SMOOTH PROJECTIVE SURFACES WITH PSEUDO-EFFECTIVE TANGENTBUNDLESJIA JIA, YONGNAM LEE, AND GUOLEI ZHONG

Abstract. Let S be a non-uniruled (i.e., non-birationally ruled) smooth projective surface. We showthat the tangent bundle TS is pseudo-eﬀective if and only if the canonical divisor KS is nef and thesecond Chern class vanishes, i.e., c2 (S) = 0. Moreover, we study the blow-up of a non-rational ruled
surface with pseudo-eﬀective tangent bundle.

1. Introduction
We work over the ﬁeld C of complex numbers. A smooth projective variety comes naturally equippedwith the tangent bundle, which is the dual of its sheaf of Kähler diﬀerentials, and the related properties ofsuch objects can be applied to classify algebraic varieties. A well-known theorem of Mori [Mor79] assertsthat if the tangent bundle of a smooth projective variety is ample, then it is a projective space, which
gives a solution to Hartshorne’s conjecture. Since then, the study of smooth projective varieties whose
tangent bundles admit some positivity properties has attracted a lot of attention and such properties areusually expected to impose strong restrictions on the geometry of the underlying varieties.
Deﬁnition 1.1. Let X be a smooth projective variety. Given a vector bundle E on X, we denoteby P(E) the Grothendieck projectivisation of E with OP(E) (1) denoting the relative hyperplane section
bundle. Recall that E is ample (resp. nef, big, pseudo-eﬀective) if OP(E) (1) is ample (resp. nef, big,
pseudo-eﬀective) on P(E).
Following the program of Campana and Peternell, a smooth Fano variety with nef tangent bundle
is conjectured to be a rational homogeneous space, and this conjecture has been intensively studied(see [CP91, DPS94, MOSCWW15, Kan17]). Starting from this aspect, it is natural to classify smoothprojective varieties with other positivity properties, e.g., with big or pseudo-eﬀective tangent bundles.
In the past few years, there are many beautiful results in this direction, especially when X is a smooth
Fano variety, i.e., the anti-canonical divisor −KX is ample. For example, Höring, Liu and Shao showedin [HLS22, Theorem 1.2] that the tangent bundle of a smooth del Pezzo surface (i.e., a Fano surface) of2
is big (resp. pseudo-eﬀective) if and only if d ≥ 5 (resp. d ≥ 4). We refer the reader
degree d := KX
to [FL22, GW20, HLS22, HL23, Hsi15, KKL22, Kim23, Liu23, Mal21, Sha23] and the references therein
for more information involving projective varieties with big tangent bundles.
As smooth projective varieties with big tangent bundles are known to be uniruled ([Miy87b, Corollary 8.6]; cf. [GW20, Proposition 4.6]), one may ask if there exist many non-uniruled projective varieties
sitting in the “boundary”, i.e., admitting a pseudo-eﬀective but non-big tangent bundle. Apart from thetrivial example of abelian varieties, a product of an abelian variety and any smooth projective variety
becomes another example coming to our mind (cf. Lemma 2.5). To the best knowledge of ourselves, up
to a ﬁnite étale cover, there seems no more other example which has been explored before. The main
result of our paper is to conﬁrm this expectation in dimension 2.
Theorem 1.2. Let S be a non-uniruled (i.e., non-birationally ruled) smooth projective surface. Thenthe following assertions are equivalent.
(1) The tangent bundle TS is pseudo-eﬀective;
2020 Mathematics Subject Classiﬁcation. 14E30, 14J26, 14J27, 14J60.
Key words and phrases. tangent bundle, pseudo-eﬀective, projective surface, elementary transformation.

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JIA JIA, YONGNAM LEE, AND GUOLEI ZHONG(2) S is minimal and the second Chern class (as an element in H 4 (S, Z)) vanishes, i.e., c2 (S) = 0.
Moreover, if one of the above equivalent conditions holds, then the Kodaira dimension κ(P(TS ), O(1)) =1 − κ(S) ∈ {0, 1}, and there is a ﬁnite étale cover S ′ → S such that S ′ is either an abelian surface or a
product E × F where E is an elliptic curve and F is a smooth curve of genus ≥ 2.
We refer the reader to [HP21, Proposition 1.10 and Theorem 1.12] for a more general result on
non-uniruled projective varieties with pseudo-eﬀective tangent bundles by using the foliation theory(see Remark 1.6). However, our result here is obtained by applying more elementary tools like surface
ﬁbration theory [Ser96], Maruyama’s elementary transformation method [Mar82] and the stability ofcotangent bundles [Bog79] (cf. [Eno87]), which are well-known to algebraic geometers.
From Theorem 1.2, the pseudo-eﬀectiveness of the tangent bundle forces the surface to be minimal,
i.e., the canonical divisor is nef. However, this is no longer true in the higher dimensional case (see Example 2.6). Besides, as the second Chern class of a smooth projective surface of general type is always
positive (see e.g. [BHPV04, VII, (2.4) Proposition]), our result excludes the possibility of a general typesurface having pseudo-eﬀective tangent bundle; see Proposition 3.2 and Remark 3.3 for the case of higher
dimensional varieties. As a consequence of Theorem 1.2, we obtain the following corollary.
Corollary 1.3. Let S be a non-uniruled smooth projective surface. If the tangent bundle TS is pseudoeﬀective, then there is some integer m such that H 0 (S, Symm TS ) ̸= 0; equivalently, the tautological line
bundle of P(TS ) is Q-linearly equivalent to an eﬀective divisor.
We note that Corollary 1.3 is known up to dimension 3 (see [HP21, Prposition 4.11 and Proof of
Theorem 1.12], [PT13, Theorem D] and [Tou08, Theorem 1.2]). In general, the pseudo-eﬀectiveness ofthe tangent bundle is conjecturally expected to give rise to the non-vanishing H 0 (X, Symm TX ) ̸= 0 forsome m-th symmetric power.
Before moving to the second part of this section, we give the following remark on a possible extensionof Theorem 1.2 to higher dimensional projective manifolds.
Remark 1.4. Let X be a non-uniruled smooth projective variety of dimension n.
(1) In the previous version of our paper, we asked whether such a variety X with vanishing top Chernclass ctop (X) ∈ H 2n (X, Z) and non-vanishing augmented irregularity q ◦ (X) has pseudo-eﬀective
tangent bundle TX . Here, the augmented irregularity q ◦ (X) is deﬁned to be the supremum of q(X ′ ) :=h1 (X ′ , OX ′ ) where X ′ → X runs over all the ﬁnite étale covers of X (see [NZ09, Deﬁnition 4.1]),
and the top Chern class of a product variety X = Y × Z satisﬁes ctop (X) = ctop (Y ) × ctop (Z).
(2) A recent paper [BL23] gives a negative answer to this question (see [BL23, Page 11]): in fact, if
one considers a product variety X = Y × Z where Y is a simply connected Calabi-Yau variety withvanishing top Chern class (see [KS00, Fig 1] for the existence of such a Calabi-Yau threefold) and
Z is of general type with q(Z) > 0, then the tangent bundles TY and TZ are not pseudo-eﬀective(see [HP19, Theorem 1.6], [Dru18, Corollary 6.5] and Remark 3.3); hence TX is not pseudo-eﬀective,
either (see [BL23, Lemma 4]).
(3) On the other hand, by [Lie78], if the variety X admits a global holomorphic vector ﬁeld, then up to aﬁnite étale cover, it will split into a product of an abelian variety and a projective variety admitting
no global holomorphic vector ﬁeld, in which case, the top Chern class ctop (X) vanishes and theirregularity q(X) is positive. We refer to [BL23, Proposition 10] for a further extension in this line.
In [Kim23], Kim proved that a projective bundle PC (E) over a smooth curve C has a big tangent
bundle if and only if either C ∼= P1 or E is not semi-stable. Indeed, as kindly pointed out by Kim,
according to the proof of [Kim23], any projective bundle (of arbitrary rank) over a smooth projective
curve always has a pseudo-eﬀective tangent bundle; see Proposition 6.1. We slightly summarise the resultin Section 6. In terms of the non-rational ruled surface, by analysing the position of a single blow-up, weshow the following proposition. In particular, compared with the non-uniruled surfaces, there does exist
a non-relatively-minimal non-rational uniruled surface with pseudo-eﬀective but non-big tangent bundle.

TANGENT BUNDLES

3

Proposition 1.5. Let f : S = PC (E) → C be a P1 -bundle over a smooth curve C with the genus g(C) ≥ 1.
Suppose the tangent bundle TS is pseudo-eﬀective but not big. Then the blow-up of S along a point p hasa pseudo-eﬀective tangent bundle if and only if there exist some positive integer m and some line bundle⊗m
L which is numerically equivalent to the relative tangent bundle TS/C such that H 0 (S, mm) ̸= 0,p ⊗Lwhere mp is the maximal ideal of the local ring OS,p .
Making Proposition 1.5 as an initial point, we would like to give a rather clean description of nonrational uniruled projective surfaces admitting pseudo-eﬀective but non-big tangent bundles.
We summarise the organisation of our paper. In Section 2, we prepare some preliminary results for theconvenience of later use. In Section 3, we prove Theorem 1.2 for the cases κ(S) = 0 and 2. In Section 4,
we study the case κ(S) = 1 when S is minimal, and the minimality of such S with pseudo-eﬀective
tangent bundle will be shown in Section 5. Finally, we prove Proposition 1.5 in Section 6.
Let us end up the introduction with the following remark on the comparison with other papers.
Remark 1.6. In [Mat22, Theorem 1.1], Matsumura establishes the minimal model program of a projective klt variety with strongly pseudo-eﬀective (reﬂexive) tangent sheaf and ends up with a quasi-étale
quotient of an abelian variety; cf. [HIM22, Theorem 1.1]. Here, a vector bundle E on a smooth projective variety X is said to be strongly pseudo-eﬀective if OP(E) (1) is pseudo-eﬀective and the restrictedbase locus does not dominate X (see [BDPP13, Deﬁnition 7.1]). Since our assumption is weaker than
the assumption in [Mat22], even in the surface case, we could not expect to obtain a similar result
like [Mat22, Theorem 1.1]. The main reason is that, when running the minimal model program, although our (weak) pseudo-eﬀectiveness of the tangent bundle descends along any birational contraction
or ﬂips (see Lemma 2.5), it could not be preserved under a Fano contraction any more (see [Kim23] andProposition 1.5; cf. [Mat22, Proposition 3.2]).
After the present paper was ﬁnished, we were informed by Professor Andreas Höring on his previous
joint work [HP21] which gives a splitting structure of the tangent bundles of non-uniruled projective
manifolds which are pseudo-eﬀective by using a deep foliation theory (see [HP21, Proposition 1.10]). As
a consequence of [HP21], Theorem 1.2 follows. Compared with [HP21], the proof of our result here is moreelementary and more self-contained to algebraic-geometers. Furthermore, based on [Ser96] and [HP20],
the techniques we summarized and developed in this paper (see e.g. Notation 2.2 and Notation 4.2) havetheir own independent interests for the study of other positivity of tangent bundles of projective surfaces
equipped with a ﬁbration structure (see [IMZ23, Claim 6.7 and Question 7.4]).
Acknowledgements. We would like to thank Professor Andreas Höring for informing us his previous
joint work [HP21] and the valuable comments, and Doctor Jeong-Seop Kim for improving Proposition 6.1
and useful comments. We would also like to thank the referee for the very careful reading and suggestions
to improve the paper. The ﬁrst author is partially supported by a President’s Graduate Scholarship from
NUS and partially supported by Shuimu Tsinghua Scholar Program and China Postdoctoral Science
Foundation (2023TQ0172) The second and the third authors are supported by the Institute for Basic
Science (IBS-R032-D1-2022-a00).
2. PreliminaryFirst of all, we ﬁx the following notation throughout the paper.
Notation 2.1. Let X be a projective variety.
(1) The symbol ∼ (resp. ∼Q , ≡) denotes the linear equivalence (resp. Q-linear equivalence, numericalequivalence) on Cartier divisors (resp. Q-Cartier divisors). Let f : X → Y be a morphism of projective
varieties. We denote by ∼f the relative (or f -) linear equivalence of Cartier divisors, i.e., for two
Cartier divisors D1 and D2 on X, D1 ∼f D2 if and only if there is some Cartier divisor E on Y suchthat D1 − D2 ∼ f ∗ E.
(2) Denote by NS(X) the Néron-Severi group of X. Let N1 (X) := NS(X) ⊗Z R be the space of RCartier divisors modulo numerical equivalence and ρ(X) := dimR N1 (X) the Picard number of X.

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JIA JIA, YONGNAM LEE, AND GUOLEI ZHONGLet N1 (X) be the dual space of N1 (X) consisting of 1-cycles. Denote by Nef(X) (resp. PE(X)) thecone of nef divisors (resp. pseudo-eﬀective divisors) in N1 (X) and NE(X) the dual cone consistingof pseudo-eﬀective 1-cycles in N1 (X). In particular, when X is a smooth projective surface, we have
N1 (X) = N1 (X) and NE(X) = PE(X).
(3) For a smooth projective variety X, we denote by KX the canonical divisor and κ(X) = κ(X, KX )the Kodaira dimension of X.
(4) Let f : X → C be a surjective morphism between normal projective varieties. We say that f is aﬁbration if f∗ OX = OC or equivalently, the general ﬁbre of f is connected. A ﬁbration is said to
be isotrivial if all the smooth ﬁbres are isomorphic to each other; otherwise, we say that it is nonisotrivial. We say that f is trivial if there exist another projective variety F and an isomorphismX∼= F × C such that f is the natural projection. We say that f is locally trivial (or a ﬁbre bundle)
if each point c ∈ C is contained in a small open neighbourhood U having the property that f −1 (U )is trivial over U . See Notation 2.2 for a more detailed description on surface ﬁbrations.
(5) Let H be a nef and big divisor on X. Let F be a torsion free coherent sheaf on X. The slope of F
with respect to H is deﬁned to be the rational numberµH (F) :=c1 (F) · H dim(X)−1
rank(F)

where c1 is the ﬁrst Chern class. A torsion free coherent sheaf E is said to be µ-semi-stable if for any
non-zero subsheaf F ⊆ E, the slopes satisfy the inequality µH (F) ≤ µH (E).
Next, we recall and develop some basic properties on the ﬁbration of surfaces. Most of them comefrom [Ser96, Section 3] and [HP20, Section 5].
Notation 2.2 (Surface ﬁbration).
(1) Let S be a smooth projective surface and f : S → C a ﬁbration with F a general ﬁbre of f .
(2) For a closed point c ∈ C, we denote by Sc its ﬁbre over c.
(3) Let W := P(TS ) be the Grothendieck projectivisation of the tangent bundle of S with π : W → S
the natural projection, and let ξ := OW (1) be the corresponding tautological class.
(4) Let {νi Ei }i∈I be the set of all components of non-multiple ﬁbres of f , where the νi ’s denote theircorresponding multiplicities within their ﬁbres. Let {mj Fj }j∈J be the set of multiple ﬁbres of f .
P
Clearly, for each j ∈ J and c ∈ C, we have mj Fj ≡ i∈I,f (Ei )=c νi Ei ≡ F .
(5) LetXX
X
X
E = ES :=f ∗ c − (f ∗ c)red =(νi − 1)Ei +(mj − 1)Fj = E0 +(mj − 1)Fj
c∈C

i∈I

j∈J

j∈J

P

where E0 := i∈I (νi − 1)Ei comes from the non-multiple non-reduced ﬁbres. This decomposition isP
indeed the Zariski decomposition where j∈J (mj − 1)Fj is the nef part and E0 is the ﬁxed part.
(6) Applying [Ser96, Section 3], we have an exact sequence
0 −→ TS/C −→ TS −→ f ∗ TC −→ F −→ 0,
where TS , TC are the tangent bundles of S and C respectively, F is a torsion sheaf and TS/C := Ω∨S/C isthe relative tangent sheaf of f , which is locally free. Let JS/C be the torsion-free image of TS → f ∗ TC .
(7) By [Ser96, (3.0.3) and Proposition 3.1], we have−1∨∨
JS/C = JS/C
⊗ IΓ = KS−1 ⊗ TS/C
⊗ IΓwhere IΓ is an ideal sheaf and the support of Γ consists of points s ∈ S such that the reduced
structure f −1 (f (s))red is singular at s. In addition, we have TS/C = −KS + f ∗ KC + E where E isdeﬁned in (5). In particular, we have the following short exact sequence(†)

0 −→ TS/C −→ TS −→ (−f ∗ KC − E) ⊗ IΓ −→ 0.

TANGENT BUNDLES

5(8) Let Y := PS ((−f ∗ KC − E) ⊗ IΓ ) ∼= PS (IΓ ) ⊆ W which is a prime divisor on W . Note that Y isisomorphic to the blow-up of S along the ideal sheaf IΓ since Γ is locally generated by a regularsequence (see [Ser96, Proposition 3.1 (iii)] and [HP20, § 3.10]). However, since the length(OΓ,s ) at
each point s ∈ Γ is not necessarily 1, such Y in general is not necessarily smooth.
(9) Denote by Exc(π|Y ) the exceptional divisor of π|Y . By the short exact sequence (†) and [Har77,
Chapter II, Lemma 7.9 and Proposition 7.13], we have ξ|Y = OW (1)|Y = OY (1) where
OY (1) = OY (− Exc(π|Y )) ⊗ (π|Y )∗ (−f ∗ KC − E).
Moreover, from [BHPV04, IV, (10.5) Lemma] and its proof, we know that ξ − Y ∼ π ∗ TS/C and(π|Y )∗ OY (1) = (−f ∗ KC − E) ⊗ IΓ .
Now, we recall Maruyama’s elementary transformation which will be heavily used in our proofs.
Notation 2.3 (Elementary transformation for projective bundles over surface blow-ups).
(1) Let h : S2 → S1 be a blow-up between smooth projective surfaces with the exceptional (−1)-curvei : D ,→ S2 . Let ξi be the tautological divisor of the projective bundle P(TSi ) and πi : P(TSi ) → Si
the natural projection.
(2) There is a natural short exact sequence
0 −→ TS2 −→ h∗ TS1 −→ i∗ TD (D) −→ 0.
∗
e := PD (TD (D)) is a projective subbundle of D′ := πHere, TD (D) ≃ OD (1) and D
f1 D = PD (h∗ TS1 )deﬁned by the restriction of the following exact sequence

0 → K = OD (−1) → h∗ TS1 |D → OD (1) → 0,
where the kernel K = det(h∗ TS1 |D ) ⊗ OD (1)∨ = OD (−1) by the projection formula.
(3) Applying Maruyama’s elementary transformation [Mar82, Theorem 1.4 and (1.7)], we have the fole and α is the blow-down
lowing commutative diagram, where β is the blow-up of P(h∗ TS1 ) along D∗∗′
of BlDe (P(h TS1 )) along the β-strict transform of D = π
f1 D.

P(TS1 ) o

e
h

π
f
1

π1


S1 o

BlDe (P(h∗ TS1 ))
NNN
o
NNNα
β ooo
o
NNN
oo
o
NN'
o
woo
∗
_
_
_
_
_
_
_
_
_
_
_
_/ P(TS )P(h TS1 )
2

h


S2

π2


S2

(4) By [Mar82, Theorem 1.4 and (1.7)], we have β ∗ eh∗ ξ1 ∼ α∗ ξ2 + G where G is the β-exceptional divisor.
ee(5) The β-blown-up section D satisﬁes D ∼ c0 + e in D′ where c0 = eh∗ ξ1 |D′ is a tautological divisor of′e ̸⊆ eD → D and e is a ﬁbre (see [Har77, Chapter V, Proposition 2.6]). In particular, Dh∗ Y1 for any
section Y1 ∈ |ξ1 | by the above linear equivalence.
In what follows, we collect several results to be used in the subsequent sections.
Lemma 2.4 ([HLS22, Lemma 2.2]; cf. [Dru18, Lemma 2.7]). Let X be a projective variety, E a vector
bundle on X, and H a big Q-Cartier Q-divisor on X. Then E is pseudo-eﬀective if and only if for allc > 0 there exist suﬃciently divisible integers i, j ∈ N such that i > cj andH 0 (X, Symi E ⊗ OX (jH)) ̸= 0.
Lemma 2.5. Let E ⊆ F be an injection between two vector bundles over a projective variety X. If E ispseudo-eﬀective, then so is F.

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JIA JIA, YONGNAM LEE, AND GUOLEI ZHONG

Proof. Fix a big Q-Cartier Q-divisor H on X. By Lemma 2.4, we only need to show for all c > 0, there
exist suﬃciently divisible integers i, j ∈ N such that i > cj andH 0 (X, Symi F ⊗ OX (jH)) ̸= 0.
This follows from the injection Symi E ⊆ Symi F and the pseudo-eﬀectiveness of E (see Lemma 2.4). □As a consequence of the above lemma, a typical diﬀerence between nefness and pseudo-eﬀectivenessis that the quotient bundle of a pseudo-eﬀective vector bundle is not pseudo-eﬀective any more. For
example, a rank two vector bundle E = O ⊕ O(−1) over a smooth rational curve is pseudo-eﬀective byLemma 2.5 while its quotient O(−1) is not (cf. [Mat22, Proposition 3.4]).
The following example shows that we could not expect the variety equipped with pseudo-eﬀective
tangent bundle to be minimal in the higher dimensional case.
Example 2.6. Let X := E × S be a product of an elliptic curve E and a non-minimal smooth projective
surface S which contains some (−1)-curve. Let p : X → E be the natural projection. Applying Lemma 2.5
and considering the natural injection 0 → p∗ OE → TX , we see that TX is pseudo-eﬀective. However, it
is clear that KX is not nef.
Lemma 2.7 ([HLS22, Corollary 2.4]). Let π : X ′ → X be a birational morphism between smooth projective varieties. If the tangent bundle TX ′ is pseudo-eﬀective, then so is TX .
We end up this section with the following lemma, which will be used to conﬁrm the existence ofsections of some symmetric power Symm TS in Theorem 1.2 by taking an étale base change.
Lemma 2.8 ([Uen75, Theorem 5.13]). Let f : X → Y be a surjective morphism of projective varietiesand D a Cartier divisor on Y . Then κ(Y, D) = κ(X, f ∗ D).
3. The case κ(S) = 0 or 2
In this section, we prove Theorem 1.2 when the Kodaira dimension κ(S) = 0 or 2. Let us begin witha smooth projective surface of Kodaira dimension zero.
Proposition 3.1. Let S be a smooth projective surface of κ(S) = 0. Suppose that the tangent bundle TS
is pseudo-eﬀective. Then KS is nef, i.e., S is minimal. Moreover, S is an étale quotient of an abeliansurface and thus the second Chern class c2 (S) = 0 vanishes. In this case, κ(P(TS ), O(1)) = 1.
Proof. First, assuming the nefness of KS , we prove the second half of the proposition. From the abundance, we know that KS ≡ 0. Applying [BHPV04, VI, Table 10] to S, we see that S has to be one of
the following: an Enriques surface, a bi-elliptic surface, a K3 surface or an abelian surface. Since thetangent bundles of Enriques surfaces and K3 surfaces are not pseudo-eﬀective (see [Nak04, Chapter VI,
Theorem 4.15], or more generally, [HP19, Theorem 1.6]), our S is covered by an abelian surface. For⊕2
a ﬁnite étale morphism π : A → S from an abelian surface A, we have π ∗ TS = TA = OA. The laststatement follows immediately from Lemma 2.8 and the fact that h0 (A, Symm TA ) = m + 1 for eachpositive integer m.
Now, we prove the nefness of KS . In the view of Lemma 2.7, we only need to exclude the casewhen S := S2 is a blow-up of a smooth minimal surface S1 with KS1 ≡ 0. Suppose to the contrary
that the tangent bundle TS2 is pseudo-eﬀective. By the ﬁrst paragraph of our proof, there is an étale
cover S1′ → S1 from an abelian surface S1′ . Let S2′ := S1′ ×S1 S2 be the ﬁbre product, which is also asmooth projective surface with κ(S2′ ) = 0. Note that TS2′ (as the pullback of TS2 ) is also pseudo-eﬀective.
Replacing Si with Si′ , we may assume that S1 is an abelian surface and thus TS1 ≃ OS⊕2.
1
Let us consider the elementary transformation in Notation 2.3 and use the notation therein. Note
f1 ,that the tautological divisors ξ1 and eh∗ ξ1 can be taken as irreducible horizontal sections Y1 and Yff
and both of them are nef. Note also that Y1 |Y1 ≡ 0 and Y1 |Yf1 ≡ 0, since Y1 (resp. Y1 ) is a ﬁbre ofP(TS1 ) ∼= S1 × P1 → P1 (resp. P(h∗ TS1 ) ∼= S2 × P1 → P1 ). Moreover, ξ2 is pseudo-eﬀective if and only if
so is α∗ ξ2 = β ∗ eh∗ ξ1 −G (cf. e.g. [Nak04, Chapter II, Lemma 5.6]). On the other hand, since β ∗ eh∗ ξ1 −G isTANGENT BUNDLES

7pseudo-eﬀective and eh∗ ξ1 is nef, by taking a suﬃciently ample divisor A on P(h∗ TS1 ) such that β ∗ A − G
is ample (see e.g. [KM98, Proposition 1.45]), we have0 ≤ (β ∗ eh∗ ξ1 − G) · (β ∗ A − G) · β ∗ eh∗ ξ 1= (β ∗ eh∗ ξ 1 · β ∗ A − β ∗ eh∗ ξ 1 · G − G · β ∗ A + G 2 ) · β ∗ eh∗ ξ 1= G2 · β ∗ eh∗ ξ1 = β∗ (G|G ) · eh∗ ξ 1 .
e is a ruled surface and G|G ∼β OG (1) ∼β −C0 where C0 is some horizontal section,
Recall that G → D
e = −1 < 0h∗ ξ1 = −eh∗ ξ 1 · Dnoting that G|G may contain some β-ﬁbre component. In particular, G2 · β ∗ e(see Notation 2.3), which gives us the desired contradiction.
□
In the higher dimensional case, by the Beauville-Bogomolov decomposition [Bea03], if a smooth projective variety X has numerically trivial canonical divisor KX ≡ 0, then up to an étale cover, it splits into
a product of an abelian variety, a simply connected (strict) Calabi-Yau variety and a simply connected
symplectic variety. In this case, it follows from [HP19, Theorem 1.6] and [Nak04, Chapter II, Lemma
5.6] that TX is pseudo-eﬀective if and only if the augmented irregularity q ◦ (X) is positive. However,as we observed in Example 2.6, in the higher dimensional case, the pseudo-eﬀectiveness of the tangent
bundle of a non-uniruled smooth projective variety does not imply that it is minimal.
In the second part of this section, we deal with the case when the variety is of general type.
Proposition 3.2. Let S be a smooth projective surface of general type. Then TS is not pseudo-eﬀective.
Proof. By Lemma 2.7, we only need to exclude the case when S is minimal (and hence KS is nef)and has pseudo-eﬀective tangent bundle. By the semi-stability of the cotangent bundle ΩS (see [Bog79,
Section 13.1]), the tangent bundle TS is also semi-stable with respect to KS (cf. [Eno87, Corollary 1.2]).
We claim that
H 0 (S, Symi TS ⊗ OS (jKS )) = 0,
for any i > 2j. Suppose the claim for the time being. Then it follows from Lemma 2.4 that TS is not
pseudo-eﬀective which concludes our proposition.
To prove the claim, suppose to the contrary that H 0 (S, Symi TS ⊗ OS (jKS )) ̸= 0 for some i > 2j. Fix
a non-zero section s of Symi TS ⊗ OS (jKS ) which deﬁnes an injection 0 → OS → Symi TS ⊗ OS (jKS ).
On the other hand, we havec1 (Symi TS ⊗ OS (jKS )) = c1 (Symi TS ) + rank(Symi TS ) · c1 (OS (jKS ))i(i + 1)
· c1 (TS ) − (i + 1) · j · c1 (TS )2i− j c1 (S)= (i + 1) ·2=and hence c1 (Symi TS ⊗ OS (jKS )) · c1 (KS ) < 0. This contradicts the semi-stability of TS and our claim
is thus proved.
□
Indeed, with the same argument as above, we get the following result in the higher dimensional case.
Remark 3.3. Let X be a normal Q-factorial projective variety which is of general type and has at worst
klt singularities. We shall show that the reﬂexive tangent sheaf TX := Ω∨X is not pseudo-eﬀective inthe sense of [HP20, Deﬁnition 3.5]. In the view of [HP20, Corollary 4.3] and Lemma 2.4, after running
the special minimal model program with scaling by [BCHM10], we may assume that X is minimal, i.e.,
KX is nef and big. By the semi-stability of the tangent sheaf TX with respect to KX (see [Eno87,
Corollary 1.2]), we have
H 0 (X, Sym[i] TX ⊗ OX (jKX )) = 0,
for i > dj, where d := dim(X) and Sym[i] TX is the reﬂexive hull of the symmetric power. By [HP20,
Deﬁnition 3.5] and a similar calculation as in Proposition 3.2, TX is not pseudo-eﬀective. Consequently,
the tangent bundle of any hyperbolic smooth projective variety is never pseudo-eﬀective.

8

JIA JIA, YONGNAM LEE, AND GUOLEI ZHONG

4. Relatively minimal elliptic fibration, the case κ(S) = 1
In this section, we study Theorem 1.2 for the case κ(S) = 1. With the further assumption that S is
minimal, we obtain the following theorem as our main result of this section.
Theorem 4.1. Let f : S → C be a relatively minimal elliptic ﬁbration from a smooth projective surface
S of Kodaira dimension κ(S) = 1. Then the following assertions are equivalent.
(1) The tangent bundle TS is pseudo-eﬀective;(2) The second Chern class vanishes, i.e., c2 (S) = 0;
(3) f is almost smooth, i.e., the only singular ﬁbres are multiples of smooth elliptic curves.
We stick to Notation 2.2 and the following additional notation throughout this section.
Notation 4.2.
(1) We use the same notation as in Notation 2.2. In addition, S is further assumed to be a smoothprojective surface of Kodaira dimension κ(S) = 1 and f is a relatively minimal elliptic ﬁbration, i.e.,free of (−1)-curves among the ﬁbres of f . In fact, such S is a minimal surface, i.e., KS is nef andc1 (S)2 = 0; see Lemma 4.3.
fc over c as follows:(2) For each point c ∈ C, we deﬁne the normalised ﬁbre S
 


X
X
1
fc := 1Se(Fj )mj Fj +1 − 1 − e(Sf (Ei ) ) νi Ei ,1212
j∈J,f (Fj )=ci∈I,f (Ei )=c

where e(`) is the Euler number of a curve `. It is clear that if Sc is a multiple of a smooth elliptic
curve, then the normalised multiple ﬁbre will vanish since e(Sc ) = 0 in this case. We shall see infc are Q-eﬀective (see Remark 4.9 for a more detailed
Lemma 4.6 that all of the normalised ﬁbre S
description).
(3) By the canonical bundle formula for relatively minimal elliptic ﬁbrations [BHPV04, V, (12.1) Theorem], we haveXωS = OS (KS ) = f ∗ (ωC ⊗ (R1 f∗ OS )∨ ) ⊗ OS
(mj − 1)Fj ,j∨where deg(R f∗ OS ) = χ(OS ) (see e.g. [BHPV04, V, (12.2) Proposition]). By Noether’s formula,χ(OS ) = c2 (S)/12, noting that c1 (S)2 = 0.
(4) For the relatively minimal elliptic ﬁbration f : S → C, recall the following invariant (see e.g. [BHPV04,
V, (12.5) Proposition])Xδ(f ) := χ(OS ) + (2g(C) − 2 +(1 − m−1
j )).
1

j∈J

Then κ(S) = 1 is equivalent to δ(f ) > 0.
Lemma 4.3. Let S → C be an elliptic ﬁbration which is relatively minimal. Suppose that κ(S) ≥ 0.
Then S is minimal, i.e., the canonical divisor KS is nef.
Proof. Note that κ(S) ≤ 1 since S → C is an elliptic ﬁbration. Let Sm be the minimal model of S. By
Notation 4.2 (3), we know that KS2 = 0. If S → S1 contracts some (−1)-curve, then KS21 > 0. Inductively,
KS2m > 0 and thus KSm is nef and big, a contradiction to κ(S) ≤ 1.
□
We recall the following lemma for the convenience of later proofs.
Lemma 4.4 ([BHPV04, III, (18.2) Theorem and V, (12.1) Theorem]). For a relatively minimal elliptic
ﬁbration f : S → C, one has χ(S, OS ) ≥ 0 with the equality holds if and only if f is isotrivial and all the
singular ﬁbres are multiple of smooth elliptic curves (i.e., f is almost smooth).
Lemma 4.5. We have the following linear and numerical equivalences
c2 (S)
F ).
12
In particular, if c2 (S) = 0, then the tangent bundle TS is pseudo-eﬀective.
ξ ∼ Y + π ∗ TS/C ≡ Y + π ∗ (E0 −TANGENT BUNDLES

9Proof. By the canonical bundle formula and Notation 2.2 (5), we haveXωS = OS (KS ) = f ∗ (ωC ⊗ (R1 f∗ OS )∨ ) ⊗ OS
(mj − 1)Fj = f ∗ (ωC ⊗ (R1 f∗ OS )∨ ) ⊗ OS (E − E0 )j∈J∨where deg((R f∗ OS ) ) = c2 (S)/12. By Notation 2.2 (5) and (7), we have TS/C ≡ E0 − c2 (S)F/12.
Together with Notation 2.2 (9), our lemma is thus proved.
□
1

In what follows, we show the normalised ﬁbres deﬁned in Notation 4.2 (2) are all Q-eﬀective. We refer
the reader to Remark 4.9 for a more detailed description.
Lemma 4.6. The divisor −TS/C ≡ c2 (S)F/12 − E0 is pseudo-eﬀective. In particular, the normalisedﬁbres deﬁned in Notation 4.2 (2) are all Q-eﬀective.
Proof. Applying [BHPV04, III, (11.4) Proposition], we have
1 X

X
1 X
c2 (S)F − E0 =e(Sc ) F − E0 ≡e(Sc )Sc −(νi − 1)Ei
12
12
12
c∈C
c∈C
i∈I
X
1 X
1 X
=
e(Fj )mj Fj +e(Sf (Ei ) )νi Ei −(νi − 1)Ei
12
12
j∈J
i∈I
i∈I

 
X
X
1 X
1
fc .
=
e(Fj )mj Fj +1 − 1 − e(Sf (Ei ) ) νi Ei =
S
12
12
j∈J

i∈I

c∈C

By the non-negativity of e(Fj ) ≥ 0, we only need to verify that for each non-multiple non-reduced
singular ﬁbre Sf (Ei ) with i ∈ I, we have1
1 − 1 − e(Sf (Ei ) ) νi ≥ 0.
12
We refer to the Kodaira’s table [BHPV04, p. 201] and check case by case, noting that the Euler number
for a singular ﬁbre of Types I∗0 , I∗b , II∗ , III∗ or IV∗ is 6, b + 6, 10, 9 or 8, respectively, and the multiplicity
of an irreducible component in a singular ﬁbre of Types I∗0 , I∗b , II∗ , III∗ or IV∗ is no more than 2, 2, 6, 4□
or 3, respectively (see [Mir89, Table IV.3.1]).
To prove Theorem 4.1, we ﬁrst treat the case when f is non-isotrivial. The proof of the following
proposition is inspired by [HP20, Proposition 5.4].
Proposition 4.7. Suppose that f is non-isotrivial. Then the tangent bundle TS is not pseudo-eﬀective.
Proof. Suppose to the contrary that TS is pseudo-eﬀective. Since f is non-isotrivial, there is a non-zero
Kodaira-Spencer class which induces the following unique non-trivial extension on an elliptic curve
0 −→ OF −→ TS |F ∼= F2 −→ OF −→ 0.
Note that the above short exact sequence is the restriction of
0 −→ TS/C −→ TS −→ f ∗ TC −→ F −→ 0
to a general ﬁbre F (cf. Notation 2.2). Let WF := π −1 (F ) = P(TS |F ), and CF := Y ∩ WF which isthe section of π|WF associated with the surjection TS |F → OF → 0. Since ξ is pseudo-eﬀective, by thedivisorial Zariski decomposition [Bou04, Theorem 3.12], we haveXξ≡ai Yi + P
(1)
where Yi are ﬁnitely many prime divisors, ai > 0, and P is a modiﬁed nef R-divisor (in the sense that
P |D is pseudo-eﬀective for every prime divisor D on W ) (see [Bou04, Proposition 2.4]). Restricting thedecomposition (1) to WF , we haveX
CF ∼ ξ|WF ≡a i Y i | W F + P | WF .
i

Here, we can choose suﬃciently general F such that Yi ∩ WF (if non-zero) are mutually distinct, notingthat some of Yi come from the pull-back of components of E on S and hence have no intersection with10

JIA JIA, YONGNAM LEE, AND GUOLEI ZHONG

WF . Since the tautological section CF is extremal in WF but P |WF is still numerically movable by notingthat {WF }F is a free family, we have P |WF = 0. Without loss of generality, we may assume that a1 = 1,
Y1 ∩ WF = CF and Yi ∩ WF = 0 for i ̸= 1. Now that Y1 ∩ WF = CF = Y ∩ WF for every suﬃcientlyP
general ﬁbre F of f , we have Y1 = Y . In particular, π ∗ TS/C ≡ ξ − Y ≡ i≥2 ai Yi + P is pseudo-eﬀective
by Lemma 4.5. By Lemma 4.6 and the Zariski decomposition on the surface S, we have E0 = 0 and□hence c2 (S) = 0, a contradiction to Lemma 4.4.
In the remaining part of this section, we shall deal with the isotrivial case, which is more troublesome.
Let us ﬁrst recall the following lemma.
Lemma 4.8 (see e.g. [PS20, Lemma 3.2]). Let f : S → C be a relatively minimal isotrivial elliptic
ﬁbration. Then the singular ﬁbre of f is either a multiple of smooth elliptic curves (of Type m I0 ) or a
non-multiple ﬁbre not being of Types Ib or I∗b for b ≥ 1.
Before moving into the proof of Theorem 4.1 for the isotrivial case, we do some preparations from
Remark 4.9 to Lemma 4.12 where most of the calculations involved might be known to experts. However,as we were not able to ﬁnd an explicit reference, we formulate them in a way which is convenient for us
to prove Theorem 4.1; we also present the explanations for the convenience of the reader and our futureuse. First, Remark 4.9 below gives a detailed computation in terms of the normalised ﬁbres and theirlocal equations.
fc (cf. Notation 4.2,
Remark 4.9. For every point c ∈ C, one can calculate the normalised singular ﬁbre S
Lemma 4.6 and [Mir89, Table IV.3.1]) as follows.
 


X
X
1
1
fc =
S
e(Fj )mj Fj +1 − 1 − e(Sf (Ei ) ) νi Ei
1212
j∈J,f (Fj )=ci∈I,f (Ei )=c1II 6 e11
 4 (e1 + e2 )
III




1

IV

 3 (e1 + e2 + e3 )1
= 2 (e2 + e3 + e4 + e5 )
I∗0



 56 e1 + 23 (e2 + e9 ) + 12 (e3 + e7 ) + 13 (e4 + e8 ) + 16 e5 II∗




1
1
3

III∗

4 (e1 + e8 ) + 2 (e2 + e5 + e7 ) + 4 (e3 + e6 )


2
1
IV∗
3 (e1 + e3 + e5 ) + 3 (e2 + e4 + e6 )For every point s ∈ Γ, we may use x, y as regular parameters of OS,s . The ideal sheaf of the singularpoints on each ﬁbre could be chosen as below (see [Ser96, Proof of Proposition 3.1]).
Table 1. Local equation of each ﬁbre
I∗0

II∗

III∗

type of Sc

II

III

IV

local equation of Sc

x3 = y 2

x4 = y 2x3 = y 3xνi y νj = 0

IΓ,s(x2 , y)(x3 , y)(x2 , y 2 )(x, y)IV∗

The following observation plays a signiﬁcant role to the proof of Theorem 4.1. Once f is not almost
smooth and if we assume ξ is pseudo-eﬀective, then the prime divisor Y would appear in the negative
part of the divisorial Zariski decomposition of ξ.
Lemma 4.10. Let f be an isotrivial ﬁbration. If c2 (S) > 0, then ξ|Y is not pseudo-eﬀective.
Proof. By Notation 2.2 (9), ξ|Y ∼ − Exc(π|Y ) − (π|Y )∗ (f ∗ KC + E). If the genus g = g(C) ≥ 1, thenf ∗ KC + E ≡ (2g − 2)F + E ≥ 0 is eﬀective. Since F is free and (π|Y )∗ E is the sum of the propertransform and some (π|Y )-exceptional curve, it follows that (π|Y )∗ (f ∗ KC + E) is eﬀective. In particular,

TANGENT BUNDLES

e2

11

e3

e4

e5

e1
2e1

I∗0

I0
2e2

6e6

e1

4e4

pt
e1

II

2e9

3e3

5e5
II∗

3e7

4e8

2e2

2e7

e1
4e4

e2

III

e1

3e3

2e5

3e6

e8

III∗
e1

e2

e1
e3

e3

e5

3e7
2e2

IV

2e4

IV∗

2e6

Figure 1. Singular ﬁbre of an isotrivial elliptic ﬁbration
ξ|Y is anti-pseudo-eﬀective. Suppose that ξ|Y is pseudo-eﬀective. Then we have ξ|Y ≡ 0, which in turnimplies g(C) = 1 and Exc(π|Y ) = 0. In other words, Γ = ∅, i.e., the reduced structure of every ﬁbre issmooth. This leads to a contradiction to χ(S, OS ) = c2 (S)/12 > 0 (see Lemma 4.4).
In the following, we may assume that C ≃ P1 . By [HP20, Corollaries 3.13 and 3.19], ξ|Y being notpseudo-eﬀective is equivalent to the vanishing
H 0 (S, IΓk ⊗ OS (−k(f ∗ KC + E))) = H 0 (S, IΓk ⊗ OS (k(KS − f ∗ KC − E − KS ))) = 0
for all k ∈ N. Since C ≃ P1 , by the canonical bundle formula, KS − f ∗ KC − E ∼ c2 (S)F/12 − E0 . Notethat KS = δ(f )F with δ(f ) > 0 (see Notation 4.2 (4)). Since f is isotrivial, by Lemma 4.8, the only
multiple ﬁbres are of Type m I0 , and hence has Euler number zero. Then Xe(Sf (Ei ) )c2 (S)F − E0 =1− 1−
ν i Ei .
1212
i∈I

Now we obtain∗

KS − f KC − E ∼ Q

X
i∈I



e(Sf (Ei ) )
1− 1−
12

 
X
fc .
ν i Ei =S
c∈D

Here, the set D ⊆ C consists of the point c such that the reduction of the scheme theoretic ﬁbre Sc is
not an elliptic curve. We are left to show that for every s ∈ Γ and every k ∈ N, the local equation ofkk S]
f (s) does not kill the ideal sheaf IΓ,s . By Lemma 4.8, the non-multiple ﬁbres are of Kodaira’s type II,∗∗∗∗
III, IV, I0 , II , III and IV . Now the proof is ﬁnished by Remark 4.9 and Table 1.
□
Lemma 4.11. If TS is pseudo-eﬀective and Y |Y is not pseudo-eﬀective, then c2 (S) = 0.
Proof. By Proposition 4.7, our f is isotrivial. Suppose to the contrary that c2 (S) > 0. Applying
Lemmas 4.4 and 4.10, we see that ξ|Y is not pseudo-eﬀective. By the divisorial Zariski decomposition[Bou04, Theorem 3.12], there exists c > 0 such thatX
ξ ≡ cY +ai Yi + P,
(2)
where ai > 0, Yi (̸= Y ) are prime divisors and P is a modiﬁed nef R-divisor (see [Bou04, Proposition2.4]). Suppose ﬁrst that c ≥ 1. Then ξ − Y is pseudo-eﬀective. Together with Lemma 4.6, we have ξ ≡ Y

12JIA JIA, YONGNAM LEE, AND GUOLEI ZHONG

and E0 ≡ c2 (S)F which implies that c2 (S) = 0, noting that E02 < 0 once it is non-zero (see [BHPV04,
III, (8.2) Lemma]), while F is free. Suppose now that c < 1. Then we have
(1 − c)Y ≡ (ξ − cY ) − π ∗ TS/C .
Restricting to Y itself, we have
(1 − c)Y |Y ≡ (ξ − cY )|Y − π ∗ TS/C |Y ,where (ξ − cY )|Y is pseudo-eﬀective by Eq. (2). Since −TS/C is pseudo-eﬀective by Lemma 4.6, it follows
that −π ∗ TS/C |Y is also pseudo-eﬀective, noting that Y is horizontal and irreducible. In particular,(1 − c)Y |Y is pseudo-eﬀective. However, this gives rise to a contradiction to our assumption.
□
In the view of Lemma 4.11, we are left to show that Y |Y is not pseudo-eﬀective. For the convenienceof our later use, we formulate the following lemma which is a direct consequence of Zariski’s lemma.
Lemma 4.12. Let f : S → C be a ﬁbration from a smooth projective surface to a smooth curve. LetXX
n j c Fj c
Sc := f −1 (c) =mic Eic +ic ∈Icjc ∈Jcbe the ﬁbre of f over c ∈ C where Eic and Fjc are (distinct) irreducible components, and mic and njc
are the corresponding multiplicities in Sc . Then

X X
X
M :=
a i c E ic −b j c Fj cc∈D

ic ∈Icjc ∈Jcis not pseudo-eﬀective for any non-empty ﬁnite set D ⊂ C and any rational numbers aic ≥ 0, bjc > 0
satisfying the following two conditions:(1) there is at least one c ∈ D such that Jc ̸= ∅;(2) for each c ∈ D, if Jc = ∅, then there exists some aic = 0 with ic ∈ Ic .
In particular, M − δF is not pseudo-eﬀective for any rational number δ > 0 if M satisﬁes condition(2), where F is a general ﬁbre of f .
Proof. Suppose to the contrary that M is pseudo-eﬀective. Let
X X
M ≡ P + N = P + N0 +d ic Eicc∈D ic ∈Ic

be the Zariski decomposition, where P is nef, N is eﬀective (dic ≥ 0) such that N0 and Eic ’s do not haveany common components. Then we have

X X
X X
X
a′ic Eic = P + N0 +d′ic Eic +bj c Fj cc∈D ic ∈Ic,1c∈D

ic ∈Ic,2

jc ∈Jcwith aic ≥ a′ic > 0, d′ic > 0, Ic,1 ∩ Ic,2 = ∅ and Ic,1 ∪ Ic,2 ⊆ Ic for each c ∈ D. Note that there exists atleast one, say c′ ∈ D, such that Ic′ ,1 ̸= ∅, since the RHS is not trivial by (1). Take the intersection with
X X

a′ic Eicc∈D ic ∈Ic,1

on both sides. Then we get the contradiction by Zariski’s lemma (see e.g. [BHPV04, III, (8.2) Lemma]):X X
2a′ic Ei,c < 0,
P + N0 +
P

X X
c∈D

ic ∈Ic,2c∈D ic ∈Ic,1d′ic Eic +X

jc ∈Jcb j c Fj c

 X X

a′ic Ei,c ≥ 0,c∈D ic ∈Ic,1′
ic ∈Ic,1 aic Eic is never a (positive) rational multiple of Sc by condition (2).

noting that
For the last assertion, note that F is numerically equivalent to any ﬁbre Sc . It is not hard to see thatM − δF satisﬁes both (1) and (2) whenever M satisﬁes (2).
□Now we come to the last part of the proof of the isotrivial case for Theorem 4.1.

TANGENT BUNDLES

13

Lemma 4.13. Suppose that f is isotrivial. If c2 (S) > 0, then Y |Y is not pseudo-eﬀective.
Proof. Since Y ≡ ξ + π ∗ (c2 (S)F/12 − E0 ) holds by Lemma 4.5, we need to calculate c (S) c (S)22
ξ + π∗F − E0 |Y = ξ|Y + (π|Y )∗F − E012
12
and show that it is not pseudo-eﬀective on Y . Recall that Y is isomorphic to the blow-up of the ideal
sheaf IΓ . By Notation 2.2 (9), OW (1)|Y ≃ OY (1) ≃ OY (−EY ) ⊗ (π|Y )∗ OS (−f ∗ KC − E), where EY isthe exceptional divisor of π|Y : Y → S. In other words, ξ|Y ∼ −EY − (π|Y )∗ (f ∗ KC + E). Therefore, c (S)2
Y |Y ≡ ξ + π ∗F − E0 | Y
12 c (S)2
F − E0∼ −EY − (π|Y )∗ (f ∗ KC + E) + (π|Y )∗12 c (S)X
1 
2= −EY + (π|Y )∗− (2g(C) − 2) −(1 −) F − 2E0
12
mj
j∈J
 X

fc − δ(f )F= −EY + (π|Y )∗ 2
S
c∈C

where the last equality is due to Lemma 4.6 and δ(f ) > 0 is deﬁned in Notation 4.2 (4). We use
fc via the blow-up π|Y equals
Remark 4.9 and the notations therein. The pullback (π|Y )∗ S

1

II

3 Y1



1

III

2 Y1,2



1

IV

 2 Y1,2,3−1 f
1(π|Y )∗ (Sc ) + 2 (Y1,2 + Y1,3 + Y1,4 + Y1,5 )
I∗0



7
5
1
1
1
3

II∗

2 Y1,2 + 6 Y2,3 + 6 Y3,4 + 2 (Y4,5 + Y6,7 ) + 3 Y6,8 + 6 Y5,6 + Y8,9



5
3
1
1

III∗

4 (Y1,2 + Y7,8 ) + 4 (Y2,3 + Y6,7 ) + 2 Y4,5 + 4 (Y3,4 + Y4,6 )
Y1,2 + Y3,4 + Y5,6 + 13 (Y2,7 + Y4,7 + Y6,7 )IV∗
fwhere (π|Y )−1∗ (Sc ) is the proper transform and Yi,j is the exceptional divisor over ei ∩ ej (schemetheoretically).
Remark 4.14. We remind the reader that, when the blown-up point s lies in the ﬁbre of Types I∗0 , II∗ ,III∗ or IV∗ , the ideal sheaf IΓ,s is reduced and the corresponding blow-up is the usual one; hence thefc along Yi,j is simply the sum of the coeﬃcients of S
fc along ei and ej . However, thecoeﬃcient of (π|Y )∗ S
blown-up point in the ﬁbre of Types II, III or IV is non-reduced (see Table 1), in which cases, the coeﬃfc along Yi,j is a bit more involved. We refer to Example 4.15 for the explicit calculationcient of (π|Y )∗ S
on the case of Type IV for the convenience of the reader but skip other cases for the organisation of ourpaper.
Now, we come back to the proof of our lemma. By the above list, within each singular ﬁbre Sc , there
exists a pair of indices i1 ̸= i2 ∈ I such that ei1 ∩ ei2 ̸= ∅ and the coeﬃcient1
coeﬀ (π|Y )∗ (Sfc ) Yi1 ,i2 ≤ .
2Therefore,X
fc −2(π|Y )∗ S
Yss∈Γ∩Scis a linear combination of some components of the ﬁbre of Y → C over c ∈ C satisfying (2) of Lemma 4.12.
P
Noting that EY = s∈Γ Ys and δ(f ) > 0, we see that the divisor c (S)XX
2
fc − δ(f )(π|Y )∗ F
Y |Y ≡ ξ|Y + (π|Y )∗Ys + 2(π|Y )∗ S
F − E0 ∼ Q
−
12
c∈C

s∈Γ∩Scis not pseudo-eﬀective by Lemma 4.12. Our lemma is thus proved.

□

14

JIA JIA, YONGNAM LEE, AND GUOLEI ZHONG

Example 4.15. In this example, we calculate the blow-up of the non-reduced point in the ﬁbre of TypeIV (see Table 1). Let f (x, y) = (y − x)(y − ζx)(y − ζ 2 x) = y 3 − x3 be the union of three lines on A2 , whereζ = exp(2iπ/3) is the primitive root of unity. Then the partial derivatives fx = −3x2 and fy = 3y 2 . So
the blown-up ideal is I = (x2 , y 2 ) (see [Ser96, Proof of Proposition 3.1]). Let [a : b] be the homogeneous
coordinates of P1 . Then the blow-up π of I is deﬁned by y 2 a − x2 b = 0 in A2 × P1 . Consider the pullbackof L, L1 and L + L1 , where L := {y − x = 0} is a component of f (x, y) = 0, and L1 := {y + x = 0)}.
These pullbacks are deﬁned by
(
(
(
2
2
2
2
y
a
−
x
b
=
0,
y
a
−
x
b
=
0,
y 2 a − x2 b = 0,(π|Y )∗ L :(π|Y )∗ L1 :and (π|Y )∗ (L + L1 ) :y − x = 0,y + x = 0,
y 2 − x2 = 0.
Their proper transforms are

2
2


y a − x b = 0,e : a − b = 0,Ly − x = 0,

f1 :
L


2
2


y a − x b = 0,a − b = 0,y + x = 0,

(
e+L
f1 :
and L

y 2 a − x2 b = 0,a − b = 0,e+L
f1 is a Cartier divisor on the blow-up of I. The π-exceptional divisor E is deﬁned by
Note that L

2
2

(

y a − x b = 0,y 2 a − x2 b = 0,
2
or juston the aﬃne chart a ̸= 0.
x = 0,
x2 = 0,y 2 = 0,
On the aﬃne chart a ̸= 0 (equivalently, a = 1), we may calculate the lengths:
C[x, y, b]/(y 2 − x2 , 1 − b, y − x, x2 ) ≃ C[x]/(x2 )has length 2;

C[x, y, b]/(y 2 − x2 , 1 − b, y + x, x2 ) ≃ C[x]/(x2 )has length 2;

C[x, y, b]/(y − x , 1 − b, x ) ≃ C[x, y]/(x , y )has length 4.

2

2

2

2

2

By [Ful98, Proposition 7.1], we havee ≤ 2,
L.E

f1 .E ≤ 2,Le+L
f1 ).E = 4.
(Le
f1 .E = 2. Assume the pullback of L is π ∗ L = L
e + tE. Then it follows from the
Therefore, L.E= L
projection formula thate + tE) · E = 2 + tE 2 .
0 = (LBy [Ful98, Page 79, the paragraph before Example 4.3.1], the anti-self-intersection −E 2 is the (Samuel)multiplicity of the blown-up point, which is 4. Therefore, we have t = 1/2 and hence π ∗ L = L + 1/2E.
Corollary 4.16. Suppose that f is isotrivial. If TS is pseudo-eﬀective, then c2 (S) = 0.
Proof. Assume that c2 (S) > 0. Then Y |Y is not pseudo-eﬀective by Lemma 4.13. On the other hand,by Lemma 4.11, we have c2 (S) = 0, a contradiction.
□
Proof of Theorem 4.1. The equivalence of (2) and (3) follows from Lemma 4.4. The implication (2) ⇒ (1)is proved in Lemma 4.5. The implication (1) ⇒ (2) follows from Proposition 4.7 and Corollary 4.16. □In the last part of this section, we study the Kodaira dimension κ(P(TS ), O(1)).
Lemma 4.17. If TS is pseudo-eﬀective, then OP(TS ) (1) is Q-linearly equivalent to an eﬀective divisor.
In particular, κ(P(TS ), O(1)) ≥ 0.
1
Proof. By Lemma 4.5, we know that ξ ∼ Y + π ∗ TS/C and TS/C ≡ E0 − 12
c2 (S)F . Since TS is pseudoeﬀective, applying Theorem 4.1, we have c2 (S) = 0 and hence E0 = 0; in particular, TS/C ≡ 0. We shallshow that TS/C ∼Q 0. Indeed, in the view of Notation 2.2 (7), Notation 4.2 (3) and Serre duality, we
haveXTS/C = −KS + f ∗ KC +(mj − 1)Fj + E0 = −f ∗ ((R1 f∗ OS )∨ ) = −f ∗ (f∗ ωS/C ).

TANGENT BUNDLES

15Since deg(f∗ ωS/C ) = χ(S, OS ) = (c1 (S)2 + c2 (S))/12 = 0, by [BHPV04, III, (18.3) Proposition], f∗ ωS/C
and hence TS/C are Q-trivial. Consequently, ξ ∼Q Y and thus κ(P(TS ), O(1)) ≥ 0.
□
Lemma 4.18. Let f : S → C be a locally trivial elliptic ﬁbration over a smooth projective curve C ofgenus g = g(C) ≥ 1. Then κ(P(TS ), O(1)) = 1 − κ(C).
Proof. Since χ(S, OS ) = 0, after replacing S by a further étale cover, we may assume that the relative
tangent bundle TS/C is trivial (see [BHPV04, III, (18.3) Proposition] and Lemma 2.8). We consider thesymmetric power of the following short exact sequence
0 → OS → TS → f ∗ TC → 0,noting that TS/C = OS by our reduction. For each positive integer m, we have
0 → Symm−1 TS → Symm TS → f ∗ (−mKC ) → 0.
Hence, we haveh0 (S, Symm TS ) ≤ 1 +m
X

h0 (S, f ∗ (−iKC )).

i=1

In particular, if g(C) ≥ 2, then h (S, Sym TS ) ≤ 1 for any m. On the other hand, since f issmooth, it follows from Theorem 4.1 that TS is pseudo-eﬀective. Together with Lemma 4.17, we haveh0 (S, Symm TS ) = 1 for any m whenever g(C) ≥ 2. As a result, κ(P(TS ), O(1)) = 0 when g(C) ≥ 2.
If g(C) = 1, then by Notation 4.2 (4), the vanishing of δ(f ) implies κ(S) = 0. Since S is also minimal,
our lemma follows from Proposition 3.1.
□0m

As a corollary, we show our main theorem when S is assumed to be minimal.
Corollary 4.19. Let S be a smooth minimal projective surface, i.e., KS is nef. Then the tangent bundle
TS is pseudo-eﬀective if and only if the second Chern class c2 (S) = 0. Moreover, after a Galois étale
cover, S admits a locally trivial elliptic ﬁbration onto a smooth projective curve C with κ(S) = κ(C). In
particular, after a further étale cover, S is either an abelian surface or isomorphic to a product of anelliptic curve and a smooth curve of genus ≥ 2.
Proof. The ﬁrst statement follows from Propositions 3.1 and 3.2 and Theorem 4.1. Let f : S → C be
an elliptic ﬁbration. Note that κ(S) = 0 or 1 and c2 (S) = 0 if and only if the only singular ﬁbres of f
are multiple of elliptic curves. If C ≃ P1 , then f has at least three multiple ﬁbres, since δ(f ) ≥ 0 by
[BHPV04, V, (12.5) Proposition]. Applying [GMM21, Lemma 1.1.9], we get a ﬁnite Galois étale map
gS : S ′ → S which is induced by a ramiﬁed base change g : C ′ → C and f ′ : S ′ → C ′ has no multiple
ﬁbre. That is, f ′ is a locally trivial elliptic ﬁbration. Note that when κ(S) = 1, one has δ(f ′ ) > 0 andhence g(C ′ ) ≥ 2; when κ(S) = 0, one has 0 ≤ κ(C ′ ) ≤ κ(S ′ ) = 0 and hence κ(C ′ ) = 0. Finally, the lastpart of our corollary follows from [Har10, Corollary 26.5].
□
5. Proof of Theorem 1.2
In this section, we prove Theorem 1.2. In the view of Propositions 3.1 and 3.2 and Theorem 4.1,
our main task left is to exclude the non-minimal surface which is of Kodaira dimension one and has
pseudo-eﬀective tangent bundle. The following theorem is our main result of this section.
Theorem 5.1. Let S be a smooth projective surface of κ(S) = 1. If TS is pseudo-eﬀective, then KS isnef, i.e., S is minimal.
Proof. In the view of Lemmas 2.7 and 4.3, we only need to exclude the case when S := S2 is a blow-up ofa smooth minimal surface S1 . Suppose to the contrary that the tangent bundle TS2 is pseudo-eﬀective.

16

JIA JIA, YONGNAM LEE, AND GUOLEI ZHONG

Let us consider the following commutative diagram as in Notation 2.3
P(TS1 ) oehP(h∗ TS1 ) _ _ _ _ _ _/ P(TS2 )
π
f
1

π1


S1 o

h

r
rrr
r
r
f1
rr
 xrrrr f2
C

π2


S2


S2

where h : S2 → S1 is a single blow-up between smooth projective surfaces with an exceptional (−1)-curveD, f1 : S1 → C is an elliptic ﬁbration onto a smooth curve C, f2 := f1 ◦ h, and πi : P(TSi ) → Si is thenatural projection. Let ξi be the tautological divisor of the projective bundle P(TSi ) and let Yi be thedivisor in P(TSi ) deﬁned in Notation 2.2 (8). Then we have ξi ∼ Yi + πi∗ TSi /C (see Notation 2.2 (9)).
We apply Corollary 4.19 to get an étale morphism S1′ → S1 , which is induced by a (possibly ramiﬁed)base change g : C ′ → C, such that S1′ admits a locally trivial elliptic ﬁbration over a smooth curve C ′ ofgenus ≥ 2. Then we have the following commutative diagram induced by the base change
S2′

g2

h′


S1′

h

g1


/ S1

g


/C

f1′


C′

/ S2

f1

where h′ is the blow-up along smooth (reduced) points and g2 is étale. Then TS2′ = g2∗ TS2 is pseudoeﬀective (see Lemma 2.8). Replacing Si with Si′ , we may assume that f1 is locally trivial with g(C) ≥ 2.
Since the divisor ξ2 is pseudo-eﬀective, it follows from the divisorial Zariski decomposition [Bou04,
Theorem 3.12] thatX
ξ2 ≡ai Pi + N,
where ai > 0, Pi are prime divisors and N is a modiﬁed nef R-divisor. Since TS2 is pseudo-eﬀective,
applying Lemma 2.7 and Corollary 4.19, we know that c2 (S1 ) = 0 (indeed, we can even make S1 a
product variety); hence, TS1 /C ≡ 0 by Lemmas 4.5 and 4.6. We calculate the diﬀerence (cf. Notation 2.2(7))
h∗ TS1 /C − TS2 /C = −h∗ KS1 + KS2 + h∗ ES1 − ES2 = D + h∗ ES1 − ES2 = D,
where ESi is deﬁned in Notation 2.2 (5). Therefore, by Notation 2.2 (9), we haveX
Y2 − π2∗ D ∼ ξ2 ≡ai Pi + N.
Without loss of generality, we may assume that P1 = Y2 with a1 possibly being zero. Since 0 ̸= −π2∗ D
is anti-pseudo-eﬀective, we see that 0 ≤ a1 < 1. From the numerical equivalenceX(1 − a1 )Y2 |Y2 ≡ai Pi |Y2 + N |Y2 + (π2 |Y2 )∗ D,i≥2

the pseudo-eﬀectiveness of Y2 |Y2 follows, noting that π2∗ D|Y2 is eﬀective. On the other hand, by Notation 2.2 (9), we have the following equivalence
Y2 |Y2 ≡ (ξ2 + π2∗ D)|Y2 ∼ − Exc(π2 |Y2 ) + (π2 |Y2 )∗ (−f2∗ KC + D).
Since g(C) ≥ 2, after moving one free ﬁbre from f ∗ KC to f2−1 (f2 (D)), our Y2 |Y2 is anti-pseudo-eﬀective
and non-zero, which contradicts Y2 |Y2 being pseudo-eﬀective. So we ﬁnish the proof of our theorem. □Proof of Theorem 1.2. In the view of Propositions 3.1 and 3.2, we may assume that κ(S) = 1. Then, theﬁrst half follows from Theorems 4.1 and 5.1. The second half follows from Lemma 4.18, Proposition 3.1,
and Corollary 4.19.
□

TANGENT BUNDLES

17

6. Ruled surface over non-rational base, Proof of Proposition 1.5
In the last section, we study non-rational uniruled surfaces with pseudo-eﬀective tangent bundles.
First, we show that any projective bundle (of arbitrary rank) over a smooth curve always has a pseudoeﬀective tangent bundle, which slightly extends [Kim23].
Proposition 6.1 (cf. [Kim23]). The tangent bundle TX of any projective bundle f : X = PC (E) → C
over a smooth curve C is pseudo-eﬀective. In particular, TX is pseudo-eﬀective but non-big if and only
if E is semi-stable and C ̸≃ P1 .
Proof. In the view of [Kim23], we only need to show that when E is semi-stable, the tangent bundle TX
is pseudo-eﬀective.
Now E is semi-stable, so are E ∨ and hence E ∨ ⊗ E. Since the determinant det(E ∨ ⊗ E) ≃ OC , thesemi-stable vector bundle E ∨ ⊗ E is nef by [Laz04, Proposition 6.4.11]. We consider the following relative
Euler sequence
0 −→ OX −→ f ∗ E ∨ ⊗ OX (1) −→ TX/C −→ 0,
where the relative tangent bundle TX/C := Ω∨
X/C . Then the following composite map (see [Har77,
Chapter II, Proposition 7.11(2)])
f ∗ (E ∨ ⊗ E) −→ f ∗ E ∨ ⊗ OX (1) −→ TX/C
is a surjection, which implies that TX/C is nef (see e.g. [Laz04, Proposition 6.1.2]) and hence pseudoeﬀective. So our proposition follows from Lemma 2.5.
□
Now we are in the position of proving Proposition 1.5.
Proof of Proposition 1.5. Let h : S2 → S1 be the blow-up of S1 = S along p. We may assume that E
is normalised (see [Har77, Chapter V, Notation 2.8.1]). Since TS1 is pseudo-eﬀective but not big, by
Proposition 6.1, E is semi-stable. Together with E being normalised, we have deg det E ≥ 0. Denote by
C0 the tautological divisor on S1 = PC (E), which is not necessarily eﬀective. Then NE(S1 ) = Nef(S1 ) =R+ (2C0 − f1∗ det E) + R+ F , where F is a ﬁbre of f1 = f : S1 = S → C by [Miy87a, Proposition 3.1].
Consider the following commutative diagram as in Notation 2.3.
P(TS1 ) oehP(h∗ TS1 ) _ _ _ _ _ _/ P(TS2 )
π
f
1

π1


S1 o

h

r
rrr
r
r
f1
rr
 xrrrr f2
C


S2

π2


S2

Let D be the h-exceptional (−1)-curve, f2 := f1 ◦ h the induced composite map, ξi the tautologicaldivisor of the projective bundle P(TSi ), and Yi the divisor in P(TSi ) deﬁned in Notation 2.2 (8). Then
Notation 2.2 (9) implies
ξi ∼ Yi + πi∗ TSi /C .
Let us calculate the diﬀerenceh∗ TS1 /C − TS2 /C = −h∗ KS1 + KS2 + h∗ ES1 − ES2 = D + h∗ ES1 − ES2 = D,
where ESi is deﬁned in Notation 2.2 (5). Here, since f1 is smooth and the only singular ﬁbre of f2 isreduced, we have h∗ ES1 = ES2 = 0 and
TS1 /C = −KS1 + f1∗ KC ∼ 2C0 − f1∗ det E.
Claim 6.2. The divisor TS2 /C = h∗ TS1 /C − D is pseudo-eﬀective if and only if there exist some positive⊗m
integer m and some line bundle L ≡ TS1 /C such that H 0 (S1 , mm
) ̸= 0, where mp is the maximalp ⊗L
ideal of OS1 ,p .

18

JIA JIA, YONGNAM LEE, AND GUOLEI ZHONG

Proof of Claim 6.2. One direction is clear. Let us assume that h∗ TS1 /C − D is pseudo-eﬀective. ThenPP
we have the Zariski decomposition h∗ TS1 /C − D ≡i ai Pi + N , where N is nef andi ai Pi has
negative deﬁnite intersection matrix. Since (h∗ TS1 /C − µD)2 < 0 for any µ > 0, there exists at least
one, say P1 , which is not h-exceptional such that P12 < 0. Pushing this forward to S1 , we obtainP
TS1 /C ≡ i ai h∗ Pi + h∗ N . Since TS1 /C is extremal, TS1 /C ≡ th∗ P1 for some t > 0. Suppose to the⊗mcontrary that for any positive integer m and any line bundle L ≡ TS1 /C , one has h0 (S1 , mm
) = 0.
p ⊗L∗
By our assumption, h (h∗ P1 ) = P1 + sD with st < 1; see [Har77, Chapter V, Proposition 3.6]. Hence,h∗ TS1 /C − D ≡ tP1 − (1 − st)D which is not pseudo-eﬀective by noting that P1 is extremal and notparallel to D. This leads to a contradiction to our assumption.
□⊗m
We come back to the proof of Proposition 1.5. If H 0 (S1 , mm
) ̸= 0 for some positive integerp ⊗L∗m and some line bundle L ≡ TS1 /C , then TS2 /C = h TS1 /C − D is pseudo-eﬀective and thus TS2 ispseudo-eﬀective, noting that ξ2 ∼ Y2 + π2∗ TS2 /C holds by Lemma 2.5.
⊗m
Now we assume that H 0 (S1 , mm
) = 0 for any positive integer m and any line bundle L ≡p ⊗LTS1 /C . Suppose that ξ2 is pseudo-eﬀective. It follows from the divisorial Zariski decomposition [Bou04,
Theorem 3.12] thatX
Y2 + π2∗ TS2 /C ∼ ξ2 ≡ai Pi + N,

where ai > 0, Pi are prime divisors and N is a modiﬁed nef R-divisor. By Notation 2.2 (9), ξ2 |Y2 =− Exc(π2 |Y2 ) − (π2 |Y2 )∗ f2∗ KC , which is anti-pseudo-eﬀective and non-zero. Without loss of generality,
we may assume that P1 = Y2 with a1 > 0. Note that now π2∗ TS2 /C ∼ ξ2 − Y2 is not pseudo-eﬀective by
Claim 6.2 (cf. [Nak04, Chapter II, Lemma 5.6]). Consequently, we deduce that 0 < a1 < 1. Consider
the following restriction(ξ2 − a1 Y2 )|Y2 = ((1 − a1 )ξ2 + a1 π2∗ TS2 /C )|Y2
= (1 − a1 )(− Exc(π2 |Y2 ) − (π2 |Y2 )∗ f2∗ KC ) + a1 (π2 |Y2 )∗ TS2 /C
which is a pseudo-eﬀective divisor on Y2 . Pushing this forward to S2 along the birational morphism
π2 |Y2 , we have the pseudo-eﬀectiveness of(π2 |Y2 )∗ ((ξ2 − a1 Y2 )|Y2 ) = −(1 − a1 )f2∗ KC + a1 TS2 /C ,while the RHS is never pseudo-eﬀective whenever 0 < a1 < 1 and g(C) ≥ 1. In particular, our assumption
is absurd and ξ2 is thus not pseudo-eﬀective. So we ﬁnish the proof of our proposition.
□
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Department of Mathematics, National University of Singapore, Singapore 119076, Republic of Singapore
Current address: Yau Mathematical Sciences Center, Jingzhai, Tsinghua University, Haidian District, Beijing, China,
100084
Email address: jia_jia@u.nus.edu, mathjiajia@tsinghua.edu.cn
Center for Complex Geometry, Institute for Basic Science (IBS), 55 Expo-ro, Yuseong-gu, Daejeon, 34126,
Republic of Korea, andDepartment of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Deajeon, 34141, Republic of
Korea
Email address: ynlee@kaist.ac.kr, ynlee@ibs.re.kr
Center for Complex Geometry, Institute for Basic Science (IBS), 55 Expo-ro, Yuseong-gu, Daejeon, 34126,
Republic of Korea
Email address: zhongguolei@u.nus.edu, guolei@ibs.re.kr