On Projective Manifolds With Pseudo-Effective Tangent Bundle

Abstract

In this paper, we develop the theory of singular Hermitian metrics on vector bundles. As an application, we give a structure theorem of a projective manifold X with pseudo-effective tangent bundle; X admits a smooth fibration $X \to Y$ to a flat projective manifold Y such that its general fibre is rationally connected. Moreover, by applying this structure theorem, we classify all the minimal surfaces with pseudo-effective tangent bundle and study general nonminimal surfaces, which provide examples of (possibly singular) positively curved tangent bundles.

Examples of Fano manifolds with non‐pseudoeffective tangent bundle

“…Paris [39, Proposition 3.2.3 and Proposition 3.2.4] and Hosono, Iwai and Matsumura [19, Proposition 4.5 and Proposition 4.8] obtained similar results for the tangent bundle of rational surfaces, but their definition of pseudoeffectiveness is (at least a priori) strictly stronger than ours. Note that Mallory proved, with completely different techniques, that $$T_X$$ is not big if $$d \leqslant 4$$ [34, Corollary 5.7].…”

“…An elementary computation using (19) shows that this intersection number is −8, a contradiction. Hence, by Corollary 2.6 there exists a unique prime divisor ⊂ ℙ(𝑇 𝑋 ) generating the extremal ray ℝ + 𝜁.…”

“…Hence, by Corollary 2.6 there exists a unique prime divisor ⊂ ℙ(𝑇 𝑋 ) generating the extremal ray ℝ + 𝜁. Given any 0 < 𝜀 ≪ 1, we let 𝑋 be a general member in its deformation family such that 𝜁 + ( An elementary computation using (19) shows that this intersection product is −49 6 if 𝜀 = 0. Thus, we have reached a contradiction by choosing 𝜀 small enough.…”

“…Peternell [40, Theorem 1.3] proved that $$T_X$$ is always generically ample, that is, the restriction $$T_X|_C$$ to a general complete intersection of sufficiently ample divisors is ample. On the other hand it is clear that $$T_X$$ will not admit some positive metric as this quickly leads to strong restrictions on the geometry of $$X$$, see [35, Theorem 1.1], [25, Corollary 1.3] and [19, Theorem 1.1]. In this paper we will consider a strictly weaker positivity property.…”

Structure theorem for projective klt pairs with nef anti-canonical divisor

“…On the other hand, the pioneering studies ( [Mor79,SY80]) have demonstrated that the structures of varieties with "semipositive" curvature have a certain rigidity and are closely related to the geometry of rational curves. Based on this philosophy, several structure theorems and uniformization theorems have been established for Albanese maps or maximal rationally connected (MRC) fibrations of varieties, under various positivity assumptions of the curvature; including non-negative holomorphic bisectional curvatures ([HSW81, MZ86,Mok88]), nonnegative holomorphic sectional curvatures ([Yan16, HW20, Mat18]), nef tangent bundles ([CP91, DPS94]), pseudo-effective tangent bundles ( [HIM19]), and nef anti-canonical divisors ([Cao19, CH19, DLB20]). In particular, the study of projective manifolds with nef anti-canonical divisor covers a large range of varieties with "non-negative" curvature; moreover, the study of such varieties is more natural from the viewpoint of birational geometry than that of varieties with positive tangent bundles.…”