Universal algorithm for line bundles with n=0 on symplectic 4-manifolds with c1(K)^2>0

Develop a general algorithm that, given any compact symplectic 4-manifold X with positive square of the canonical first Chern class (c1(K_X)·c1(K_X) > 0), constructs a complex line bundle L → X such that the spectral flow of the symmetric first-order operator associated to the massive Vafa–Witten equations remains bounded along the corresponding |ω|-divergent reducible solution sequence. In the construction analyzed in this paper, bounded spectral flow is equivalent to ensuring that the integer n in the cup-product pairing c1(L)·[K] = q n + e (where [K] is the Euler class of the oriented 2-plane bundle orthogonal to the self-dual part of the harmonic curvature form and q parametrizes the sequence) satisfies n = 0.

Background

The paper constructs |ω|-divergent sequences of reducible solutions to the massive Vafa–Witten equations on compact oriented 4‑manifolds using complex line bundles L with harmonic curvature. It then analyzes the spectral flow of the associated symmetric first-order operator. A central result (Theorem 1.1 and Proposition 5.15) shows that, for these sequences, the spectral flow is bounded if and only if an integer n arising from a cohomological pairing vanishes.

For symplectic 4‑manifolds with non-positive square of the canonical first Chern class and sufficient second Betti numbers, the author exhibits choices of line bundles L for which n can be zero (bounded spectral flow) and others for which it is not. However, when the canonical class has positive square, a general procedure to find line bundles achieving n = 0 is not known, and the Kähler-case construction does not directly apply. The open problem is to devise a universal, constructive method to select such L on symplectic 4‑manifolds with c1(K_X)^2 > 0, thereby guaranteeing bounded spectral flow along the associated reducible solution sequences.