Entire holomorphic curves into projective spaces intersecting a generic hypersurface of high degree
aa r X i v : . [ m a t h . AG ] N ov Entire holomorphic curves into projective spacesintersecting a generic hypersurface of high degree
Dinh Tuan HUYNH, Duc-Viet VU and Song-Yan XIE
Abstract
In this note, we establish the following Second Main Theorem type estimate for every algebraicallynondegenerate entire curve f : C → P n ( C ) , in presence of a generic divisor D ⊂ P n ( C ) of sufficientlyhigh degree d ≥ n + 1) n n : for every r outside a subset of R of finite Lebesgue measure and every realpositive constant δ , we have T f ( r ) ≤ N [1] f ( r, D ) + O (cid:0) log T f ( r ) (cid:1) + δ log r, where T f ( r ) and N [1] f ( r, D ) stand for the order function and the -truncated counting function in Nevanlinnatheory. This inequality quantifies recent results on the logarithmic Green–Griffiths conjecture. Keywords:
Nevanlinna theory, Second Main Theorem, holomorphic curve, Green–Griffiths’ conjecture, alge-braic degeneracy
Mathematics Subject Classification 2010:
We first recall the standard notation in Nevanlinna theory. Let E = P i α i a i be a divisor on C where α i ≥ , a i ∈ C and let k ∈ N ∪ {∞} . Denote by ∆ t the disk { z ∈ C , | z | < t } . Summing the k -truncated degrees ofthe divisor on disks by n [ k ] ( t, E ) := X a i ∈ ∆ t min { k, α i } ( t > , the truncated counting function atlevel k of E is then defined by taking the logarithmic average N [ k ] ( r, E ) := Z r n [ k ] ( t, E ) t d t ( r > . When k = ∞ , we write n ( t, E ) , N ( r, E ) instead of n [ ∞ ] ( t, E ) , N [ ∞ ] ( r, E ) . Let f : C → P n ( C ) be an entirecurve having a reduced representation f = [ f : · · · : f n ] in the homogeneous coordinates [ z : · · · : z n ] of P n ( C ) . Let D = { Q = 0 } be a divisor in P n ( C ) defined by a homogeneous polynomial Q ∈ C [ z , . . . , z n ] ofdegree d ≥ . If f ( C ) D , we define the truncated counting function of f with respect to D as N [ k ] f ( r, D ) := N [ k ] (cid:0) r, ( Q ◦ f ) (cid:1) , where ( Q ◦ f ) denotes the zero divisor of Q ◦ f .The proximity function of f for the divisor D is defined as m f ( r, D ) := Z π log (cid:13)(cid:13) f ( re iθ ) (cid:13)(cid:13) d k Q k (cid:12)(cid:12) Q ( f )( re iθ ) (cid:12)(cid:12) d θ π , where k Q k is the maximum absolute value of the coefficients of Q and (cid:13)(cid:13) f ( z ) (cid:13)(cid:13) = max {| f ( z ) | , . . . , | f n ( z ) |} . Since (cid:12)(cid:12) Q ( f ) (cid:12)(cid:12) ≤ k Q k · k f k d , one has m f ( r, D ) ≥ . 1astly, the Cartan order function of f is defined by T f ( r ) : = 12 π Z π log (cid:13)(cid:13) f ( re iθ ) (cid:13)(cid:13) d θ = Z r d tt Z ∆ t f ∗ ω n + O (1) , where ω n is the Fubini–Study form on P n ( C ) .With the above notations, the Nevanlinna theory consists of two fundamental theorems (for a comprehensivepresentation, see Noguchi-Winkelmann [19]). First Main Theorem.
Let f : C → P n ( C ) be a holomorphic curve and let D be a hypersurface of degree d in P n ( C ) such that f ( C ) D . Then for every r > , the following holds m f ( r, D ) + N f ( r, D ) = d T f ( r ) + O (1) , whence (1.1) N f ( r, D ) ≤ d T f ( r ) + O (1) . Hence the First Main Theorem gives an upper bound on the counting function in terms of the order function.On the other side, in the harder part, so-called Second Main Theorem, one tries to establish a lower bound forthe sum of certain counting functions. Such types of estimates were given in several situations.Throughout this note, for an entire curve f, the notation S f ( r ) means a real function of r ∈ R + such thatthere is a constant C for which S f ( r ) ≤ C T f ( r ) + δ log r for every positive constant δ and every r outside of a subset (depending on δ ) of finite Lebesgue measure of R + .A holomorphic curve f : C → P n ( C ) is said to be algebraically (linearly) nondegenerate if its image is notcontained in any hypersurface (hyperplane). A family of q ≥ n + 1 hypersurfaces { D i } ≤ i ≤ q in P n ( C ) is ingeneral position if any n + 1 hypersurfaces in this family have empty intersection: ∩ i ∈ I supp( D i ) = ∅ ( ∀ I ⊂ { ,...,q } , | I | = n +1) . We recall here the following classical result [3], with truncation level n . Cartan’s Second Main Theorem.
Let f : C → P n ( C ) be a linearly nondegenerate holomorphic curve and let { H i } ≤ i ≤ q be a family of q > n + 1 hyperplanes in general position in P n ( C ) . Then the following estimateholds ( q − n − T f ( r ) ≤ q X i =1 N [ n ] f ( r, H i ) + S f ( r ) . In the one-dimensional case, Cartan recovered the classical Nevanlinna theory for nonconstant meromor-phic functions and families of q > distinct points. Since then, many author tried to extend the result of Cartanto the case of (possible) nonlinear hypersurface. Eremenko-Sodin [12] established a Second Main Theoremfor q > n hypersurfaces D i ( ≤ i ≤ q ) in general position in P n ( C ) and for any nonconstant holomorphiccurve f : C → P n ( C ) whose image is not contained in ∪ ≤ i ≤ q supp( D i ) . Keeping the same assumption on q > n + 1 hypersurfaces, Ru [23] proved a stronger estimate for algebraically nondegenerate holomorphiccurves f : C → P n ( C ) . He then extended this result to the case of algebraically nondegenerate holomorphicmappings into an arbitrary nonsingular complex projective variety [24]. Note that it remains open the questionof truncating the counting functions in the above generalizations of Cartan’s Second Main Theorem. Someresults in this direction are obtained recently but one requires the presence of more targets, see for instance [1],[27].In the other context, Noguchi-Winkelmann-Yamanoi [20] established a Second Main Theorem for alge-braically nondegenerate holomorphic curves into semiabelian varieties intersecting an effective divisor. Ya-manoi [28] obtained a similar result in the case of abelian varieties with the best truncation level , which isextended to the case of semiabelian varieties by Noguchi-Winkelmann-Yamanoi [21].2n the qualitative aspect, the (strong) Green-Griffiths conjecture stipulates that if X is a complex projectivespace of general type, then there exists a proper subvariety Y ( X containing the image of every nonconstantentire holomorphic curve f : C → X .Following a beautiful strategy of Siu [25], Diverio, Merker and Rousseau [11] confirmed this conjecturefor generic hypersurface D ⊂ P n +1 of degree d ≥ n . Berczi [2] improved the degree bound to d ≥ n n .Demailly [8] gave a new degree bound d ≥ n (cid:18) n log( n log(24 n )) (cid:19) n . In the logarithmic case, namely for the complement of a hypersurface D ⊂ P n ( C ) , there is another variantof this conjecture, so-called the logarithmic Green-Griffiths conjecture, which expects that for a generic hyper-surface D ⊂ P n ( C ) having degree d ≥ n + 2 , there should exist a proper subvariety Y ⊂ P n ( C ) containingthe image of every nonconstant entire holomorphic curve f : C → P n ( C ) \ D . Darondeau [5] gave a positiveanswer for this case with effective degree bound d ≥ (5 n ) n n . In this note, we show that the current method towards the Green-Griffiths conjecture can yield not onlyqualitative but also quantitative result , namely a Second Main Theorem type estimate in presence of only onegeneric hypersurface D of sufficiently high degree with the truncation level . Main Theorem.
Let D ⊂ P n ( C ) be a generic divisor having degree d ≥ n + 1) n n . Let f : C → P n ( C ) be an entire holomorphic curve. If f is algebraically nondegenerate, then the followingestimate holds T f ( r ) ≤ N [1] f ( r, D ) + S f ( r ) . For background and standard techniques in Nevanlinna theory, we use the book of Noguchi-Winkelmann[19] as our main reference. The proof of the existence of logarithmic jet differentials in the last part of this noteis based on the work of Darondeau [5].
Acknowledgments
The authors would like to thank Jo¨el Merker for his encouragements and his comments that greatly improvedthe manuscript. We would like to thank Nessim Sibony for very fruitful discussions on the paper [22]. We wantto thank Junjiro Noguchi, Katsutoshi Yamanoi, Yusaku Tiba and Yuta Kusakabe for their interests in our workand for listening through many technical details. We would like to thank the referee for his/her careful readingof the manuscript and helpful suggestions. The first author is supported by the fellowship of the Japan Societyfor the Promotion of Science and the Grant-in-Aid for JSPS fellows Number 16F16317. k -jet bundle The general strategy to prove the logarithmic Green-Griffiths conjecture consists of two steps. The first one is toproduce many algebraically independent differential equations that all holomorphic curve f : C → P n ( C ) \ D must satisfy. The second step consists in producing enough jet differentials from an initial one such that from thecorresponding algebraic differential equations, one can eliminate all derivative in order to get purely algebraicequations.The central geometric object corresponding to the algebraic differential equations is the logarithmic Green-Griffiths k -jet bundle constructed as follows. Let X be a complex manifold of dimension n . For a point x ∈ X , consider the holomorphic germs ( C , → ( X, x ) . Two such germs are said to be equivalent if theyhave the same Taylor expansion up to order k in some local coordinates around x . The equivalence class of3 holomorphic germ f : ( C , → ( X, x ) is called the k -jet of f , denote j k ( f ) . A k -jet j k ( f ) is said to beregular if f ′ (0) = 0 . For a point x ∈ X , denote by j k ( X ) x the vector space of all k -jets of holomorphic germs ( C , → ( X, x ) . Set J k ( X ) := ∪ x ∈ X J k ( X ) x and consider the natural projection π k : J k ( X ) → X. Then J k ( X ) is a complex manifold which carries the structure of a holomorphic fiber bundle over X , whichis called the k -jet bundle over X . When k = 1 , J ( X ) is canonically isomorphic to the holomorphic tangentbundle T X of X .For an open subset U ⊂ X , for a section ω ∈ H ( U, T ∗ X ) , for a k -jet j k ( f ) ∈ J k ( X ) | U , the pullback f ∗ ω is of the form A ( z ) dz for some holomorphic function A . Since each derivative A ( j ) ( ≤ j ≤ k − ) iswell defined, independent of the representation of f in the class j k ( f ) , the holomorphic -form ω induces theholomorphic map(2.1) ˜ ω : J k ( X ) | U → C k ; j k ( f ) → (cid:0) A ( z ) , A ( z ) (1) , . . . , A ( z ) ( k − (cid:1) . Hence on an open subset U , a local holomorphic coframe ω ∧· · ·∧ ω n = 0 yields a trivialization H ( U, J k ( X )) → U × ( C k ) n by giving new nk independent coordinates: σ → ( π k ◦ σ ; ˜ ω ◦ σ, . . . , ˜ ω n ◦ σ ) , where ˜ ω i are defined as in (2.1). The components x ( j ) i ( ≤ i ≤ n , ≤ j ≤ k ) of ˜ ω i ◦ σ are called jet-coordinates. In a more general setting where ω is a section over U of the sheaf of meromorphic -forms, theinduced map ˜ ω is meromorphic.Now, in the logarithmic setting, let D ⊂ X be a normal crossing divisor on X . This means that at eachpoint x ∈ X , there exist some local coordinates z , . . . , z ℓ , z ℓ +1 , . . . , z n ( ℓ = ℓ ( x ) ) centered at x in which D isdefined by D = { z . . . z ℓ = 0 } . Following Iitaka [14], the logarithmic cotangent bundle of X along D , denoted by T ∗ X (log D ) , corresponds tothe locally-free sheaf generated by d z z , . . . , d z ℓ z ℓ , z ℓ +1 , . . . , z n in the above local coordinates around x .A holomorphic section s ∈ H ( U, J k ( X )) over an open subset U ⊂ X is said to be a logarithmic k -jetfieldif ˜ ω ◦ s are holomorphic for all sections ω ∈ H ( U ′ , T ∗ X (log D )) , for all open subsets U ′ ⊂ U , where ˜ ω areinduced maps defined as in (2.1). Such logarithmic k -jet fields define a subsheaf of J k ( X ) , and this subsheafis itself a sheaf of sections of a holomorphic fiber bundle over X , called the logarithmic k -jet bundle over X along D , denoted by J k ( X, − log D ) [18].The group C ∗ acts fiberwise on the jet bundle as follows. For local coordinates z , . . . , z ℓ , z ℓ +1 , . . . , z n ( ℓ = ℓ ( x )) centered at x in which D = { z . . . z ℓ = 0 } , for any logarithmic k -jet field along D represented by some germ f = ( f , . . . , f n ) , if ϕ λ ( z ) = λz is the homothety with ratio λ ∈ C ∗ , the action is given by ((cid:0) log( f i ◦ ϕ λ ) (cid:1) ( j ) = λ j (cid:0) log f i (cid:1) ( j ) ◦ ϕ λ (1 ≤ i ≤ ℓ ) , (cid:0) f i ◦ ϕ λ (cid:1) ( j ) = λ j f ( j ) i ◦ ϕ λ ( ℓ +1 ≤ i ≤ n ) . Now we are ready to introduce the Green-Griffiths k -jet bundle [13] in the logarithmic setting. A logarith-mic jet differential of order k and degree m at a point x ∈ X is a polynomial Q ( f (1) , . . . , f ( k ) ) on the fiberover x of J k ( X, − log D ) enjoying weighted homogeneity: Q ( j k ( f ◦ ϕ λ )) = λ m Q ( j k ( f )) . E GGk,m T ∗ X (log D ) x the vector space of such polynomials and set E GGk,m T ∗ X (log D ) := ∪ x ∈ X E GGk,m T ∗ X (log D ) x . By Fa`a di bruno’s formula [4], [15], E GGk,m T ∗ X (log D ) carries the structure of a vector bundle over X , calledlogarithmic Green-Griffiths vector bundle. A global section P of E GGk,m T ∗ X (log D ) locally is of the followingtype in jet-coordinates x ( j ) i : X α ,...,α k ∈ N n | α | +2 | α | + ··· + k | α k | = m A α ,...,α k (cid:18) ℓ Y i =1 (cid:0) (log x i ) (1) (cid:1) α ,i n Y i = ℓ +1 (cid:0) ( x i ) (1) (cid:1) α ,i (cid:19) . . . (cid:18) ℓ Y i =1 (cid:0) (log x i ) ( k ) (cid:1) α k,i n Y i = ℓ +1 (cid:0) ( x i ) ( k ) (cid:1) α k,i (cid:19) , where α λ = ( α λ, , . . . , α λ,n ) ∈ N n (1 ≤ λ ≤ k ) are multi-indices of length | α λ | = X ≤ i ≤ n α λ,i , and where A α ,...,α k are locally defined holomorphic functions.By the following classical result, the first step to prove the Green-Griffiths conjecture reduces to findinglogarithmic jet differentials valued in the dual of some ample line bundle. Fundamental vanishing theorem. ([9], [10]) Let X be a smooth complex projective variety and let D ⊂ X be a normal crossing divisor on X . If P is a nonzero global holomorphic logarithmic jet differential along D vanishing on some ample line bundle A on X , namely if = P ∈ H (cid:0) X, E
GGk,m T ∗ X (log D ) ⊗ A − (cid:1) , then all nonconstant holomorphic curves f : C → X \ D must satisfy the associated differential equation (2.2) P (cid:0) j k ( f ) (cid:1) ≡ . In the compact case, the existence of such global sections has been proved recently, first by Merker [15] forthe case of smooth hypersurfaces of general type in P n ( C ) , and later for arbitrary general projective variety byDemailly [7]. Adapting this technique in the logarithmic setting, Darondeau [5] obtained a similar result forsmooth hypersurface in projective space, provided that the degree is high enough compared with the dimension. Proposition 2.1. ([5, Th. 1.2]) Let c ∈ N be a positive integer and let D ⊂ P n ( C ) be a smooth hypersurfacehaving degree d ≥ c + 2) n n . For jet order k = n , for weighted degrees m ≫ d big enough, the vector space of global logarithmic jetdifferentials along D of order k and weighted degree m vanishing on the m -th tensor power of the ample linebundle O P n ( C ) ( c ) has positive dimension: dim H (cid:0) P n ( C ) , E GGn,m T ∗ P n ( C ) (log D ) ⊗ O P n ( C ) ( c ) − m (cid:1) > . Let D ⊂ P n ( C ) be a smooth hypersurface in P n ( C ) . Let f : C → P n ( C ) be an entire holomorphic curve, notnecessary in the complement of D . If there exists a global logarithmic jet differential P which does not satisfy(2.2), then the fundamental vanishing theorem guarantees that the curve f must intersect the hypersurface D .Furthermore, in the quantitative aspect, based on the proof of the fundamental vanishing theorem, it is knownthat a Second Main Theorem type estimate T f ( r ) ≤ C N f ( r, D ) + S f ( r ) P . There are several variants of the above estimate,see for instance in [22], [26]. Here we provide more information about the constant C and truncation of thecounting function.Before going to introduce the main result of this section, we need to recall the following lemma on loga-rithmic derivative which is a crucial tool in Nevanlina theory. Logarithmic derivative Lemma.
Let g be a nonzero meromorphic function on C . For any integer k ≥ ,we have m g ( k ) g ( r ) := m g ( k ) g ( r, ∞ ) = S g ( r ) . We refer to [19, Lem. 4.7.1] for a more general version of the above Lemma. Here is our main result in thissection.
Theorem 3.1.
Let f : C → P n ( C ) be an entire curve and let D ⊂ P n ( C ) be a smooth hypersurface. Let ˜ m bea positive integer. If there exists a global logarithmic jet differential P ∈ H (cid:0) P n ( C ) , E GGk,m T ∗ P n ( C ) (log D ) ⊗ O P n ( C ) (1) − e m (cid:1) such that P (cid:0) j k ( f ) (cid:1) , then the following Second Main Theorem type estimate holds: T f ( r ) ≤ m e m N [1] f ( r, D ) + S f ( r ) . Proof.
Our proof is partly based on [19, Lem. 4.7.1] and [9]. Let Q be the irreducible homogeneous polynomialdefining D . By assumption, P (cid:0) j k ( f ) (cid:1) is a nonzero meromorphic section of f ∗ O P n ( C ) (1) − e m . Let D P ,f be thepole divisor of P (cid:0) j k ( f ) (cid:1) .Let (cid:0) V, φ ) be a small enough local chart of P n ( C ) such that φ : P n ( C ) → C n is a rational map and D isgiven by D = { z = 0 } , where z = ( z , · · · , z n ) are the natural coordinates on C n . Put f j := φ ( f ) , (3.1)which is a meromorphic function on C for ≤ j ≤ n . Then f is written in the local chart V as ( f , · · · , f n ) on f − ( V ) . Observe that f /Q ( f ) is a nowhere vanishing holomorphic function on f − ( V ) . Recall that on V ,the section P (cid:0) j k ( f ) (cid:1) can be written as P (cid:0) j k ( f ) (cid:1) = X α ,...,α k ∈ N n | α | +2 | α | + ··· + k | α k | = m A α ,...,α k k Y ℓ =1 (cid:18)(cid:0) (log f ) ( ℓ ) (cid:1) α ℓ, n Y j =2 (cid:0) f ( ℓ ) j (cid:1) α ℓ,j (cid:19) , (3.2)where A α ,...,α k are holomorphic functions on f − ( V ) and α j = ( α j, , · · · , α j,n ) for ≤ j ≤ n . Hence, thesupport of D P ,f is a subset of the zero set of Q ◦ f on C . Furthermore, since for each ≤ ℓ ≤ k , the poleorder of (log f ) ( ℓ ) at any point z ∈ C is at most ℓ min { ord z f , } (hence at most ℓ min { ord z Q ( f ) , } ) andsince the degree of P is m , we get D P ,f ≤ m X z ∈ C min { ord z ( Q ◦ f ) , } z. Let h be the pullback by f of the Fubini-Study form ω n on P n ( C ) . Using the Poincar´e-Lelong formula, wehave dd c log k P (cid:0) j k ( f ) (cid:1) k h ≥ e mf ∗ ω − [ D P ,f ] , where [ D P ,f ] is the integration current of D P ,f . Combining this fact with the above inequality, we obtain e m T f ( r ) + O (1) ≤ Z ∂ ∆ r log k P (cid:0) j k ( f ) (cid:1) k h d θ π + m N [1] f ( r, D ) . Z ∂ ∆ r log k P (cid:0) j k ( f ) (cid:1) k h d θ π = S f ( r ) . Using a partition of unity on P n ( C ) , the problem reduces to proving that(3.4) Z ∂ ∆ r log | χ ( f ) P (cid:0) j k ( f ) (cid:1) | d θ π = S f ( r ) , where χ is a smooth positive function compactly supported on a local chart V as above. Using the followingelementary observations with s, s , · · · , s N ∈ R ∗ + : log s = log + s − log + s ≤ log + s log + N X i =1 s i ≤ N X i =1 log + s i + log N log + N Y i =1 s i ≤ N X i =1 log + s i , where log + denotes max { log , } , we get Z ∂ ∆ r log | χ ( f ) P (cid:0) j k ( f ) (cid:1) | d θ π ≤ X α ,...,α k ∈ N n | α | +2 | α | + ··· + k | α k | = m k X ℓ =1 (cid:18) Z ∂ ∆ r log + (cid:0) χ ( f ) | (log f ) ( ℓ ) | α ℓ, (cid:1) d θ π + n X j =2 Z ∂ ∆ r log + (cid:0) χ ( f ) | f ( ℓ ) j | α ℓ,j (cid:1) d θ π (cid:19) + O (1) , (3.5)Recall from (3.1) that f j are meromorphic functions on C for ≤ j ≤ n . Hence applying the logarithmicderivative Lemma to f , we infer that Z ∂ ∆ r log + (cid:0) χ ( f ) | (log f ) ( ℓ ) | α ℓ, (cid:1) d θ π = S f ( r ) . (3.6)Therefore, it suffices to show that this property still holds for the remaining terms in the right-hand side of (3.5).Continuing to apply the logarithmic derivative Lemma, we obtain Z ∂ ∆ r log + (cid:0) χ ( f ) | f ( ℓ ) j | α ℓ,j (cid:1) d θ π ≤ c Z ∂ ∆ r log + (cid:0) χ ( f ) | f (1) j | (cid:1) d θ π + S f ( r ) , for some constant c which is independent of r, f . Hence it remains to check Z ∂ ∆ r log + (cid:0) χ ( f ) | f (1) j | (cid:1) d θ π = S f ( r ) . This can be done by using the similar arguments as in [19, p. 149]. For the reader’s convenience, we presentthe idea here. Since χ is compactly supported on V , there exists a bounded positive function B for which χ d z j ∧ d¯ z j ≤ B ( z ) ω n on V for ≤ j ≤ n . This yields χ ( f ) | f (1) j | d z ∧ d¯ z = f ∗ ( χ d z j ∧ d¯ z j ) ≤ B ( f ) f ∗ ω n . The pullback f ∗ ω n is of the form B d z ∧ d¯ z . Hence we deduce from the above inequality that Z ∂ ∆ r log + (cid:0) χ ( f ) | f (1) j | (cid:1) d θ π ≤ π Z ∂ ∆ r log + B ( f ) d θ + 12 π Z ∂ ∆ r log + B d θ ≤ π Z ∂ ∆ r log + B d θ + supp z ∈ V | B ( z ) | . Estimating R ∂ ∆ r log + B is done by following the same arguments as in the proof of the logarithmic derivativeLemma, see [19, (3.2.8)]. The proof is finished. 7 Existence of logarithmic jet differentials
Let f : C → P n ( C ) be an algebraically nondegenerate holomorphic curve. Following the second step in Siu’sstrategy to prove the Green-Griffith conjecture, morally, if we can produce enough logarithmic jet differentialsvalued in the dual of some ample line bundle on P n ( C ) , then among them, we can choose at least one such that f does not satisfy the algebraic differential equation (2.2). Theorem 4.1.
Let c be a positive integer with c ≥ n − . Let D ⊂ P n ( C ) be a generic smooth hypersurfacein P n ( C ) having degree d ≥ c + 2) n n . Let f : C → P n ( C ) be an entire holomorphic curve. If f is algebraically nondegenerate, then for jet order k = n and for weighted degrees m > d big enough, there exists an integer ≤ ℓ ≤ m and a globallogarithmic jet differential P ∈ H (cid:0) P n ( C ) , E GGn,m T ∗ P n ( C ) (log D ) ⊗ O P n ( C ) (1) − cm + ℓ (5 n − (cid:1) such that P (cid:0) j n ( f ) (cid:1) . (4.1)The rest of this section is devoted to proving Theorem 4.1 whose proof is based on [11, 5]. Let S := P H (cid:0) P n ( C ) , O ( d ) (cid:1) be the projective parameter space of homogeneous polynomials of degree d in P n ( C ) which identifies with the projective space P N d ( C ) of dimension N d = dim P H (cid:0) P n ( C ) , O P n ( C ) ( d ) (cid:1) = (cid:18) n + dd (cid:19) − . We then introduce the universal hypersurface
H ⊂ P n ( C ) × S parametrizing all hypersurfaces of fixed degree d in P n ( C ) , defined by the equation X α ∈ N n +1 A α Z α in the following two collections of homogeneous coordinates Z = [ Z : · · · : Z n ] ∈ P n ( C ) ,A = [( A α ) α ∈ N n +1 , | α | = d ] ∈ P N d ( C ) , where α = ( α , . . . , α n ) ∈ N n +1 are multiindices. Since H is a smooth hypersurface on P n ( C ) × S , we canconstruct the logarithmic k -jet bundle J k ( P n ( C ) × S , − log H ) over P n ( C ) × S along H .Now, let η be be the natural projection from J k ( P n ( C ) × S , − log H ) to P n ( C ) × S . Let pr and pr be thenatural projections from P n ( C ) × S to the first and second part, respectively. Let V H ,k be the analytic subset of J k ( P n ( C ) × S , − log H ) consisting of all vertical logarithmic jetfields oforder k which, by definition, are jets j k ( f ) such that f lies entirely in some fiber of the second projection pr .Denote by V reg H ,k the open subset consisting of all regular jets. By [6, p. 571-572] (see also [16, p. 1088]), V reg H ,k is smooth manifold. Following the method of producing new jet differentials developed by Siu [25], inthe logarithmic setting, one needs to construct low pole order frames on V reg H ,k . Proposition 4.1. ([6, Main Theorem]) For jet order k ≥ , for degree d ≥ k , the twisted tangent bundle T V H ,k ⊗ η ∗ (cid:0) O P n ( C ) (5 k − ⊗ O S (1) (cid:1) is generated over V reg H ,k \ η − H by its global holomorphic sections.
8n fact, those global sections mentioned in the above Proposition are global vector fields on the wholelogarithmic k -jet bundle and satisfy the canonical tangential conditions described as in [6, 16]. Hence they aretrue vector fields on the smooth part of V H ,k . Moreover by the constructions in [6, 16, 26], the coefficients ofthose vector fields are polynomials in local logarithmic jet coordinates.Let Z be the subset of S consisting of all s whose corresponding hypersurface D s is not smooth. Observethat Z is a proper analytic subset of S .From now on we work with the fixed jet order k = n . Since pr − s = P n ( C ) × { s } and since D s is smoothfor every s outside Z , one can define J n (pr − s, − log D s ) for any s ∈ S \ Z . Let us set L := [ s ∈ S \ Z J n (pr − s, − log D s ) , L reg := [ s ∈ S \ Z J reg n (pr − s, − log D s ) . Observe that L has a natural structure of holomorphic fiber bundle over P n ( C ) × ( S \ Z ) . Note also that L , L reg are open subsets of V H ,n and V reg H ,n , respectively. Set E := [ s ∈ S \ Z E GGn,m T ∗ pr − s (log D s ) , then E carries the structure of holomorphic vector bundle over P n ( C ) × ( S \ Z ) . This fact allows us to extendholomorphically a nonzero jet differential provided by Proposition 2.1. Let us enter the details. Lemma 4.1.
Let c ≥ n − be a positive integer. For degree d ≥ c + 2) n n , for weighted degree m ≫ d ,there exists a proper analytic subset Z of S containing Z such that for every s ∈ S \ Z , we can find a Zariskiopen neighborhood U s of s in S \ Z and a nonzero holomorphic section P of the twisted vector bundle E ⊗ pr ∗ O P n ( C ) (1) − cm over pr − U s .Proof. By construction, for any s ∈ S \ Z , the restriction of E to pr − s = P n ( C ) × { s } coincides with E GGn,m T ∗ pr − s (log D s ) . Hence Proposition 2.1 guarantees the existence of a nonzero global section P s ∈ H (cid:0) P n ( C ) × { s } , E ⊗ pr ∗ O P n ( C ) (1) − cm | pr − s (cid:1) of the restriction of the twisted vector bundle E ⊗ pr ∗ O P n ( C ) (1) − cm to pr − s . By the semi-continuity theorem(c.f. [17, p. 50]), there exists a proper Zariski closed subset Z of S containing Z such that for any s ∈ S \ Z ,the natural restriction map H (cid:0) pr − U s , E ⊗ pr ∗ O P n ( C ) (1) − cm (cid:1) −→ H (cid:0) pr − s, E ⊗ pr ∗ O P n ( C ) (1) − cm | pr − s (cid:1) is onto for some Zariski open subset U s ⊂ S \ Z containing s. As a consequence, the above section P s can beextended holomorphically to a section P of E ⊗ pr ∗ O P n ( C ) (1) − cm over pr − U s . Proof of Theorem 4.1.
Let Z , P be as in Lemma 4.1. Let us first describe precisely the generic assumption of D in the statement. By this, we mean that if D corresponds to the element s ∈ S (i.e. D = D s ), then s liesoutside Z ∪ H S , where H S is a fixed arbitrary hyperplane of S . Here the condition s H S is given in order toget rid of η ∗ O S (1) in Proposition 4.1 because the line bundle O S (1) is trivial on S \ H S . From now on, we fix s ∈ S \ Z and D = D s .Applying Proposition 4.1 for jet order k = n , the twisted tangent bundle T V H ,n ⊗ η ∗ (cid:0) O P n ( C ) (5 n − ⊗ O S (1) (cid:1)
9s generated by its global holomorphic sections over V reg H ,n \ η − H . Moreover, the coefficients of those sectionsare polynomials in the logarithmic n -jet coordinates associated with the canonical coordinates of P n ( C ) × S .In particular, the restriction of the bundle T V H ,n ⊗ η ∗ O P n ( C ) (5 n − to η − Y , where Y := pr − ( U s \ H S ) \ H is generated on ( V reg H ,n ∩ η − Y ) by its global sections whose coefficients are polynomials in the logarithmic jetcoordinates as above.For ≤ ℓ ≤ m , let v , . . . , v ℓ be sections of T V H ,n ⊗ η ∗ O P n ( C ) (5 n − over the open subset L ⊂ V H ,n .As explained below, the significance of those sections is that they allow to construct new global logarithmic jetdifferentials. Indeed, we can view P as a holomorphic mapping P : L| pr − U s → pr ∗ O P n ( C ) (1) − cm | pr − U s , which is locally a homogeneous polynomial of degree m . It follows that the Lie derivative ( v · · · v ℓ ) · P isalso a holomorphic map from L| pr − U s to pr ∗ O P n ( C ) (1) − cm + ℓ (5 n − and is locally a homogeneous polynomial of the same degree m . The fact that the derivative of P along v j preserves the degree m can be deduced from the fact that the coefficients of v j are polynomials in the logarith-mic jet coordinates and by the chain rule of derivatives, the degree of those polynomials should compensatethe losses of degree due to the differentiation with respect to v j , see [26, Sec. 3.7]. In summary, we obtain aholomorphic map ( v · · · v ℓ ) · P : L| pr − U s → pr ∗ O P n ( C ) (1) | − cm + ℓ (5 n − − U s . (4.2)By composing f with the inclusion P n ( C ) ֒ → P n ( C ) × { s } ⊂ P n ( C ) × S , we can consider f as a holomorphic curve into Y ⊂ P n ( C ) × S because s ∈ U s \ H S and f is not includedin D. Let { P = 0 } ⊂ pr − U s be the zero divisor of P , where we view P as a holomorphic section of E ⊗ pr ∗ O P n ( C ) (1) − cm over pr − U s . Since f is algebraically nondegenerate, there exists z ∈ C such that f ′ ( z ) = 0 and f ( z ) D ∩ { P = 0 } . Consequently, we get j n ( f )( z ) ∈ ( L reg ∩ η − Y ) . (4.3)Now proceeding as in [11], we can show that there exist global slanted vector fields v , . . . , v ℓ for some ≤ ℓ ≤ m such that ( v · · · v ℓ ) · P (cid:0) j n ( f ) (cid:1) = 0 . For reader’s convenience, we briefly recall the idea. Denoted by P s the restriction of P to P n ( C ) × { s } .Consider the logarithmic jet coordinates ( z, z (1) , . . . , z ( n ) ) ∈ C n ( n +1) around j n ( f )( z ) of L| P n ( C ) ×{ s } . Usinga linear change of coordinates, we obtain modified logarithmic jet coordinates ( z ′ , z ′ (1) , . . . , z ′ ( n ) ) in which j n ( f )( z ) is the origin. Since P s is locally a homogeneous polynomial in logarithmic jet coordinates whosecoefficients are holomorphic functions on local charts of P n ( C ) , so it is in the new logarithmic jet coordinates.By the choice of z , there exists a coefficient A α ,...,α n ( z ) of P s (see (3.2)) for which A α ,...,α n ( f ( z )) = 0 .Let A α ,...,α n ( z ) (cid:0) z ′ (1) (cid:1) α · · · (cid:0) z ′ ( n ) (cid:1) α n be the monomial of P s associated with A α ,...,α n , where α j ∈ N n for ≤ j ≤ n and | α | + 2 | α | + · · · + n | α n | ≤ m . We then choose local vector fields v ′ , . . . v ′ ℓ around the origin j n ( f )( z ) for which ( v ′ · · · v ′ ℓ ) · P (cid:0) j n ( f )( z ) (cid:1) = A α ,...,α n (cid:0) f ( z ) (cid:1) = 0 . As we mentioned above, these vector fields v ′ , . . . , v ′ ℓ can be generated by global vector fields v , . . . , v ℓ on ( L reg ∩ η − Y ) . Combining this with (4.3), we get ( v · · · v ℓ ) · P (cid:0) j n ( f )( z ) (cid:1) = 0 . This together with (4.2)implies (4.1). The proof of Theorem 4.1 is completed.10 orollary 4.1. Let c be a positive integer with c ≥ n − . Let D ⊂ P n ( C ) be a generic hypersurface in P n ( C ) having degree d ≥ c + 2) n n . Let f : C → P n ( C ) be an entire holomorphic curve. If f is algebraically nondegenerate, then the followingSecond Main Theorem type estimate holds: T f ( r ) ≤ c − n + 2 N [1] f ( r, D ) + S f ( r ) . In particular, choosing c = 5 n − , one obtains the Main Theorem.Proof. This is a direct application of Theorem 3.1 to global logarithmic jet differential P supplied by Theorem4.1, where ˜ m = mc − ℓ (5 n − ≥ m ( c − n + 2) ≥ m. eferences [1] Do Phuong An, Si Duc Quang, and Do Duc Thai. “The second main theorem for meromorphic mappingsinto a complex projective space”. In: Acta Math. Vietnam.
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