Let f:X→Delta be a semistable one-parameter family of complex projective varieties over a holomorphic disk, with X smooth and central fiber X_0=f^-1(0) a reduced simple normal crossing divisor. For t∈Delta^, let X_t=f^-1(t) be a smooth fiber, and let T be the monodromy operator on H^1(X_t,Z). Then the natural restriction map H^1(X,Z)longrightarrow H^1(X_t,Z)^T is surjective.

Integral local invariant cycles in degree one

The problem asks whether, for a semistable one-parameter family, every monodromy-invariant integral class in $H^1(X_t,\mathbb{Z})$ extends from $H^1(X,\mathbb{Z})$ ; equivalently, whether the natural map to $H^1(X_t,\mathbb{Z})^T$ is surjective.

May 2026 claimed proof; related degree- $2$ failure

A proofQED entry claims surjectivity, using the Wang sequence and residue calculations, but it is not independently verified. A separate semistable-family construction shows failure of the analogous integral statement in degree $2$ ; it does not settle degree $1$ .

Current status (as of May 2026): Surjectivity in degree $1$ is claimed in a repository proof but remains unverified; no counterexample or corroborated proof was found.

Integral local invariant cycles in degree one

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