A converse of Hörmander's $L^2$-estimate and new positivity notions for vector bundles

Positivity of holomorphic vector bundles in terms of $L^p$-conditions of $\bar\partial$

“…Proof. We prove the theorem by modifying the idea in [17,10]. For the same reason as in the proof of Theorem 3.1, we may assume that there is a strictly smooth plurisubharmonic function on X.…”

“…The conditions defined in Definition 1.1, 1.2 for trivial line bundles were studied in [10]. The multiple coarse L p -extension condition for vector bundles with singular Finsler metrics was introduced in [11], and the multiple coarse L 2 -estimate condition for Hermitian vector bundles was introduced in [17], which was named as the twisted Hörmander condition there. A property (called "minimal extension property") that is related to the optimal L 2 -extension condition was introduced in [15].…”

“…The case that p = 2 and h is Hölder continuous for Theorem 1.2 was proved in [17], by showing that the multiple coarse L 2 -estimate condition implies the multiple coarse L 2 -extension condition and then applying [11,Theorem 1.2]. The case that E is a trivial line bundle was proved in [10].…”

“…The case that E is a trivial line bundle was proved in [10]. Theorem 1.2 is proved by modifying the technique in [17,10].…”

“…Theorem 1.3 and Theorem 1.4 show that the optimal L 2extension condition and the multiple coarse L 2 -extension condition imply Griffiths positivity of (E, h). It is observed in [17] that the optimal L 2 -extension condition and the multiple coarse L 2extension condition are strictly weaker than the Nakano positivity in some cases. Motivated by this and the above theorems, we believe that the optimal L 2 -estimate condition and the multiple coarse L 2 -estimate condition are equivalent to the Nakano positivity, while, the optimal L 2 -extension condition and the multiple coarse L 2 -extension condition are equivalent to the Griffiths positivity.…”