Methods for Kodaira-Dimension Superadditivity

Definitions and mathematical statement

Formulation unavailable from the supplied excerpt. The record concerns Kodaira-dimension superadditivity, but contains commentary referring to Conjectures 1, 2, and 3 without stating them. Consequently, no precise conjectural inequality, quantified mathematical statement, or complete set of hypotheses can be recovered from this record.

The notation $\kappa(Z)$ denotes the Kodaira dimension of a variety $Z$; the phrase "of general type" means $\kappa(Z)=\dim Z$ in a setting where this invariant is defined. The excerpt describes $Y$ as a base, $F$ as a fiber, and also uses $Y$ for a compactification of $V$. It mentions $X$ without defining it. No morphism, ground field, smoothness or projectivity assumptions, boundary divisor, or relationship among these objects is supplied. These roles do not determine the missing hypotheses.

Source-reported cases and qualifications. The following are retained as reports about the absent conjectures, not as independently verified theorems or substitutes for their statements:

• Conjectures 1 and 3 hold when $Y$ is an abelian variety or a curve, or when $V$ compactifies to an abelian variety; the excerpt refers to [1].
• Conjecture 3 holds when the fiber $F$ is of general type; the excerpt refers to [4].
• When $X$ is of general type, Conjecture 2 holds. Under that same condition on $X$, Conjecture 1 also holds if a compactification $Y$ of $V$ is not uniruled; the excerpt includes $\kappa(Y)\geq 0$ as a sufficient condition for this additional assumption and refers to [4]. The ambient hypotheses needed for these assertions are absent.
• The speaker reports that [4] proves that Conjecture 2 follows from conjectures of the minimal model program (MMP), without specifying those conjectures here. The speaker also reports receiving, during the conference, a preprint from F. Campana claiming the corresponding implication for Conjecture 3. The latter is explicitly a reported preprint claim.

The fragment begins mid-sentence with "and [4]". It attributes methods to Hodge module techniques considered in [3] and earlier hyperbolicity methods of Kovács and Viehweg--Zuo, among others. It describes analytic techniques using singular metrics on direct images of pluricanonical bundles, as in work of Berndtsson and Păun--Takayama, as essential when $\kappa(Y)=0$. It also identifies a theorem of Campana--Păun detecting varieties of general type as an important ingredient. No precise statements or full references for these ingredients are supplied.

The supplied registry marks the statement as unrecoverable: its verification field reports that the official report confirms this extraction is commentary on methods and known cases, and that no separate clean statement is promoted. That verification report is supplied provenance, not an independent verification performed for this draft. The missing conjectures must be recovered with their hypotheses before this record can support a precise mathematical problem.

Short English statement

This record discusses methods and reported cases of three unstated Kodaira-dimension superadditivity conjectures. Their formulas and hypotheses are missing, so a precise open problem cannot be reconstructed. The supplied partially_solved label and literature assessment are dated curation, not independently verified mathematical conclusions.

Sources

• Local UnsolvedMath registry, permanent ID $30005170$, title "Methods for Kodaira-Dimension Superadditivity," exact source problem code catalogue entry. The supplied record lists proposed year 2022 and no named proposer. Its original excerpt is retained for provenance.
• Supplied report citation: Algebraic Geometry: Moduli Spaces, Birational Geometry and Derived Aspects (2023), Oberwolfach Reports, pp. 1805--1864, DOI https://doi.org/10.4171/owr/2022/32. The extracted passage bears page number 1821. The literature-source label is "Oberwolfach Report 32/2022." The supplied 2023 citation and 2022 report designation are retained without reconciliation.
• Supplied literature source: Popa, Conjectures on the Kodaira dimension, https://people.math.harvard.edu/~mpopa/papers/kodaira.pdf.
• Literature assessment checked at 2026-08-22, according to the supplied record: further cases of the superadditivity conjectures are proved, some follow from MMP conjectures, and the general statements are not known. This is supplied dated curation derived from the OWR record, not independently verified here; time-sensitive status and cited claims require independent verification.
• The excerpt's reference labels [1], [3], and [4] have no accompanying bibliographic entries in the supplied record. The report citation supplies provenance, but the supplied excerpt does not supply an authoritative self-contained statement of the intended conjectures.