Uniform RC-positivity of direct image bundles

The concept of RC-positivity and uniform RC-positivity is introduced by Xiaokui Yang to solve a conjecture of Yau on projectivity and rational connectedness of a compact Kähler manifold with positive holomorphic sectional curvature. Some main theorems in Yang’s proof hold under a weaker condition called weak RC-positivity. It is therefore natural to ask if (uniform) weak RC-positivity implies (uniform) RC-positivity. Another motivation for studying this problem is to understand the relation between rational connectedness of $X$ and (uniform) RC-positivity of the holomorphic tangent bundle $TX$. In this paper, we obtain results in this direction. In particular, we show that if a vector bundle $E$ is uniformly weakly RC-positive, then $S^kE\otimes \det E$ is uniformly RC-positive for any $k\geq 0$, and $S^kE$ is uniformly RC-positive for $k$ large. We also discuss an approach that might lead to a solution to the question of whether weak RC-positivity of $E$ implies RC-positivity of $E$.

💡 Research Summary

This paper, titled “Uniform RC-positivity of direct image bundles” by Kuang-Ru Wu, makes significant progress on a fundamental question in complex geometry concerning the relationship between different notions of positivity for vector bundles. The central concepts are RC-positivity and its uniform variant, introduced by Xiaokui Yang to solve Yau’s conjecture on the projectivity and rational connectedness of compact Kähler manifolds with positive holomorphic sectional curvature. A key observation is that some of Yang’s main theorems hold under a strictly weaker condition called “weak RC-positivity.” This naturally raises the question of whether (uniform) weak RC-positivity actually implies (uniform) RC-positivity. The paper provides affirmative results in the “uniform” setting and discusses a potential approach for the general case.

The author’s primary achievement is Theorem 3. It states that if a holomorphic vector bundle E over a compact Kähler manifold is uniformly weakly RC-positive, then for any integer k ≥ 0, the twisted symmetric power bundle S^kE ⊗ det E is uniformly RC-positive. Furthermore, for all sufficiently large k, the symmetric power S^kE itself is uniformly RC-positive. This result is motivated not only by the aforementioned question but also by the problem of understanding the precise relationship between the rational connectedness of a manifold X and the (uniform) RC-positivity of its holomorphic tangent bundle TX.

The proof strategy is elegant and powerful. Instead of attacking Theorem 3 directly, the author first proves a more general result, Theorem 4, within the framework of direct image bundles. Consider a proper holomorphic submersion p: X → Y between complex manifolds (with X Kähler and Y compact) and a Hermitian line bundle (L, h) over X. The direct image bundle V = p_*(L ⊗ K_{X/Y}) carries a natural L^2 metric H. Theorem 4 asserts that if the curvature Θ of h is positive on every fiber X_t, and if for every base point t ∈ Y there exists a tangent vector v ∈ T_tY such that Θ(ṽ, ṽ) > 0 for every possible lift ṽ of v to the total space X, then the Hermitian bundle (V, H) is uniformly RC-positive.

The proof of Theorem 4, detailed in Section 3, is an adaptation of techniques pioneered by Berndtsson for proving positivity of direct images. The core of the argument involves taking a local holomorphic section u of V extending a given non-zero vector u0 at a fixed point t0, and computing the complex Hessian ∂∂̅H(u, u) in two different ways. One expression involves the curvature Θ_V of V, leading to H(Θ_V u0, u0)(v0, v̅0). The other is an integral expression over the fiber X_t0 derived from the metric h. A crucial Lemma 5 shows that the condition on lifts of v is equivalent to the positivity of the horizontal component of Θ, denoted Θ_H(v, v̅). By carefully choosing a representative for u and comparing the two expressions for the Hessian, the author demonstrates that H(Θ_V u0, u0)(v0, v̅0) must be positive, establishing uniform RC-positivity.

Theorem 3 then follows as a corollary of Theorem 4 by applying it to the specific fibration p: P(E*) → X. Choosing the line bundle L appropriately (as O(r+k) or O(k) ⊗ K^{-1}) makes the direct image bundle V isomorphic to S^kE ⊗ det E or S^kE, respectively.

In the final section, the author provides a characterization of weak RC-positivity (Lemma 6) and briefly discusses how a variant of this lemma might pave the way toward resolving the original, more challenging question of whether non-uniform weak RC-positivity implies RC-positivity. Overall, the paper successfully constructs uniformly RC-positive metrics on important tensor bundles derived from a uniformly weakly RC-positive bundle, offering new tools and insights for studying positivity and rational connectedness in complex geometry.