MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION Nguyen Thi Nhung VALUE DISTRIBUTION OF MEROMORPHIC MAPPINGS FROM KHLER MANIFOLDS INTO PROJECTIVE VARIETIES AND ITS APPLICATION Major: Geometry and Topology Code: 9.46.01.05 SUMMARY OF MATHEMATICS DOCTOR THESIS Ha Noi - 2019 The thesis was done at: Ha Noi National University of Education The suppervisors: Asso Prof Dr Si Duc Quang Referee 1: Prof Dr Pham Hoang Hiep, Institute of Mathematics - VAST Referee 2: Prof Dr Nguyen Quang Dieu, Ha Noi National University of Education Referee 3: Asso Prof Dr Nguyen Thac Dung, Hanoi University of Science - VNU This Dissertation will be examined by Examination Board At: Ha Noi National University of Education On hour minute, day .month 2019 The thesis can be found at - Library of Hanoi National University of Education - National Library of Vietnam WORKS RELATED TO THE THESIS [1] S D Quang, N T Q Phuong and N T Nhung (2017), Non-intergrated defect relation for meromophic maps from a Kahler manifold intersecting hypersurfaces in subgeneral of P n (C), Journal of Mathematical Analysis and Application, 452 (2017), 4341452 [2] N T Nhung and L N Quynh, Unicity of Meromorphic Mappings From Complete Kă ahler Manifolds into Projective Spaces, Houston Journal of Mathematics, 44(3) (2018), 769-785 [3] N T Nhung and P D Thoan, On Degeneracy of Three Meromorphic Mappings From Complete Kă ahler Manifolds into Projective Spaces, Comput Methods Funct Theory, 19(3) (2019), pp 353–382 [4] S D Quang, L N Quynh and N T Nhung, Non-integrated defect relation for meromorphic mappings from a Kăahler manifold with hypersurfaces of a projective variety in subgeneral position, submitting INTRODUCTION Rationale Nevanlinna theory begins with the study on the value distribution of meromorphic functions In 1926, R Nevanlinna extended the classical little Picard’s theorem by proving the two elegant theorems called the First and the Second Main Theorem The work of Nevanlinna has evoked a very strong interest of research in his theory and a number of important papers have been published Recently, many mathematicians have generalized Nevanlinna theory to the case of meromorphic mappings from Kăaher manifolds into projective varieties In 1985, H Fujimoto studied value distribution theory to the case of a meromorphic map of a complete Kăaher manifold M whose universal covering is biholomorphic to a ball B(R0 ) in Cm The difference is that there is no parabolic exhaustion on gereral Kăaher manifolds Therefore, notions of divisor counting function, characteristic function as well as proximility function for meromorphic mappings can not be defined In order to overcome this difficulty, using property that distances on base spaces are less than or equal covering spaces, H Fujimoto transfered problems for meromorphic mapping f from B(R0 ) into projective space Pn (C) He also introduced new notions as well as new methods to solve the diferent cases when applying Nevanlinna theory on a ball B(R0 ) comparing with Cm In more details, He introduced the notion of the nonintegrated defect and obtained some results analogous to the classical defect relation In 2012, T V Tan and V V Truong gave a non-integrated defect relation for the family of hypersurfaces in subgeneral position However, their definition of ”subgeneral position” is quite special, which has an extra condition on the intersection of these q hypersurfaces Independently, M Ru and S Sogome generalized Fujimoto’s result to the case of meromorphic mappings intersecting a family of hypersurfaces in general position After that, some au- thors such as Q.Yan, D D Thai and S D Quang extended the result of M Ru and S Sogome by considering the case of hypersurfaces in subgeneral position However, the above results not completely extend the results of H Fujimoto as well as M Ru and S Sogome Thus, it raises a natural question ”Are there any ways to establish a better non-integrated defect relation for the family of hypersurfaces in subgeneral position?” In the thesis, we will give a new method to answer this question Since R Nevanlinna proved five points theorem, which is also called uniqueness theorem, many authors have extended this one to the case of meromorphic mappings from Cm into Pn (C) The first results were obtained by H Fujimoto and L Smiley L Smiley showed that if two meromorphic mappings f and g have the same inverse images of 3n + hyperplanes without counting multiplicities, the codimmension of the intersection of inverse images of any two different hyperplanes is at least 2, f and g coincide on the inverse images of these hyperplanes then f = g This condition of Smiley helps us have better estimation on counting function and many results that improve Smiley’s theorem have been given Some of the best results in this direction were given by Z Chen and Q Yan, H H Giang, L N Quynh and S Quang In 1986, after succeeding in establishing non-integrated defect relation, H Fujimoto proved uniqueness theorem for meromorphic mappings of M into Pn (C)) intersecting family of hyperplanes However, H.Fujimoto’s result is not the one in direction of L.Smiley, consequently it doesn’t generalize the mentioned results when we restrict it in case M = Cm Therefore, the next our purpose is giving a theorem that both extends Fujimoto’s result and generalizes ones on Cm Beside uniqueness problem, Algebraic dependences of meromorphic mappings have also been intensively studied by many authors This direction started with S Ji’s work and there have been a number of results released Some of the best results belong to Z Chen and Q Yan, S D Quang, S D Quang and L N Quynh Thus, the following question arises naturally: ”Is it possible to extend algebraic dependence theorem of meromorphic mapping f from Cm to the case f from M into Pn (C)?” We note that although many authors have generalized Fujimoto’s uniqueness theorem, generalizations of the dependency theorems have not been obtained yet In the last chapter of the thesis, we introduced some new techniques to give a positive answer for this question From the above questions, we choose the topic Value distribution of meromorphic mappings from Kă ahler manifolds into projective varieties and its application” to investigate non-integrated defect relation for meromorphic mappings intersecting hypersurfaces in subgeneral position, uniqueness problems as well as algebraic dependence ones for meromorphic mappings intersecting hyperplanes Objectives of research The first aim of this thesis is establishing non-integrated defect relation for meromorphic mappings from Kăahler manifolds into projective varieties intersecting hypersurfaces in subgeneral position The next one is studying uniqueness problems and the last one is examining algebraic dependence theorems of meromorphic mappings from Kăahler manifolds into projective spaces intersecting hyperplanes in general or subgeneral position Object and scope of research Research objects: non-integrated defect relation, uniqueness problems and algebraic dependence problems for meromorphic mappings from Kăahler manifolds into projective varieties Research scope: Nevanlinna theory for meromorphic mappings from Kăahler manifolds Methodology In order to solve problem given in the thesis, we use methods in value distribution and complex theory Besides using traditional techniques, we also introduce new techniques to achieve aims of the thesis Scientific and practical significances The thesis gives more developed results on non-integrated defect relation for meromorphic mappings from Kăahler manifolds into projective varieties intersecting hypersurfaces in subgeneral position It also proves more deepened theorems on uniqueness problem for meromorhic mappings from Kăahler manifolds In addition, it presents some new results on algebraic dependence problems for meromorphic mappings from Kăahler manifolds This thesis acts as a helpful reference to bachelor, master as well as PhD students majoring in Nevanlinna theory Structure The thesis consists of four chapters (excluding the Introduction and Conclusion): Chapter I Overview Chapter II Non-integrated defect relation for meromorphic mappings intersecting hypersurfaces in subgeneral position Chapter III Unicity of meromorphic mappings sharing hyperplanes Chapter IV Algebraic dependences of meromorphic mappings sharing few hyperplanes Chapter OVERVIEW In this chapter, we deeply pay attention to analylizing history and results of previous authors as well as our new results that we achieved regarding nonintegrated defect problems, uniqueness problems and algebraically dependent problems for meromorphic mappings from Kăahler M into projective space Pn (C), where the universal covering of M is biholomorphic to a ball in Cm I Non-integrated defect relation for meromorphic mappings intersecting hypersurfaces in subgeneral position Let M be a complete Kăahler manifold of dimension m Let f : M −→ Pn (C) be a meromorphic mapping and Ωf be the pull-back of the Fubini-Study form Ω on Pn (C) by f Definition 1.0.1 For ρ ≥ we say that f satisfies the condition (Cρ ) if there exists a nonzero bounded continuous real-valued function h on M such that ρΩf + ddc logh2 ≥ √ Ricω, where Ωf is the full-back of the Fubini-Study −1 form Ω on Pn (C), ω = ahler form on M , Ricω = i,j hi¯j dzi ∧ dz j is Kă ddc log(det(hij )), d = ∂ + ∂ and dc = (∂ − ∂) 4π Definition 1.0.2 For a positive integer µ0 and a hypersurface D of degree d in Pn (C) with f (M ) ⊂ D, we denote by νf (D)(p) the intersection multiplicity of the image of f and D at f (p) The non-integrated defect of f with respect to D [µ ] truncated to level µ0 by δf := − inf{η ≥ : η satisfies condition (∗)} Here, the condition (*) means that there exists a bounded non-negative continuous function h on√M whose order of each zero is not less than min{νf (D), µ0 } such −1 ¯ that dηΩf + ∂ ∂logh2 ≥ [min{νf (D), µ0 }] 2π Definition 1.0.3 Let V be a subvariety of Pn (C) of dimension k > Let N ≥ n and q ≥ N + Let Q1 , , Qq be hypersurfaces in Pn (C) The hypersurfaces Q1 , , Qq are said to be in N -subgeneral position with respect to V if Qj1 ∩ · · · ∩ QjN +1 ∩ V = ∅ for every ≤ j1 < · · · < jN +1 ≤ q If N = n then we say that Q1 , , Qq are in general position w.r.t V In 1985, H Fujimoto established non-integrated defect relation for meromorphic mappings intersecting hyperplanes in general position as follows Theorem A Let M be an m-dimensional complete Kăahler manifold and be a Kă ahler form of M Assume that the universal covering of M is biholomorphic to a ball in Cm Let f : M → Pn (C) be a meromorphic map which is linearly nondegenerate and satifies condition Cρ If H1 , · · · , Hq be hyperplanes of Pn (C) [n] in general position then qi=1 δf (Hi ) ≤ n + + ρn(n + 1) In 2012, M Ru-S Sogome generalized Theorem A to the case of meromorphic mappings intersecting a family of hypersurfaces in general position as follows Theorem B Let M be an m-dimensional complete Kăahler manifold and be a Kă ahler form of M Assume that the universal covering of M is biholomorphic to a ball in Cm Let f be an algebraically nondegenerate meromorphic map of M into Pn (C) and satifies condition Cρ Let Q1 , , Qq be hypersurfaces in Pn (C) of degree PIj , in k-subgeneral position in Pn (C) Let d = l.c.m.{Q1 , , Qq } ρu(u − 1) [u−1] Then, for each > 0, we have qj=1 δf (Qj ) ≤ n + + + , where d u ≤ 2n +4n en d2n (nI(ε−1 ))n and here, for a real number x, we define I(x) := min{a ∈ Z ; a > x} After that, Q Yan extended Theorem A to the case of the family of hypersurfaces in subgeneral position He proved the following Theorem C Let M be an m-dimensional complete Kăahler manifold and be a Kă ahler form of M Assume that the universal covering of M is biholomorphic to a ball in Cm Let f be an algebraically nondegenerate meromorphic map of M into Pn (C) and satifies condition Cρ Let Q1 , , Qq be hypersurfaces in Pn (C) of degree PIj , in k-subgeneral position in Pn (C) Then, for each > 0, ρu(u − 1) K0 +n [u−1] q we have (Qj ) ≤ N (n + 1) + + , where u = ≤ j=1 δf n d (3eN dI( −1 ))n (n + 1)3n and K0 = 2N dn2 (n + 1)2 I( −1 ) The above result of Q Yan does not completely extend the results of H Fujimoto and M Ru-S Sogome Indeed, when the family of hypersurfaces is in general position, i.e., k = n, the first term in the right hand side of the defect relation inequality is n(n + 1), which is bigger than (n + 1) as usual In usual principle, to deal with the case of family of hypersurfaces in subgeneral position, we need to generalize the notion of Nochka weights However, for the case of hypersurfaces, there are no Nochka weights constructed In order to overcome this difficulty, we will use a technique ”replacing hypersurfaces” proposed by S D Quang Our main idea to avoid using the Nochka weights is that: each time when we estimate the auxiliary functions, we will replace N+1 hypersurfaces by n+1 other new hypersurfaces in general position so that this process does not change the estimate By using this technique, we prove the following theorem Theorem 2.2.4 Let M be an m-dimensional complete Kăahler manifold and be a Kă ahler form of M Assume that the universal covering of M is biholomorphic to a ball in Cm Let f be an algebraically nondegenerate meromorphic map of M into Pn (C) and satifies condition Cρ Let Q1 , , Qq be hypersurfaces in Pn (C) of degree dj , in k-subgeneral position in Pn (C) Let d = l.c.m.{d1 , , dq } ρu(u − 1) [u−1] Then, for each > 0, we have qj=1 δf (Qj ) ≤ p(n + 1) + + , where d L +n p = N − n + 1, u = 0n ≤ en+2 (dp(n + 1)2 I( −1 ))n and L0 = (n + 1)d + p(n + 1)3 I( −1 )d Then we see that, if the family of hypersurfaces is in general position, i.e., k = n, then our result implies the results of H Fujimoto and also of M Ru-S Sogome In the above theorems, f is assumed to be algebraically nondegenerate In order to deal with cases that f may be algebraically degenerate, we need to establish non-integrated defect relation for f from M into variety V of Pn (C) intersecting hypersurfaces in subgeneral position We continue to use technique ”replacing hypersurfaces”, we extend Therem ?? to the case of hypersurfaces in subgeneral position with respect to a projective subvariety of Pn (C) Our main theorem is stated as follows Theorem 2.2.10 Let M be an m-dimensional complete Kăahler manifold and be a Kă ahler form of M Assume that the universal covering of M is biholomorphic to a ball in Cm Let f be an algebraically nondegenerate meromorphic map of M into a subvariety V of dimension k in Pn (C) and satifies condition (b) f (z) = g(z) on q −1 j=1 f (Hj ) In algebraic dependence problems of three meromorphic mappings, we find conditions such that f , f , f ∈ F(f, {Hi }qi=1 , d) is algebraically dependent , namely we give conditions such that set {(f (z), f (z), f (z)), z ∈ Cm } is included in a proper algebraic subset of Pn (C) × Pn (C) × Pn (C) Algebraic dependence of f , f , f can be obtained by proving a stronger result f ∧f ∧f ≡ or showing mapping f ×f ×f is algebraically degenerate In 2015, S D Quang and L N Quynh proved the algebraic dependence theorem of three meromorphic mappings sharing less than 2n + hyperplanes in general position as follows Theorem F Let f , f , f be three linearly nondegenerate meromorphic mappings of Cm into Pn (C) Let {Hi }qi=1 be a family of q hyperplanes of Pn (C) in general position with dim f −1 (Hi ) ∩ f −1 (Hj ) m − (1 i < j q) Assume that the following conditions are satisfied: (a) min{ν(f ,Hi ) , n} = min{ν(f ,Hi ) , n} = min{ν(f ,Hi ) , n} (1 i q), (b) f = f = f on qi=1 (f )−1 (Hi ) √ 2n + + 28n2 + 20n + If q > then one of the following assertions holds: (i) There exist 3q + hyperplanes such that (f u , Hi[ q ]+1 ) (f u , Hi1 ) (f u , Hi2 ) = v = ··· = v , v (f , Hi1 ) (f , Hi2 ) (f , Hi[ q ]+1 ) (ii) f ∧ f ∧ f ≡ Theorem F showed algebraic dependence of three mappings with the number of hyperplanes q less than 2n + but truncation d was still n In 2018, S D Quang proved the following result of algebraic dependence of three mappings in which the number of hyperplanes q was 2n + and truncation was d n or q was 2n + and truncation was Theorem G Let f be a linearly nondegenerate meromorphic mapping of Cm into Pn (C) Let H1 , , H2n+1 be 2n+1 hyperplanes of Pn (C) in general position such that dim f −1 (Hi ) ∩ f −1 (Hj ) m − (1 i < j 2n + 2) Then the map f × f × f of Cm into Pn (C) × Pn (C) × Pn (C) is linearly degenerate for every three mappings f , f , f ∈ F(f, {Hi }2n+1 i=1 , p) 10 Theorem H If the three mappings f , f , f belong to F(f, {Hi }2n+2 i=1 , 1), then m f ∧ f ∧ f ≡ on C It implies that f , f and f are algebraically dependent over C By considering new auxiliary functions and rearranging hyperplanes into groups, we will generalize Theorem F, G and H for meromorphic mappings f from Kăahler manifolds M into Pn (C) Moreover, we also extend the results from hyperplanes in general position to subgeneral position Concretely, we prove the following theorems Theorem 4.2.3 Let M be a complete and connected Kăahler manifold whose universal covering is biholomorphic to B(R0 ) ⊂ Cm , where < R0 ∞ Let n f , f , f : M → P (C) be three linearly nondegenerate meromorphic mappings which satisfy the condition (Cρ ) for some nonnegative constant ρ and there are q hyperplanes H1 , H2 , , Hq of Pn (C) in N -subgeneral position such that dim f −1 (Hi ) ∩ f −1 (Hj ) m − (1 i < j q) Suppose that we have the following conditions: (a) min{ν(f ,Hi ) , n} = min{ν(f ,Hi ) , n} = min{ν(f ,Hi ) , n} (1 (b) f = f = f trn i q), q −1 i=1 (f ) (Hi ) If q > 2N − n + + ρn(n + 1) + 3nq then one of the following assertions 2q + 3n − holds: (i) There exist q + hyperplanes such that (f u , Hi[ q ]+1 ) (f u , Hi1 ) (f u , Hi2 ) = v = ··· = v , v (f , Hi1 ) (f , Hi2 ) (f , Hi[ q ]+1 ) (ii) f ∧ f ∧ f ≡ on M Theorem 4.2.6 Let M , f , f , f and H1 , , Hq be as in Theorem ?? Let n and p n be a positive integer Assume that the following assertions are satisfied: (a) min{ν(f ,Hi ) , p} = min{ν(f ,Hi ) , p} = min{ν(f ,Hi ) , p} (1 (b) f = f = f on i q), q −1 i=1 (f ) (Hi ) q(2n + p) then the map f × f × f of M 2q − + 3p n n n into P (C) × P (C) × P (C) is linearly degenerate If q > 2N − n + + ρn(n + 1) + Theorem 4.2.4 Let M , f , f , f and H1 , , Hq be as in Theorem ?? Assume 11 that the following assertions are satisfied: (a) min{ν(f ,Hi ) , 1} = min{ν(f ,Hi ) , 1} = min{ν(f ,Hi ) , 1} (1 (b) f = f = f on i q), q −1 i=1 (f ) (Hi ) 3nq , then f ∧ f ∧ f ≡ on M It 2q + 2n − implies that three mappings f , f and f are algebraically dependent on M If q > 2N − n + + ρn(n + 1) + 12 Chapter NON-INTEGRATED DEFECT RELATION FOR MEROMORPHIC MAPPINGS INTERSECTING HYPERSURFACES IN SUBGENERAL POSITION As we presented in the Overview, the main goal of chapter two is establishing defect relation for meromorphic mappings from Kăahler manifolds into projective varieties intersecting hypersurfaces in subgeneral position By using technique ”replacing hypersurfaces” proposed by S D Quang, we succeeded in gereralizing the theorem of M Ru and S Sogome to the case of hypersurfaces in subgeneral position and extended the results of previous authors as well Chapter two is written based on article [1] and [4] (in Works related to the thesis) 2.1 Basic notions In this section, first we present some basic notions and important results in Nevanlinnas theory related to thesis such as: Nevanlinna’s basic functions, Nevanlinna’s defect, Wronskian, First Main Theorem and Lemma on logarithmic derivative Then, we recall definition as well as properties of non-integrate defect Last, we present Chow weight, Hilbert weight and some properties which will be used later 13 2.2 Non-integrate defect relation for meromorphic mappings In this section, we prove two main theorems about non-integrate defect relation for meromorphic mappings from M into projective varieties and projective spaces intersecting hypersurfaces in subgeneral position We start with recalling some auxiliary lemmas Lemma 2.2.1 and Lemma 2.2.2 give us important statements which are: From a family of hypersurfaces in subgeneral position (with respect to V ), we construct a family of hypersurfaces in general position (with respect to V ) which each of these hypersurfaces can represent linearly through given hypersurfaces This is a basic idea of the technique ”replacing hypersurfaces” mentioned in the Rationale as well as in the Overview It is also a key technique to find new results when we establish non-integrate defect relation for hypersurfaces in subgeneral position Lemma 2.2.1 Let V be a smooth projective subvariety of Pn (C) of dimension k Let Q1 , , QN +1 be hypersurfaces in Pn (C) of the same degree d ≥ 1, such N +1 that i=1 Qi ∩ V = ∅ Then there exists k hypersurfaces P2 , , Pk+1 of the −k+t ctj Qj , ctj ∈ C, t = 2, , k +1, such that forms Pt = N j=2 where P1 = Q1 k+1 t=1 Pt ∩V = ∅, When V = Pn (C), Lemma 2.2.1 is stated as follows Lemma 2.2.2 Let Q1 , , Qk+1 be hypersurfaces in Pn (C) of the same degree k+1 d ≥ 1, such that i=1 Qi = ∅ Then there exist n hypersurfaces P2 , , Pn+1 of the forms Pt = where P1 = Q1 k−n+t ctj Qj , j=2 ctj ∈ C, t = 2, , n + 1, such that n+1 t=1 Pt = ∅, Lemma 2.2.3 Let {Qi }i∈R be a family of hypersurfaces in Pn (C) of the common degree d and let f be a meromorphic mapping of Cm into Pn (C) Assume that i∈R Qi = ∅ Then, there exist positive constants α and β such that α||f ||d ≤ maxi∈R |Qi (f )| ≤ β||f ||d Theorem 2.2.4 Let M be an m-dimensional complete Kăahler manifold and be a Kă ahler form of M Assume that the universal covering of M is biholomorphic to a ball in Cm Let f be an algebraically nondegenerate meromorphic map of M into Pn (C) and satify condition Cρ for ρ ≥ Let Q1 , , Qq be hypersurfaces in Pn (C) of degree dj , in N -subgeneral position in Pn (C) Let [u−1] d = l.c.m.{d1 , , dq } Then, for each > 0, we have qj=1 δf (Qj ) ≤ p(n + 14 ρu(u − 1) , where p = N − n + 1, u = d and L0 = (n + 1)d + p(n + 1)3 I( −1 )d L0 +n n 1) + + In the above theorem, letting = + we obtain the following corollary with ≤ en+2 (dp(n + 1)2 I( > and then letting −1 ))n −→ 0, Corollary 2.2.5 With the assumption of Theorem 2.2.4, we have ρu(u − 1) [u−1] q δ (Q ) ≤ p(n + 1) + + , where p = N − n + 1, j j=1 f d L +n u = 0n ≤ en+2 (dp(n + 1)2 )n and L0 = (n + 1)d(1 + p(n + 1)2 ) In order to prove Theorem 2.2.4 we need to prepare some following lemmas Now, for a positive integer L, we denote by VL the vector subspace of C[x0 , , xn ] which consists of all homogeneous polynomials of degree L and zero polynomial We see that L0 is divisible by d Hence, for each (i) = (i1 , , in ) ∈ Nn0 with σ(i) = ns=1 is ≤ Ld0 , we set j1 jn PI1 · · · PIn · VL0 −dσ(j) I W(i) = (j)=(j1 , ,jn )≥(i) Lemma 2.2.6 Let (i) = (i1 , , in ), (i) = (i1 , , in ) ∈ Nn0 Suppose that (i ) I W(i) I follows (i) in the lexicographic ordering and defined m(i) = dim I Then, we W(i) I n have m(i) = d , provided dσ(i) < L0 − nd I I I We assume that VN = W(i) ⊃ W(i) ⊃ · · · ⊃ W(i) , where (i)s = (i1s , , ins ), K I I W(i) follows W(i) in the ordering and (i)K = ( Nd , 0, , 0) We see that K is s+1 s the number of n-tuples (i1 , , in ) with ij ≥ and i1 + · · · + in ≤ Ld0 We define WI mIs = dim W I(i)s for all s = 1, , K − and set mIK = (i)s+1 Lemma 2.2.7 For L0 = (n + 1)d + p(n + 1)3 I( −1 )d as in the assumption, we have puL0 (a) ≤ (N − n + 1)(n + 1) + , db n (b) u ≤ en+2 dp(n + 1)2 I( −1 ) Proposition 2.2.8 b q j=1 νQj (f ) − pνW α (φs (f )) ≤ b q i=1 min{u − 1, νQj (f ) } Theorem 2.2.9 With the assumption of Theorem 2.2.4 and suppose that M = Bm (R0 ) Then, we have q (q − p(n + 1) − )Tf (r, r0 ) ≤ i=1 15 [u−1] N (r) + S(r), d Qi (f˜) where S(r) ≤ K(log+ R01−r + log+ Tf (r, r0 )) for all < r0 < r < R0 outside a set E ⊂ [0, R0 ] with E Rdt < ∞ −t Theorem 2.2.10 Let M be an m-dimensional complete Kăahler manifold and be a Kă ahler form of M Assume that the universal covering of M is biholomorphic to a ball in Cm Let f be an algebraically nondegenerate meromorphic map of M into a subvariety V of dimension k in Pn (C) satifying condition (Cρ ) for ρ ≥ Let Q1 , , Qq be hypersurfaces in Pn (C) of degree dj , in N -subgeneral position with respect to V Let d = l.c.m.{d1 , , dq } ρεM0 (M0 − 1) , d where p = N − k + 1, M0 = dk +k deg(V )k+1 ek pk (2k + 4)k lk ε−k + and l = (k + 1)q! Then, for each ε > 0, we have [M0 −1] q (Qj ) j=1 δf ≤ p(k + 1) + ε + In the case of the family of hypersurfaces is in general position, we get N = n Moreover, since (n−k+1)(k+1) ≤ ( n2 +1)2 for every ≤ k ≤ n, letting ε = 1+ε with ε > and then letting ε −→ from the above theorem, we obtain the following corollary for the case f may be algebraically degenerate Corollary 2.2.11 Let f : M → Pn (C) be a meromorphic mapping and let {Qi }qi=1 be hypersurfaces in Pn (C) of degree di , located in general position Then, for every ε > 0, we get q [M0 −1] δf (Qj ) ≤ j=1 n +1 2 +1+ ρM0 (M0 − 1) , d for some positive integer M0 In order to prove Theorem 2.2.10, we need the following one Theorem 2.2.12 With the assumption of Theorem 2.2.10 Then, we have q (q − p(k + 1) − ε)Tf (r, r0 ) ≤ i=1 [M0 −1] N (r) + S(r), d Qi (f˜) where S(r) is evaluated as follows: + log+ Tf (r, r0 )), for all < R0 − r dt r0 < r < R0 outside a set E ⊂ [0, R0 ] with E < ∞ and K is a positive R0 − t constant (i) In the case R0 < ∞, S(r) ≤ K(log+ (ii) In the case R0 = ∞, S(r) ≤ K(logr + log+ Tf (r, r0 )), for all < r0 < r < ∞ outside a set E ⊂ [0, ∞] with E dt < ∞ and K is a positive constant 16 Chapter UNICITY OF MEROMOPHIC MAPPINGS SHARING FEW HYPERPLANES Chapter three concentrates on extending the uniqueness theorem of H Fujimoto for meromorphic mappings from Kăahler manifold M into Pn (C) and generalizing uniqueness theorem in the direction of L Smiley for meromorphic mappings from Cm into Pn (C) In H Fujimoto’s result, he considered a family of hyperplanes {Hj }qj=1 in general position and L Smiley added condition of dimention for {Hj }qj=1 , dim f −1 (Hi ) ∩ f −1 (Hj ) ≤ m − 2, (1 ≤ i < j ≤ q) We replace k+1 the above condition with the following dim f −1 ≤ m − (1 ≤ j=1 Hij i1 < · · · < ik+1 ≤ q)(∗) Then, when the family of hyperplanes are in general position then condition (∗) is always satified and when k = then (∗) is exactly Smiley’s condition Thus, our result not only extend Fujimoto’s theorem but also generalize results when we restrict to the case M = Cm Chapter three is written based on article [2] (in Works related to the thesis) 3.1 Second main theorem for meromorphic mapping from a ball and hyperplanes in general position In this section, we prove some auxiliary lemmas that will be used in the following section Lemma 3.1.1 Let f be a linearly nondegenerate meromorphic mapping from B(R0 ) into Pn (C) and H1 , , Hq be q hyperplanes of Pn (C) in general position Set l0 = |α1 | + · · · + |αn+1 | and take t, p with < tl0 < p < Then, for 17 < r0 < R0 there exists a positive constant K such that for r0 < r < R < R0 , z , Fn+1 ) t σm ≤ K L1 (F ) · · · Ln+1 (F ) α1 +···+αn+1 Wα1 , ,αn+1 (F1 , · · · R2m−1 TF (R, r0 ) R−r p S(r) Lemma 3.1.2 Let f be a linearly nondegenerate meromorphic mapping from B(R0 ) into Pn (C) and H1 , , Hq be q hyperplanes of Pn (C) in general position Then we have q [n] (q − n − 1)Tf (r, r0 ) ≤ NHj (f ) (r, r0 ) + Sf (r), j=1 where Sf (r) is evaluated as follows: + log+ Tf (r, r0 )) for all < R0 − r dt r0 < r < R0 outside a set E ⊂ [0, R0 ) with E < ∞, where K is a R0 − t positive constant (i) In the case R0 < ∞, Sf (r) ≤ K(log+ (ii) In the case R0 = ∞, Sf (r) ≤ K(log+ Tf (r, r0 )+logr) for all < r0 < r < ∞ outside a set E ⊂ [0, ∞) with E dt < ∞, where K is a positive constant 3.2 Uniqueness theorem for meromorphic mappings sharing few hyperplanes Theorem 3.2.1 Let M be a complete, connected Kăahler manifold whose universal covering is biholomorphic to B(R0 ) ⊂ Cm Let f, g : M → Pn (C) be linearly nondegenerate meromorphic mappings Assume that f and g satisfy the condition (Cρ ) for some ρ ≥ and there are q hyperplanes H1 , , Hq of Pn (C) in general position such that (i) dim f −1 k+1 j=1 Hij ≤ m − (1 ≤ i1 < < ik+1 ≤ q), (ii) f = g on ∪qj=1 f −1 (Hj ) ∪ g −1 (Hj ) 2nkq > n+1+ q + 2nk − 2k ρn(n + 1) or q < 2(n + 1)k and q > (n + 1)(k + 1) + ρn(n + 1) Then we have f ≡ g if either q ≥ 2(n + 1)k and q − From the above theorem, we have the following remarks 18 1) If k = and q > 3n + + ρn(n + 1) then q− 2nkq ≥ q − 2n > n + + ρn(n + 1) q + 2nk − 2k So the assumption of Theorem 3.2.1 is satisfied Hence, our result is valid for q > 3n + + ρn(n + 1) 2) If k = n then the condition (i) in Theorem 3.2.1 is alway true since the family of hyperplanes is in a general position Let q > (n + 1)2 + ρn(n + 1) It is easy to see that 2q • If n = then q > + 2ρ implies q ≥ and q − > + 2ρ q n−1 • If n ≥ and ρ < then 2(n + 1)n > q > (n + 1)2 + ρn(n + 1) n Thus, the assumption of Theorem 3.2.1 is satisfied Therefore, in the above cases our theorem implies Theorem D of Fujimoto 3) In the case M = Cm , by taking ρ = 0, we see that • If k = then q = (n + 1)k + n + satifies the conditions q ≥ 2(n + 1)k 2nkq and q − > n + q + 2nk − 2k • If k ≥ then q = (n + 1)k + n + satifies the conditions q < 2(n + 1)k and q > (n + 1)(k + 1) in Theorem 3.2.1 Hence, our result is also a generalization of the above conclusion of Giang-Quynh-Quang and the result of Z Chen and Q Yan as well 19 Chapter ALGEBRAIC DEPENDENCES OF MEROMORPHIC MAPPINGS SHARING FEW HYPERPLANES In Chapter four, we study algebraic dependences of three meromorphic mappings from Kăahler M , namely we generalize results in which meromorphic mappings are from Cm into Pn (C) By giving new methods basing on rearrange hyperplanes into groups and construct appropriate auxiliary lemmas, we succeeded in generalizing algebraically dependent theorems on Kăahler M Chapter four is written based on article [3] (in Works related to the thesis) 4.1 Second main theorem for meromorphic mapping from a ball and hyperplanes in subgeneral position In this section, we prove the second main theorem for meromorphic mapping from a ball into projective space intersecting hyperplanes in subgeneral position Proposition 4.1.1 Let H1 , , Hq (q > 2N − n + 1) be hyperplanes in Pn (C) located in N -subgeneral position Then there exists a function ω : {1, , q} → (0, 1] called a Nochka weight and a real number ω ˜ ≥ called a Nochka constant satisfying the following conditions: (i) If j ∈ {1, · · · , q}, then < ω(j)˜ ω ≤ q (ii) q − 2N + n − = ω ˜ ( j=1 ω(j) − n − 1) (iii) For R ⊂ {1, , q} with |R| = N + 1, then i∈R ω(i) ≤ n + −n+1 (iv) Nn ≤ ω ˜ ≤ 2Nn+1 (v) Given real numbers λ1 , , λq with λj ≥ for ≤ j ≤ q and given any R ⊂ {1, , q} and |R| = N + 1, there exists a subset R1 ⊂ R such that 20 |R1 | = rank {Hi }i∈R1 = n + and ω(j) ≤ λj λi i∈R1 j∈R We note that Lemma 4.1.2 and Lemma 4.1.3 generalize correspondingly Lemma 3.1.1 and Lemma 3.1.2 in Chapter three for case hyperplanes in general position Lemma 4.1.2 Let f be a linearly nondegenerate meromorphic mapping from B(R0 ) into Pn (C) and H1 , , Hq be q hyperplanes of Pn (C) in N − subgeneral position Set l0 = |α0 | + · · · + |αn | and take t, p with < tl0 < p < Then, for < r0 < R0 there exists a positive constant K such that for r0 < r < R < R0 , z α0 +···+αn Wα0 αn (f ) (f, H1 )ω(1) · · · (f, Hq )ω(q) t q j=1 f ω(j)−n−1 t σm R2m−1 p K Tf (R, r0 ) , R−r S(r) where ω(j) are Nochka weights with respect to Hj , j q Lemma 4.1.3 Let f be a linearly nondegenerate meromorphic mapping from B(R0 ) into Pn (C) and let H1 , H2 , , Hq be q hyperplanes of Pn (C) in N subgeneral position Then we have q [n] (q − 2N + n − 1)Tf (r, r0 ) N(f,Hj ) (r, r0 ) + Sf (r), j=1 where Sf (r) is evaluated as follows: + log+ Tf (r, r0 )) for all < R0 − r r0 < r < R0 outside a set E ⊂ [0, R0 ) with E Rdt < ∞, where K is a −t positive constant (i) In the case R0 < ∞, Sf (r) K(log+ (ii) In the case R0 = ∞, Sf (r) K(log+ Tf (r, r0 )+logr) for all < r0 < r < ∞ outside a set E ⊂ [0, ∞) with E dt < ∞, where K is a positive constant 4.2 Algebraic dependences of three meromorphic function sharing some hyperplanes Lemma 4.2.1 Let f , f , f be three meromorphic mappings from B(R0 ) into Pn (C) and H1 , H2 , , Hq are q hyperplanes in Pn (C) satifying 21 f = f = f on ∪qj=1 (f )−1 (Hj ) ∪ (f )−1 (Hj ) ∪ (f )−1 (Hj ) Suppose that there exist s, t, l ∈ {1, , q} such that   (f , Hs ) (f , Ht ) (f , Hl )   P := Det  (f , Hs ) (f , Ht ) (f , Hl )  ≡ (f , Hs ) (f , Ht ) (f , Hl ) Then we have q ( {ν(f u ,Hi ) (z)} − νP (z) i=s,t,l u [1] ν(f ,Hi ) (z)) [1] ν(f ,Hi ) (z), ∀z ∈ / S +2 i=1 Lemma 4.2.2 Let M , f , f , f and H1 , H2 , , Hq be as in Theorem 4.2.3 Let P be a holomorphic function on M and β be a positive real number such that q [n] ν(f u ,Hv ) (z) βνP (z) (4.1) u=1 v=1 for all z outside an analytic subset of codimension two If |P β | with some positive constants C and α, then q C( f f2 f )α 2N − n + + ρn(n + 1) + α Theorem 4.2.3 Let M be a complete and connected Kăahler manifold whose universal covering is biholomorphic to B(R0 ) ⊂ Cm , where < R0 ∞ Let f , f , f : M → Pn (C) be three linearly nondegenerate meromorphic mappings which satisfy the condition (Cρ ) for some nonnegative constant ρ and there are q hyperplanes H1 , H2 , , Hq of Pn (C) in N -subgeneral position such that dim f −1 (Hi ) ∩ f −1 (Hj ) m − (1 i 2N − n + + ρn(n + 1) + (i) There exist i (ii) f ∧ f ∧ f ≡ on M 22 Theorem 4.2.4 Let M , f , f , f and H1 , , Hq be as in Theorem 4.2.3 Assume that the following assertions are satisfied: (a) min{ν(f ,Hi ) , 1} = min{ν(f ,Hi ) , 1} = min{ν(f ,Hi ) , 1} (1 (b) f = f = f on i q), q −1 i=1 (f ) (Hi ) 3nq , then f ∧ f ∧ f ≡ on M It 2q + 2n − 2 implies that three mappings f , f and f are algebraically dependent on M If q > 2N − n + + ρn(n + 1) + Besides these above lemmas, in order to prove Theorem 4.2.3 (Theorem 4.2.4) we need to prove Lemma 4.2.5 (Lemma 4.2.7) Assertions in these lemma are important to rearrange hyperplanes into groups and they helps us have neccesary transformation in the main theorems This is the foundation for the new technique that the thesis introduced in the previous parts in order to solve dificulties in algebraically dependent problems for meromorphic mappings from Kăahler manifolds into projective spaces Lemma 4.2.5 Let q, N be two integers satisfying q ≥ 3(N2+3) , N and q mod = Let {a1 , , aq } be a family of vectors in a 3-dimensional vector space such that rank {aj }j∈R = for any subset R ⊂ Q = {1, , q} with |R| = N + For each fixed i0 ∈ Q, put Ri0 = {j ∈ Q : aj ∧ ai0 = 0} q/3 q If |Ri0 | j=1 Ij of {1, , q} for all i0 ∈ Q, then there exists a partition satisfying |Ij | = and rank {ai }i∈Ij = for all j = 1, , q/3 Theorem 4.2.6 Let M , f , f , f and H1 , , Hq be as in Theorem 4.2.3 Let n and p n be a positive integer Assume that the following assertions are satisfied: (a) min{ν(f ,Hi ) , p} = min{ν(f ,Hi ) , p} = min{ν(f ,Hi ) , p} (1 (b) f = f = f on i q), q −1 i=1 (f ) (Hi ) q(2n + p) then the map f × f × f of M 2q − + 3p n n n into P (C) × P (C) × P (C) is linearly degenerate If q > 2N − n + + ρn(n + 1) + Lemma 4.2.7 Let q, N be two integers satisfying q 2N + 2, N and q be even Let {a1 , a2 , , aq } be a family of vectors in a 3-dimensional vector space such that rank {aj }j∈R = for any subset R ⊂ Q = {1, , q} with cardinality q/2 |R| = N + Then there exists a partition j=1 Ij of {1, , q} satisfying |Ij | = and rank {ai }i∈Ij = for all j = 1, , q/2 23 CONCLUSION AND RECOMMENDATIONS Conclusions The main results of the thesis: • Proving non-integrate defect relation theorems for meromorphic mappings from Kăahler manifolds into projective spaces as well as projective varieties intersecting hypersurfaces in subgeneral position • Proving uniqueness theorem for meromorphic mappings from Kăahler manifolds into projective spaces intersecting hyperplanes in general position • Proving algebraic dependent theorems for three meromorphic mappings from Kăahler manifolds into projective spaces intersecting hyperplanes in subgeneral position Recommendations for further research We continue to investigate the following problems • Studying uniqueness problems for meromorphic mappings from Kăahler manifolds into projective spaces or projective varieties intersecting hypersurfaces in general position or subgeneral position • Studying algebraic dependent problems for meromorphic mappings from Kăahler manifolds into projective spaces or projective varieties intersecting hypersurfaces in general position or subgeneral position • Studying problems in value distribution theory for meromorphic mappings from more general Kăahler manifolds rather than Kăahler manifolds like one in the thesis in which the universal covering is biholomorphic to a ball in Cm 24 ... position, submitting INTRODUCTION Rationale Nevanlinna theory begins with the study on the value distribution of meromorphic functions In 1926, R Nevanlinna extended the classical little Picard’s... + 1)3 I( −1 )d L0 +n n 1) + + In the above theorem, letting = + we obtain the following corollary with ≤ en+2 (dp(n + 1)2 I( > and then letting −1 ))n −→ 0, Corollary 2.2.5 With the assumption... position, we get N = n Moreover, since (n−k+1)(k+1) ≤ ( n2 +1)2 for every ≤ k ≤ n, letting ε = 1+ε with ε > and then letting ε −→ from the above theorem, we obtain the following corollary for the case