Let (A,Theta) be a principally polarized abelian variety of dimension g, and let X⊂ A be a subvariety of dimension d with 1≤ d≤ g-2. The conjecture asserts that X represents the minimal cohomology class if and only if its ideal sheaf twisted by the principal polarization is a generic-vanishing sheaf: [X]=frac[Theta]^g-d(g-d)!quad Longleftrightarrowquad I_X otimes O_A(Theta) is GV, where the generic-vanishing condition means that, for every i>0, operatornamecodim_operatornamePic^0(A)alpha| H^i(A, I_X otimes O_A(Theta)otimes alpha)≠0≥ i. The full Pareschi–Popa formulation further relates these conditions to the Jacobian and Fano-surface classifications in the relevant dimensions.

Pareschi–Popa generic vanishing conjecture

Let $(A,\Theta)$ be a principally polarized abelian variety of dimension $g$ , and let $X\subset A$ be a subvariety of dimension $d$ with $1\le d\le g-2$ . The conjecture asserts that $X$ represents the minimal cohomology class if and only if its ideal sheaf twisted by the principal polarization is a generic-vanishing sheaf:

$[X]=\frac{[\Theta]^{g-d}}{(g-d)!}\quad\Longleftrightarrow\quad \mathcal I_X\otimes\mathcal O_A(\Theta)\text{ is }\mathrm{GV}$ ,

where the generic-vanishing condition means that, for every $i>0$ , $\operatorname{codim}_{\operatorname{Pic}^0(A)}\{\alpha\mid H^i(A,\mathcal I_X\otimes\mathcal O_A(\Theta)\otimes\alpha)\ne0\}\ge i$ . The full Pareschi–Popa formulation further relates these conditions to the Jacobian and Fano-surface classifications in the relevant dimensions.

References

Primary source

arXiv

Additional references

Generic vanishing conjecture for divisors — arXiv — Fanjun Meng

abelian-varieties algebraic-geometry

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed progress

A new paper claims progress for divisors, but the original conjecture in higher codimension remains open.

Pareschi and Popa formulated the conjecture in 2006, asserting equivalences between minimal cohomology classes, generic-vanishing conditions, and Jacobian or Fano-surface classifications for subschemes of principally polarized abelian varieties.

Known results

The original paper proves $(2)\Longleftrightarrow(4)\Rightarrow(3)$ and $(2)\Rightarrow(1)$ .
For $d=1$ or $d=g-2$ , condition $(2)$ gives the Jacobian and Abel–Jacobi classifications.
In dimension $4$ , $(1)$ , $(2)$ , $(4)$ , and $(5)$ are equivalent, and imply $(3)$ .
The full higher-codimension equivalence was explicitly left conjectural.

September 2026 codimension-one result

Fanjun Meng’s preprint claims a codimension-one analogue for divisors, extending the framework but not proving the original conjecture. This is claimed progress, not a verified resolution.

Current status (as of September 2026): The classical partial implications are known, and a new divisor analogue is claimed, but the original conjecture in codimension greater than one remains open.

Sources

No solutions have been posted yet.