Quasi-positive orbifold cotangent bundles ; Pushing further an example by Junjiro Noguchi
aa r X i v : . [ m a t h . AG ] A ug QUASI-POSITIVE ORBIFOLD COTANGENT BUNDLES ;PUSHING FURTHER AN EXAMPLE BY JUNJIRO NOGUCHI. by Lionel Darondeau & Erwan Rousseau
Abstract . —
In this work, we investigate the positivity of logarithmic and orbifold cotan-gent bundles along hyperplane arrangements in projective spaces. We show that a veryinteresting example given by Noguchi (as early as in 1986) can be pushed further to a verygreat extent. Key ingredients of our approach are the use of Fermat covers and the produc-tion of explicit global symmetric di ff erentials. This allows us to obtain some new resultsin the vein of several classical results of the literature on hyperplane arrangements. Theseseem very natural using the modern point of view of augmented base loci, and working inCampana’s orbifold category. As an application of our results, we derive two new orbifoldhyperbolicity results, going beyond some classical results of value distribution theory.
0. IntroductionPositive and quasi-positive cotangent bundles. —
In recent years, families of varietieswith ample cotangent bundles have attracted a lot of attention (see e.g. [ Deb05, Xie18,BD18a, Den20, Moh18, CR20, Ete19 ]), and there have been significant progress in thisarea (even though finding an explicit surface with ample cotangent bundle in P is stilla tremendous challenge). With the development of our understanding, the enrichingof techniques, and in connection with hyperbolicity problems, some variations of thisproblem have started to emerge. For instance, in [ BD18b ], the authors have beeninterested in the determination of the augmented base locus of logarithmic cotangentbundles along normal crossing divisors in projective spaces. The stable base locus B ( L ) ⊆ X of a line bundle L on a projective variety X is defined as the intersection ofthe base loci of all multiples of L . Then, the augmentedbaselocus (or non-amplelocus) B + ( L ) ⊆ X is B + ( L ) ≔ \ q ∈ N B ( qL − A ) , for any ample line bundle A → X . The augmented base locus of a line bundle is ageometric measure of the positivity of its sheaf of global sections. In particular, it isdi ff erent from the base variety when the line bundle is big, and it is empty when theline bundle is ample. For vector bundles, one studies the augmented base locus of theSerre line bundle on their projectivizations. The idea of augmented base loci for vectorbundles can be traced back to [ Nog77 ], where it was already used in connection tohyperbolicity (see below).
Key words and phrases . —
Ampleness, symmetric di ff erential forms, orbifold cotangent bundles, hyper-plane arrangements, Fermat covers, value distribution theory.E. R. was partially supported by the ANR project “FOLIAGE”, ANR-16-CE40-0008. UASI-POSITIVE ORBIFOLD COTANGENT BUNDLES In various cases, one does not really need this augmented base locus to be empty inorder to obtain interesting geometric consequences, and in many interesting settings,such as logarithmic and orbifold setting, one actually cannot expect the augmented baselocus to be empty. This leads to the definition of several notions of positivity, whereone only ask for a certain geometric control of the non-ample locus. For example a linebundle L is said to be ample modulo a divisor D when B + ( L ) ⊆ D . Note that if L isample modulo D , then L is necessarily big.Denote by p : X ′ ≔ P ( Ω ( X , D )) → X the projectivized bundle of rank 1 quotients ofthe logarithmic cotangent bundle of a smooth logarithmic pair ( X , D ). In this work, wewill use the following definition. Definition 0.1 . — We say that the cotangent bundle of ( X , D ) is amplemoduloboundaryif p (cid:16) B + ( O X ′ (1)) (cid:17) ⊆ D . It is a weaker positivity property that the one introduced in [
BD18b ]. Consider thevarious residue exact sequences coming with a simple normal crossing divisor in P n .One gets a lot of trivial quotients supported on the boundary components. Then, theprojectivizations of these trivial quotients give subvarieties in the projectivized logarith-mic cotangent bundle, that constitute obstructions to the ampleness of the logarithmiccotangent bundle (see [ BD18b , Sect. 2.3]). In particular, one has always D ⊆ p (cid:16) B + ( O X ′ (1)) (cid:17) . One can hence view Definition 0.1 as asking the projection of the augmented baselocus to be minimal. Brotbek and Deng define Ω ( X , D ) to be “almost ample” whenthe augmented base locus B + ( O X ′ (1)) itself (and not its projection) is minimal. Thismeans that the augmented base locus corresponds exactly to the trivial quotients ofthe cotangent bundle given by the residue short exact sequence. Then, one has thefollowing ([ BD18b , Theo. A]).
Theorem 0.2 (Brotbek–Deng) . —
Let Y be a smooth projective variety of dimension n, witha very ample line bundle H → Y. For c > n and d > (4 n ) n + , the logarithmic cotangent bundlealong the sum D = D + · · · + D c of c general hypersurfaces D , . . . , D c ∈ | H d | is “almost ample”. This result is optimal concerning the number of components of the boundary divisor([
BD18b , Prop. 4.1]).
Proposition 0.3 (Brotbek–Deng) . —
The logarithmic cotangent bundle along a simple nor-mal crossing divisor with c < n irreducible components in P n is never big. The e ff ective degree bounds in [ BD18b ] being quite large, it is a natural question toask what would be the optimal degree bound (when one relaxes the condition on thenumber of components) ? An associated problem is to find some low degree examplesof pairs with ample cotangent bundles modulo boundary.To the best of our knowledge, before [
BD18b ], the only example of such quasi-positivity of the cotangent bundle is due to Noguchi [
Nog86 ]. It is an example givenas early as in 1986, in the paper in which he defined logarithmic jet bundles. Noguchiintroduced the following positivity property.
Definition 0.4 . — Let ( X , D ) be a smooth logarithmic pair. Denote V ≔ X \ D . A vectorbundle E on X is said “quasi-negative” over V if there is a proper morphism ϕ : E → C N to an a ffi ne space, such that ϕ is an isomorphism from E | V \ O to ϕ ( E ) \ ϕ ( E | D ), where O denotes the zero section.Then, one has the following ([ Nog86 ]).
UASI-POSITIVE ORBIFOLD COTANGENT BUNDLES Theorem 0.5 (Noguchi) . —
The logarithmic tangent bundle along a general arrangement A of lines in P is “quasi-negative” over P \ A . The rough idea of the proof is that using an explicit basis of the logarithmic cotan-gent sheaf along an arrangement of lines in general position, one is able to constructan immersive Kodaira map (under some further explicit genericity condition). Somecombinatorial work allows one to identify this supplementary genericity condition asasking that in the dual projective space parametrizing hyperplanes, the points of thearrangement do not all lie in a single quadric.Now, one has:
Lemma 0.6 . —
Let L be a globally generated line bundle. If | L | defines an immersive map onX \ V, then B + ( L ) ⊆ V.Proof . — According to [
BCL14 , Theo. A], the augmented base locus B + ( L ) is the smallestclosed subset V of X such that the linear system | qL | defines an isomorphism of X \ V onto its image for su ffi ciently large q .For q large enough, the Stein factorization of φ | L | is given by X | qL | −→ φ | qL | ( X ) ν q −→ φ | L | ( X ) , for some finite morphism ν q ([ Laz04 , Lemma 2.1.28]). Now, since | L | defines an immer-sive map on X \ V , the fibers of φ | L | are discrete. An immediate consequence is that onthis set φ | qL | has discrete and connected fibers. In other words, for su ffi ciently large q the linear system | qL | defines an isomorphism of X \ V onto its image. (cid:3) This lemma allows us to reformulate the result of Noguchi as follows.
Theorem 0.7 (Noguchi) . —
The logarithmic cotangent bundle along an arrangement A ofd > lines in P in general position with respect to hyperplanes and to quadrics is amplemodulo A . As mentioned above, in this smooth logarithmic setting, one cannot expect the orb-ifold cotangent bundle to be plainly ample, and we have explained that amplenessmodulo boundary is somehow optimal. Concerning the optimal number of lines, com-bining Noguchi’s result with Theorem 0.11 below, we now that it can only be 5 or 6.It is not clear yet how to prove that for 5 lines one cannot expect ampleness moduloboundary of the logarithmic cotangent bundle.
Hyperbolicity of complements of hypersurfaces. —
A very connected research areais the one of complex hyperbolicity. Indeed, the following result is now classical(see [
Nog77 ] for the compact case). Given a logarithmic symmetric di ff erential form ω on a smooth logarithmic pair ( X , D ), which vanishes on an ample divisor, all entiremaps f : C → X \ D lands in the zero locus of ω . In other words, f ( C ) ⊆ p ( B + ( O X ′ (1))).If Ω ( X , D ) is ample modulo boundary, one immediately gets that all these curves areconstant. One says that the pair ( X , D ) is Brodyhyperbolic. We see that here, there is noneed to have global ampleness in order to obtain interesting geometric applications.It is thus an interesting companion question to ask about the hyperbolicity of comple-ments of hypersurfaces. Concerning this question, a very interesting setting seems tobe the classical setting of hyperplane arrangements, for which optimal degree boundsare reached.To sum up some classical results of value distribution theory: in the case of hyperplanearrangements, the (conjectural) optimal degree bounds are reached. Conjecture 0.8 (Kobayashi) . —
The complement of a general high degree hypersurface in P n is Brody-hyperbolic. UASI-POSITIVE ORBIFOLD COTANGENT BUNDLES Theorem 0.9 (Zaidenberg [ Zai87, Zai93 ] ) . — For a general hypersurface D in P n of degree n, there is a line in P n meeting D in at most two points. Theorem 0.10 (Bloch, Cartan, Green [ Gre72 ] ) . — The complement of an arrangement of n + hyperplanes in general position in P n is Brody-hyperbolic. Theorem 0.11 (Snurnitsyn [ Snu86, Zai93 ] ) . — For any arrangement A of n hyperplanesin P n , there is a line in P n meeting A in only two points. And these results have also their counterparts concerning weak hyperbolicity.
Conjecture 0.12 (Green–Gri ffi ths–Lang) . — On a logarithmic pair ( X , D ) of logarithmicgeneral type, there is a proper subvariety Exc( X ) ( X containing the images of all non-constantentire maps f : C → X \ D. Theorem 0.13 (Borel,Green [ Gre72 ] ) . — The maps f : C → P n \ A with values in thecomplement of an arrangement of n + hyperplanes are linearly degenerate. Here, the condition d > n + Kob98 ] or [
NW14 ]). One ofthe key tools is so-called Cartan’s Second Main Theorem, which allows one to studynot only entire curves in complements but also entire curves intersecting the boundarydivisor with prescribed multiplicities (see e.g. [
Kob98 , Coro 3.B.46]). A complementarymodern point of view on these orbifold curves is also given by the theory of Cam-pana’s orbifolds [
Cam04 ]. An alternative approach to Nevanlinna theory for orbifoldhyperbolicity is developed in [
CDR20 ]. We will pursue these ideas here, studying theaugmented base loci of orbifold cotangent bundles along hyperplane arrangements. Inthis direction, to the best of our knowledge, there are no existing results in the literaturebefore this work.
Main results of the paper. —
The common thread of this work is to push furtherTheorem 0.7. We obtain three main new results in this direction (Theorems A, B, C).Then, we derive two new hyperbolicity results (Theorems D and E).We generalize the result of Noguchi to higher dimensions. We prove the following.
Theorem A . —
The logarithmic cotangent bundle along an arrangement A of d > (cid:0) n + (cid:1) hyperplanes in P n in general position with respect to hyperplanes and to quadrics is amplemodulo A . We extend the result of Noguchi to the geometric orbifold category introduced byCampana.
Theorem B . —
The orbifold cotangent bundle along an arrangement A of d > (cid:0) n + (cid:1) hyper-planes in P n in general position with respect to hyperplanes and to quadrics, with multiplicitiesm > n + , is ample modulo A . Theorem 0.7 amongs to n = UASI-POSITIVE ORBIFOLD COTANGENT BUNDLES Lastly, we prove the positivity of orbifold cotangent bundles in all dimensions withlow degrees and very low multiplicities.
Theorem C . —
For n > , the orbifold cotangent bundle along an arrangement A of d > n ( nm − + hyperplanes in P n with multiplicity m > is big. Theorem C is weaker concerning positivity of the cotangent bundle but is spectacularconcerning multiplicities. Remark also that taking m linear in n , one gets a linear lowerbound on the degree.Next, we derive two hyperbolicity results from Theorem B (and from its reformulationin terms of Fermat covers). The first result is in the vein of several classical results inthe literature on Fermat covers (see [ Kob98, Dem97 ]).
Theorem D . —
The Fermat cover associated to an arrangement A of d > (cid:0) n + (cid:1) hyperplanes in P n in general position with respect to hyperplanes and to quadrics, with ramification m > n + ,is Kobayashi-hyperbolic. The second result could be seen as a strong hyperbolicity counterpart of the classical(weak) hyperbolicity results derived from Cartan’s Second Main Theorem.
Theorem E . —
Consider an arrangement A of d hyperplanes H , . . . , H d in P n in generalposition with respect to hyperplanes and to quadrics, with respective orbifold multiplicities m i ,and the associated orbifold divisor ∆ ≔ P di = (1 − / m i ) · H i . If d > (cid:0) n + (cid:1) and m i > n + , theorbifold pair ( P n , ∆ ) is Kobayashi-hyperbolic. It is noteworthy that, even if both results are in the same flavor, Theorem E is not aconsequence of Theorem D. To the best of our knowledge, both hyperbolicity resultsare new.
Organization of the paper. —
The paper is organized as follows. In §1, we generalizethe result of Noguchi to higher dimensions and prove Theorem A, using an explicitcohomological method, in the spirit of the original approach by Noguchi.In §2, we introduce precise definitions for various notions of positivity of orbifoldcotangent bundles.In §3, we extend the result of Noguchi to the orbifold category introduced by Campanaand prove Theorem B, using a quite di ff erent explicit cohomological method. Werephrase the approach of explicit ˇCech cohomology on complete intersections by Brotbekin the context of what we call Fermat covers. Computations would tend to be quicklyintractable when dimension grow. However, we are able to use the assumption ofgeneral type with respect to quadrics brought out in the study of the logarithmic casein order to tame a little the computations and find a quick way to the proof.In §4, we investigate the existence of orbifold symmetric forms for low multiplicitiesand prove Theorem C, using a non-explicit cohomological method. We derive thesought result from works by Brotbek and by Coskun–Riedl, using again Fermat covers.In §5, we focus on hyperbolicity questions and we prove Theorems D and E, buildingon the results of Sect. 3.
1. Ampleness modulo boundary of the logarithmic cotangent bundle
This section is devoted to prove the following generalization of Noguchi’s example.
Theorem A . —
The logarithmic cotangent bundle along an arrangement A of d > (cid:0) n + (cid:1) hyperplanes in P n in general position with respect to hyperplanes and to quadrics is amplemodulo A . UASI-POSITIVE ORBIFOLD COTANGENT BUNDLES Proof . — Consider an arrangement A of d = n + + k hyperplanes H , . . . , H n + k ingeneral linear position. Choose homogeneous coordinates Z , . . . , Z n of P n in such waythat H , . . . , H n are given by the equations Z i =
0, and that H n + j is given by the equation a j Z + a j Z + · · · + a jn Z n = , for some complex coe ffi cients a ji , for j = , . . . , k .In the dual projective space parametrizing hyperplanes, consider the coordinatepoints parametrizing H , . . . , H n and the points ( a j , . . . , a jn ) paramatrizing H n + , . . . , H n + k .The arrangement A is in general position with respect to hyperplanes if ( n +
1) of thesepoints never lie in a single hyperplane, and A is in general position with respect toquadrics if (cid:0) n + (cid:1) of these points never lie in a quadric. Recall that (cid:0) n + (cid:1) − P n determine a unique quadric in the dual projective spaceparametrizing hyperplanes.Very concretely, in our setting, the arrangement A is in general position with respectto hyperplanes when the minors (of any size) of the ( n + × k coe ffi cient matrix A ≔ hh a ji ii i n j k are non-zero. Moreover, for k > (cid:0) n + (cid:1) , the arrangement A is in general position withrespect to quadrics if all the maximal minors of the (cid:0) n + (cid:1) × ( n + + k ) matrix of all degree 2monomials in the equation coe ffi cients are non-zero. Putting the squares in first position,and taking the coordinate points as the first n + (cid:0) n + (cid:1) × k matrix of products a ji a ji (in lexicographic order) A [2] ≔ hh a ji a ji ii i < i n j k are non-zero. We will use this fact at the end of the proof.Outside of A , one can work on the a ffi ne chart Z ,
0. The equations of the k lasthyperplanes become a j + a j z + · · · + a jn z n = , in the inhomogeneous coordinates z j ≔ Z j / Z . Then a local frame of the logarithmictangent sheaf Ω ∨ ( P n , A ) around the origin in U is given by z ∂∂ z , . . . , z n ∂∂ z n , and if wedenote (for obvious reason) z n + j ≔ a j + a j z + · · · + a jn z n , a basis of the space of global sections H (cid:16) P n , Ω ( P n , A ) (cid:17) is given byd z z , . . . , d z n z n , d z n + z n + , . . . , d z n + k z n + k . The Kodaira map associated to |O P ( Ω ( P n , A )) (1) | , maps a point( z , [ ξ ]) = ( z , . . . , z n ; [ V z ∂/∂ z + · · · + V n z n ∂/∂ z n ]) ∈ P ( Ω ( P n , A )) , to the point ϕ ( z , [ ξ ]) ≔ [ V : . . . : V n : ϕ ( z , V ) : . . . : ϕ k ( z , V )] ∈ P n + k − , where: ϕ j ( z , V ) ≔ a j V z + · · · + a jn V n z n a j + a j z + · · · + a jn z n . We will prove that under the assumptions of the theorem, ϕ gives an immersion.Then, we obtain the result by Lemma 0.6.The coordinates V i cannot be simultaneously zero. Regarding the symmetries of ϕ ,it is su ffi cient to prove that ϕ is immersive on one a ffi ne chart V i ,
0. Let us thus work
UASI-POSITIVE ORBIFOLD COTANGENT BUNDLES on the chart V ,
0, in a ffi ne coordinates v i = V i / V , and in the a ffi ne chart “ Z ,
0” in P n + k − . One has then: ϕ ( z , [ ξ ]) = ( v , . . . , v n , ϕ ( z , v ) , . . . , ϕ k ( z , v ))and ϕ j ( z , v ) ≔ a j z + a j v z + · · · + a jn v n z n a j + a j z + · · · + a jn z n . The Jacobian matrix of ϕ with respect to the coordinates ( z , v ) is the matrix: . . . ... · · · ... · · · . . . ∗ . . . ∗ ∂ϕ i /∂ z j ... · · · ... ∗ . . . ∗ . Its rank is thus n − + rank( J ), where J ≔ (cid:16) ∂ϕ i /∂ z j (cid:17) .Let us write by convention v = ∂ϕ j /∂ z i = a j a ji v i + a ji a j ( v i − v ) z + · · · + a ji a jn ( v i − v n ) z n ( a j + a j z + · · · + a jn z n ) . shows that this matrix can be written as a matrix product J = M · A [2] / ( z n + · · · z n + k ) . Herethe columns of M are M i = v i E i for i = , . . . , n and then M ( i , i ) = ( v i − v i )( z i E i − z i E i ),where E , . . . , E n is the canonical basis of C n for 1 i < i n (in lexicographic order).E.g. for n = M ≔ v v − v ) z ( v − v ) z v v − v ) z v − v ) z v v − v ) z ( v − v ) z Points where ϕ is not an embedding are those where rank( M · A [2] ) < n . We claim thatrank( M ) = n . If not, considering the first minor | M . . . M n | , one infers that at least one ofthe v i has to be 0. Assume thus that v p + , . . . , v n are zero but v , . . . , v p are not. Note that p >
1, since v =
1. The minor | M . . . M p M ( p , p + . . . M ( p , n ) | is then v · · · v p ( − v p z p ) n − p + which is not zero since z p ,
0. This is a contradiction. Since k > (cid:0) n + (cid:1) , the matrix A [2] hasmore columns than rows. By the general position assumption, it is of maximal row rank.Therefore rank( J ) = rank( M · A [2] ) = rank( M ) = n . This ends the proof of Theorem A. (cid:3) Remark 1.1 . — Observe that for 6 lines in P , we retrieve the generic condition broughtout by Noguchi, by elementary linear algebra manipulations (in [ Nog86 ]’s convention, a = a = a = a = a = Remark 1.2 . — We do not really need the general position assumption for d > (cid:0) n + (cid:1) , butwe only need that at least (cid:0) n + (cid:1) of the d hyperplanes satisfy it. Question . — For the critical degree d = (cid:0) n + (cid:1) , is there an obstruction to positivity oflogarithmic cotangent bundles if all hyperplanes lie in a single quadric ? UASI-POSITIVE ORBIFOLD COTANGENT BUNDLES
2. Positivity of orbifold cotangent bundles2.1. Campana’s orbifold category. —
Before proceeding to the proof, let us first makesome recall.A smooth orbifold pair is a pair ( X , ∆ ), where X is a smooth projective variety andwhere ∆ is a Q -divisor on X with only normal crossings and with coe ffi cients between0 and 1. In analogy with ramification divisors, it is very natural to write ∆ = X i ∈ I (1 − / m i ) ∆ i , with multiplicities m i = a i / b i in Q > ∪ { + ∞} . If b i =
0, by convention a i =
1. The multi-plicity 1 corresponds to empty boundary divisors. The multiplicity + ∞ corresponds toreduced boundary divisors. We denote | ∆ | ≔ P i ∈ I ∆ i (it could be slightly larger than thesupport of ∆ because of possible multiplicities 1).Such pairs ( X , ∆ ) are studied using their orbifold cotangent bundles ([ CP15 ]). Fol-lowing the presentation used notably in [
Cla15 ], it is natural to define these bundles oncertain Galois coverings, the ramification of which is partially supported on ∆ . A Ga-lois covering π : Y → X from a smooth projective (connected) variety Y will be termedadapted for the pair ( X , ∆ ) if– for any component ∆ i of | ∆ | , π ∗ ∆ i = p i D i , where p i is an integer multiple of a i and D i is a simple normal crossing divisor;– the support of π ∗ ∆ + Ram( π ) has only normal crossings, and the support of thebranch locus of π has only normal crossings.There always exists such an adapted covering ([ Laz04 , Prop. 4.1.12]).Let π : Y → X be a ∆ -adapted covering. For any point y ∈ Y , there exists an openneighbourhood U ∋ y invariant under the isotropy group of y in Aut( π ), equipped withcentered coordinates w i such that π ( U ) has coordinates z i centered in π ( y ) and π ( w , . . . , w n ) = ( z p , . . . , z p n n ) , where p i is an integer multiple of the coe ffi cient a i of ( z i = z i is not involved in the local definition of ∆ then a i = b i = ∆ = | ∆ | ), for any ∆ -adapted covering π : Y → X , wedenote Ω ( π, ∆ ) ≔ π ∗ Ω X (log ∆ ) . For arbitrary multiplicities, the orbifold cotangent bundle is defined to be the vectorbundle Ω ( π, ∆ ) fitting in the following short exact sequence:(1) 0 → Ω ( π, ∆ ) ֒ → Ω ( π, | ∆ | ) res −→ M i ∈ I : m i < ∞ O π ∗ ∆ i / m i → . Here the quotient is the composition of the pullback of the residue map π ∗ res : π ∗ Ω X (log | ∆ | ) → M i ∈ I : m i < ∞ O π ∗ ∆ i with the quotients O π ∗ ∆ i ։ O π ∗ ∆ i / m i ([ Cla15 , loc. cit. ]).Alternatively, the sheaf of orbifold di ff erential forms adapted to π : Y → ( X , ∆ ) is thesubsheaf Ω ( π, ∆ ) ⊆ Ω ( π, | ∆ | ) locally generated (in coordinates as above) by the elements w p i / m i i π ∗ (d z i / z i ) = w − p i (1 − / m i ) i π ∗ (d z i ) . Note that if the multiplicities m i ’s are integers and if the cover π is strictlyadapted (i.e. p i = m i ), then Ω ( π, ∆ ) identifies with Ω Y via the di ff erential map of π . UASI-POSITIVE ORBIFOLD COTANGENT BUNDLES The direct image of the sheaf of Aut( π )-invariant sectionsof S N Ω ( π, ∆ ) S [ N ] Ω ( X , ∆ ) ≔ π ∗ (( S N Ω ( π, ∆ ))) Aut( π ) ⊆ S N Ω X (log | ∆ | ) , is a subsheaf of logarithmic symmetric di ff erentials which does not depend on thechoice of π . Note that in almost all situations S [ N ] Ω ( X , ∆ ) , S N Ω ( X , ∆ ). The sheaves S [ N ] Ω ( X , ∆ ) are independently defined and cannot be seen as symmetric powers. Onehas merely a morphism S p S [ N ] Ω ( X , ∆ ) → S [ pN ] Ω ( X , ∆ ) given by multiplication. However,the philosophy in the framework of Campana’s orbifolds is to study orbifold pairsthrough adapted covers, and we will.We would like to relate positivity properties of the orbifold cotangent bundle withsome positivity properties of Ω ( π, ∆ ), for some adapted cover π . The definition forbigness is quite clear. Definition 2.1 . — We say that ( X , ∆ ) has a big cotangent bundle if Ω ( π, ∆ ) is big forsome (hence for all) adapted cover π . Equivalently, the orbifold cotangent bundle of thepair ( X , ∆ ) is big if for some / any ample integral divisor A ⊆ X , there exists an integer N such that H ( X , S [ N ] Ω ( X , ∆ ) ⊗ A ∨ ) , { } .To define ampleness, we will use augmented base loci, or rather their natural pro-jections. In the spirit of [ MU19 ], in which augmented base loci of vector bundles arestudied, we define the orbifold augmented base locus of the cotangent bundle to the pair( X , ∆ ), as follows. Recall that the base locus of a vector bundle E is defined in [ MU19 ] asBs( E ) ≔ n x ∈ X . H ( X , E ) → E x is not surjective o . Definition 2.2 . — The orbifoldaugmentedbaselocus of Ω ( X , ∆ ) is B + ( Ω ( X , ∆ )) ≔ \ p / q ∈ Q \ N > Bs( S [ Nq ] Ω ( X , ∆ ) ⊗ ( A ∨ ) ⊗ Np ) , for an integral ample divisor A → X .Before proceeding to the definition, observe first the following. For an adapted cover π : Y → ( X , ∆ ), we use the notation Y ′ ≔ P ( Ω ( π, ∆ )) → Y . Proposition 2.3 . —
Over X \ | ∆ | , the image of the augmented base locus B + ( O Y ′ (1)) by thenatural projection Y ′ ։ Y ։ X does not depend on π . Indeed, it actually coincides with theorbifold augmented base locus B + ( Ω ( X , ∆ )) | X \| ∆ | .Proof . — (1) We claim that for any adapted cover π , in order to compute the augmentedbase locus of Ω ( π, ∆ ), it is su ffi cient to consider Aut( π )-invariant sections.Notice first that because of the relative ampleness of O Y ′ (1), one can assume thatthe ample line bundle in the definition of B + ( O Y ′ (1)) is the pull-back of an ample linebundle on X , and in particular is invariant under Aut( π ). Then observe that B + ( O Y ′ (1))is Aut( π )-invariant. Indeed, for any global section σ and for any element of the Galoisgroup α , the Galois transform σ α is also a global section. We deduce that for each orbitof Aut( π ), either all points are in the augmented base locus, or none.Let B G + ( O Y ′ (1)) denote the base locus obtained by considering only Aut( π )-invariantsections. If v ∈ B + ( O Y ′ (1)), then obviously v ∈ B G + ( O Y ′ (1)). Conversely, consider v < B + ( O Y ′ (1)). By the preceding considerations, the (finite) orbit of v stays outside B + ( O Y ′ (1)). By Noetherianity, B + ( O Y ′ (1)) can be realized as a single base locus. One canthen find a divisor in the associated linear system that avoids all the points in the orbitof v . In other words, one can find a global section σ which does not vanish at any pointof the orbit of v . Moreover, this section can be made invariant after multiplication by itsGalois conjugates. To conclude, B + ( O Y ′ (1)) = B G + ( O Y ′ (1)). UASI-POSITIVE ORBIFOLD COTANGENT BUNDLES (2) Now remark that there is a natural morphism π ∗ S [ N ] Ω ( X , ∆ ) → S N ( Ω ( π, ∆ )) which isan injection of sheaves and an isomorphism outside | ∆ | . Combining with the precedingequality of base loci, one obtains that B + ( O Y ′ (1)) has a projection on X \| ∆ | which dependsonly on the sheaves S [ N ] Ω ( X , ∆ ). Namely (reasoning as in [ MU19 ]), it is the restrictionof B + ( Ω ( X , ∆ )). (cid:3) There are many interesting situations where one cannot expect global ampleness of Ω ( π, ∆ ) but where bigness is not su ffi cient for applications (see below). Therefore, weshall introduce an intermediate positivity property. Definition 2.4 . — We say that ( X , ∆ ) has an amplecotangentbundlemoduloboundaryif its orbifold augmented base locus is contained in the boundary.Equivalently, ( X , ∆ ) has an ample cotangent bundle modulo boundary, if for some(hence for all) adapted cover π , the orbifold cotangent bundle Ω ( π, ∆ ) is ample modulothe Aut( π )-invariant closed subset living over the boundary. This definition will beused in practice. Remark 2.5 . — As a consequence of Proposition 2.3, the “ampleness modulo boundary”of Ω ( π, ∆ ) does not depend on π . Ampleness of orbifold cotangent bundles has beenrecently studied in the PhD thesis of Tanuj Gomez where it is shown by a di ff erentmethod that for strictly adapted covers ramifying exactly on | ∆ | , the (global) amplenessof Ω ( π, ∆ ) does not depend on the cover. It would be interesting to check to whichextent ampleness of Ω ( π, ∆ ) is equivalent to the triviality of H q ( X , S [ N ] Ω ( X , ∆ ) ⊗ A ⊗ p ), forsome A ample, any p , q >
0, and N large enough. Remark 2.6 . — In general, one cannot expect that there exists a strictly adapted cov-ering ramifying exactly over the boundary divisor. But if π : Y → ( X , ∆ ) is a strictlyadapted cover ramifying exactly over ∆ , a convenient way to prove that the orbifoldcotangent bundle Ω ( X , ∆ ) is ample modulo boundary is to prove that the orbifold cotan-gent bundle Ω ( π, ∆ ) ≃ Ω Y is ample modulo its ramification locus and it is actuallyequivalent. Positivity of cotangent bundles of projec-tive manifolds or log-cotangent bundles of pairs has been investigated by many authors(see e.g. [ Deb05, Xie18, BD18a, Den20, CR20, Ete19, Nog86, BD18b ]). In the orbifoldsetting, much less seems to be known. Nevertheless, results of [
Som84 ] can be inter-preted as the study of ampleness of orbifold cotangent bundles associated to orbifolds( P , ∆ ) corresponding to arrangements of lines in P . In particular, [ Som84 , Theo. 4.1]characterizes exactly which arrangements have ample orbifold cotangent bundles. Aninteresting consequence of this result is that when the orbifold ( P , ∆ ) is smooth (i.e.when the lines are in general position), the orbifold cotangent bundle is never ample.This is due to the following fact. Let C be any irreducible component of π − ( | ∆ | ) then Ω Y | C (cid:27) Ω C ⊕ N ∗ C ([ Som84 , p. 217]), and deg N ∗ C = − C
0. In other words, eachcomponent of the boundary carries a negative quotient.This can be generalized as follows.
Lemma 2.7 . —
Let ( P n , ∆ ) be a smooth orbifold pair with integer (or infinite) coe ffi cients.Then, for any strictly adapted covering π the cotangent bundle Ω ( π, ∆ ) has negative quotientssupported on each boundary component with finite multiplicity, and trivial quotients supportedon each boundary component with infinite multiplicity.Proof . — Let ∆ = (1 − / m ) ∆ + ∆ ′ , where the multiplicity of ∆ in ∆ ′ is zero.If m = ∞ , the residue exact sequence Ω ( π, | ∆ ′ | ) ֒ → Ω ( π, | ∆ | ) ։ O π ∗ ∆ restricts to Ω ( π, ∆ ′ ) ֒ → Ω ( π, ∆ ) ։ O π ∗ ∆ . UASI-POSITIVE ORBIFOLD COTANGENT BUNDLES We get the sought trivial quotient on | π ∗ ∆ | If m < ∞ , let D ≔ π ∗ ∆ / m . Note that this is a reduced divisor. By (1), one has: Ω ( π, ∆ ) ֒ → Ω ( π, | ∆ | ) ։ O D ⊕ M i ∈ I : m i < ∞ O π ∗ ∆ ′ i / m i , and Ω ( π, ∆ ′ ) ֒ → Ω ( π, | ∆ | ) ։ O m D ⊕ M i ∈ I : m i < ∞ O π ∗ ∆ ′ i / m i . One infers Ω ( π, ∆ ′ ) ֒ → Ω ( π, ∆ ) ։ O m D (cid:30) O D . Let I denote the ideal sheaf of D in Y . The quotient above is isomorphic to I (cid:30) I m .Composing with the quotient I / I m ։ I / I ≃ N ∗ D , we deduce that Ω ( π, ∆ ) has anegative quotient supported on | D | (and namely the conormal bundle of D ). (cid:3) Therefore, starting with smooth orbifold pairs associated to hyperplane arrangementsin projective spaces, the best one can hope for is ampleness modulo the boundary.
3. Ampleness modulo boundary of the orbifold cotangent bundle
This section is devoted to prove the following extension of Theorem A.
Theorem B . —
The orbifold cotangent bundle along an arrangement A of d > (cid:0) n + (cid:1) hyper-planes in P n in general position with respect to hyperplanes and to quadrics, with multiplicitiesm > n + , is ample modulo A . We keep the setting and notation of Sect. 1.
Remark 3.1 . — One can accept di ff erent multiplicities for the hyperplanes. Indeed,lowering all multiplicities to the lowest one (still assumed at least 2 n + Considering the k linear relations between the hyperplanes: H n + j = a j H + · · · + a jn H n , we identify the projective space P n with the linear subspace of P N ≔ P n + k cut out bythe k linear equations Z n + j = a j Z + · · · + a jn Z n . in homogeneous coordinates Z , . . . , Z N , for N ≔ n + k . We also define the completeintersection Y in P N of the k Fermat hypersurfaces Z mn + j = a j Z m + · · · + a jn Z mn . The map π : [ Z i ] [ Z mi ] realizes Y as a cover of P n ramifying exactly over the hyper-planes H i , with multiplicity m . In other words, Y is a (strictly) adapted cover of theorbifold pair ( P n , ∆ ), where ∆ = (1 − / m )( H + · · · + H N ). We call π : Y → ( P n , ∆ ) theFermatcover of ( P n , ∆ ).The cotangent bundle of the orbifold pair ( P n , ∆ ) is ample modulo boundary whenthe cotangent bundle of its Fermat cover is ample modulo its ramification locus.An obvious obstruction to ampleness of the cotangent bundle is the presence ofrational lines. The following remark gives another nice justification that we need totake at least 2 n + UASI-POSITIVE ORBIFOLD COTANGENT BUNDLES Remark 3.2 . — Recall that each Fermat hypersurface of degree m (without zero coef-ficient) in P n + contains a n − { , . . . , n + } in r subsetswith cardinalities i , . . . , i r >
2, there is a rational map P n + d P i − × · · · × P i r − , thefibers of which are linear subspaces P r − . Its restriction to the Fermat hypersurfaceyields a rational map onto a product of lower dimensional Fermat hypersurfaces (oftotal dimension n + − r ). Each fiber of this map contains a (2 r − Lemma 3.3 . —
There is no standard line in a generic complete intersection of k Fermat hyper-surfaces in P n + k i ff k > n.Proof . — Now, we consider a complete intersection of k > P n + k with generic coe ffi cients, and we consider only partitions of { , . . . , n + k } in subsetswith cardinalities at least 1 + k (otherwise the intersection of the complete intersectionwith the linear subspace would be generically empty). There is no nontrivial suchpartition as soon as n + + k < + k ), i.e. k > n . (cid:3) ff erentials on Fermat covers. — In [
Bro16 ], Brotbek hasdescribed a way to produce global twisted symmetric di ff erentials on complete inter-sections Y in P N . The following is a slight adaptation to the particular setting of Fermatcovers of [ Bro16 ] (see also [
Xie18, Dem20 ]) ; this could appear not so obvious due tosome redaction shortcuts. We could have made the proof (slightly) more heuristic withan approach involving a N × ( N +
1) matrix in the spirit of [
Bro16 ], but here we haveprefered compactness.
Lemma 3.4 . —
For any subset of pairwise distinct integers { j , . . . , j n } in { n + , . . . , n + k } ,there is a global section of S n Ω Y (2 n + − m ) given on Z , by: σ ≔ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a j − n ( z z ′ j − z ′ z j ) . . . a j − nn ( z n z ′ j − z ′ n z j ) ... ... a j n − n ( z z ′ j n − z ′ z j n ) . . . a j n − nn ( z n z ′ j n − z ′ n z j n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ⊗ Z n + − m , where z i ≔ Z i / Z denote the standard a ffi ne coordinates on the chart Z , .Proof . — We would like to underline that the most interesting part of the lemma is the“extra vanishing” of order m − Dem20 , Sect. 12D] foran analog construction for higher order jet di ff erentials).The proof relies on the following very basic fact of linear algebra. Consider a n × ( n + n × n minors areequal. This is more or less Cramer’s rule. Let us denote by det c the minor obtained byremoving column c from such a matrix.For any j = n + , . . . , n + k , one has: a j − n + a j − n z m + · · · + a j − nn z mn = z mj and a j − n z m − z ′ + · · · + a j − nn z m − n z ′ n = z m − j z ′ j . Therefore: a j − n ( z z ′ j − z ′ z j ) . . . a j − nn ( z n z ′ j − z ′ n z j ) ... ... a j n − n ( z z ′ j n − z ′ z j n ) . . . a j n − nn ( z n z ′ j n − z ′ n z j n ) z m − ... z m − n = . UASI-POSITIVE ORBIFOLD COTANGENT BUNDLES where we denote z ≔ z ′ ≔ σ = det a j − n ( z z ′ j − z ′ z j ) . . . a j − nn ( z n z ′ j − z ′ n z j ) ... ... a j n − n ( z z ′ j n − z ′ z j n ) . . . a j n − nn ( z n z ′ j n − z ′ n z j n ) ⊗ Z n + − m . One immediately infers some alternative expressions of σ on the intersections ( Z Z i , Z Z , instead of det (the same reasoning holds for Z , · · · , Z n ) : σ = − z m − z m − det a j − n ( z z ′ j − z ′ z j ) . . . a j − nn ( z n z ′ j − z ′ n z j ) ... ... a j n − n ( z z ′ j n − z ′ z j n ) . . . a j n − nn ( z n z ′ j n − z ′ n z j n ) ⊗ Z n + − m . Let y i = z i / z denote the standard a ffi ne coordinates on ( Z , z =
1. Weget the following expression for σ : σ = − det a j − n ( y y ′ j − y ′ y j ) . . . a j − nn ( y n y ′ j − y ′ n y j ) ... ... a j n − n ( y y ′ j n − y ′ y j n ) . . . a j n − nn ( y n y ′ j n − y ′ n y j n ) ⊗ Z n + − m . Here we use ( z i z ′ j − z ′ i z j ) = z ( y i y ′ j − y ′ i y j ) . (2) Let us now consider the intersection ( Z Z n + , Z n + , . . . , Z n + k . Here we have to enlarge the matrix by considering a z . . . a n z n a j − n ( z z ′ j − z ′ z j ) . . . a j − nn ( z n z ′ j − z ′ n z j ) 0 ... ... ... a j n − n ( z z ′ j n − z ′ z j n ) . . . a j n − nn ( z n z ′ j n − z ′ n z j n ) 0 z m − ... z m − n − z mn + = . One has σ = ( − n det a z . . . a n z n a j − n ( z z ′ j − z ′ z j ) . . . a j − nn ( z n z ′ j − z ′ n z j ) 0 ... ... ... a j n − n ( z z ′ j n − z ′ z j n ) . . . a j n − nn ( z n z ′ j n − z ′ n z j n ) 0 ⊗ Z n + − m . Using det n + instead of det , one gets the alternative expression: σ = − z m − z mn + det a j − n z . . . a j − nn z n a j − n ( z z ′ j − z ′ z j ) . . . a j − nn ( z n z ′ j − z ′ n z j ) ... ... a j n − n ( z z ′ j n − z ′ z j n ) . . . a j n − nn ( z n z ′ j n − z ′ n z j n ) ⊗ Z n + − m . UASI-POSITIVE ORBIFOLD COTANGENT BUNDLES Let y i = z i / z denote the standard a ffi ne coordinates on ( Z n + , σ = − det a j − n y . . . a j − nn y n a j − n ( y y ′ j − y ′ y j ) . . . a j − nn ( y n y ′ j − y ′ n y j ) ... ... a j n − n ( y y ′ j n − y ′ y j n ) . . . a j n − nn ( y n y ′ j n − y ′ n y j n ) ⊗ Z n + − mn + . This ends the proof. (cid:3)
Remark 3.5 . — Note that the zero locus of σ does not depend on m . However, one willneed m > n + ff erential vanishing on an ample divisor. Let V ⊂ Y be the Aut( π )-invariant open subset livingabove X \ | ∆ | . In other words V = ( Z · · · Z N , Theorem 3.6 . —
When m > n + , the projection of the augmented base locus of O P ( Ω Y ) (1) does not intersect the open V.Proof . — In this proof, we use repeatedly that we work on Z · · · Z N ,
0, and we willnot necessarily mention it anymore.Let us denote: B ≔ a ( z z ′ n + − z ′ z n + ) . . . a n ( z n z ′ n + − z ′ n z n + ) ... ... a k ( z z ′ n + k − z ′ z n + k ) . . . a kn ( z n z ′ n + k − z ′ n z n + k ) , where z , . . . , z N are the standard extrinsic a ffi ne coordinates on ( Z i ,
0) (for some i ∈ { , . . . , n } ), and where z ′ , . . . , z ′ N are the standard extrinsic homogeneous coordinateson P ( Ω Y ) ⊂ P ( Ω P N ). By convention z i = , z ′ i = B < n . Indeed, sincethe first column is a non-zero linear combination of the last n columns, it is equivalentto say that the rank of the n last column is less than n . But by the previous lemma, n × n -minors in the last n columns are global sections of S n Ω Y (2 n + − m ). Here, it isalso useful to notice that O (1) is relatively ample on P ( Ω Y ). Therefore, one can definethe augmented base locus of O (1) using the pullback of an ample line bundle on Y .(2) In the spirit of the proof in the logarithmic case, we will write B as a product involvingthe matrix A [2] . Denote b ji the coe ffi cients of the matrix B . For j = , . . . , k , using theequations of P ( Ω Y ), one has: z m − n + j b ji = z m − n + j a ji ( z i z ′ n + j − z ′ i z n + j ) = n X i = a ji a ji ( z i z ′ i − z ′ i z i ) . Therefore, B = diag(1 / z m − n + , . . . , / z m − n + k ) · A T [2] · W , where W is a (cid:0) n + (cid:1) × ( n + i , i ) of which is ( z i z ′ i − z ′ i z i )( E i − E i ) (denoting E , . . . , E n the canonical basis of C n + ). For example, for n = W = ( z z ′ − z ′ z ) ( z z ′ − z ′ z ) 0( z z ′ − z ′ z ) 0 ( z z ′ − z ′ z )0 ( z z ′ − z ′ z ) ( z z ′ − z ′ z ) . One infers that rank B = rank A T [2] W . Moreover, under the assumption that A [2] is fullrow rank (which also means that A T [2] is full column rank), one has rank( A T [2] W ) = rank W .Hence: rank B = rank W . UASI-POSITIVE ORBIFOLD COTANGENT BUNDLES (3) Now, we claim that W is of rank at least n , from which one deduces the result of thetheorem, by the first two points of the proof.Indeed, we will exhibit a non-zero n × n minor in W . We work with the standarda ffi ne coordinates on ( Z , w i , i for z i z ′ i − z ′ i z i . If z ′ = · · · = z ′ n =
0, using the equations of Ω Y , one would immediately get that all firstderivatives are simultaneously zero, which is not possible. Assume therefore that atleast one of these first derivatives is non zero, say z ′ = w , . For i = , . . . , n , onehas: z i w , = ( z w , i − z w , i ). As a consequence, at least one of w , i or w , i is non-zero.Let us call it w ⋆, i for convenience. Recall that the rows of W are indexed by couples( i < i ) ∈ { , . . . , n } . Consider the n × n minor made of columns 1 , . . . , n and of rows(0 , ⋆, , . . . , ( ⋆, n ). It is (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w , . . . . . . ∗ w ,⋆ . . . ...... . . . . . . ...... ... . . . . . . ∗ . . . w n ,⋆ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ( − n w , w ⋆, · · · w ⋆, n , . This proves our claim and therefore ends the proof. (cid:3)
Theorem B is then a plain corollary, because the Fermat cover π : Y → ( P n , ∆ ) is anadapted cover such that Ω ( π, ∆ ) ≃ Ω Y . Remark 3.7 . — One could deduce Theorem A from the same proof, but we have thefeeling that the natural role of the coe ffi cient matrix A [2] would be less highlighted inthis way. Remark 3.8 . — The proof above may look disappointingly simple, but it is actually thesynthesis of very hard computational explorations. The matrix A [2] , brought out by thelogarithmic case, is the key of the proof and it was a turning point when we were ableto involve it in the proof for n =
2. All barriers quickly came down after that. We invitethe reader to forget its existence for the fun and to try to find some genericity conditionfor A already in the cases n = n =
4. Bigness of the orbifold cotangent bundle with low multiplicities
We are not able yet to generalize the strategy of Noguchi to the full orbifold category.Indeed, it seems very di ffi cult to produce explicit global sections for low multiplicities ,even with a lot of components in the boundary. This is quite surprising in view ofTheorem C, that we recall below. Theorem C . —
For n > , the orbifold cotangent bundle along an arrangement A of d > n ( nm − + hyperplanes in P n with multiplicity m > is big. For n =
2, it was proved in [
CDR20 ] that for m > Ω ( P , ∆ ) is big if d >
11. Here we generalize this statement to higher dimension for anymultiplicity m > Remark 4.1 . — m = ffi cult. This is illustrated by thefollowing vanishing theorem proven in [ CDR20 ]. If D is a (reduced) smooth divisor(with an arbitrary large degree ) in P n and if m n , then there is no non-zero global orbifoldsymmetric di ff erential for the pair ( P n , (1 − / m ) D ). Actually, there is even no non-zeroglobal jet di ff erential of any order (higher jet order analogs of symmetric di ff erentials). UASI-POSITIVE ORBIFOLD COTANGENT BUNDLES Proof ot Theorem C . — The proof relies on a theorem by Brotbek [
Bro14 ] improved byCoskun and Riedl [
CR20 ] on cotangent bundles of complete intersections, and on ouruse of Fermat covers.We keep the setting and notation of previous sections. Consider the Fermat cover π : Y → ( P n , ∆ ) where ∆ is the orbifold divisor ∆ ≔ P n + ki = (1 − / m ) H i on P n . Showingthat Ω ( P n , ∆ ) is big is equivalent to showing that Ω ( π, ∆ ) ≃ Ω Y is big. In order to provethe bigness of Ω Y , we apply Theorem 2.7 in [ CR20 ] which gives that a smooth completeintersection of dimension n in P N and type ( d , . . . , d c ), with c > n , has big cotangentbundle if d i > n N − n + + . In our situation, N = n + k , and the complete intersection has type ( m , . . . , m ). (cid:3) Remark 4.2 . — The methods used in [
CDR20 ] (Riemmann–Roch) and in [
CR20 ] (Morseinequalities) do not provide any explicit global symmetrc di ff erential. Hence the exis-tence of a lot a global sections does not provide any precise geometric information onthe augmented base locus. On the counterpart, the orbifold multiplicity in Theorem Cis extremely low, and there is no genericity assumption on A .
5. Applications to complex hyperbolicity5.1. Entire curves in Fermat covers. —
Hyperbolicity properties of Fermat hypersur-faces have been studied by several people. One can find in [
Kob98 , Example 3.10.21]the following result.
Theorem 5.1 (Kobayashi) . —
Consider the Fermat hypersurface of degree mF ( n , m ) ≔ { z m + · · · + z mn + = } ⊆ P n + . If m > ( n + then every entire curve f : C → F ( n , m ) lies in a linear subspace of dimension atmost ⌊ n / ⌋ . The proof of [
Kob98 ] consists in using the fact that F ( n , m ) is a Fermat cover of P n ramified over ( n +
2) hyperplanes H i with multiplicity m . Then the result is aconsequence of the truncated defect of Cartan (see [ Kob98 , 3.B.42]) which gives thelinear degeneracy of orbifold entire curves f : C → ( P n , P n + i = (1 − / m ) H i ) provided that P n + i = (1 − n / m ) + > n + Dem97 , Ex. 11.20], algebraic degeneracy of entire curves in F ( n , m ) is also obtainedusing jet di ff erentials. It gives the same degree estimate but not the second assertion onthe linear subspace of dimension ⌊ n / ⌋ containing the image of the entire curve. Remark 5.2 . — The dimension of the linear subspace in Theorem 5.1 is (at least) almostoptimal. In the setting of Remark 3.2, if instead of considering rational lines, one nowconsiders entire curves as in Theorem 5.1, and one takes r = ⌊ n / ⌋ , one infers that thedimension of the linear subspaces needed for some curves in Theorem 5.1 cannot beless than ⌊ n / ⌋ − Theorem D . —
The Fermat cover associated to an arrangement A of d > (cid:0) n + (cid:1) hyperplanes in P n in general position with respect to hyperplanes and to quadrics, with ramification m > n + is Kobayashi-hyperbolic. UASI-POSITIVE ORBIFOLD COTANGENT BUNDLES Proof . — Let π : Y → ( P n , ∆ ) be the associated Fermat cover. Since Y is compact, it issu ffi cient to prove that Y is Brody hyperbolic. Let f : C → Y be an entire curve. Theo-rem B implies that f ( C ) is contained in the ramification locus of π : Y → ( P n , ∆ ). Nowwe remark that the ramification locus has a natural structure of Fermat cover associatedto an induced arrangement A of d > (cid:0) n + (cid:1) -1 hyperplanes in P n − with multiplicity m .Up to removing some members, one can still assume that this arrangement is in gen-eral position with respect to hyperplanes and to quadrics, by Lemma 5.3 below. Usinginductively Theorem B, we obtain the hyperbolicity of the Fermat cover associated to A . (cid:3) Lemma 5.3 . —
Let A be an arrangement of d > (cid:0) n + (cid:1) hyperplanes in P n in general positionwith respect to hyperplanes and to quadrics. Let A I be the arrangement obtained by removing | I | hyperplanes, indexed by I. In T i ∈ I H i ≃ P n −| I | , there is a subarrangement A ′ of A I with atleast (cid:0) n −| I | + (cid:1) members, such that A ′ ∩ P n −| I | is in general position with respect to hyperplanesand to quadrics.Proof . — The situation being symmetric, we can safely assume that we are in the settingand notation of Sect. 1, and that we have removed the | I | last coordinate hyperplanes.Clearly, any subarrangement is still in linear position, because if n + − | I | hyperplanesof A I would satisfy a single linear equation in P n −| I | then these hyperplanes togetherwith the | I | last coordinate hyperplanes would satisfy the same linear equation in P n .Now, we want to prove that at least one subarrangement is also in general position withrespect to quadrics. For any (cid:0) n + (cid:1) − | I | hyperplanes in A I , containing the n + − | I | firstcoordinate hyperplanes, we have seen that the general position with respect to quadricsis equivalent to the non-vanishing of the determinant of the matrix A [2] . We split therows of A [2] in two blocks: those involving the | I | last coordinates, and those that doesnot. By Laplace expansion with respect to these blocks, there is at least one minor inthe second block that is not zero. We take the (cid:0) n −| I | + (cid:1) hyperplanes corresponding to thecolumns involved in one such minor, and the ( n − | I | +
1) first hyperplane coordinates,and we get (cid:0) n −| I | + (cid:1) hyperplanes in general position with respect to hyperplanes and toquadrics in P n −| I | . (cid:3) A complete intersection of general Fermat hypersurfaces cannot be reduced to aFermat cover. Moreover, we cannot use openness of ampleness in families withoutadditional e ff orts (see [ BD18b ]). However, it is most likely that the results obtainedin the present work for Fermat covers would generalize to complete intersections ofgeneral Fermat hypersurfaces. We even think that this problem should be accessibleusing the technics involved in this work. As an example, we were able to prove thatgeneral complete intersection surfaces of Fermat type in P + k have ample cotangentbundles modulo ramification for k >
3, under some explicit algebraic condition ontheir coe ffi cients. For higher dimensions, computations become tedious, and we wouldprobably need to use (explicit) resultant theory in order to conclude. This is far beyondthe scope of this work. Let us hence formulate the expected results as questions. Question . — Do general n -dimensional complete intersections of Fermat type withsu ffi ciently large codimension (e.g. k > (cid:0) n + (cid:1) ) and ramification order (e.g. m > n + Question . — When are these general complete intersections of Fermat type Kobayashi-hyperbolic?
Let ( X , ∆ ) be an orbifold with ∆ = P i ∈ I (1 − / m i ) ∆ i .Following [ CW09 ] it is natural to define (Kobayashi) hyperbolicity of ( X , ∆ ) consideringholomorphic maps h : D → ( X , ∆ ) from the unit disk D to X satisfying the two conditions: UASI-POSITIVE ORBIFOLD COTANGENT BUNDLES – h ( D ) | ∆ | .– mult x ( h ∗ ∆ i ) > m i for all i and x ∈ D with h ( x ) ∈ | ∆ i | .Such maps are called orbifold maps D → ( X , ∆ ). Orbifold entire curves C → ( X , ∆ ) aredefined mutadis mutandis .Then, one defines the orbifoldKobayashipseudo-distance d ( X , ∆ ) as the largest pseudo-distance on X \⌊ ∆ ⌋ such that every orbifold map from the unit disk is distance-decreasingwith respect to the Poincaré distance on the unit disk. A pair ( X , ∆ ) is said Kobayashi-hyperbolic if d ( X , ∆ ) is a distance on X \ ⌊ ∆ ⌋ .Besides, a pair ( X , ∆ ) is said Brody-hyperbolic if it does not admit any non-constantorbifold entire curve C → ( X , ∆ ). Kobayashi-hyperbolicity implies Brody hyperbolicity.Brody’s theorem characterizes Kobayashi-hyperbolicity in terms of Brody-hyperbolicity.We will now give an orbifold version of this result.We have the following proposition which slightly refines [ CW09 ]. Proposition 5.4 . —
Let (cid:16) X , ∆ ≔ P di = (1 − / m i ) ∆ i (cid:17) be an orbifold. Assume that a sequence oforbifold maps h p : D → ( X , ∆ ) from the unit disk converges locally uniformly to a holomorphicmap h : D → X. Let X h ≔ T h ( D ) ⊆ ∆ i ∆ i ⊆ X, and let ∆ h ≔ P h ( D ) * ∆ j (1 − / m j ) ∆ j ∩ X h . Then,h is an orbifold map D → ( X h , ∆ h ) .Proof . — Suppose that h (0) ∈ | ∆ | . Consider a neighbourhood V of h (0) in X such that | ∆ | ∩ V is locally defined by a holomorphic function Q f i , where f i = ∆ i ∩ V . If h ( D ) * ∆ j , one can assume that f j ◦ h has no zero in V except at 0.Apply the classical theorem of Rouché to a sequence of holomorphic function { f j ◦ h p } .For all su ffi ciently large p the multiplicity at 0 of f j ◦ h equals the sum of all multiplicitiesof all zeroes in V of f j ◦ h p . Therefore this multiplicity is at least m j because h p areorbifold maps. (cid:3) As an immediate consequence, reasoning exactly as in [
CW09 , Sect. 13], we obtainthe following result.
Theorem 5.5 (orbifold Brody’s criterion) . —
Consider a smooth orbifold pair X , ∆ ≔ d X i = (1 − / m i ) ∆ i . For a subset I of { , . . . , d } , let X I ≔ ∩ i ∈ I ∆ i , and let ∆ I ≔ P j < I (1 − / m j ) ∆ j ∩ X I . If all pairs ( X I , ∆ I ) are Brody-hyperbolic, then the pair ( X , ∆ ) is Kobayashi-hyperbolic. Now we are in position to derive from Theorem B anhyperbolicity result for the orbifold pair P n , ∆ ≔ d X i = (cid:18) − m (cid:19) H i . We will use Fermat cover, in the opposite direction as Kobayashi did in Theorem 5.1.Remark however that this orbifold hyperbolicity is not directly implied by Theorem D,because orbifold curves do not lift in general to the Fermat cover. Nevertheless, tech-niques introduced in [
CDR20 ] will permit to use it. Indeed, Corollary 3.7 of [
CDR20 ]yields the following Proposition.
Proposition 5.6 (Fundamental vanishing theorem) . —
Let ( X , ∆ ) be a smooth orbifoldpair. Then any orbifold entire curve is contained in B + ( Ω ( X , ∆ )) . We obtain the following.
UASI-POSITIVE ORBIFOLD COTANGENT BUNDLES Theorem E . —
Consider an arrangement A of d hyperplanes H , . . . , H d in P n in generalposition with respect to hyperplanes and to quadrics, with respective orbifold multiplicities m i ,and the associated orbifold divisor ∆ ≔ P di = (1 − / m i ) · H i . If d > (cid:0) n + (cid:1) and m i > n + , thenthe orbifold pair ( P n , ∆ ) is hyperbolic.Proof . — By Theorem 5.5, if ( P n , ∆ ) is not hyperbolic, then either there exists a non-constant orbifold entire curve f : C → ( P N , ∆ ) or an orbifold curve in the boundarydivisor. Theorem B and Proposition 5.6 imply that all orbifold entire curves f : C → ( P n , ∆ ) are constant. So, we are left with the second possibility. In this case, accordingto Theorem 5.5, f is a non-constant orbifold map with respect to an orbifold structure( P n −| I | , ∆ I ) induced by the arrangement A I of (cid:0) n + (cid:1) − | I | > (cid:0) n + −| I | (cid:1) hyperplanes. Weconclude by induction, using Lemma 5.3. (cid:3) Remark 5.7 . — It follows that ( P n , ∆ ) is Brody-hyperbolic. Actually, one can exclude theexistence of non-constant orbifoldcorrespondences on varieties with orbifold cotangentbundles that are ample modulo boundary. Cf. [ CDR20 ] for a definition. These are themorphisms that one would naturally consider to generalize entire curves in the orbifoldcategory (and these are much more numerous).
Acknowledgments . — L.D. and E.R. would like to thank
Joël Merker for interesting dis-cussions on explicit orbifold sections and around resultant which helped a lot to findthe right attack angle for our problem.L.D. would like to thank
Henri Guenancia for his help on augmented base loci andparticularly around Lemma 0.6, which plays an important role in the reformulationof Noguchi’s result. L.D. would also like to thank
Mikhail Zaidenberg for many inter-esting discussions over the years and for making him aware of Theorem 0.11. Theseinteractions took place during the conference Alkage hosted by
Jean-Pierre Demailly ,which gave L.D. a great opportunity to present a preliminary version of this work to adistinguished audience.E.R. would like to thank
Stefan Kebekus and
Tanuj Gomez for discussions on the positiv-ity of orbifold cotangent bundles, and
Eric Riedl for discussions on bigness of cotangentbundles.L.D. would like to thank
Frédéric Han for identifying the condition of general positionwith respect to quadrics.Lastly, L.D. would like to warmly thank
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