Endomorphisms of projective bundles over a certain class of varieties
aa r X i v : . [ m a t h . AG ] S e p Endomorphisms of projective bundles over acertain class of varieties
Ekaterina Amerik, Alexandra KuznetsovaSeptember 24, 2018
Introduction
During the last 20 years, the question which smooth projective varieties haveendomorphisms of degree greater than one (which we shall sometimes simply call“endomorphisms”, as opposed to automorphisms) has attracted some attentionfor both geometric and dynamical reasons (see e.g. [ARV], [Bea], [N], [NZ] -this is only a beginning of the list). Though in this generality it is still farfrom being solved, there is a number of partial results suggesting that varietieswith such endomorphisms generally come from two obvious cases (tori and toricvarieties) by means of simple geometric constructions such as taking a productwith another smooth projective variety or taking a quotient by a finite freelyacting group. For instance, Nakayama proved in the beginning of 2000’s that arational smooth projective surface with endomorphisms must be toric. Aroundthe same time, one of the authors of the present note has considered the caseof a projective bundle X over a projective base B , p : X → B , and proved thatif X has an endomorphism commuting with the projection onto the base, then X must be a quotient of a product B ′ × P r by a finite freely acting group. Asimple remark on endomorphisms of projective bundles X = P ( E ), where E isa vector bundle ([A], p. 18) is that a power of any f : X → X sends fibers tofibers and thus must be over an endomorphism of the base; so if by any chancewe know that all endomorphisms of B are of finite order - for instance when B is of general type - then this result describes the situation completely.The argument (the “only if” part, the “if” part being rather standard) pro-ceeds as follows. One considers the space of all morphisms R m ( P ( V )) from P n = P ( V ) to itself given by degree m polynomials (well-known to be an affinevariety) and its quotient M by P GL ( V ) (that is, the spectrum of the ring ofthe invariants). It turns out that for m big enough, P GL ( V ) acts with finitestabilizers, so M is the geometric quotient (i.e. actually parameterizes the orbitsof the action). Now let X = P ( E ) be a projective bundle over B . An endomor-phism f of X over B naturally induces a morphism from B to M . Its imagemust be a point since B is projective and M is affine. Let t be a lift of this pointto R m ( P ( V )). Over a suitably fine open covering ( U α ) of B we have f α = h α · t ,where h α is in P GL n +1 ( O U α ). Denote by g αβ the transition functions of our1rojective bundle, it follows that h − α g αβ h β ∈ Stab ( t ), in other words, by chang-ing the trivialization we make the transition functons of X constant with valuesin a finite group, q. e. d..In general, for an endomorphism f of P ( E ) we may suppose that f is overan endomorphism g of the base; there are then two cases to be treated: thecase where f induces isomorphisms of fibers (considered as exceptional; when X = P ( E ) it means that g ∗ E is a shift of E by a line bundle) and the case wherethe degree of f on the fibers is greater than one. In [A], only the rank-two case(that of projective line bundles) was considered. It was established that either X is a finite quotient of a product or E has a subbundle. This last statementhas been pursued further to yield that E must split into a direct sum of linebundles after a finite, not necessarily ´etale, base change ([A], theorem 2); froma different point of view, one can restrict to a specific class of bases to obtain astronger statement. For instance, if B satisfies the condition H ( B, L ) = 0 forany line bundle L , then having a subbundle is equivalent to splitting for rank-twobundles. It therefore follows from the results of [A] that if B is simply connectedand H ( B, L ) = 0 for any line bundle L on B , then an X with endomorphisms ofdegree greater than one on fibers must be the projectivization of a split rank-twobundle.The purpose of the present note is to prove this result in the case of arbitraryrank projective bundles over such specific bases. Theorem 1.
Let B be a simply-connected projective variety such that for anyline bundle L its first cohomology H ( B, L ) = 0 . Let E be a vector bundle ofrank n + 1 on B . If there exists a fiberwise endomorphism P ( E ) φ / / π (cid:15) (cid:15) P ( E ) π (cid:15) (cid:15) B Φ / / B (1) of degree greater than one on the fibers, then E splits into a direct sum of linebundles: E = n +1 ⊕ i =0 L i (2)What we show is in fact slightly more general, as in [A]. Theorem 2.
Let B be as in the previous theorem and E and F vector bundlesof rank n + 1 on B . If there exists a morphism φ : P ( E ) → P ( F ) (3) over B which is of degree greater than one, then E and F both split into a directsum of line bundles. Obviously, theorem 1 follows from this statement: consider the endomor-phism φ as a morphism P ( E ) → P (Φ ∗ ( E )), and apply the theorem 2.2n the ideal situation, one would like to prove the statement of Theorem1 for an arbitrary toric base B . The reason is that the projectivization of avector bundle over a toric base is itself toric if and only if the bundle is split([D]). This would therefore strongly support the principle that varieties withendomorphisms are closely related to toric varieties or tori. However few toricbases (e.g. P n , n ≥
2, or products of such) actually satisfy the cohomologyvanishing condition as above; so more work is needed to obtain such a result.It is certainly related to the fact that we never make use of a condition like F = g ∗ E in Theorem 2. Let V and W be vector spaces of dimension n + 1. Denote by R m ( P ( V ) , P ( W ))the set of all morphisms between P ( V ) and P ( W ) given by homogeneous poly-nomials of degree m without a common zero except at (0 , , . . . , y = f ( x ,x , . . . , x n ) y = f ( x ,x , . . . , x n ) · · · y n = f n ( x ,x , . . . , x n ) (4)This is an affine variety, indeed the complement to the hypersurface definedby the resultant of the f i in the projective space P ( Hom ( W ∗ , S m V ∗ )), with theaction of P GL ( V ) × P GL ( W ) given by(( g, h ) · f )( x ) = h − ( f ( g ( x ))) . (5)The quotient M of R m ( P ( V ) , P ( W )) by this action (i.e. the spectrum ofthe ring of invariants), in contrast with the case of the action of P GL ( V ) when V = W ([A]) is not a geometric quotient: indeed some points have infinitestabilizers, and all the adherent orbits give the same point on the quotient.Let us denote by M the “bad subset” of M (by definition it consists of pointscorresponding to orbits not separated by the invariants).When some fiber of a vector bundle E over B is identified with V and thatof F with W , a morphism of projective bundles P ( E ) → P ( F ) over a base B gives, in the same way as in [A], a map from B to M , which must be constantas soon as B is projective. If the image point is not in M we conclude asbefore that P ( E ) and P ( F ) trivialize after a finite unramified base change. If B is simply-connected, this yields that these are already trivial on B , and inparticular they split into a direct sum of line bundles. So the interesting case iswhen the image point lands in M . In this situation, we strive to deduce someinformaion about the geometry of our morphism. We aim to show that E and F have subbundles E ′ and F ′ such that the inverse image of P ( F ′ ) is P ( E ′ ) andthat the map f induces a morphism on the quotients. This shall enable us to3onclude by induction in the case when the cohomological condition on B issatisfied.Let us also remark that replacing our original endomorphism φ of P ( E ) bya power, we may assume that m is greater than the rank n + 1 of the vectorbundles E and F , as we shall for the computations in the next section. In this section we consider two vector spaces V and W of dimension n + 1 anda morphism f between their projectivisations of degree d = m n . First of allassume f is stabilized by an infinite subgroup Stab ( f ) in P GL ( V ) × P GL ( W ).Recall from [A]: Lemma 1 ([A], Lemma 1.2) . If m > n + 1 , then a unipotent element u of P GL ( V ) × P GL ( W ) does not stabilize any element of R m ( P ( V ) , P ( W )) . By this lemma the subgroup
Stab ( f ) ⊂ P GL ( V ) × P GL ( W ) consists ofsemisimple elements. Take any of these elements and consider the minimalsubgroup in the stabilizer that contains this element. The connected componentof the unity of this subgroup is an algebraic torus or trivial. If it is trivial forany element in Stab ( f ), then the stabilizer is discrete and therefore is finite. If Stab ( f ) is infinite, it contains a subgroup isomorphic to G m . Lifting its actionon P ( V ) and P ( W ) to an action on V and W we assume that it is given by g b,c : G m → GL ( V ) × GL ( W ) g b,c ( λ ) = ( diag ( λ c , λ c , . . . , λ c n ); diag ( λ b , λ b , . . . , λ b n )) (6)in appropriate coordinates on V and W .In these coordinates, let the morphism f be given by ( f , f , . . . , f n ) with y = f ( x , . . . , x n ) = X | I | = m a ,I x I y = f ( x , . . . , x n ) = X | I | = m a ,I x I · · · y n = f n ( x , . . . , x n ) = X | I | = m a n,I x I (7)Here I = ( i , i , . . . , i n ) is a multiindex and | I | = i + i + · · · + i n .Applying an element of the diagonal group in g b,c ( λ ) ∈ Stab ( f ), we get thefollowing formulae for g b,c · f : y j = X λ h c,I i− b j a j,I x I (8)4ere h− , −i denotes the scalar product between multiindexes: F ( I ) := h c, I i = n X j =0 c j i j (9)Since g b,c stabilizes f there exists a constant C , such that for any j, I with a j,I = 0 h c, I i − b j = C (10)Consider the n + 1)–dimensional lattice Λ ∼ = Z n +1 ⊂ R n +1 = Λ ⊗ R . Denoteby p i ∈ Λ the vertex corresponding to the i − th base vector (0 , . . . , , m, , . . . , { p i , . . . , p i k } denote by ∆( p i , . . . , p i k ) ⊂ Λ the simplex of di-mension k − p i , . . . , p i k . Set∆ = ∆( p , p , . . . , p n ) ⊂ Λ (11) p = (0 , , m ) p = (0 , m, p = ( m, , i i i Figure 1: The simplex ∆ in the case n = 2Equations (10) define n + 1 hyperplanes in R n (not necessarily distinct). Letus denote them by Π j .Now let us consider the Newton polyhedron of f j : N P ( f j ) := Conv { I ∈ Λ | a j,I = 0 } (12)and prove some easy facts about Newton polyhedra of the morphism f . Proposition 1. If f has infinite stabilizer then N P ( f j ) ⊂ Π j ∩ ∆ . roof. As the degree of f j equals m , the polyhedron N P ( f j ) lies in the simplex∆. By the previous calculation we see that if g b,c stabilizes f , then (10) holdsand consequently the multi-indices of the monomials of f j lies in the hyperplaneΠ j . Lemma 2. If f is a morphism of projective spaces then every vertex of ∆ iscontained in one of the hyperplanes Π j .Proof. Assume the vertex p = ( m, , . . . ,
0) does not lie in any Π j . Conse-quently no polynomial f j contains the monomial x m . Then all f j vanish at thepoint (1 : 0 : · · · : 0) ∈ P ( V ), so f is not a morphism. Lemma 3.
Each hyperplane Π j contains some vertex of ∆ . Moreover a hy-perplane repeated exactly k + 1 times (i.e. corresponding to the polynomials f , . . . , f k , up to renumbering) contains exactly k + 1 vertices of ∆ .Proof. Since all the hyperplanes Π j are parallel, if they contain a common vertexthey coincide. There is a natural partition of the set H ( f ) of equations H ⊔ H ⊔ · · · ⊔ H l = H ( f ) (13)where a subset H i consists of equations corresponding to the same hyperplaneΠ i , as well as of the set of vertices V (∆) = V ⊔ V ⊔ · · · ⊔ V l (14)where V i consists of vertices lying in Π j .Since | V (∆) | = n + 1 = | H ( f ) | it follows that either the statement of thelemma is true or k + 1 = | V i | > | H i | = s + 1 for some i .Assume | V i | > | H i | . The polynomials f i indexed by H i contain monomi-als depending only on the variables indexed by V i , but the others do not: upto renumbering, f s +1 , . . . , f n are zero as soon as x k +1 = . . . x n = 0. Then f , . . . , f s define a regular map of the subspace of P ( V ) given by the vanishingof x k +1 , . . . x n to the subspace of P ( W ) given by the vanishing of y s +1 , . . . , y n ,but this is impossible since the dimension of the source would then be greaterthan that of the target.From these assertions we deduce the following statement. Proposition 2.
Let f be a morphism between P ( V ) and P ( W ) with infinitestabilizer in P GL ( V ) × P GL ( W ) . There exist V ′ ⊂ V (∆) and H ′ ⊂ H ( f ) suchthat | V ′ | = | H ′ | < n + 1 and N P ( f j ) ⊂ ∆( V ′ ) (15) for any f j ∈ H ′ . Π Π Π = Π Π Figure 2: Two types of Newton polyhedra of f , f and f in the case n = 2. Proof.
Let us recall the function F from (9). Denote M ′ = max { F ( p i ) } , where p i runs through the set of vertices of ∆. Set H ′ = { f j | F | Π j = M ′ } (16)As F is not constant on ∆, ∅ ( H ′ ( H ( f ). Denote by V ′ the set of vertices onthe hyperplane corresponding to the equations in H ′ . By the previous lemma | V ′ | = | H ′ | . Obviously, Π j ∩ ∆ = ∆( V ′ ) and so the polynomials f j ∈ H ′ dependonly on the variables corresponding to the vertices in V ′ .So far, we have discussed the morphisms of projective spaces with infinitestabilizer in P GL ( V ) × P GL ( W ). But our goal is to study the morphisms f with non-closed orbits under the group action. By a generalization of theHilbert–Mumford criterion ([Bir] Theorem 4.2), we reach the boundary of theorbit ( P GL ( V ) × P GL ( W )) · f while acting on f by one-parameter subgroups g b,c ( G m ) as in (6). As earlier, the map g b,c ( λ ) · f is given by the equations (8).Let us introduce a new notation K j = min { I | a j,I =0 } {h c, I i − b j } (17)Set K = min j { K j } . Then we can describe the limit of g b,c ( λ ) · f when λ goesto zero. Lemma 4.
Denote ¯ f = lim λ → ( g b,c ( λ ) · f ) , then ¯ f j ( x , . . . , x n ) = X h c,I i− b j = K a j,I x I (18) and the original map was of type: f j = X h c,I i− b = K a j,I x I + X h c,I i− b >K a j,I x I (19)7he proof is a straightforward calculation.Obviously, the group G b,c stabilize the morphism ¯ f , so ¯ f has infinite stabi-lizer and in Proposition 2 we have a description of its Newton polyhedron. Nowconsider the set of half-spacesΠ + j = { I ∈ Λ ⊗ R | h c, I i − b j ≥ K ⊂ R n } (20)Lemma 4 implies that N P ( f j ) = Π + j ∩ ∆. From the proof of Proposition 2 wesee that there is always a hyperplane Π j intersecting our simplex ∆ by a faceand such that the rest of the simplex is below Π j . Thus the following holds. Proposition 3. If f is an unstable morphism between P ( V ) and P ( W ) , thenthere are nonempty sets V ′ ⊂ V (∆) and H ′ ⊂ H ( f ) such that | V ′ | = | H ′ | < n +1 and N P ( f j ) ⊂ ∆( V ′ ) (21) for any f j ∈ H ′ . N P ( f )Π Π Π N P ( f )Π Π Π N P ( f )Π Π Π Figure 3: Here are Newton polyhedra of f i in the case n = 2. On each picturewe highlight N P ( f i ) with a blue colour. Proof.
Actually, consider the set H ′ from the previous lemma. As for anyΠ j ∈ H ′ , the restriction of function F to Π j equals M ′ = max { F ( p i ) } , then∆ ⊂ { I | F ( I ) ≤ M ′ } (22)Therefore for any Π j ∈ H ′ the half-space Π + j also intersects ∆ by ∆( V ′ ).In the language of equations this means that the first s + 1 equations dependonly on the first s + 1 variables. From the previous section we deduce a useful corollary about morphisms be-tween projective bundles: 8 orollary 1.
Assume φ : P ( E ) → P ( F ) is a morphism over the base B ofdegree d > , such that its restriction to a fiber corresponds to an unstable orbitin R m ( P ( V ) , P ( W )) . Then there are subbundles E ( E and F ( F , such that φ − ( P ( F )) = P ( E ) (23) and < rk ( E ) = rk ( F ) < rk ( E ) = rk ( F ) .Proof. By the results in the previous section, in any fiber of P ( F ) there arecoordinates in which for any 0 ≤ j ≤ sy j = f j ( x , . . . , x s ) (24)We claim that the preimage of the subspace H = { y = · · · = y s = 0 } is thesubspace { x = · · · = x s = 0 } . Indeed the last subspace is certainly containedin the preimage of the first one. If there is another point P = ( p : . . . p s : p s +1 : · · · : p n ) in that preimage, consider the projective subspace generated by P andthe last n − s base vectors: its dimension is n − s , so it must have nonemptyintersection with the subvariety given by the equations f s +1 ( x , . . . , x n ) = · · · = f n ( x , . . . , x n ) = 0 (25)which has dimension at least s . Any point in this intersection must be anindeterminacy point of f , a contradiction.These subspaces H fit together in a subbundle F ( F . The same happento their preimages, giving a subbundle φ − ( F ) = E ( E .To complete the proof of the theorem let us consider a linear mapping in-duced by the morphism φ : φ ∗ : F ∗ → S m E ∗ (26)As we have shown we have subbundles E and F , such that the followingdiagram commutes:0 / / ( F/F ) ∗ / / ( φ/φ ) ∗ (cid:15) (cid:15) F ∗ φ ∗ (cid:15) (cid:15) / / F ∗ φ ∗ (cid:15) (cid:15) / / / / ( S m E/S m E ) ∗ / / S m E ∗ / / S m E ∗ / / S m E/S m E ) ∗ and write( S m E/S m E ) ∗ ∼ = ⊕ m − i =0 S i E ∗ ⊗ S m − i ( E/E ) ∗ (28)In particular there is a projection( S m E/S m E ) ∗ pr −−→ S m ( E/E ) ∗ (29)and pr ◦ ( φ/φ ) ∗ = ψ ∗ induces a map between projective bundles P ( E/E )and P ( F/F ) given by degree m polynomials. In fact this map is regular, that9s, a morphism. To check this one observes that one may view ( x : · · · : x s )and ( y : · · · : y s ) from corollary 1 as coordinates on the projectivization of thequotients, and the map of these projectivizations is then given by f , . . . f s . Tosay that this map has no indeterminacy point ( p : · · · : p s ) is the same as tosay that the preimage of P ( F ) from corollary 1 contains nothing but P ( E ). Proof of the Theorem 2.
We argue by induction on n + 1 = rkE . If rkE = 1then E is already linear, so the base of induction is trivial.Suppose now, that for all ranks less then n + 1 the statement is true. Therestriction of the morphism φ to a fiber gives us an element in R m ( P ( V ) , P ( W )) / ( P GL ( V ) × P GL ( W )).If this element corresponds to a stable orbit in R m ( P ( V ) , P ( W )), then theargument in the proof of theorem 1 in [A] proves that after a finite ´etale basechange both P ( E ) and P ( F ) trivialize. As the variety B is simply-connected,there are no nontrivial ´etale base changes, so both P ( E ) and P ( F ) are trivialand hence split.If we get an unstable orbit, then by corollary 1 the bundles E and F sit inshort exact sequences: 0 → E → E → E/E → → F → F → F/F → m > E , F , E/E and F/F , namely φ : P ( E ) → P ( F ) ψ : P ( E/E ) → P ( F/F ) (31)By the inductive assumption all these bundles must split into direct sums of linebundles. Since for any line bundle L on B , its first cohomology H ( B, L ) = 0,we see that Ext ( E/E , E ) = Ext ( F/F , F ) = 0 (32)So the extensions are trivial too. Consequently E and F split into a direct sumof line bundles. Acknowledgements:
This paper has been prepared within the frameworkof a subsidy granted to the NRU HSE Laboratory of Algebraic Geometry bythe Government of the Russian Federation for the implementation of the GlobalCompetitiveness Program. The first-named author was partially supported bythe Young Russian Mathematics award.
References [A] Amerik, E.: On endomorphisms of projective bundles, Manuscripta Math.111 (2003), no. 1, 17–28 (2003) 10ARV] Amerik, E.; Rovinsky, M.; Van de Ven, A. A boundedness theorem formorphisms between threefolds. Ann. Inst. Fourier (Grenoble) 49 (1999), no.2, 405–415.[Bea] Beauville, A. Endomorphisms of hypersurfaces and other manifolds. In-ternat. Math. Res. Notices 2001, no. 1, 53–58.[Bir] Birkes, D.: Orbits of linear algebraic groups. Annals of Math. 93 (1971),459 – 475[D] Druel, S. Structures de contact sur les vari´et´es toriques. Math. Ann. 313(1999), no. 3, 429 –435.[N] Nakayama, N. Ruled surfaces with non-trivial surjective endomorphisms.Kyushu J. Math. 56 (2002), no. 2, 433–446.[NZ] Nakayama, Noboru; Zhang, De-Qi Building blocks of ´etale endomorphismsof complex projective manifolds. Proc. Lond. Math. Soc. (3) 99 (2009), no. 3,725–756