Holomorphic curves into algebraic varieties

Transcription

1 Annals of Matheatics, 69 29, Holoorphic curves into algebraic varieties By Min Ru* Abstract This paper establishes a defect relation for algebraically nondegenerate holoorphic appings into an arbitrary nonsingular coplex projective variety V rather than just the projective space intersecting possible nonlinear hypersurfaces, extending the result of H. Cartan.. Introduction and stateents Let f : C P n C be a linearly nondegenerate holoorphic ap, and H j, j q, be hyperplanes in P n C in general position. In 933, H. Cartan [Ca] proved the defect relationor a Second Main Theore δ f H j n +. Since then, researches of higher diensional Nevanlinna theory have been carried out along these two directions: i study the algebraically nondegenerate holoorphic appings into an arbitrary nonsingular coplex projective variety V ; ii replace targets of the hyperplanes appearing in Cartan s result by curvilinear divisors. Recently, the author see [Ru3] established a defect relation for algebraically nondegenerate holoorphic curves f : C P n C intersecting curvilinear hypersurfaces, which settled a long-standing conjecture of B. Shiffan see [Shi]. This paper further extends the above entioned result to holoorphic curves f : C V intersecting hypersurfaces, where V is an arbitrary nonsingular coplex projective variety. To get a ore precise stateent, we first introduce soe standard notation in Nevanlinna theory: Let f : C P N C be a holoorphic ap. Let f = f,..., f N be a reduced representative of f, where f,..., f N are entire functions on C and have no coon zeros. The Nevanlinna-Cartan character- *The author is supported in part by NSA grant H The United State Governent is authorized to reproduce and distribute reprints notwithstanding any copyright notation hereon.

2 256 MIN RU istic function T f r is defined by where T f r = 2π log fre iθ dθ fz = ax{ f z,..., f N z }. The above definition is independent, up to an additive constant, of the choice of the reduced representation of f. Let D be a hypersurface in P N C of degree d. Let Q be the hoogeneous polynoial for of degree d defining D. The proxiity function f r, D is defined as f r, D = log fre iθ d Q Q fre iθ dθ 2π, where Q is the axiu of the absolute values of the coefficients of Q. To define the counting function, let n f r, D be the nuber of zeros of Q f in the disk z < r, counting ultiplicity. The counting function is then defined by r n f t, D n f, D N f r, D = dt + n f, D log r. t The Poisson-Jensen forula iplies: The first ain theore. Let f : C P N C be a holoorphic ap, and let D be a hypersurface in P N C of degree d. If fc D, then for every real nuber r with < r <, f r, D + N f r, D = dt f r + O, where O is a constant independent of r. Let V P N C be a sooth coplex projective variety of diension n. Let D,..., D q be hypersurfaces in P N C, where q > n. Also, D,..., D q are said to be in general position in V if for every subset {i,..., i n } {,..., q},. V supp D i supp D in =, where supp D eans the support of the divisor D. A ap f : C V is said to be algebraically nondegenerate if the iage of f is not contained in any proper subvarieties of V. The ain result of this paper is as follows. Theore The ain result. Let V P N C be a sooth coplex projective variety of diension n. Let D,..., D q be hypersurfaces in P N C of degree d j, located in general position in V. Let f : C V be an algebraically nondegenerate holoorphic ap. Then, for every ε >, d j f r, D j n + + εt f r,

3 DEFECT RELATION 257 where the inequality holds for all r, + except for a possible set E with finite Lebesgue easure. Define the defect of f, with respect to a hypersurface D of degree d, f r, D δ f D = li inf r + dt f r. Then we have the following defect relation. Corollary Defect relation. Let V P N C be a sooth coplex projective variety. Let D,..., D q be hypersurfaces in P N C, located in general position in V. Let f : C V be an algebraically nondegenerate holoorphic ap. Then δ f D j di V +. Note that our result could be easily extended to eroorphic aps f : C V. This paper is otivated by the analogy between Nevanlinna theory and Diophantine approxiation, discovered by C. Osgood, P. Vojta and S. Lang. etc.. It is now known that Cartan s Second Main Theore corresponds to Schidt s subspace theore. In 994, G. Faltings and G. Wüstholz [FW] extended Schidt s result to the systes of Diophantine inequalities to be solved in algebraic points of an arbitrary projective variety. Whereas Schidt s proof of his subspace theore is based on techniques fro Diophantine approxiation and the geoetry of nubers, Faltings and Wüstholz developed a totally different ethod, based on Faltings Product Theore cf. [FW, Ths. 3., 3.3]. Moreover they introduced a probability easure on R whose expected value is the crucial tool in the proof of their ain result. R. G. Ferretti see [F], [F2] later observed that their expected value can be reforulated in ters of the Chow weight of X or Muford s degree of contact. In fact, for every N-tuple c = c,..., c N where c,..., c N are integers with c c N, e c X R.G. Ferretti observed that E c, = dix + degx, where E c, is the Faltings-Wüstholz expected value with respect to c and e c X is the Chow weight of X with respect to c. Ferretti s observation brought the geoetric invariant theory Muford s degree of contact is a birational invariant often considered in Geoetric Invariant Theory see [Mu], [Mo] into the study of Diophantine approxiation. Later, J.H. Evertse and R. Ferretti cf. [EF], [EF2] further developed this technique and derived a quantitative version of Faltings and Wüstholz s result directly fro Schidt s quantitative subspace theore. They also extended Schidt s subspace theore with polynoials

4 258 MIN RU of arbitrary degree see also [CZ]. The Main Theore in this paper can be viewed as the counterpart of Corollary.2 of [EF2] in Nevanlinna theory. 2. Chow weights and Hilbert weights Let X P N be a projective variety i.e., a geoetrically irreducible Zariski-closed subset of diension n and degree. In this section, we shall give the definition of Chow weight of X and the definition of the -th Hilbert weight of X. We then recall a theore due to Evertse and Ferretti see Theore 4. in [EF] which gives an explicit lower bound of the -th noralized Hilbert weight of X in ters of the noralized Chow weight of X see Theore 2. below. This lower bound is sufficient for our purpose in proving the Main Theore. 2.. To X we can associate, up to a constant scalar, a unique polynoial F X u,..., u n = F X u,..., u N ;... ; u n,..., u nn in n + blocks of variables u i = u i,..., u in, i =,..., n, which is called the Cayley-Bertini-van der Waerden-Chow for of X, with the following properties: F X is irreducible in C[u,..., u nn ]; F X is hoogeneous of degree in each block u i, i =,..., n,; and F X u,..., u n = if and only if X H u H un, where H ui, i =,..., n, are the hyperplanes given by u i x = u i x + + u in x N = Let F X be the Chow for associated to X. Let c = c,..., c N be a tuple of reals. Let t be an auxiliary variable. We consider the decoposition 2. F X t c u,..., t cn u N ;... ; t c u n,..., t cn u nn = t e G u,..., u n + + t er G r u,..., u n, with G,..., G r C[u,..., u N ;... ; u n,..., u nn ] and e > e > > e r. We define the Chow weight of X with respect to c by 2.2 e X c := e. For each subset J = {j..., j n } of {,..., N} with j < j < < j n we define the bracket 2.3 [J] = [J]u,..., u n := detu ijk i,k=,...,n, where again u i = u i,..., u in denotes the blocks of N + variables. Let N + J,..., J β with β = be all subsets of {,..., N} of cardinality n +. n + Then, fro [HP, p. 4, Th. IV], the Chow for F X of X can be written as a

5 DEFECT RELATION 259 hoogeneous polynoial of degree in [J ],..., [J β ]. It is easy to show that, for c = c,..., c N R N+ and for any J aong J,..., J β, 2.4 [J]t c u,..., t cn u N ;... ; t c u n,..., t cn u nn = t P cj [J]u,..., u N ;... ; u n,..., u nn Denote by Z N+, RN+ the set of N + -tuples consisting of nonnegative integers, nonnegative reals, respectively. For a = a,..., a N Z N+ we write x a for the onoial x a xan N. Let I = I X be the prie ideal in C[x,..., x N ] defining X. Let C[x,..., x N ] denote the vector space of hoogeneous polynoials in C[x,..., x N ] of degree including. Put I := C[x,..., x N ] I and define the Hilbert function H X of X by, for =, 2,..., 2.5 H X := di C[x,..., x N ] /I. By the usual theory of Hilbert polynoials, 2.6 H X = n n! + O n We define the -th Hilbert weight S X, c of X with respect to a tuple c = c,..., c N R N+ by H X 2.7 S X, c := ax a i c, where the axiu is taken over all sets of onoials x a,..., x ah X whose residue classes odulo I for a basis of C[x,..., x N ] /I. i= 2.5. According to Muford [Mu, Prop. 2.], S X, c = e X c Together with 2.6, this iplies that 2.8 li H X S X, c = n+ n +! + On. n + e Xc. We call H S X X, c the -th noralized Hilbert weight and n+ e Xc the noralized Chow weight of X with respect to c Muford s identity above is not sufficient for our purpose. To prove our ain result, we need to copute the explicit constants appearing in Muford s identity. However, a lower of S X, c with explicit constants should be sufficient for our purpose.

6 26 MIN RU Theore 2.. Let X P N C be an algebraic variety of diension n and degree. Let > be an integer and let c = c,..., c N R N+. Then H X S X, c n + e 2n + Xc ax c i. i=,...,n Theore 2. here is the special case when K = C of Theore 4. of [EF]. Readers ay consult with [EF] for the proof. 3. Proof of the ain result To prove our ain result, we need the following general for of the Second Main Theore for holoorphic curves intersecting hyperplanes, due to P. Vojta see [V2]. The theore is also stated and proved in [Ru, Th. 2.]. Theore 3.. Let f : C P n C be a holoorphic ap whose iage is not contained in any proper linear subspace of P n C. Let H,..., H q be arbitrary hyperplanes in P n C. Let L j, j q, be the linear fors defining H,..., H q. Then, for every ε >, 3. ax log K j K fre iθ L j dθ L j fre iθ 2π n + + εt f r, where the inequality holds for all r outside of a set E with finite Lebesgue easure, f = f,..., f n is a reduced representation of f, the axiu is taken over all subsets K of {,..., q} such that #K = n + and the linear fors L j, j K, are linearly independent, and L j is the axiu of the absolute values of the coefficients in L j. We now prove our ain result. Proof. Let D,..., D q be the hypersurfaces in P N C, located in general position on V. Let Q j, j q, be the hoogeneous polynoials in C[X,..., X N ] of degree d j defining D j. Replacing Q j by Q d/dj j if necessary, where d is the l.c. of d j s, we can assue that Q,..., Q q have the sae degree of d. For every b = [b : : b N ] P N C, consider the function b, D j = Q jb b d Q j, where b = ax j N b j and Q j is the axiu of the absolute values of the coefficients of Q j. At each point b V, by the in general position condition, b, D j can be zero for no ore than n indicies j {,..., q}. For the reaining indices j, we have b, D j > and by the continuity of these functions and the copactness of V, there exists C > such that b, D j > C

7 DEFECT RELATION 26 for all b V and all D j, except for at ost n of the. holoorphic ap f : C V, 3.2 f r, D j = = dθ log fre iθ, D j 2π log q fre iθ d Q j dθ Q j fre iθ 2π { n log fre } iθ d Q ik Q ik fre iθ ax {i,...,i n} k= Hence, for any dθ 2π + O, where f = f,..., f N is a reduced representation of f. Define a ap φ : x V [Q x : : Q q x] P q C and let Y = φv. By the in general position assuption, φ is a finite orphis on V and Y is a coplex projective subvariety of P q C. We also have di Y = n and deg Y =: d n degv. For every a = a,..., a q Z q, denote ya = y a yaq q. Let be a positive integer. Put q n := H Y, q :=. Consider the Veronese ebedding 3.4 φ : P q C P q C : y [y a : : y aq ], where y a,..., y aq are the onoials of degree in y,..., y q, in soe order. Denote by Y the sallest linear subvariety of P q C containing φ Y. Then, clearly, a linear for q i= γ i z i vanishes identically on Y if and only if q i= γ i y ai, as a polynoial of degree, vanishes identically on Y. In other words, there is an isoorphis C[y,..., y q ] /I Y Y : y ai z i, i =,..., q, where I Y is the prie ideal in C[y,..., y q ] defining Y, I Y is the vector space of hoogeneous polynoials of degree in I Y, and Y is the vector space of linear fors in C[z,..., z q ] odulo the linear fors vanishing identically on Y. Hence Y is an n -diensional linear subspace of P q C where n = H Y. Since Y is an n -diensional linear subspace of P q C, there are linear fors L,..., L q C[w,..., w n ] such that the ap 3.5 ψ : w P n C [L w : : L q w] Y is a linear isoorphis fro P n C to Y. Thus ψ φ is an injective ap fro Y into P n C. Let f : C V be the given holoorphic ap and let F = ψ φ φ f : C P n C. Then F is a holoorphic ap. Furtherore,

8 262 MIN RU since f is algebraically nondegenerate, F is linearly nondegenerate. For every z C, let c z = c,z,..., c q,z where 3.6 c j,z := log fz d Q j Q j fz, j =,..., q. Obviously, c j,z for j =,..., q. By the definition of the Hilbert weight, for every z C, there is a subset I z of {,..., q } with #I z = n + = H Y such that {y ai : i I z } is a basis of C[y,..., y q ] /I Y and 3.7 S Y, c z = i I z a i c z. Note that, for every w P n C, we have L j w = y aj, j =,..., q, where L j, j q, are the linear fors obtained in 3.5. Hence, by the definition of F and 3.7, we have 3.8 log i I z L i L i F z = log i I z Q fz ai, Q q fz + OH Y ai,q = log [ fz d ai, Q i I z Q fz fz d ai,q ] Q q Q q fz dh Y log fz + OH Y = i I z a i c z dh Y log fz + OH Y = S Y, c z dh Y log fz + OH Y, where F is a reduced representation of F, and the ter OH Y does not depend on z. Hence 3.9 S Y, c z ax log L j J L j F z + dh Y log fz + OH Y = ax log F z L j J L j F + dh Y log fz z n + log F z + OH Y, where the axiu is taken over all J {,..., q } such that #J = n + and L j, j J, are linearly independent. By Theore 2., 3. H Y S Y, c z n + e 2n + Y c z ax c i,z. i q

9 DEFECT RELATION 263 Hence 3. n + e Y c z H Y +d log fz + ax log J 2n + F z L j L j F n + log F z z ax c i,z + O/. i q Our next step is to derive a lower bound for the Chow weight e Y c. To do so, we need the following lea. Lea 3.2. Let Y be a subvariety of P q C of diension n and degree. Let c = c,..., c q be a tuple of positive reals. Let {i,..., i n } be a subset of {,..., q} such that 3.2 Y {y i =,..., y in = } =. Then 3.3 e Y c c i + + c in. Lea 3.2 is nearly identical to Lea 5. of [EF2], except that the base field here is C instead of Q a. Since the proof is short, we enclose it here for the sake of copleteness. Proof. For a subset J = {j..., j n } of {,..., q} with j < j < < j n recall that we defined see 2.2 the bracket [J] = [J]u,..., u n := detu i,jk i,k=,...,n, where again u i = u i,..., u iq denotes the blocks of q variables. Let J,..., J β q with β = be all subsets of {,..., q} of cardinality n +. Then the n + Chow for F Y of Y can be written as a hoogeneous polynoial of degree in [J ],..., [J β ]: 3.4 F Y = a A Ca[J ] a [J β ] aβ, where A is the set of tuples of nonnegative integers a = a,..., a β with a + + a β =. For each bracket [J], [J]t c u,..., t cq u q ;... ; t c u n,..., t cq u nq ] = t P cj [J]u,..., u q ;... ; u n,..., u nq.

10 264 MIN RU This, together with 3.4, iplies that 3.5 F Y t c u,..., t cq u q ;... ; t c u n,..., t cq u nq = β CatP ajp i J j c i [J ] a [J β ] aβ. a A Put e :=,,...,, e 2 :=,,...,,..., e q :=,,...,. Write {i,..., i n } =: J. By 3.2 we have F Y e i,..., e in. Further, [J ]e i,..., e in =, [J]e i,..., e in =, for J J. Hence in expression 3.4 there is a ter C [J ] with C, and if we substitute u j = e ij, j =,..., n, in 3.5 we obtain C t ci + +cin. That is, one of the nubers e i in 2. is equal to c i + + c in. Hence e Y c c i + + c in. This proves Lea 3.2 Now, we continue the proof of the Main Theore. By Lea 3.2, for any {i,..., i n } {,..., q}, since D,..., D q are in general position in V so that 3.2 is satisfied, 3.6 e Y c z c i,z + + c in,z. By the definition of c z, we get 3.7 c i,z + + c in,z = log Cobining 3., 3.6 and 3.7 gives 3.8 log = fz d Q i Q i fz n + ax log H Y J +dn + log fz + n + H Y fz d Q in Q in fz ax log J +dn + log fz + fz d Q i Q i fz fz d Q in Q in fz. F z L j L j F n + log F z z 2n + n + ax c i,z + O/ i q F z L j L j F n + log F z + O z 2n + n + ax log fz Q j j q Q j fz.

11 DEFECT RELATION 265 Therefore 3.9 ax log i,...,i n fz d Q i n + H Y Q i fz ax log J +dn + log fz + fz d Q in Q in fz F z L j L j F n + log F z + O z 2n + n + ax log fz Q j j q Q j fz. Applying integration on the both sides of 3.9 and using the First Main theore yield 3.2 ax log i,...,i n n + H Y fre iθ d Q i Q i fre iθ +dn + T f r + n + H Y +dn + T f r + ax log J 2n + n + ax log J fre iθ d Q in dθ Q in fre iθ 2π F re iθ L j dθ L j F re iθ 2π n + T F r f r, D j + O j q F re iθ L j dθ L j F re iθ 2π n + T F r, 2n + n + q T f r + O here we note that various constants in the O ter above depend only on Q,..., Q q, not on f and z. For the ε > given in the Main Theore, take large enough so that 3.2 n + H Y < ε 2n + n + q, and < ε/3. 3d Fix such an. Applying Theore 3. with ε = to holoorphic ap F and linear fors L,..., L q, we obtain that 3.22 ax log J F re iθ L j dθ L j F re iθ 2π n + 2T F r

12 266 MIN RU holds for all r outside of a set E with finite Lebesgue easure. By cobining 3.2, 3.2, 3.2 and 3.22, we get 3.23 f r, D j ε 3d T F r + dn + T f r + ε/3t f r + O where the inequality holds for all r outside of a set E with finite Lebesgue easure. By the definition of the characteristic function, we have T F r dt f r. Hence 3.23 becoes 3.24 f r, D j dn + + 2ε/3T f r + C where the inequality holds for for all r outside of a set E with finite Lebesgue easure, and where C is a constant, independent of r. Take r big enough so that we can ake C ε/3t f r. Thus we have f r, D j dn + + εt f r where the inequality holds for all r outside of a set E with finite Lebesgue easure. This proves our ain result. University of Houston, Houston, TX E-ail address: inru@ath.uh.edu [CG] [Ca] References J. Carlson and P. Griffiths, A defect relation for equidiensional holoorphic appings between algebraic varieties, Ann. of Math , H. Cartan, Sur les zeros des cobinaisions linearires de p fonctions holoorpes donnees, Matheatica 7 933, 8 3. [CZ] P. Corvaja and U. Zannier, On a general Thue s equation, Aer. J. Math , [EF] [EF2] [FW] [F] J.-H. Evertse and R. G. Ferretti, Diophantine inequalities on projective varieties, Internat. Math. Res. Notices 25 22, , A generalization of the subspace theore with polynoials of higher degree, Developents in Matheatics 6, 75 98, Springer-Verlag, New York 28. G. Faltings and G. Wüstholz, Diophantine approxiations on projective varieties, Invent. Math , R. G. Ferretti, Muford s degree of contact and Diophantine approxiations, Copositio Math. 2 2, [F2], Diophantine approxiations and toric deforations, Duke. Math. J. 8 23, [HP] W. V. D. Hodge and D. Pedoe, Methods of Algebraic Geoetry, Vol. II, Cabridge Univ. Press, Cabridge, 952.

13 DEFECT RELATION 267 [Mo] I. Morrison, Projective stability of ruled surfaces, Invent. Math , [Mu] D. Muford, Stability of projective varieties, Enseign. Math. XXIII 977, 39. [MFK] D. Muford, J. Fogarty, and J. Kirwan, Geoetric Invariant Theory, Erg. Math. Grenzgeb. 34, Springer-Verlag, New York, 999. [No] J. Noguchi, Holoorphic curves in algebraic varieties, Hiroshia Math. J , [Ru] [Ru2] [Ru3] M. Ru, On a general for of the Second Main Theore, Trans. Aer. Math. Soc , , Nevanlinna Theory and its Relation to Diophantine Approxiation, World Sci. Publ. Co., Inc., River Edge, NJ, 2., A defect relation for holoorphic curves intersecting hypersurfaces, Aer. J. Math , [Sh] I. R. Shafarevich, Basic Algebraic Geoetry, Springer-Verlag, New York, 977. [Shi] [SY] [V] B. Shiffan, On holoorphic curves and eroorphic aps in projective space, Indiana Univ. Math. J , Y.-T. Siu and S.-K. Yeung, Defects for aple divisors of Abelian varieties, Schwarz lea, and hyperbolic hypersurfaces of low degrees, Aer. J. Math , P. Vojta, Diophantine Approxiations and Value Distribution Theory, Springer- Verlag, New York, 987 [V2], On Cartan s theore and Cartan s conjecture, Aer. J. Math , 7. Received January 3, 25 Revised May 5, 27