Mathematicians chasing the shape of space often ride two different weather systems at once. In one world, the land is laid out in the familiar language of characteristic zero, where powerful tools like resolutions of singularities and vanishing theorems let us tame the wild corners of geometry. In the other, positively characteristic worlds behave differently, sometimes stubbornly resisting those same tools. A recent contribution from Shihoko Ishii at the University of Tokyo’s Graduate School of Mathematical Sciences builds a bridge between these two climates. The bridge is not a literal crossing but a precise, technical transfer that lets us translate sharp invariants of singularities from the known terrain of characteristic zero into the trickier terrain of positive characteristic. The payoff is a clearer picture of how bad singularities can be and how their fingerprints behave as we move between mathematical worlds.
Lead author Shihoko Ishii and colleagues show that certain numerical fingerprints of singularities, called log discrepancies and log canonical thresholds, behave in a surprisingly disciplined way even when the usual tools are not available. The work takes a central problem in algebraic geometry one step further: can we lift the invariants measured in positive characteristic to a realm where the theory is well developed, and then import the conclusions back? The answer, at least for the questions addressed, is yes, and the consequences are both elegant and practical for researchers trying to classify and understand singularities across dimensions. Ishii’s claim is not merely that a translation exists, but that the translation preserves enough structure to deduce discrete sets, rational numbers, and stability phenomena that people had previously proved only in characteristic zero. In short, the study makes the abstract idea of a bridge feel almost tangible to readers who care about how geometry’s rough edges can be measured and compared across different mathematical universes.
What makes this result particularly compelling is that the bridge works in high dimensions. Earlier work by Ishii and collaborators secured similar conclusions in dimensions up to three, but the leap to higher dimensions is where the landscape begins to look truly chancy. The current accomplishment shows that a carefully constructed lifting of the geometric data exists even when the source world lacks a clean resolution of singularities. It uses a choreography of blowups, liftings, and skeletons that align the characteristic p world with a counterpart in characteristic zero. The payoff is a powerful set of corollaries: the set of log discrepancies for a fixed exponent becomes discrete, the collection of log canonical thresholds of multi-ideals on a smooth variety over a positive characteristic field sits inside the corresponding set over the complex numbers, and the accumulation points of those thresholds are rational when the exponents are rational. It also proves an ACC, the ascending chain condition, for these thresholds in positive characteristic. All of this emulates, in a robust way, the behavior known in characteristic zero and codifies it for higher dimensional spaces in positive characteristic.
Ultimately, Ishii’s paper is a story about how mathematics can reconcile seemingly incompatible worlds by peeling back enough structure to make the essential features compatible. The bridge is anchored in concrete geometric operations blowups and in an abstract framework of liftings that connects algebro-geometric invariants to a common language across characteristics. The result feels less like a clever trick and more like a principle: when the heart of a problem is the singularity’s stubbornness rather than the ambient space, a carefully engineered translation can reveal a unified underlying behavior that transcends the field you start in. The study is a clear demonstration that even in the absence of a long list of tools in positive characteristic, structure can still be recovered through a well-constructed dialogue with characteristic zero.
A Global Bridge Between Characteristics
Think of a prime divisor as a kind of topographic marker on a space. It records where a surface bulges or tears in a way that matters to the geometry around it. The invariants attached to these markers, such as log discrepancies and log canonical thresholds, quantify how bad or mild those singularities are when you try to simplify or resolve them. In characteristic zero, a rich toolkit exists for chasing these markers and proving properties about them. In positive characteristic, many of those tools break or simply aren’t available. Ishii’s central move is to construct a bridge that lets the invariants from positive characteristic be studied by transporting a compatible picture from characteristic zero.
Concretely, the main theorem shows that for a smooth N-dimensional affine space over a positive characteristic field and for a prime divisor over the origin, one can find a corresponding prime divisor in the complex affine space whose numerical data mirrors the original. The mathematical language is precise: the center of the divisors aligns, the valuations of lifted ideals line up, and a delicate equality of invariants holds when you compare the two settings. The upshot is not just an abstract correspondence but a way to deduce concrete properties in positive characteristic by looking at the well-mudled waters of complex geometry. In practice this means a kind of invariance principle: a key invariant computed in positive characteristic is faithfully represented by a cousin in characteristic zero, revealing a shared structure that can be exploited to draw conclusions about discreteness, rationality, and stability of thresholds across the two worlds.
Crucially, this bridge does not rely on a resolution of singularities in the positive characteristic world, which is notoriously delicate. The technique works by constructing liftings and a sequence of blowups, a Zariski-style approach that tracks how a divisor appears along a chain of transformations. The authors then show that the critical numerical data at each step matches the data of a lifted object in the complex world. It is a careful, almost surgical, alignment of two different kinds of spaces so that the invariants you care about have a common echo across the boundary of characteristics.
From a historical vantage point, the paper sits at the intersection of two long arcs in algebraic geometry. On one side lies the characteristic zero story: the theory of singularities, resolution techniques, and the lattice of invariants that measure how far a space is from being smooth. On the other side stands positive characteristic, a fertile but more treacherous terrain where many standard tools fail. The genius of Ishii’s approach is to lean into the structural similarities rather than pretend the worlds are identical, and then to prove that the structural correspondences are strong enough to pass numerical consequences back and forth.
Why Discreteness, Rationality, and ACC Matter
Two ideas sit at the heart of the paper’s conclusions: discreteness and the ascending chain condition. Discreteness says that, for a fixed exponent, the set of log discrepancies that can occur is finite. In plain terms, among all the potential ways a singularity can behave, only finitely many “intensity levels” exist when you fix how aggressively you weigh the multi-ideals. It is a kind of coarse-grained stability: there aren’t an endless parade of increasingly subtle discrepancy values to chase. The second idea, ACC, is the assertion that there are no infinite strictly increasing sequences of log canonical thresholds within a specified framework. Together these properties give a robust sense of order to a potentially chaotic landscape: despite the wildness singularities can exhibit, there is a bound on how far and how fast their invariants can drift.
One striking corollary is that the set of log canonical thresholds on positive characteristic varieties is contained in the corresponding set over the complex numbers. In other words, the complex world provides a universal ceiling for these thresholds. If the exponents are rational, then the accumulation points—the values where thresholds cluster—are themselves rational. This last point is remarkably satisfying: rationals appear as natural endpoints of what could have been a messy, irrational accumulation. It echoes a broader theme in algebraic geometry: even when working across different mathematical climates, certain arithmetic regularities persist, offering a kind of arithmetic compass by which researchers can navigate high-dimensional singularities.
The authors further show that the set of thresholds satisfies ACC when the exponents come from a given DCC set. This strengthens the sense that there is a global rhythm to singularities: as you tighten the permissible exponents, the thresholds settle into a predictable pattern rather than chasing ever more exotic outliers. That, in turn, helps researchers formulate conjectures and test new ideas about how singularities behave in families, how they deform, and how they respond to deformations and degenerations in positive characteristic. It is not just a classification tool; it is a lens that makes the geometry’s tempo legible across a wide range of settings.
How Lifting Turns a Hard Problem into a Bridge
The technical heart of the paper lies in the art of lifting: constructing a counterpart in complex geometry that preserves the essential data of the positive characteristic world. The method proceeds in stages. First comes the idea of a skeleton, a finitely generated substructure that can be lifted from the positive characteristic world to a characteristic zero world in a controlled, Z flat way. Then, using a careful dance of compatible skeletons, you can define liftings of elements and ideals so that the basic valuation data against a divisor remains aligned under the lift. In other words, you attach to a point a full set of algebraic ingredients that can be transported across the bridge without losing the information that matters for log discrepancies and lct values.
Next comes a sequence of blowups, a sequence known in geometry as a Zariski sequence, which is a way to spread out the singularity across a tower of spaces while keeping track of how each step changes the geometry. Ishii’s work shows how to perform two extra blowups as part of the lifting stage and still preserve the alignment of the numerical data. The upshot is a one-to-one correspondence between a prime divisor over the positive characteristic origin and a divisor over the complex lift, with the right invariants matching. This correspondence then lets the author leverage the full weight of characteristic zero results to deduce properties for the positive characteristic setting, which would be hard or impossible to prove directly in that setting.
The proof is a tour of ideas that appear, at first glance, technical and abstract. It nonetheless reveals a deep pattern: even if a particular method (like resolution of singularities) is not available, a disciplined translation of structure makes it possible to import robust results. The arc-space intuition referenced in the paper provides a way to think about the invariants as measurements of how singularities behave along all possible paths through the space. The lifting framework makes those paths meaningful on both sides of the characteristic boundary and ensures that the measurement is faithful when you compare the two worlds.
What This Means for the Future of Singularities
The work is a milestone not because it solves every open problem about singularities in positive characteristic, but because it broadens the toolkit in a domain where progress has often depended on tools that don’t exist in positive characteristic. By showing that the invariants are controlled in a way that mirrors characteristic zero, Ishii has given researchers a reliable compass for navigating higher dimensions where intuition can fail and the algebra becomes unwieldy. This is the kind of advance that quietly reshapes what can be attempted: once you know the set of possible log discrepancies is discrete for fixed exponents, or that ACC holds for a broad class of lct values, you can design families and degenerations with a clearer expectation of the singularities you’ll encounter.
The practical impact for the field is incremental but meaningful. It builds a bridge that allows the community to borrow results from complex geometry to guide conjectures and proofs in positive characteristic. It also sharpens questions about the existence and behavior of log resolutions in positive characteristic by showing that, even when those resolutions don’t exist, the invariants still obey a familiar arithmetic rhythm. In a sense, the paper demonstrates that the universe of algebraic geometry is not split into two incompatible camps but rather two faces of the same mathematical landscape, connected by a carefully engineered corridor of liftings and zariski sequences.
As for who stands behind the bridge, the work is anchored at the University of Tokyo, specifically within the Graduate School of Mathematical Sciences, led by Shihoko Ishii. The study bears the mark of a scholar who has long pursued the delicate interplay between characteristic p and characteristic zero, and who has already illuminated the path in lower dimensions. This higher-dimensional generalization is, in many ways, the natural continuation of a conversation that began with the idea that singularities behave like stubborn riddles until you find a way to translate them into a language that can be read with existing tools. The bridge Ishii builds is not merely a workaround; it is a principled route toward a more unified understanding of singularities across the entire spectrum of algebraic geometry.
Looking ahead, the immediate questions are about extending these lifting techniques to other invariants and to even broader classes of spaces. The success in higher dimensions invites mathematicians to test the limits of the method, to seek comparable bridges for other birational invariants, and to explore whether similar ACC-type statements can be established in even more general settings. There is also a quiet optimism that refining the lifting framework could illuminate the longstanding goal of resolving singularities in positive characteristic, or at least of clarifying precisely where current techniques can and cannot reach. The bridge, after all, is only as strong as the ground it spans—and Ishii’s work gives us confidence that the ground on both sides shares a deeper structure than we might have guessed.
In the end, this is a story about how mathematics evolves. It reminds us that progress often comes from not just solving problems in isolation but from creating channels through which ideas can travel. The bridge across characteristics does not erase the differences between worlds; it translates them in a way that preserves their essential shape. When the translation is precise enough to recover discreteness, rationality, and ACC, it becomes a powerful lens for future exploration. The lone question that remains is how far this lens can take us and what new vistas it will reveal as we continue to chart the geometry of singularities in every characteristic we can imagine.
Lead author and institution Shihoko Ishii, University of Tokyo, Graduate School of Mathematical Sciences, leads this exploration into liftings of ideals across characteristics, expanding the frontier of singularity theory in higher dimensions.
