Horizontal generic vanishing under isomonodromy
Develop a nondecreasing function f with f(3)=1 such that, for g≥3 and a unitary irreducible nontrivial flat bundle (\mathscr{E},\nabla) with regular singularities on a marked curve (X,D), after isomonodromic deformation to a general nearby curve X′, one has H^0(X′,\mathscr{E}(Z))=0 for every effective divisor Z of degree d≤f(g) (strong form), or for a general such divisor Z (weak form).
References
Conjecture [Horizontal generic vanishing] Let $g\geq 3$, and let $(X,D)$ be a marked smooth projective curve of genus $g\geq 3$. Let $(\mathscr{E}, \nabla)$ be a flat bundle on $X$ with irreducible, non-trivial unitary monodromy, and regular singularities along $D$, whose residue matrices have eigenvalues with real parts in $[0,1)$. There exists some non-decreasing function $f$, with $f(3)=1$, such that after isomonodromic deformation to a general nearby curve $X'$, we have: (1) (weak form) $H0(X', \mathscr{E}(Z))=0$ for a general effective divisor $Z$ on $X'$ of degree $d\leq f(g)$. (2) (strong form) $H0(X', \mathscr{E}(Z))=0$ for all effective divisors $Z$ on $X'$ of degree $d\leq f(g)$.