Isoperiodic meromorphic forms with at least three simple poles

The paper proves that Per^{-1}_{Σ,R}(p) is connected for n ≥ 3 when the image of p is either non-real or real but not contained in the Q-span of peripheral periods (Theorems 1.3–1.4), using the pointed Torelli cover, a bordification that makes the period map well-behaved near boundary, and an induction that reduces to connected spherical boundary strata; Schiffer variations provide the needed isoperiodic surgeries and smoothings. These ingredients are all explicitly laid out (definition/regularity of the period map and local fiber structure; boundary stratification and product/normalization maps; spherical boundary choices in the non-real vs real regimes; the inductive connectivity proofs) . The candidate solution mirrors this blueprint closely (same Torelli/bordification setup, the same real vs non-real case split, the same boundary factorization and Schiffer moves), with only minor overstatements (e.g., stating a global submersion rather than the local fiber-regularity actually used).