Existence of strictly polystable extensions in negative half-genus range

Establish the existence of a strictly polystable extension of L^{-1} by L for a compact Riemann surface X of genus g_X ≥ 2 and a generic holomorphic line bundle L with deg(L) ≤ −(1/2)·g_X. Concretely, show that there exists J ∈ Pic^0(X) and an embedding L → J ⊕ J^{-1} whose induced section in the projective unitary flat bundle P = ℙ(J ⊕ J^{-1}) is non-locally flat, i.e., defines a strictly polystable short exact sequence 0 → L → J ⊕ J^{-1} → L^{-1} → 0.

Background

The paper studies cone spherical metrics via the algebraic framework of polystable extensions of two line bundles, and distinguishes strictly polystable extensions as those whose induced section in the associated projective unitary flat bundle is non-locally flat. Such extensions produce reducible cone spherical metrics representing effective divisors.

The authors establish existence of strictly polystable extensions in several regimes: for general genus when deg(L) ≤ −2g_X (Lemma 4.3, via ampleness), and on elliptic curves for deg(L) ≤ −1 (Theorem 4.4). The conjecture proposes extending existence to a broader “generic” range deg(L) ≤ −(1/2)·g_X, motivated by considerations involving the Riemann theta-divisor and base locus arguments.

Proving the conjecture would significantly enlarge the parameter range over which reducible metrics exist and provide a clearer picture of the algebraic subsets of Ext^1_X(L^{-1},L) supporting strictly polystable extensions.