Quotient stacks, inertia obstructions, and polarized point blowups

Definitions and mathematical statement

For an algebraic group $G$ acting on a quasi-projective scheme $U$, the quotient stack $[U/G]$ assigns to a test scheme $T$ the groupoid of $G$-torsors over $T$ with equivariant maps to $U$. The AIM problem asks whether every such quotient stack can be a moduli stack for some class of polarized manifolds. The original item does not specify affine versus arbitrary groups, ground field, polarization convention, or whether an equivalence with a full moduli substack is intended. The subsequent paragraph about interiors of canonically polarized moduli and period-domain analogies belongs to the next section, rather than adding hypotheses to this question.

For a linearly polarized manifold, interpret the data as a smooth projective variety $X$ with an ample line bundle $L$, not merely a numerical polarization class. Isomorphisms preserve $L$ up to isomorphism; scalar rigidification removes the scalar $\mathbb G_m$ acting on the line bundle. The supplied report notes that stabilizers in this convention are affine. If $A$ is a positive-dimensional abelian variety, then \[ BA=[\operatorname{Spec}k/A] \] is a quotient of a quasi-projective scheme but has non-affine stabilizer $A$. The report therefore refutes the unrestricted-group formulation for this line-bundle polarization convention. A different convention using only polarization classes requires separate treatment.

Its positive result is conditional on a fixed smooth connected projective parent $W$, a $G$-invariant quasi-projective $U\subset W$, and specified canonical-power, linearization, ampleness and full-automorphism hypotheses. For $u:T\to U$, the proposed family is the point blowup \[ W_u=\operatorname{Bl}_{\Gamma_u}(W\times T) \] with its chosen ample line bundle $L_u$. The report claims an isomorphism of relative fppf sheaves, functorially over arbitrary test schemes, \[ \operatorname{Trans}_G(u,v)\cong \operatorname{Isom}_{\mathrm{pol,rig}}((W_u,L_u),(W_v,L_v)), \qquad\operatorname{Trans}_G(u,v)(T')= \{g\in G(T'):g u_{T'}=v_{T'}\}. \] Thus $[U/G]$ maps fully faithfully to the scalar-rigidified polarized moduli stack and is identified with its full essential image. The claimed sheaf identification retains infinitesimal and nonreduced arrows, not just geometric-point automorphisms.

The summary does not actually list the canonical-power or automorphism hypotheses, the polarization formula or exponent, the parent dimension, or reconstruction proof. Accordingly this conditional special case is not a theorem realizing every affine-group quotient, nor a claim that the essential image is open or closed. In particular the point-blowup mechanism needs hypotheses ensuring the point can be recovered from the polarized blowup.

Short English statement

Ask whether quotient stacks can be realized as moduli of polarized smooth projective varieties. The report gives a non-affine-inertia obstruction for arbitrary groups, and a conditional construction using polarized point blowups for certain affine-group actions. Its positive statement concerns full relative isomorphism sheaves, but the essential construction hypotheses are missing from the supplied summary.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent ID $20000327$, source code catalogue entry; original item, following-section context and supplied research summary.
• AIM workshop "Compact moduli spaces and birational geometry", December 6--10, 2004, Problem 1.8: https://aimath.org/WWN/birational/birational.pdf. Canonical location aim-algebraic-geometry-notes.json, notes[326]; original remarks and literature field empty. The supplied verification mentions official TeX context but does not include its lead-in; no source or PDF was opened here.
• Supplied machine-generated research attempt 1: partial result; problem partially solved at the run and in the registry; rigor metadata "proved", novelty confidence low. No authoritative citation or full hypotheses for the inertia and point-blowup claims are supplied. This non-peer-reviewed aid requires independent verification before citation.