A REMARK ON BIGNESS OF THE TANGENT BUNDLE OF A SMOOTH PROJECTIVE VARIETY AND D-SIMPLICITY OF ITS
SECTION RINGS
JEN-CHIEH HSIAO
Abstract. We point out a connection between bigness of the tangent bundle of a smooth projective variety X over C and simplicity of the section rings of X as modules over their rings of differential operators. As a consequence, we see that the tangent bundle of a smooth projective toric variety or a (partial) flag variety is big. Some other applications and related questions are discussed.
1. Introduction
Let X be a smooth projective variety of dimension n over C and TX be the tangent bundle of X. A well-known theorem of Mori [Mor79] says that if TX is ample, then X is isomorphic to the projective space Pn. In [CP91], Campana and Peternell initiated a program to characterize projective manifolds whose tangent bundles are numerically effective (nef). In view of the theory of positivity of vector bundles [Laz04a, Laz04b], it is natural to consider the following
Problem 1.1. Classify projective manifolds whose tangent bundles are big.
Recall that a line bundle L on X is big if and only if the map φm : X 99K PH0(X, L⊗m)
defined by L⊗m is birational onto its image for some m > 0 [Laz04b, Definition 2.2.1]. A vector bundle E on X is big if the line bundle O(1) on the projective bundle P(E) := Proj(Sym E) is big. Except for the case where E is nef (see the discussion in Section 4), it is in general not clear how to determine the bigness of a vector bundle. Here, we propose the following criterion for bigness of the tangent bundle TX.
Theorem 1.2. If X admits an ample line bundle whose section ring S is simple as a module over its ring of C-linear differential operators DS, then TX is a big vector bundle.
In fact, it is shown that if TX is not big, then the section ring S associated to any ample line bundle on X has no differential operators of negative weight. In
2010 Mathematics Subject Classification. 13N10, 14J60.
The author was partially supported by National Science Council in Taiwan under grant 101- 2115-M-006-011-MY2.
1
particular, the maximal graded ideal of S is a DS-submodule (S is not DS-simple) and DS itself is not a simple ring.
When dim X = 1, Theorem 1.2 was proved by Levasseur [Lev89] following the results in [BGG72, Ish85]. The essential ideas due to Bernˇste˘ın, Gel0fand, Gel0fand [BGG72] show that there exist no differential operators of negative weight on
S = C[x, y, z]/hx3+ y3+ z3i
by using the geometric properties of the cubic curve Proj S. These ideas generalize readily to the higher dimensional cases (see Section 2).
As the first consequence of Theorem 1.2, we have the following corollary whose proof can be found in Section 3.
Corollary 1.3. The tangent bundle on a smooth projective toric variety or on a (partial) flag variety is big.
Another application of Theorem 1.2 is that if TX is not big and if S is a quasi- Gorenstein section ring on X, then the multiplier ideal J (S) is a DS-module (see Remark 3.4). This can be regarded as a characteristic 0 analog of Theorem 2.2(1) in [Smi95]: every test ideal is a D-module.
In the case where TX is nef, the bigness of TX is equivalent to the positivity of the Segre class sn(TX∗) [Ful98, Section 3.1]. Using this and the results in [CP91], we show in Proposition 4.1 that if TX is nef and big then X is Fano (i.e. −KX is ample). This provides higher-dimensional examples of smooth projective varieties whose section rings are not D-simple (e.g. abelian varieties or other non-Fano manifolds with nef TX), generalizing the classical one-dimensional examples in [BGG72, Lev89]. On the other hand, Proposition 4.1 gives a partial solution to Problem 1.1 when TX is nef. In fact, following the discussions in Section 5 we expect that the tangent bundle TX on a Fano manifold X is always big. When dim X ≤ 3 and TX is nef, this is verified by computing the Segre class sn(TX∗) using the classification in [CP91]. See the paragraph after Question 4.5 for more details.
To end the introduction, we mention that inspired by the proof of Proposition 4.1, one can prove a conjecture in [CP91] that states: if TX is nef and χ(OX) = 1 then X is Fano (see Theorem4.3).
Acknowledgement. We thank Thomas Peternell and Sam Payne for the corre- spondences concerning the bigness of tangent bundles. We also thank Shin-Yao Jow, Ching-Jui Lai, and Mircea Mustat¸˘a for their interests in this work. Spe- cial thank goes to Holger Brenner, who helped to improve the presentation of the results and indicated possible directions for further developments.
2. The proof of Theorem 1.2
Except for Proposition 2.1, most parts of the following proof of Theorem 1.2 come from the arguments in [BGG72, Ish85, Lev89]. We reproduce them here for the convenience of the reader.
Fix an ample line bundle L on X and set Si := H0(X, Li). Consider the associated section ring
S = SL:= M
i∈Z≥0
Si.
Since X is smooth, it is well-known that S is a N-graded normal domain that is finitely generated over C = S0. The maximal graded ideal of S will be denoted by
m:=M
i∈N
Si.
Choose a representation T /J of S, where T = C[x1, . . . , xs] is a polynomial ring whose variable xi has weight di and J is a homogeneous prime ideal. Thanks to the ampleness of L, the graded ring S satisfies the condition (]) in [Ish85] which states that there exists l0 such that for every l ≥ l0, S(l) = L
i∈NSil is generated by Sl = [S(l)]1 over S0 = C.
Let DSbe the ring of C-linear differential operators on S in the sense of Grothendieck.
By definition, the ring DS has an increasing filtration {Dm}m≥0 by the order of the differential operators, where D0 = S and for m > 0
Dm = {δ ∈ HomC(S, S) | [δ, s] ∈ Dm−1 for all s ∈ S}.
Following [BGG72] or [Nak70], we decompose DS as DS = SM [
m≥1
Derm(S) where
Derm(S) := {δ ∈ DS | ord δ ≤ m and δ(1) = 0}
is the set of all derivations of order ≤ m. An element δ ∈ Derm(S) is called homogeneous of weight l (l ∈ Z) if δ(Si) ⊆ Si+l for all i. In particular, the Euler derivation
I =
s
X
i=1
dixi ∂
∂xi
∈ Der1(S)
is homogeneous of weight 0. Moreover, Derm(S) can be decomposed as Derm(S) =M
l∈Z
Derml (S)
where Derml (S) is the space of homogeneous derivations of S with weight l and order ≤ m. Notice that if
(2.1) Derlm(S) = 0 for all m ≥ 0 and l < 0,
then the maximal ideal m is a DS-submodule of S; in particular, the ring S is not DS-simple. Our goal is to show that non-bigness of TX implies the condition (2.1).
Let ˆX = Spec(S) be the cone over X and U := ˆX \{m} be the punctured spectrum.
We have the natural projection
π : U −→ X = Proj S.
Let Derm be the sheaf of derivations on U of order ≤ m and let Derml be the sheaf of derivations δ on U of order ≤ m that satisfy the condition [I, δ] = Iδ − δI = lδ and denote ∆ml := π∗Derml . Since Derm(S) is the dual of the mth-order differential module Ωm(S) = I/Im+1 where I is the diagonal ideal of S ⊗CS, it is reflexive. Moreover, since S is normal, standard facts about reflexive sheaves [Har80, Proposition 1.6] implies that Γ(U,Derm) = Derm(S). In particular, by comparing the weight of derivations we have
Γ(X, ∆ml ) = Γ(U,Derml ) = Derml (S).
For |l| ≥ l0 in condition (]), one can check locally that ∆ml ∼= ∆m0 ⊗ Ll (m ≥ 0) where this isomorphism is compatible with the natural embedding ∆ml ,→ ∆nl (m < n) [Ish85, Lemma 6]. In particular, we have the following exact sequence (2.2) 0 −→ ∆m−1l −→ ∆ml −→ σm⊗ Ll−→ 0 (m ≥ 2, |l| ≥ l0).
Here σm := ∆m0 /∆m−10 (m ≥ 2) and we set σ1 = ∆10.
By [Ish85, Lemma 6(3)], we have the following short exact sequence that comes from the Euler derivation
(2.3) 0 −→ OX
−→ σ·I 1 −→ TX −→ 0.
Taking symmetric power and using the facts that σm = Symmσ1 [Ish85, Lemma 6(3)] and TX are locally free, we have the exact sequence [Ish85, Lemma 7]
(2.4) 0 −→ σm−1 −→ σm −→ SymmTX −→ 0 (m ≥ 2).
On the other hand, we need the following Proposition 2.1. If TX is not big, then
H0(X, SymmTX ⊗ Ll) = 0 for m ≥ 1 and l < 0.
Proof. This can be achieved by a similar argument as in [Laz04b, Example 6.1.23]:
Suppose H0(X, SymmTX ⊗ A−1) 6= 0 for some m and some ample line bundle A on X. Since O(1) := OP(TX)(1) is π-ample, O(1) ⊗ π∗(A⊗k) is ample for k sufficiently large. Denote ξ = c1(O(1)) and α = c1(A). By the projection formula, π∗(O(m) ⊗ π∗A−1) = SymmTX ⊗ A−1. So H0(P(TX), O(m) ⊗ π∗A−1) 6= 0 and hence mξ − π∗α is an effective class. For k sufficiently large,
(km + 1)ξ = [kmξ − π∗(kα)] + [ξ + π∗(kα)]
is a sum of an effective class and an ample class. Therefore, ξ is big. From now, assume that TX is not big. By Proposition 2.1, we have
H0(X, TX ⊗ Ll) = 0 for l < 0.
The global evaluation of (2.3) tensored with Lltogether with the Kodaira vanishing theorem imply that
Der1l(S) = H0(X, ∆1l) = 0 (l < 0).
Moreover, the global evaluation of (2.4) tensored with Ll together with the van- ishing
H0(X, SymmTX ⊗ Ll) = 0 (m ≥ 2, l < 0) imply that
H0(X, σm⊗ Ll) = H0(X, σ1⊗ Ll) = H0(X, ∆1l) = 0 (m ≥ 1, l < 0).
Therefore, it follows from the global evaluation of (2.2)⊗Ll that Derlm(S) = H0(X, ∆ml ) = H0(X, ∆1l) = 0 (m ≥ 1, l ≤ −l0).
Now, for any δ ∈ Derlm(S) (m ≥ 1, l < 0), we have δk ∈ Derkmkl (S) = 0 for k sufficiently large. This forces δ = 0, and the proof of Theorem 1.2 is finished.
3. Some applications
The following fact ( [Smi95, Proposition 3.1] or [LS89, Proposition 3.5]) says that D-simplicity is preserved when considering pure subrings.
Proposition 3.1. Let S and T be arbitrary commutative algebra over a commu- tative ring A. Suppose that S ,→ T and that this inclusion splits as a map of S-modules. If T is a simple DA(T )-module, then S is a simple DA(S)-module.
In particular, if S admits a split embedding into a polynomial algebra C[x1, . . . , xt], then S is DS-simple.
We are now ready to prove the main corollary in the introduction which states: the tangent bundle on a smooth projective toric variety or on a (partial) flag variety is always big.
Proof of Corollary 1.3. For the toric case, this follows from the facts that every section ring of a smooth projective toric variety is a normal semigroup ring [Ful93, Section 3.4] and that every positive normal semigroup ring admits a split embed- ding into a polynomial algebra [BH93, 6.1.10]. Alternatively, one can check directly that every normal semigroup ring S is DS-simple (see e.g. [Hsi12, Theorem 3.7]).
For the flag case, note that the homogeneous coordinate ring of a (partial) flag variety under the Pl¨ucker embedding is isomorphic to the ring of invariants RG of a polynomial algebra R under the action of certain linearly reductive group G [Ful97, Section 9.2]. The existence of a Reynolds operator ρ : R → RG [BH93, Section 6.5] guarantees that the embedding RG → R is split. Remark 3.2. The flag case of Corollary 1.3 recovers the bigness of TP(T
P2) in [SCW04, Example 2].
Remark 3.3. It is pointed out by Mustat¸˘a that the toric case of Corollary 1.3 can also be recovered by using the Euler exact sequence. On the other hand, the pseudoeffective cone of a projectivized toric vector bundle P(F) on a toric variety X is described using Klyachko filtration [GHPS12]. In the case where F is the tangent bundle TX, the pseudoeffective cone of P(TX) is generated by the classes DH, for hypersurfaces H in PF (the π-fiber over 1T), and the classes Di
(the preimage under π of the torus invariant prime divisors on X). As Payne
mentioned to us, the results in [GHPS12] might be helpful to re-establish the bigness of tangent bundles on smooth projective toric varieties.
We mention another consequence of Theorem 1.2.
Remark 3.4. For a normal standard graded ring S that is quasi-Gorenstein and has an isolated singularity, the multiplier ideals J (S) of S is a power of the maximal ideal m [Smi00b, Example 3.7]. Therefore, if the section ring S in Theorem 1.2 is quasi-Gorenstein, then J (S) is a DS-submodule of S. This can be regarded as a characteristic 0 analog of Theorem 2.2(1) in [Smi95]: every test ideal is a D-module.
4. The case where TX is nef
Recall that a vector bundle E of rank r + 1 on X is numerically effective (nef) if the line bundle O(1) on P(E) is nef, i.e. c1(O(1)) · C ≥ 0 for all curves C in P(E).
In this case, the vector bundle E is big if and only if the volume of OP(E)(1) Z
P(E)
c1(OP(E)(1))n+r = Z
X
sn(E∗) > 0,
where sn(E∗) is the n-th Segre class of the dual of E [Laz04a, Section 2.2.C], [Ful98, Section 3.1]. The reader is cautioned that in [Ful98], P(E) denotes the projective bundle of lines in E.
The study of projective manifolds with nef tangent bundle was initiated in [CP91]. In general, if TX is nef, then the Chern classes
ci := ci(X) = ci(TX) ≥ 0.
In this case, we have (−KX)n= cn1 ≥ 0 and the inequality is strict if and only if X is Fano (i.e. −KX is ample) [CP93, Theorem 1.2]. Moreover, the Schur polynomial [Ful98, Example 12.1.7]
∆λ = det (cλi+j−i)1≤i,j≤n ≥ 0
for any partition λ = (λ1, λ2, . . . , λn) of n with n ≥ λ1 ≥ λ2 ≥ · · · ≥ λn≥ 0. Note that by [Ful98, Lemma A.9.2]
sn(TX∗) = ∆(1n), where (1n) = (1, 1, . . . , 1),
so TX is big if and only if ∆(1n) > 0. On the other hand, for a partition µ = (µ1, . . . , µn) of n with n ≥ µ1 ≥ · · · ≥ µn≥ 0, denote
cµ = cµ1 · cµ2 · · · cµn.
It follows from the Pieri’s formula that [Ful97, Section 2.2]
(4.1) cµ=X
λ
Kλµ∆λ,
where Kλµ is the Kostka number and the sum runs over all partitions of n. We note that Kλλ= 1 and that
Kλµ 6= 0 if and only if µ E λ (i.e. µ1+ · · · + µi ≤ λ1+ · · · + λi for all i).
Proposition 4.1. If TX is nef and big, then X is Fano.
Proof. Form the discussion above, it suffices to show cn1 = 0 =⇒ ∆(1n)= 0.
By (4.1),
cn1 = c(1n) =X
λ
Kλ(1n)∆λ = 0.
Since (1n) E λ for all partition λ, we have Kλ(1n) > 0 and hence ∆λ = 0 for all λ.
In particular, ∆(1n) = 0.
Remark 4.2. Proposition 4.1 provides new examples where DS is not simple, gen- eralizing the classical examples in [BGG72, Lev89].
Inspired by the proof of Proposition 4.1, we prove the following theorem that was conjectured in [CP91] and was verified for n = dim X ≤ 4 [CP93].
Theorem 4.3. Let X be a projective manifold with TX nef. If χ(OX) 6= 0, then X is Fano.
Proof. It suffices to show that cn1 = 0 implies χ(OX) = 0. By Hirzebruch–
Riemann–Roch [Ful98, Corollary 15.2.2], χ(OX) =
Z
X
tdn(TX).
Since the n-th Todd polynomial tdn(TX) of TX is a Q-linear combination of the cλ’s [Ful98, Example 3.2.4], we only have to show that cλ = 0 for all λ ∈ Λn. Again, it follows from (4.1) that
[cn1 = 0] =⇒ [∆λ = 0 for all λ ∈ Λn] =⇒ [cλ = 0 for all λ ∈ Λn].
Example 4.4. We give some explicit examples to illustrate (4.1).
1. When n = 1, c1 = ∆(1).
2. When n = 2, write λ1 = (1, 1) and λ2 = (2, 0). The Segre classes
∆(1,1)= c21− c2 = cλ1 − cλ2 and ∆(2,0) = c2 = cλ2. We have
(cλ1, cλ2) = (∆λ1, ∆λ2) ·1 0 1 1
.
3. When n = 3, write λ1 = (1, 1, 1), λ2 = (2, 1, 0) and λ3 = (3, 0, 0). We have
(cλ1, cλ2, cλ3) = (∆λ1, ∆λ2, ∆λ3) ·
1 0 0 2 1 0 1 1 1
.
4. When n = 4, write λ1 = (1, 1, 1, 1), λ2 = (2, 1, 1, 0), λ3 = (2, 2, 0, 0), λ4 = (3, 1, 0, 0), and λ5 = (4, 0, 0, 0). We have
(cλ1, cλ2, cλ3, cλ4, cλ5) = (∆λ1, ∆λ2, ∆λ3, ∆λ4, ∆λ5) ·
1 0 0 0 0 3 1 0 0 0 2 1 1 0 0 3 2 1 1 0 1 1 1 1 1
.
Conversely, we ask the following
Question 4.5. If X is Fano with nef TX, is it true that TX is big?
Using the classification in [CP91], Question 4.5 can be verified for n ≤ 3 by com- puting the Segre class ∆(1n) or quoting the known results.
(1) When n = 1, the only Fano curve is P1 and TP1 is ample.
(2) When n = 2, the only Fano surfaces with nef TX are P2 and P1×P1. These are toric varieties and hence have big tangent bundles.
(3) When n = 3, the only Fano threefolds with nef TX are P3, P1× P2, P1× P1 × P1, Q3, and P(TP2). The first three are toric varieties. The last two have big tangent bundles by [SCW04, Example 1,2].
A stronger version of Question 4.5 (dropping the nefness assumption) has to do with D-simplicity of the section ring S (see Question 5.2).
5. Questions on D-simplicity
Let Y be an affine algebraic variety over C. When Y is smooth, it is well-known that its ring of differential operators DY is simple and Noetherian. The example in [BGG72] shows that DY may not be simple nor Noetherian. On the other hand, it is shown in [LS89] that DY is simple if Y is a geometric invariant quotient of classical algebraic group acting on an affine space. It is asked in [LS89] whether the same result holds for other linear algebraic groups. See also Conjectures 1.1 and 1.2 of [SVdB97], in which the conjectures are proved in positive characteristics.
Unfortunately, in view of Proposition 3.1, Theorem 1.2 is not helpful in finding counterexamples to this problem.
The results in [LS89] also suggest that DY might have nice properties when Y has only rational singularities. Although this is not true in general, it was proposed in [Smi95] that Gorensteinness should play a role in answering this question. In fact, the main theorem in [Smi95] shows that in positive characteristics a domain is strongly F -regular if and only if it is F -split and D-simple. In view of the correspondence between strong F -regularity and log terminal singularities and the fact that Gorenstein log terminal singularities are rational singularities, it is natural to ask the following question.
Question 5.1. Let Y be an affine variety over C that is Gorenstein. Is it true that Y has at worst rational singularities if and only if OY is a simple DY-module?
By [Smi00a, Proposition 6.2(2)], a normal projective variety is Fano and has ra- tional singularities if and only if it admits a Gorenstein section ring with rational singularities. Therefore, Theorem 1.2 suggests the following question in Fano ge- ometry.
Question 5.2. Does there exist a smooth Fano variety over C whose tangent bundle is not big?
A positive answer to Question 5.2 will provide a counterexample to the necessary part of Question 5.1.
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Department of Mathematics, National Cheng Kung University. Tainan 70101, Taiwan.
E-mail address: jhsiao@mail.ncku.edu.tw
