Canonical Centers in Birational Geometry

• Canonical Centers are subvarieties defined as the images of divisors with discrepancy -1, marking the exact log canonical zones on a variety.
• They underlie adjunction and inversion of adjunction processes by transferring birational data from ambient pairs to lower-dimensional centers in both algebraic and analytic settings.
• Extending to mixed-characteristic and perfectoid frameworks, these centers capture intrinsic birational invariants and connect to results on seminormality, depth, and L2 extension problems.

In birational geometry, the basic center construction attached to a log canonical pair $(X,\Delta)$ is the log canonical center: the image of a divisor with discrepancy $-1$. For a normal variety $X$ with an effective boundary $\Delta$ and $K_X+\Delta$ $\mathbb Q$-Cartier, these centers record the loci where the pair is exactly log canonical rather than Kawamata log terminal, and they organize adjunction, inversion of adjunction, and inductive arguments in the minimal model program (Hacon, 2012). Minimal centers admit subadjunction and inherit klt structures (Fujino et al., 2010), higher-codimensional centers carry intrinsic birational data through sources and springs (Kollár, 2011), and analytic and mixed-characteristic analogues connect the theory to $L^2$ extension problems and perfectoid purity (Kim, 2021, Fayolle, 22 Apr 2025).

1. Definitions and basic configurations

Let $X$ be a normal variety and $\Delta$ an effective $\mathbb Q$- or $-1$0-divisor with $-1$1 $-1$2-Cartier or $-1$3-Cartier. On a birational model $-1$4, one writes the discrepancy formula and calls $-1$5 log canonical if all discrepancies are at least $-1$6, and klt if they are strictly greater than $-1$7. A subvariety $-1$8 is a log canonical center if $-1$9 is log canonical at the generic point of each irreducible component of $X$0 and there exists a prime divisor $X$1 with $X$2 and $X$3 (Prelli, 2015). In Kollár’s formulation, a log center is a subvariety with $X$4, and an lc center is the special case $X$5 (Kollár, 2011).

Two extremal classes are standard. A minimal lc center is minimal with respect to inclusion among lc centers (Han et al., 2019). In the analytic setting of a complex manifold with a quasi-plurisubharmonic weight $X$6, maximal lc centers are precisely the irreducible components of the non-klt locus

$X$7

(Kim, 2021). These two notions serve different purposes: minimal centers are the usual targets of subadjunction, while maximal centers control the geometry of non-klt loci and the behavior of the Ohsawa measure.

In the log-smooth case the geometry is completely explicit. If $X$8 is a smooth compact Kähler manifold and $X$9 is a reduced simple-normal-crossing divisor, then every non-empty intersection

$\Delta$0

is an lc center of codimension $\Delta$1 (Chan et al., 2023). This model case underlies both the algebro-geometric and analytic formalisms: the centers are smooth, compact Kähler, and accessible via repeated residue constructions.

2. Inversion of adjunction in arbitrary codimension

The central higher-codimension theorem is Hacon’s inversion of adjunction for log canonical centers. Let $\Delta$2 be a normal variety with $\Delta$3, and let $\Delta$4 be an lc center. Hacon associates to $\Delta$5 an adjoint $\Delta$6-divisor $\Delta$7 on birational models of $\Delta$8, and proves that

$\Delta$9

if and only if

$K_X+\Delta$0

(Hacon, 2012). The theorem generalizes Kawakita’s codimension-one result to lc centers of arbitrary codimension.

When $K_X+\Delta$1 has codimension one, the adjoint $K_X+\Delta$2-divisor reduces to the usual different on the normalization of $K_X+\Delta$3. In that case one recovers the familiar statement that if $K_X+\Delta$4 is lc, then $K_X+\Delta$5 is lc near $K_X+\Delta$6 (Hacon, 2012). The higher-codimensional extension requires replacing the divisor-theoretic different by a genuine $K_X+\Delta$7-divisor whose coefficients are determined birationally from lc thresholds on strata lying above $K_X+\Delta$8.

The proof uses three ingredients emphasized in the paper. First, one passes to a $K_X+\Delta$9-factorial dlt model $\mathbb Q$0, writing $\mathbb Q$1 with $\mathbb Q$2 dlt. Second, one decomposes $\mathbb Q$3, where $\mathbb Q$4 is the unique coefficient-one component dominating $\mathbb Q$5, and runs the relative $\mathbb Q$6-MMP with scaling over $\mathbb Q$7. Third, one combines Kawamata–Viehweg vanishing with a surjectivity argument: any failure of lc on $\mathbb Q$8 along $\mathbb Q$9 would force a failure of lc on $L^2$0, contradicting base-point-freeness produced by the MMP with scaling (Hacon, 2012).

This theorem is the precise form of the principle that the ambient pair is controlled by the pair induced on the center, provided the center is equipped with the correct adjoint birational data.

3. Subadjunction, generalized pairs, and birational structures on centers

Subadjunction is the complementary process: instead of testing lc singularities near a center, it transfers the canonical divisor and boundary to the center itself. Fujino and Gongyo prove that if $L^2$1 is normal projective, $L^2$2, $L^2$3 is lc, and $L^2$4 is a minimal lc center, then there exists an effective divisor $L^2$5 on $L^2$6 such that

$L^2$7

and $L^2$8 is klt. In particular, $L^2$9 has only rational singularities (Fujino et al., 2010). Their proof factors through a dlt blow-up, restriction to a minimal center upstairs, Ambro’s canonical bundle formula for a connected-fiber morphism, and a canonical bundle formula for a generically finite morphism.

For generalized pairs, Han and Liu extend the same philosophy. If $X$0 is an lc generalized pair and $X$1 is an lc center, then on the normalization $X$2 there exist an effective divisor $X$3 and a nef $X$4-divisor $X$5 such that

$X$6

and $X$7 is an lc generalized pair; if $X$8 is minimal, the induced generalized pair is klt (Han et al., 2019). Here the moduli part is intrinsic to the generalized-pair setting and records nef birational data invisible in ordinary pairs.

Kollár’s theory of sources and springs refines higher-codimensional adjunction further. For an lc center $X$9, one chooses a $\Delta$0-factorial dlt crepant model $\Delta$1 and a minimal lc center $\Delta$2 mapping onto $\Delta$3. The source of $\Delta$4 is the crepant birational equivalence class of

$\Delta$5

while the spring of $\Delta$6 is the finite normal cover $\Delta$7 occurring in the Stein factorization $\Delta$8 (Kollár, 2011). The point is that the source depends only on $\Delta$9 and $\mathbb Q$0, not on the chosen dlt model. This produces a canonical higher-codimensional adjunction package, including a divisor $\mathbb Q$1 on the normalization $\mathbb Q$2 with

$\mathbb Q$3

and a Poincaré-residue description of reflexive pluricanonical sheaves (Kollár, 2011).

These constructions make lc centers into lower-dimensional carriers of intrinsic birational data rather than merely images of discrepancy $\mathbb Q$4 divisors.

4. Existence of prescribed centers

The theory is not only classificatory; it also supports realization results. de Fernex and Kollár prove that if $\mathbb Q$5 is a $\mathbb Q$6-Gorenstein normal variety and $\mathbb Q$7 is any irreducible subvariety not contained in $\mathbb Q$8, then there exists an effective boundary divisor $\mathbb Q$9 such that $-1$00 is a log canonical center of $-1$01 (Prelli, 2015).

The construction proceeds by embedding $-1$02 as a component of a reduced complete intersection

$-1$03

If $-1$04 is the normalization of the blow-up of $-1$05 along $-1$06, and $-1$07 with $-1$08, then the exceptional divisors corresponding to the irreducible components of $-1$09 acquire discrepancy $-1$10; the divisor lying over the chosen component $-1$11 realizes $-1$12 as an lc center (Prelli, 2015).

A stronger theorem addresses the ambient singularities. If $-1$13 is Gorenstein with lc singularities and $-1$14 is a reduced irreducible special complete intersection with lc singularities, not contained in $-1$15, then one can choose a boundary $-1$16 such that $-1$17 is lc and $-1$18 is an lc center (Prelli, 2015). The proof replaces the generators of $-1$19 by generic invertible linear combinations, verifies Du Bois properties for partial intersections, and applies the Graf–Kovács criterion.

A direct corollary is that every subvariety of $-1$20, and more generally of any smooth variety $-1$21, is an lc center of some pair $-1$22 (Prelli, 2015). This shows that the center formalism is flexible enough to encode arbitrary subvarieties once the boundary is allowed to vary.

5. Regularity, seminormality, depth, and analytic criteria

Although lc centers can be singular, several regularity theorems constrain their pathology. Kollár proves that if $-1$23 is lc and $-1$24 are log centers with

$-1$25

then the union $-1$26 is seminormal. If instead

$-1$27

then every irreducible component of $-1$28 is again a log center, with minimal log discrepancy bounded by the same sum (Kollár, 2011). In the relative setting, if $-1$29 is lc with $-1$30 $-1$31-Cartier and $-1$32, then

$-1$33

is seminormal relative to $-1$34 (Kollár, 2011). The deformation-theoretic corollary is that in one-parameter lc families, boundary components with coefficients $-1$35 are flat over the base with reduced fibers.

In positive and mixed characteristic, Arvidsson and Posva prove that for a log canonical threefold over an excellent base whose residue fields at closed points are perfect of characteristic $-1$36, every minimal lc center is normal (Arvidsson et al., 2023). They also show that, for standard coefficients and sufficiently large characteristic, the reduced union of all lc centers is seminormal. Their paper includes a characteristic-$-1$37 cone example in which an lc center fails $-1$38 and is not seminormal, demonstrating that the characteristic restrictions are substantive (Arvidsson et al., 2023).

For isolated lc centers, Chou gives a depth criterion in terms of the reduced discrepancy-$-1$39 divisor $-1$40 on a log resolution. If $-1$41 is an isolated lc center of dimension $-1$42, then for every $-1$43,

$-1$44

In particular, $-1$45 is Cohen–Macaulay at $-1$46 if and only if $-1$47 for all $-1$48 (Chou, 2013). The proof combines Kovács vanishing, Grothendieck duality, and local/Matlis duality.

The analytic theory identifies lc centers as the geometric content behind $-1$49-extendability. Kim proves that if $-1$50 is an lc pair, $-1$51 its non-klt locus, and $-1$52, then $-1$53 is locally $-1$54 with respect to the Ohsawa measure $-1$55 if and only if it vanishes on every non-maximal lc center contained in $-1$56 and on every maximal lc center that admits more than one lc place on a resolution (Kim, 2021). Moreover, if a maximal lc center $-1$57 has at least two distinct lc places, then the restriction of the Ohsawa measure to $-1$58 is infinite (Kim, 2021). In the compact Kähler snc setting, this lc-center technology feeds directly into injectivity theorems: if the zero locus of a section $-1$59 contains no lc center of $-1$60, then the multiplication map

$-1$61

is injective for all $-1$62 under the curvature hypotheses stated in the theorem (Chan et al., 2023).

6. Mixed-characteristic analogue: centers of perfectoid purity

Fayolle introduces centers of perfectoid purity as a mixed-characteristic analogue of log canonical centers in characteristic $-1$63 and centers of $-1$64-purity in positive characteristic (Fayolle, 22 Apr 2025). The local setup fixes a complete Noetherian local ring $-1$65 with perfect residue field of characteristic $-1$66, a Cohen ring $-1$67, and a module-finite map

$-1$68

After perfectoid base change and perfectoidization one obtains $-1$69, and for $-1$70 an ideal $-1$71 is called $-1$72-compatible if $-1$73. A prime $-1$74 is a center of perfectoid purity if $-1$75 is perfectoid-pure and $-1$76 is uniformly perfectoid-compatible; the finite set of all such primes is denoted $-1$77 (Fayolle, 22 Apr 2025).

The relation to classical center theories is explicit. If $-1$78 is a quasi-Gorenstein normal local $-1$79-algebra with log-canonical divisor $-1$80, then the log-canonical centers of $-1$81 coincide with the centers of perfectoid purity after reduction mod $-1$82 and passage to $-1$83 (Fayolle, 22 Apr 2025). If $-1$84 has characteristic $-1$85, is $-1$86-finite and $-1$87-pure, then these primes are exactly Schwede’s centers of $-1$88-purity (Fayolle, 22 Apr 2025). In this sense, centers of perfectoid purity interpolate between lc centers and $-1$89-pure centers.

Several structural theorems parallel the older theories. If $-1$90 is perfectoid-pure, then there are only finitely many uniformly perfectoid-compatible ideals, hence $-1$91 is finite (Fayolle, 22 Apr 2025). The conductor $-1$92 of the normalization $-1$93 is uniformly perfectoid-compatible, so the absence of nontrivial compatible ideals forces normality. There is also a largest uniformly perfectoid-compatible ideal

$-1$94

which satisfies $-1$95 if and only if $-1$96 is perfectoid-pure; when $-1$97 is perfectoid-pure, $-1$98 is the unique maximal center and $-1$99 is a normal domain (Fayolle, 22 Apr 2025). Fayolle also proves étale stability: $X$00 for finite-étale maps and localizations of étale maps (Fayolle, 22 Apr 2025).

The basic mixed-characteristic example is

$X$01

Here $X$02 is perfectoid-pure, the two height-one primes $X$03 and $X$04 are uniformly perfectoid-compatible, and

$X$05

Moreover $X$06, and

$X$07

is the nodal curve in characteristic $X$08 (Fayolle, 22 Apr 2025). This example exhibits nontrivial mixed-characteristic “canonical” centers in exact analogy with the node’s lc and $X$09-pure center pictures.