (cid:18) sup ≤ s ≤ t log 1 k f ( X s ) , a k > λ (cid:19) . Similar to Theorem 2.3, we can show that b T ψ ( t, ω F S ) = b m ψ ( t, a ) + b N ψ ( t, a ) + O (1) . Define e T ( t, ψ ) := e m ( t, ψ, ∞ ) + e N ( t, ψ, ∞ ) , where e m ( t, ψ, ∞ ) = E o (cid:2) log + | ψ ( X t ) | (cid:3) , e N ( t, ψ, ∞ ) = lim λ →∞ λ P o (cid:18) sup ≤ s ≤ t log + | f ( X s ) | > λ (cid:19) . Since e N ( t, ψ, ∞ ) = b N ψ ( t, ∞ ) and e m ( t, ψ, ∞ ) = b m ψ ( t, ∞ ) + O (1) , whence(10) e T ( t, ψ ) = b T ψ ( t, ω F S ) + O (1) , e T ( t, ψ − a ) = e T ( t, ψ ) + O (1) . On P ( C ) = C ∪ {∞} , we define a singular metricΦ = 1 | ζ | (1 + log | ζ | ) √− π dζ ∧ dζ. A direct computation shows that(11) Z P ( C ) Φ = 1 , mπ ψ ∗ Φ ∧ α m − α m = k∇ M ψ k | ψ | (1 + log | ψ | ) . Define e T ψ ( t, Φ) = 12 E o (cid:20)Z t e ψ ∗ Φ ( X s ) ds (cid:21) , e ψ ∗ Φ ( x ) = 2 m ψ ∗ Φ ∧ α m − α m . According to (11), we obtain(12) e T ψ ( t, Φ) = 12 π E o (cid:20)Z t k∇ M ψ k | ψ | (1 + log | ψ | ) ( X s ) ds (cid:21) . Lemma 3.2.
Assume that e T ( t, ψ ) < ∞ for < t < ∞ , then e T ψ ( t, Φ) ≤ e T ( t, ψ ) + O (1) . Proof.
By Fubini theorem and the First Main Theorem e T ψ ( t, Φ) = 12 E o (cid:20)Z t e ψ ∗ Φ ( X s ) ds (cid:21) = 12 Z M (cid:20)Z t p ( s, o, x ) ds (cid:21) e ψ ∗ Φ ( x ) dV ( x )= π m ( m − Z M g t ( o, x ) ψ ∗ Φ ∧ α m − = π m ( m − Z P ( C ) Φ Z M ∩ ψ − ( ζ ) g t ( o, x ) α m − = Z P ( C ) e N ( t, ψ, ζ )Φ ≤ e T ( t, ψ ) + O (1) . This completes the proof. (cid:3)
Lemma 3.3.
For any δ > , there exists a set E δ ⊂ (1 , ∞ ) of finite Lebesguemeasure such that E o (cid:20) log + k∇ M ψ k | ψ | (1 + log | ψ | ) ( X t ) (cid:21) ≤ (1 + δ ) log e T ( t, ψ ) + O (1) holds for r ∈ (1 , ∞ ) outside E δ . Proof.
By Jensen inequality, E o (cid:20) log + k∇ M ψ k | ψ | (1 + log | ψ | ) ( X t ) (cid:21) ≤ E o (cid:20) log (cid:16) k∇ M ψ k | ψ | (1 + log | ψ | ) ( X t ) (cid:17)(cid:21) ≤ log + E o (cid:20) k∇ M ψ k | ψ | (1 + log | ψ | ) ( X t ) (cid:21) + O (1) . Lemma 3.1 and Lemma 3.2 with (12) implylog + E o (cid:20) k∇ M ψ k | ψ | (1 + log | ψ | ) ( X t ) (cid:21) ≤ (1 + δ ) log + E o (cid:20)Z t k∇ M ψ k | ψ | (1 + log | ψ | ) ( X s ) ds (cid:21) ≤ (1 + δ ) log e T ( t, ψ ) + O (1) . This completes the proof. (cid:3)
Define e m (cid:18) t, k∇ M ψ k| ψ | (cid:19) = E o (cid:20) log + k∇ M ψ k| ψ | ( X t ) (cid:21) . EVANLINNA-TYPE THEORY BASED ON HEAT DIFFUSIONS 11
Proof of Theorem 1.2Proof.
We have on the one hand e m (cid:18) t, k∇ M ψ k| ψ | (cid:19) = 12 E o (cid:20) log + (cid:18) k∇ M ψ k | ψ | (1 + log | ψ | ) ( X t ) (cid:0) | ψ ( X t ) | (cid:1)(cid:19)(cid:21) ≤ E o (cid:20) log + k∇ M ψ k | ψ | (1 + log | ψ | ) ( X t ) (cid:21) + 12 E o (cid:2) log + (cid:0) | ψ ( X t ) | (cid:1)(cid:3) ≤ E o (cid:20) log + k∇ M ψ k | ψ | (1 + log | ψ | ) ( X t ) (cid:21) + E o (cid:20) log (cid:16) + | ψ ( X t ) | + log + | ψ ( X t ) | (cid:17)(cid:21) . Lemma 3.3 implies that E o (cid:20) log + k∇ M ψ k | ψ | (1 + log | ψ | ) ( X t ) (cid:21) ≤ (1 + δ ) log e T ( t, ψ ) + O (1) . On the other hand, by Jensen inequality E o (cid:20) log (cid:16) + | ψ ( X t ) | + log + | ψ ( X t ) | (cid:17)(cid:21) ≤ log E o (cid:20) + | ψ ( X t ) | + log + | ψ ( X t ) | (cid:21) ≤ log (cid:0) e m ( t, ψ, ∞ ) + e m ( t, ψ, (cid:1) + O (1) ≤ log e T ( t, ψ ) + O (1) . Combining the above, we are led to the assertion. (cid:3) Second Main Theorem and defect relations
Preparation.
Let ψ : M → P n ( C ) be a meromorphic mapping, and let [ w : · · · : w n ] bethe homogeneous coordinate system of P n ( C ) . Assume that w ◦ ψ = 0 . Let i : C n ֒ → P n ( C ) be an inclusion defined by( z , · · · , z n ) [1 : z : · · · : z n ] . Clearly, the Fubini-Study form ω F S = dd c log k w k on P n ( C ) induces a (1,1)-form i ∗ ω F S = dd c log( | ζ | + | ζ | + · · · + | ζ n | ) on C n , where ζ j := w j /w for ≤ j ≤ n. The characteristic function of ψ with respect to i ∗ ω F S is definedby b T ψ ( t, ω F S ) = 14 E o h Z t ∆ M log (cid:16) n X j =0 | ζ j ◦ ψ ( X s ) | (cid:17) ds i . Clearly, b T ψ ( t, ω F S ) ≤ e T ψ ( t, ω F S ) = 14 E o h Z t ∆ M log k ψ ( X s ) k ds i . As is noted, the Dynkin formula works for a set of singularities which is polar.Note that the indeterminacy set and pole divisors of ζ j ◦ ψ for 1 ≤ j ≤ n arepolar. Assume that o is not in both indeterminacy set and pole set. Hence, b T ψ ( t, ω F S ) = 12 E o h log (cid:16) n X j =0 | ζ j ◦ ψ ( X t ) | (cid:17)i −
12 log (cid:16) n X j =0 | ζ j ◦ ψ ( o ) | (cid:17)b T ζ j ◦ ψ ( t, ω F S ) = 12 E o (cid:2) log (cid:0) | ζ j ◦ ψ ( X t ) | (cid:1)(cid:3) −
12 log (cid:0) | ζ j ◦ ψ ( o ) | (cid:1) . Theorem 4.1.
We have max ≤ j ≤ n e T ( t, ζ j ◦ ψ ) + O (1) ≤ b T ψ ( t, ω F S ) ≤ n X j =1 e T ( t, ζ j ◦ ψ ) + O (1) . Proof.
On the one hand, b T ψ ( t, ω F S ) = E o h log (cid:16) n X j =0 | ζ j ◦ ψ ( X t ) | (cid:17)i −
12 log (cid:16) n X j =0 | ζ j ◦ ψ ( o ) | (cid:17) ≤ n X j =1 (cid:16) E o (cid:2) log (cid:0) | ζ j ◦ ψ ( X t ) | (cid:1)(cid:3) − log (cid:0) | ζ j ◦ ψ ( o ) | (cid:1)(cid:17) + O (1)= n X j =1 e T ( t, ζ j ◦ ψ ) + O (1) . On the other hand, e T ( t, ζ j ◦ ψ ) = b T ζ j ◦ ψ ( t, ω F S ) + O (1)= 14 E o h Z t ∆ M log (cid:0) | ζ j ◦ ψ ( X s ) | (cid:1) ds i + O (1) ≤ E o h Z t ∆ M log (cid:16) n X j =0 | ζ j ◦ ψ ( X s ) | (cid:17) ds i + O (1)= b T ψ ( t, ω F S ) + O (1) . The claim is certified. (cid:3)
EVANLINNA-TYPE THEORY BASED ON HEAT DIFFUSIONS 13
Corollary 4.2.
We have max ≤ j ≤ n e T ( t, ζ j ◦ f ) ≤ e T ψ ( t, ω F S ) + O (1) . Let V be a complex projective algebraic variety and C ( V ) be the field ofrational functions defined on V over C . Let
V ֒ → P N ( C ) be a holomorphicembedding, and let H V be the restriction of hyperplane line bundle H over P N ( C ) to V. Denoted by [ w : · · · : w N ] the homogeneous coordinate systemof P N ( C ) and assume that w = 0 without loss of generality. Notice that therestriction { ζ j := w j /w } to V gives a transcendental base of C ( V ) . Thereby,any φ ∈ C ( V ) can be represented by a rational function in ζ , · · · , ζ N (13) φ = Q ( ζ , · · · , ζ N ) . Theorem 4.3.
Let f : M → V be an algebraically non-degenerate holomor-phic mapping such that e T f ( t, H V ) < ∞ as < t < ∞ . Then for φ ∈ C ( V ) , there is constant C > such that e T ( t, φ ◦ f ) ≤ C e T f ( t, H V ) + O (1) . Proof.
Assume f
0. Pulling back (13) by f, φ ◦ f = Q ( ζ ◦ f, · · · , ζ N ◦ f ) . Since Q j is rational, then there exists a constant C ′ > e T ( t, φ ◦ f ) ≤ C ′ N X j =1 e T ( t, ζ j ◦ f ) + O (1) . By Corollary 4.2, e T ( t, ζ j ◦ f ) ≤ e T f ( t, H V ) + O (1) . This proves the theorem. (cid:3)
Corollary 4.4.
Let f : M → V be an algebraically non-degenerate holo-morphic mapping such that e T f ( t, ω ) < ∞ as < t < ∞ . Let ω be a positive (1 , -form on V. Then for φ ∈ C ( V ) , there is constant C > such that e T ( t, φ ◦ f ) ≤ C e T f ( t, ω ) + O (1) . Proof.
Since V is compact, then for two positive (1,1)-forms ω , ω on V, wehave c ω ≤ ω ≤ c ω for two constants c , c > . Hence, the claim followsfrom Theorem 4.3. (cid:3)
Second Main Theorem.
Let M be a K¨ahler manifold with K¨ahler metric g = X i,j g ij dz i ⊗ dz j . The Ricci curvature tensor Ric = P i,j R ij dz i ⊗ dz j on M is given by R ij = − ∂ ∂z i ∂z j log det( g st ) . A well-known theorem by S. S. Chern asserts that the
Ricci curvature form R M := − dd c log det( g st ) = √− π X i,j R ij dz i ∧ dz j is a real and closed smooth (1,1)-form which represents the first Chern classof M in de Rham cohomology group H ( M, R ) . Let s M denote the Ricciscalar curvature of M, which is defined by s M = X i,j g ij R ij , where ( g ij ) is the inverse of ( g ij ) . Combining the above, we have s M = −
14 ∆ M log det( g st ) . Let (
L, h ) → N be a positive line bundle over a compact complex manifold N of complex dimension n, it defines a smooth volume form Ω = ∧ n c ( L, h )on N. Let D = P qj =1 D j ∈ | L | be the union of irreducible components so that D has only simple normal crossings. Endowing each L D j for 1 ≤ j ≤ q withHermitian metric such that the induced Hermitian metric on L = ⊗ qj =1 L D j is h. Take s j ∈ H ( V, L D j ) with ( s j ) = D j and k s j k < . On N, one definesa singular volume form(14) Φ = Ω Q qj =1 k s j k . Set f ∗ Φ ∧ α m − n = ξα m . Note that α m = m ! det( g ij ) m ^ j =1 √− π dz j ∧ dz j . A direct computation leads to dd c log ξ ≥ f ∗ c ( L, h ) − f ∗ RicΩ + R M − Supp f ∗ D in the sense of currents, where RicΩ is the Ricci form of Ω . This follows that dd c log ξ ∧ α m − α m ≥ f ∗ c ( L, h ) ∧ α m − α m − f ∗ RicΩ ∧ α m − α m + R M ∧ α m − α m − Supp f ∗ D ∧ α m − α m . Thus, 14 E o (cid:20)Z t ∆ M log ξ ( X s ) ds (cid:21) (15) ≥ e T f ( t, L ) + e T f ( t, K N ) + e T ( t, R M ) − e N f ( t, D ) + O (1) . Proof of Theorem 1.3
EVANLINNA-TYPE THEORY BASED ON HEAT DIFFUSIONS 15
Proof.
Identify N with a complex projective algebraic manifold. To use Ru-Wong’s argument (see [17], Page 231-233), there exists a finite open covering { U λ } of N and rational functions w λ , · · · , w λn on N for every λ such that w λ , · · · , w λn are holomorphic on U λ satisfying dw λ ∧ · · · ∧ dw λn ( y ) = 0 , ∀ y ∈ U λ ; U λ ∩ D = (cid:8) w λ · · · w λh λ = 0 (cid:9) , ∃ h λ ≤ n. In addition, one may require L D j | U λ ∼ = U λ × C for λ, j. On U λ , we haveΦ = φ λ | w λ | · · · | w λh λ | n ^ k =1 √− π dw λk ∧ dw λk , where Φ is given by (14) and φ λ > f λk = w λk ◦ f ,then(16) f ∗ Φ = φ λ ◦ f | f λ | · · · | f λh λ | n ^ k =1 √− π df λk ∧ df λk on each U λ . Set f ∗ Φ ∧ α m − n = ξα m , we have (15). Again, set(17) f ∗ c ( L, h ) ∧ α m − = ̺α m . It follows that(18) ̺ = 12 m e f ∗ ω . For each λ and any x ∈ f − ( U λ ) , take a local holomorphic coordinate system z around x. Since N is compact, then it is not very hard to compute by (16)and (17) that ξ is bounded from above by P λ , where P λ is a polynomial in ̺, g ij ∂f λk ∂z i ∂f λk ∂z j . | f λk | , ≤ i, j ≤ m, ≤ k ≤ n. This yields thatlog + ξ ≤ O log + ̺ + X k log + k∇ M f λk k| f λk | ! + O (1)on each f − ( U λ ) . Let { φ λ } be a partition of unity subordinate to { f − ( U λ ) } . Then we have 0 ≤ φ λ ≤ φ λ log + ( k∇ M f λk k / | f λk | ) = 0 outside f − ( U λ )for all k, λ. Hence, we have in furtherlog + ξ = X λ φ λ log + ξ (19) ≤ O log + ̺ + X k,λ φ λ log + k∇ M f λk k| f λk | + O (1) ≤ O log + ̺ + X k,λ log + k∇ M f λk k| f λk | + O (1) on M, where all f λk in the last inequality are understood as global functionson M since all w λk are globally defined on N. By means of Dynkin formula,12 E o (cid:20)Z t ∆ M log ξ ( X s ) ds (cid:21) = E o (cid:2) log ξ ( X t ) (cid:3) − log ξ ( o ) . It yields from (15) that12 E o (cid:2) log ξ ( X t ) (cid:3) ≥ e T f ( t, L ) + e T f ( t, K N ) + e T ( t, R M )(20) − e N f ( t, D ) + 12 log ξ ( o ) . On the other hand, by (19) and Theorem 1.2 we have12 E o (cid:2) log ξ ( X t ) (cid:3) ≤ O X k,λ E o (cid:20) log + k∇ M f λk k| f λk | ( X t ) (cid:21) + O (cid:16) E o (cid:2) log + ̺ ( X t ) (cid:3) (cid:17) + O (1) ≤ O X k,λ e m (cid:18) t, k∇ M f λk k| f λk | (cid:19) + O (cid:16) log + E o [ ̺ ( X t )] (cid:17) + O (1) ≤ O (cid:16) X k,λ log e T ( t, f λk ) (cid:17) + O (cid:16) log + E o [ ̺ ( X t )] (cid:17) + O (1) ≤ O (cid:0) log e T f ( t, L ) (cid:1) + O (cid:16) log + E o [ ̺ ( X t )] (cid:17) + O (1) , where the last inequality is due to Corollary 4.4. Indeed, Lemma 3.1 and(18) implylog + E o (cid:2) ̺ ( X t ) (cid:3) ≤ (1 + δ ) log + E o (cid:20)Z t ̺ ( X s ) ds (cid:21) = (1 + δ )2 m log + E o (cid:20)Z t e f ∗ c ( L,h ) ( X s ) ds (cid:21) ≤ (1 + δ ) m log e T f ( t, L ) + O (1) . By this with (20), the theorem is proved. (cid:3)
Defect relations.
Let L , L be holomorphic line bundles over a compact complex manifold N, we set (cid:20) c ( L ) c ( L ) (cid:21) = sup (cid:8) a ∈ R : L > aL (cid:9) , (cid:20) c ( L ) c ( L ) (cid:21) = inf (cid:8) a ∈ R : L < aL (cid:9) . EVANLINNA-TYPE THEORY BASED ON HEAT DIFFUSIONS 17
It is clear that(21) (cid:20) c ( L ) c ( L ) (cid:21) ≤ lim inf t →∞ e T f ( t, L ) e T f ( t, L ) ≤ lim sup r →∞ e T f ( t, L ) e T f ( t, L ) ≤ (cid:20) c ( L ) c ( L ) (cid:21) . Let f : M → N be a differentiably non-degenerate holomorphic mappingfrom a complete and stochastically complete K¨ahler manifold M into N suchthat e T f ( t, L ) → ∞ as t → ∞ . Let (
L, h ) → N be a positive line bundle over N. The defect of f with respect to D is defined by e δ f ( D ) = 1 − lim sup t →∞ e N f ( t, D ) e T f ( t, L ) . Then e δ f ( D ) = 1 if f omits D. Another defect e Θ f ( D ) is defined by e Θ f ( D ) = 1 − lim sup t →∞ e N f ( t, D ) e T f ( t, L ) , where e N f ( t, D ) = e N ( t, Supp f ∗ D ) . We have 0 ≤ e δ f ( D ) ≤ e Θ f ( D ) ≤ . In what follows, we give some conditions for e T f ( t, L ) < ∞ as 0 < t < ∞ and e T f ( t, L ) → ∞ as t → ∞ provided that f is differentiably non-degenerateand L > . Lemma 4.5.
Each of the following conditions ensures that e T f ( t, L ) < ∞ as < t < ∞ . (i) f has finite energy, i.e., R M e f ∗ c ( L,h ) ( x ) dV ( x ) < ∞ ;(ii) the energy density function e f ∗ c ( L,h ) ( x ) is bounded ;(iii) R M ( x ) ≥ − k ( r ( x )) for a nondecreasing function k ≥ on [0 , ∞ ) with k ( r ) /r → as r → ∞ and (1);(iv) R M ( x ) ≥ − k for a constant k ≥ with Z ∞ e − ǫr sup x ∈ B o ( r ) e f ∗ c ( L,h ) ( x ) dr < ∞ for any ǫ > . Proof. (i) and (ii) are obvious. (iii) can be verified by applying the estimateof p ( t, o, x ) due to Li-Yau [15]. (iv) follows from (iii) since the boundedness ofRicci curvature implies that Vol( B o ( r )) has at most the exponential growth.The arguments here is also referred to Proposition 6 in [3]. (cid:3) Lemma 4.6.
Each of the following conditions ensures that e T f ( t, L ) → ∞ as t → ∞ . (i) there exists no nonconstant bounded harmonic functions on M ;(ii) M is parabolic, i.e., there exists no nonconstant bounded plurisubhar-monic functions on M ;(iii) Ric M ≥ . Proof.
Since
L > , we can identify N with an algebraic subvariety of P K ( C )for some integer K > . Let H N be the restriction of hyperplane line bundle H over P K ( C ) to N. Note that(22) C e T f ( t, H N ) ≤ e T f ( t, L ) ≤ C e T f ( t, H N )for some constants C , C > . Denoted by [ w : · · · : w K ] the homogeneouscoordinate system of P K ( C ) . Assuming w ◦ f u := log(1 + | ζ ◦ f | + · · · + | ζ K ◦ f | )is a plurisubharmonic function outside ( w ◦ f = 0), where ζ j = w j /w for1 ≤ j ≤ K. Condition (i) implies that b T f ( t, H N ) = 14 E o (cid:20)Z t ∆ u ( X s ) ds (cid:21) → ∞ as t → ∞ . Since b T f ( t, H N ) ≤ e T f ( t, H N ) and (22), we have (i) holds. (ii) and(iii) follow from (i). (cid:3) Theorem 4.7 (defect relation) . Assume the same conditions as in Theorem . and e T f ( t, L ) → ∞ as t → ∞ . Then e Θ f ( D ) ≤ (cid:20) c ( K ∗ N ) c ( L ) (cid:21) − (cid:20) R M f ∗ c ( L ) (cid:21) . Proof.
By Theorem 1.3, it follows that1 − e N f ( t, D ) e T f ( t, L ) ≤ e T f ( t, K ∗ N ) e T f ( t, L ) − e T ( t, R M ) e T f ( t, L ) . Let t → ∞ , we have the claim. (cid:3) Corollary 4.8.
Assume the same conditions as in Theorem . . If s M ≥ , then e Θ f ( D ) ≤ (cid:20) c ( K ∗ N ) c ( L ) (cid:21) . Proof.
Since s M = − ∆ M log det( g ij ) ≥ R M = − dd c log det( g ij ) ≥ , then the conclusion holds. (cid:3) EVANLINNA-TYPE THEORY BASED ON HEAT DIFFUSIONS 19
Corollary 4.9.
Let D j ∈ | L | for ≤ j ≤ q such that P qj =1 D j has onlysimple normal crossings. Assume the same conditions as in Theorem . . If s M ≥ , then q X j =1 e Θ f ( D j ) ≤ q (cid:20) c ( K ∗ N ) c ( L ) (cid:21) . Corollary 4.10.
Let D , · · · , D q be hypersurfaces in P n ( C ) of degree d , · · · , d q such that P qj =1 D j has only simple normal crossings. Assume that f : M → P n ( C ) (dim C M ≥ n ) is a differentiably non-degenerate holomorphic map-ping such that e T f ( t, ω F S ) < ∞ for < t < ∞ . If s M ≥ satisfies E o (cid:20)Z t s M ( X s ) ds (cid:21) < ∞ for < t < ∞ . Then q X j =1 d j Θ f ( D j ) ≤ n + 1 . Proof.
Since s M ≥ R M ≥ , then it follows R M ≥ . Thereby, e T f ( t, L ) → ∞ as t → ∞ due to Lemma 4.6. Note that0 ≤ T ( t, R M ) = − E o (cid:20)Z t ∆ M log det( g ij )( X s ) ds (cid:21) = E o (cid:20)Z t s M ( X s ) ds (cid:21) < ∞ . It is an immediate consequence by using the facts c ( K ∗ P n ( C ) ) = ( n + 1)[ ω F S ]and c ( L D j ) = d j [ ω F S ] . (cid:3) Corollary 4.11.
Let a , · · · , a q be different points in a compact Riemannsurface S of genus g. Assume that f : M → S is a differentiably non-degenerate holomorphic mapping such that e T f ( t, L a ) < ∞ for < t < ∞ . If s M ≥ satisfies E o (cid:20)Z t s M ( X s ) ds (cid:21) < ∞ for < t < ∞ . Then q X j =1 Θ f ( a j ) ≤ − g. Proof.
Apply c ( K ∗ S ) = (2 − g ) c ( L a ) , we can prove the assertion. (cid:3) In the case when M is non-parabolic, we have Theorem 4.12.
Let M be a parabolic complete K¨ahler manifold. Let L → N be a positive line bundle over a compact complex manifold N, and D ∈ | L | such that D has only simple normal crossings. Assume that f : M → N (dim C M ≥ dim C N ) is a differentiably non-degenerate holomorphic map-ping. If (23) Z M s − M ( x ) dV ( x ) < ∞ . Then we have ( i ) Let R M ( x ) ≥ − cr ( x ) − c for a constant c > . If f has finite energy,i.e., E ( f ) := R M e f ∗ c ( L,h ) ( x ) dV ( x ) < ∞ , then e Θ f ( D ) ≤ (cid:20) c ( K ∗ N ) c ( L ) (cid:21) + 2 R M s − M ( x ) dV ( x ) E ( f ) . ( ii ) Let R M ( x ) ≥ − k ( r ( x )) for a nondecreasing function k ≥ such that k ( r ) /r → as r → ∞ , and satisfies (1) . If f has infinite energy, then e Θ f ( D ) ≤ (cid:20) c ( K ∗ N ) c ( L ) (cid:21) . Proof. ( i ) From Lemma 4.6, note that e T f ( t, L ) → ∞ as t → ∞ . Ricci curva-ture assumption implies that M is stochastically complete and parabolicityassumption implies that ratio ergodic theorem holds (see [16]). Apply ratioergodic theorem, we get e T ( t, R M ) e T f ( t, L ) = 2 E o hR t s M ( X s ) ds i E o hR t e f ∗ c ( L,h ) ( X s ) ds i → R M s M ( x ) dV ( x ) R M e f ∗ c ( L,h ) ( x ) dV ( x )= 2 R M s M ( x ) dV ( x ) E ( f ) < ∞ as t → ∞ . Thus, e T ( t, L ) < ∞ for t < ∞ and − (cid:20) R M f ∗ c ( L ) (cid:21) ≤ R M s − M ( x ) dV ( x ) E ( f ) . Invoking Theorem 4.7, ( i ) holds. For ( ii ) , we first note that T f ( t, L ) makessense by Lemma 4.5. Assertion ( ii ) follows by applying ratio ergodic theoremprovided E ( f ) = ∞ . (cid:3) In the case when M is non-parabolic, we have EVANLINNA-TYPE THEORY BASED ON HEAT DIFFUSIONS 21
Theorem 4.13.
Let M be a non-parabolic complete K¨ahler manifold withRicci curvature satisfying (23) and R M ( x ) ≥ − k ( r ( x )) for a nondecreasingfunction k ≥ such that k ( r ) /r → as r → ∞ . Let L → N be a positiveline bundle over a compact complex manifold N, and D ∈ | L | such that D hasonly simple normal crossings. Assume that f : M → N (dim C M ≥ dim C N ) is a differentiably non-degenerate holomorphic mapping satisfying (1) and e T f ( t, L ) → ∞ as t → ∞ . Then e Θ f ( D ) ≤ (cid:20) c ( K ∗ N ) c ( L ) (cid:21) . Proof.
The non-parabolicity implies that | e T ( t, R M ) | ≤ E o (cid:20)Z ∞ s − M ( X t ) dt (cid:21) ≤ E o (cid:20)Z ∞ R − M ( X t ) dt (cid:21) < ∞ . Thus, the assertion follows from Theorem 4.7. (cid:3)
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