1 · The essentials
The manuscripts study complex algebraic shapes near a singular point, where the shape is not smooth. They focus on a particular class of such shapes and attach a numerical measure to each point. The main manuscript claims that this measure cannot exceed a specific ceiling, determined by the dimension. It also claims that reaching the ceiling identifies exactly one local type of singularity, called an ordinary double point. The statement covers every dimension from two upward, under the stated restrictions. A companion manuscript gives the four-dimensional case, where the ceiling is 162. The main result is not formalized in Lean and has not been formally verified.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine a restricted collection of objects being scored under a common rule. The claim gives a maximum score and says that only objects of one particular type can attain it; the mathematical restrictions determine which objects belong in the collection.
Why it matters
The claimed result combines a numerical bound with an exact description of the equality case. Thus, within its hypotheses, the largest allowed normalized volume singles out the ordinary double point. The main manuscript presents this as the ordinary-double-point volume-gap conjecture in every dimension covered by the statement.
2 · Going further
The precise problem concerns a singular complex algebraic klt germ of dimension , with zero boundary. The main manuscript claims that its normalized volume is at most . Equality is claimed to hold if and only if the germ is an analytic ordinary double point. The klt and zero-boundary conditions are explicit hypotheses, not optional qualifications.
The source calls this statement the ordinary-double-point volume-gap conjecture. Its two parts are an upper bound for all singular germs in the specified class and a characterization of those attaining that bound. The supplied abstracts state the conclusions but do not describe the argument, so no method or intermediate step can be reported from this material.
The companion manuscript treats complex algebraic klt fourfold germs with zero boundary. It claims the bound 162 and the same analytic equality characterization. This agrees with the general expression at , since . The abstracts therefore present an all-dimensional statement and its four-dimensional counterpart, but give no information about how their arguments depend on one another.
Useful notions
- Complex algebraic shapes and their dimension
- Smooth points and singular points
- Local study through germs
- Normalized volume
- The klt and zero-boundary hypotheses
What was known before / what is new
Before
The source identifies the target statement as the ordinary-double-point volume-gap conjecture. It does not specify earlier bounds, previously established dimensions, or the history of the conjecture.
New
The main manuscript claims the bound for every singular boundary-zero complex algebraic klt germ of dimension , together with equality precisely for an analytic ordinary double point. The companion manuscript claims the four-dimensional statement with bound 162 and describes it as resolving the conjecture in that dimension.
Limits and assumptions
The statements apply only to singular complex algebraic klt germs with zero boundary, in dimension at least two; the companion statement is restricted to dimension four. The equality characterization is analytic. The supplied abstracts contain neither an argument nor definitions of the numerical invariant and the singularity classes. The main result is NOT formalized in Lean and has NOT been formally verified. The source identifies no remaining open questions and makes no claim about cases outside these hypotheses.
Mini glossary
- Germ —
- A germ records the local shape of a space near a specified point.
- Normalized volume —
- Normalized volume is the numerical invariant bounded in these manuscripts; the supplied abstracts do not define its construction.
- Klt —
- Klt names the class of singularities to which the stated bounds apply; its technical definition is not supplied in the source.
- Zero boundary —
- Zero boundary is the explicit restriction that the germs under consideration have no nonzero boundary in the statement.
- Analytic ordinary double point —
- An analytic ordinary double point is the local singularity type that the manuscripts claim characterizes equality in the volume bound.
Keywords : Normalized volume · Ordinary double point · Klt singularity
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Proves the ordinary-double-point volume-gap conjecture: every singular complex algebraic klt germ of dimension n ≥ 2, with zero boundary, has normalized volume at most . Equality holds precisely for an analytic ordinary double point.
To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.
Manuscripts in this family
The ordinary-double-point gap in every dimension
24/09/2026We prove the ordinary-double-point gap conjecture for boundary-zero complex algebraic klt germs in every dimension: a singular n-dimensional germ has normalized volume at most , with equality precisely at an analytic ordinary double point.
The normalized-volume gap in dimension four
24/09/2026We prove that every singular complex algebraic klt fourfold germ with zero boundary has normalized volume at most 162, with equality precisely for an analytic ordinary double point. This resolves the ordinary-double-point volume-gap conjecture in dimension four.