Implications Among Conjectures on Principally Polarized Abelian Varieties

Definitions and mathematical statement

The supplied record describes a proposed relationship between two conjectures, but does not specify a complete implication. In particular, the meaning of its proposed $\epsilon$-strengthening is unresolved.

For clarity, the following definitions describe a conventional complex interpretation of the terminology. The source record does not specify the ground field; this interpretation therefore requires confirmation before it can become a definitive formulation. Let $A$ be a complex abelian variety of dimension $g$, and let $\Theta$ be an effective divisor defining a principal polarization. The principally polarized abelian variety $(A,\Theta)$ is indecomposable if it is not isomorphic, as a polarized abelian variety, to a product of two principally polarized abelian varieties of positive dimension. Write $\widehat A=\operatorname{Pic}^{0}(A)$. For a coherent sheaf $\mathcal F$ on $A$, set \[ V^{i}(\mathcal F)=\{\alpha\in\widehat A:H^{i}(A,\mathcal F\otimes\alpha)\ne0\}. \] The generic-vanishing condition is \[ \operatorname{codim}_{\widehat A}V^{i}(\mathcal F)\ge i\qquad\text{for every integer }i>0, \] with the empty locus assigned infinite codimension. For a closed subvariety $X\subset A$, write $\mathcal I_X(\Theta)=\mathcal I_X\otimes\mathcal O_A(\Theta)$, where $\mathcal I_X$ is its ideal sheaf.

For a reduced irreducible complex subvariety $X$ of dimension $d$, the usual geometric nondegeneracy condition says that the kernel of restriction \[ H^{0}(A,\Omega_A^{d})\longrightarrow H^{0}(X_{\mathrm{sm}},\Omega_{X_{\mathrm{sm}}}^{d}) \] contains no nonzero decomposable form $\omega_1\wedge\cdots\wedge\omega_d$, with $\omega_j\in H^{0}(A,\Omega_A^{1})$. This definition does not resolve the record's missing assumptions on the scheme structure, irreducibility, or dimension range of $X$.

For $x\in\Theta$, if $f$ is a local equation for $\Theta$ in the regular local ring $\mathcal O_{A,x}$, define \[ \operatorname{mult}_{x}\Theta=\max\{m\ge0:f\in\mathfrak m_{A,x}^{m}\}. \] In the complex interpretation, a pure $d$-dimensional subvariety has minimal cohomology class when \[ [X]=\frac{[\Theta]^{g-d}}{(g-d)!}\quad\text{in }H^{2(g-d)}(A,{\mathbb{Q}}). \] For a smooth projective complex curve $C$, $J(C)=\operatorname{Pic}^{0}(C)$, and $W_d(C)$ denotes the image of the effective divisors of degree $d$ in $\operatorname{Pic}^{d}(C)$, transported to $J(C)$ by a choice of translation. In the customary cubic-threefold interpretation, the Fano surface parametrizes lines on a smooth cubic threefold $T\subset{\mathbb{P}}^{4}_{{\mathbb{C}}}$, and its Abel--Jacobi image lies in the five-dimensional intermediate Jacobian $J(T)$. The record does not explicitly supply this threefold hypothesis.

Requested relationship, with unresolved qualifications. Under the supplied hypotheses that $(A,\Theta)$ is indecomposable, $X\subset A$ is geometrically nondegenerate, and $\mathcal I_X(\Theta)$ satisfies generic vanishing, identify and quantify an additional strengthening sufficient to establish \[ \forall x\in\Theta,\qquad \operatorname{mult}_{x}\Theta\le\frac{g+1}{2}. \] In the reverse direction, determine what consequences this multiplicity bound has for the proposed classification of $X$ as a minimal-class subvariety, with the supplied target models being translates of $W_d(C)$ in a Jacobian or of a Fano surface in a five-dimensional intermediate Jacobian, under suitable identifications of principally polarized abelian varieties.

The original wording calls these statements Conjecture 2 and Conjecture 4: an $\epsilon$-strengthening of Conjecture 2 is conjectured to imply Conjecture 4, and Conjecture 4 is expected to help prove important special cases of Conjecture 2. Their complete texts, the meaning and quantifiers of $\epsilon$, and the intended special cases are absent. Thus neither the strengthened hypothesis nor a precise reverse implication can be stated from this record. Equality of cohomology classes and classification by the displayed geometric models are distinct assertions; their equivalence is not asserted here.

Short English statement

The record asks how a stronger generic-vanishing classification conjecture could imply the theta-divisor bound $\operatorname{mult}_x\Theta\le(\dim A+1)/2$, and how that bound could assist classification in certain special cases. The strengthening, special cases, ground field, and several hypotheses are unspecified, so this remains a qualified statement of the requested relationship requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003900$, exact source problem code catalogue entry, entitled "Implications Among Conjectures on Principally Polarized Abelian Varieties," in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the recorded proposed year is 2018.
• Supplied report citation: Classical Algebraic Geometry (2019), Oberwolfach Reports, pp. 1983--2031, DOI 10.4171/owr/2018/33. Supplied link: https://doi.org/10.4171/owr/2018/33. The citation's 2019 label, the DOI's 2018 component, and the recorded proposed year are retained as supplied.
• Supplied literature source, labelled "OWR report": https://ems.press/content/serial-article-files/46758?nt=1.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and describes the exact record as an incomplete cross-reference between two conjectures rather than a complete standalone claim; it also reports that the referenced general classification remains unresolved. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified. Its verification note reports that the official report was checked and that the dangling conjecture numbers were replaced by generic-vanishing classification and theta-divisor multiplicity statements. This is a supplied provenance claim; no source visit or independent verification is claimed in this draft. The complete conjecture texts and a pinpoint location within the report were not supplied.