Local-Cohomology Obstructions to Regular Maps and Projections

Definitions and mathematical statement

The supplied record asks a research question rather than stating a conjecture with specified hypotheses. Its question is: can local cohomology obstruct the existence of injective or $k$-regular maps of complex varieties, or of low-dimensional projections defined by separating invariants? The following definitions explain possible formulations; the source does not determine a unique quantified mathematical statement.

For complex algebraic varieties $X,Y$, a map $f:X\to Y$ is injective on complex points if \[ \forall x,x'\in X({\mathbb{C}}),\qquad f(x)=f(x')\Longrightarrow x=x'. \] This condition alone does not assert that $f$ is a closed immersion. The record does not specify whether all maps under consideration must be algebraic morphisms, holomorphic maps, or continuous maps, nor whether injectivity is intended only on complex points or in a stronger scheme-theoretic sense.

One convention for $k$-regularity, for integers $k\geq 1$ and $N\geq 0$, requires a map $f:X({\mathbb{C}})\to{\mathbb{C}}^N$ to satisfy \[ \forall x_1,\ldots,x_k\in X({\mathbb{C}})\text{ pairwise distinct},\qquad \operatorname{rank}\begin{pmatrix}f(x_1)&\cdots&f(x_k)\end{pmatrix}=k. \] A different convention requires the images of every $k$ distinct points to be affinely independent, equivalently \[ \operatorname{rank}\begin{pmatrix}1&\cdots&1\\ f(x_1)&\cdots&f(x_k)\end{pmatrix}=k. \] The supplied record does not choose between these conventions or specify the target spaces.

To describe separating invariants conditionally, let a group $G$ act algebraically on an affine complex variety $V$, and write $A={\mathbb{C}}[V]$ and \[ A^G=\{h\in A: h(g\cdot v)=h(v)\text{ for all }g\in G,\ v\in V({\mathbb{C}})\}. \] A finite family $f_1,\ldots,f_m\in A^G$ is separating if, for all $v,w\in V({\mathbb{C}})$, \[ \left(\exists h\in A^G:\ h(v)\ne h(w)\right) \Longrightarrow \left(\exists j\in\{1,\ldots,m\}:\ f_j(v)\ne f_j(w)\right). \] It defines the invariant map $F:V\to{\mathbb{A}}^m_{{\mathbb{C}}}$, $F(v)=(f_1(v),\ldots,f_m(v))$. Separation concerns pairs distinguishable by invariant functions; it need not mean injectivity on $V({\mathbb{C}})$. Calling such a map a projection requires additional ambient data absent from the record.

For a Noetherian commutative ring $B$, an ideal $I\subseteq B$, and a $B$-module $M$, define \[ \Gamma_I(M)=\{u\in M: I^n u=0\text{ for some }n\geq1\}, \qquad H_I^i(M)=R^i\Gamma_I(M)\quad(i\geq0). \] These are local cohomology modules. An obstruction to a specified class of maps would require a proved necessary local-cohomology condition for existence, so that failure of that condition implies nonexistence. Here the ring, support ideal, module, cohomological degrees, and necessary condition are all unspecified.

Accordingly, the statement retained is the source's question about obtaining such obstructions. No particular nonvanishing criterion, numerical lower bound, universally quantified nonexistence assertion, or equivalence is supplied. The class of varieties, map category, invariant-theoretic setting, and meaning of low dimension require clarification before a precise conjecture can be extracted.

Short English statement

The problem asks whether local cohomology can obstruct injective maps, $k$-regular maps, or maps defined by separating invariants of complex varieties. The supplied record leaves the hypotheses, the convention for $k$-regularity, the relevant local cohomology, and the dimension bounds unspecified. This is a statement-only draft of that research question requiring human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $30004083$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. Registry locations: data/unsolvedmath/problems.json and the supplementary catalog catalog/problems.json. The supplied record lists proposed year 2019 and no individual proposer.
• Supplied source citation: Singularities and Homological Aspects of Commutative Algebra (2020), Oberwolfach Reports, pp. 407--464, DOI 10.4171/owr/2019/6. https://doi.org/10.4171/owr/2019/6. The supplied literature list labels this reference "Oberwolfach Report 6/2019." Both year descriptions are retained as supplied; no exact problem passage is provided.
• Supplied literature source: Görlach, Injection dimensions. https://arxiv.org/abs/1905.11306. The record supplies no theorem number or passage identifying a precise formulation or local-cohomology obstruction.
• Literature assessment checked at 2026-08-22: the supplied dated curation labels the problem open and states that primary work supplies lower bounds and examples but no general answer to this broad title-level question. This assessment is supplied dated curation, not independently verified. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry supplies statement status corrected_verified, with a note that mathematical content was preserved while notation, typography, duplicate clauses, or omitted definitions were repaired. This is a supplied editorial designation, not independent mathematical verification of this draft.