Constancy of h¹(O) in reduced Cohen–Macaulay projective families
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In a flat projective family over a smooth complex affine curve with reduced Cohen–Macaulay fibers and smooth generic fiber, must h¹(O) remain constant? The counterexample target has h¹=1 on one fiber and zero elsewhere.
Why it matters and what counts as progress
Why it matters: It identifies limits of cohomology constancy under natural fiber hypotheses.
What counts as progress: A family with the required fiber properties but a jump in h¹(O), or a local-constancy theorem under stronger fiber assumptions.
How the problem developed
- 2025 · Open problem [1]
János Kollár and Sándor Kovács Asked whether a flat projective family over a smooth curve can have reduced Cohen–Macaulay fibers, smooth generic fiber, and nonconstant h¹ of the structure sheaf.
- 20 May 2026 · Reported claim [1]
Jihao Liu; ChatGPT 5.5 Pro and Rethlas Constructed such a family, with h¹ equal to one on the special fiber and zero on the other fibers in the chosen open curve.
AI contribution
On a question of Kollár and Kovács
A flat projective family over an open subset of the complex affine line has reduced Cohen–Macaulay fibers and smooth generic fiber, yet h¹(O) is 1 at the special fiber and 0 elsewhere.
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