--- title: Canonical Centers in Birational Geometry url: https://www.emergentmind.com/topics/canonical-centers type: topic --- # Canonical Centers in Birational Geometry 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 [1202.0491]. Minimal centers admit subadjunction and inherit klt structures [1009.3996], higher-codimensional centers carry intrinsic birational data through sources and springs [1107.2863], and analytic and mixed-characteristic analogues connect the theory to \(L^2\) extension problems and perfectoid purity [2108.11934], [2504.15569]. ## 1. Definitions and basic configurations Let \(X\) be a normal variety and \(\Delta\) an effective \(\mathbb Q\)- or \(\mathbb R\)-divisor with \(K_X+\Delta\) \(\mathbb Q\)-Cartier or \(\mathbb R\)-Cartier. On a birational model \(f:Y\to X\), one writes the discrepancy formula and calls \((X,\Delta)\) log canonical if all discrepancies are at least \(-1\), and klt if they are strictly greater than \(-1\). A subvariety \(Z\subseteq X\) is a log canonical center if \((X,\Delta)\) is log canonical at the generic point of each irreducible component of \(Z\) and there exists a prime divisor \(E\subset Y\) with \(a(E;X,\Delta)=-1\) and \(f(E)=Z\) [1512.00391]. In Kollár’s formulation, a log center is a subvariety with \(\operatorname{mld}(Z,X,\Delta)<1\), and an lc center is the special case \(\operatorname{mld}(Z,X,\Delta)=0\) [1103.0528]. Two extremal classes are standard. A minimal lc center is minimal with respect to inclusion among lc centers [1905.12542]. In the analytic setting of a complex manifold with a quasi-plurisubharmonic weight \(\Psi\), maximal lc centers are precisely the irreducible components of the non-klt locus \[ N(\Psi)=\{x\in X:\mathcal J(\Psi)\text{ vanishes at }x\} \] [2108.11934]. 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 \(X\) is a smooth compact Kähler manifold and \(D=\sum_{i=1}^N D_i\) is a reduced simple-normal-crossing divisor, then every non-empty intersection \[ D_{i_1}\cap\cdots\cap D_{i_k} \] is an lc center of codimension \(k\) [2307.12025]. 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 \((X,\Delta=\sum_i\delta_i\Delta_i)\) be a normal variety with \(0\le \delta_i\le 1\), and let \(V\subset X\) be an lc center. Hacon associates to \(V\) an adjoint \(b\)-divisor \(B(V;X,\Delta)\) on birational models of \(V\), and proves that \[ (X,\Delta)\text{ is log canonical in a neighborhood of }V \] if and only if \[ (V,B(V;X,\Delta))\text{ is log canonical} \] [1202.0491]. The theorem generalizes Kawakita’s codimension-one result to lc centers of arbitrary codimension. When \(V\) has codimension one, the adjoint \(b\)-divisor reduces to the usual different on the normalization of \(V\). In that case one recovers the familiar statement that if \((\widetilde V,\mathrm{Diff}(\Delta-V))\) is lc, then \((X,\Delta)\) is lc near \(V\) [1202.0491]. The higher-codimensional extension requires replacing the divisor-theoretic different by a genuine \(b\)-divisor whose coefficients are determined birationally from lc thresholds on strata lying above \(V\). The proof uses three ingredients emphasized in the paper. First, one passes to a \(\mathbb Q\)-factorial dlt model \(\mu:Y\to X\), writing \(K_Y+\Delta_Y=\mu^*(K_X+\Delta)\) with \((Y,\Delta_Y)\) dlt. Second, one decomposes \(\Delta_Y=S+\Gamma\), where \(S\) is the unique coefficient-one component dominating \(V\), and runs the relative \((K_Y+S+\Gamma)\)-MMP with scaling over \(X\). Third, one combines Kawamata–Viehweg vanishing with a surjectivity argument: any failure of lc on \(X\) along \(V\) would force a failure of lc on \((V,B)\), contradicting base-point-freeness produced by the MMP with scaling [1202.0491]. 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 \(X\) is normal projective, \(D\ge 0\), \((X,D)\) is lc, and \(W\subset X\) is a minimal lc center, then there exists an effective divisor \(D_W\) on \(W\) such that \[ (K_X+D)|_W \sim_K K_W+D_W, \] and \((W,D_W)\) is klt. In particular, \(W\) has only rational singularities [1009.3996]. 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 \((X/S,B+M)\) is an lc generalized pair and \(W\subset X\) is an lc center, then on the normalization \(\nu:W^\nu\to W\) there exist an effective divisor \(B_{W^\nu}\) and a nef \(b\)-divisor \(M_\iota\) such that \[ K_{W^\nu}+B_{W^\nu}+M_{\iota,W^\nu}\sim_F \iota^*(K_X+B+M_X), \] and \((W^\nu/S,B_{W^\nu}+M_\iota)\) is an lc generalized pair; if \(W\) is minimal, the induced generalized pair is klt [1905.12542]. 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 \(Z\subset X\), one chooses a \(\mathbb Q\)-factorial dlt crepant model \(g:(X',\Delta')\to (X,\Delta)\) and a minimal lc center \(S\subset X'\) mapping onto \(Z\). The source of \(Z\) is the crepant birational equivalence class of \[ (S,\Delta_S),\qquad \Delta_S:=\mathrm{Diff}_S\Delta', \] while the spring of \(Z\) is the finite normal cover \(Z_s\to Z\) occurring in the Stein factorization \(S\to Z_s\to Z\) [1107.2863]. The point is that the source depends only on \((X,\Delta)\) and \(Z\), not on the chosen dlt model. This produces a canonical higher-codimensional adjunction package, including a divisor \(\mathrm{Diff}_{Z^n}\Delta\) on the normalization \(Z^n\) with \[ (K_X+\Delta)|_{Z^n}\equiv_{\mathbb Q}K_{Z^n}+\mathrm{Diff}_{Z^n}\Delta, \] and a Poincaré-residue description of reflexive pluricanonical sheaves [1107.2863]. These constructions make lc centers into lower-dimensional carriers of intrinsic birational data rather than merely images of discrepancy \(-1\) divisors. ## 4. Existence of prescribed centers The theory is not only classificatory; it also supports realization results. de Fernex and Kollár prove that if \(X\) is a \(\mathbb Q\)-Gorenstein normal variety and \(Z\subset X\) is any irreducible subvariety not contained in \(\operatorname{Sing}X\), then there exists an effective boundary divisor \(\Delta\) such that \(Z\) is a log canonical center of \((X,\Delta)\) [1512.00391]. The construction proceeds by embedding \(Z\) as a component of a reduced complete intersection \[ W=V(F_1,\dots,F_r)\subset X,\qquad r=\operatorname{codim}_X Z. \] If \(f:Y\to X\) is the normalization of the blow-up of \(X\) along \((F_1,\dots,F_r)\), and \(\Delta=D_1+\cdots+D_r\) with \(D_i=V(F_i)\), then the exceptional divisors corresponding to the irreducible components of \(W\) acquire discrepancy \(-1\); the divisor lying over the chosen component \(Z\) realizes \(Z\) as an lc center [1512.00391]. A stronger theorem addresses the ambient singularities. If \(X\) is Gorenstein with lc singularities and \(W\subset X\) is a reduced irreducible special complete intersection with lc singularities, not contained in \(\operatorname{Sing}X\), then one can choose a boundary \(\Delta\) such that \((X,\Delta)\) is lc and \(W\) is an lc center [1512.00391]. The proof replaces the generators of \(I_W\) 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 \(\mathbb P^n\), and more generally of any smooth variety \(X\), is an lc center of some pair \((X,\Delta)\) [1512.00391]. 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 \((X,\Delta)\) is lc and \(Z_1,\dots,Z_m\subset X\) are log centers with \[ \operatorname{mld}(Z_i,X,\Delta)<\tfrac16\quad\text{for all }i, \] then the union \(Z_1\cup\cdots\cup Z_m\) is seminormal. If instead \[ \sum_{i=1}^m \operatorname{mld}(Z_i,X,\Delta)<1, \] then every irreducible component of \(Z_1\cap\cdots\cap Z_m\) is again a log center, with minimal log discrepancy bounded by the same sum [1103.0528]. In the relative setting, if \((X,S+\Delta)\) is lc with \(S\) \(\mathbb Q\)-Cartier and \(\operatorname{mld}(Z_i,X,\Delta)<\tfrac12\), then \[ S\cup Z_1\cup\cdots\cup Z_m \] is seminormal relative to \(X\setminus S\) [1103.0528]. The deformation-theoretic corollary is that in one-parameter lc families, boundary components with coefficients \(>\tfrac12\) 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 \(p\notin\{2,3,5\}\), every minimal lc center is normal [2302.07329]. 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-\(3\) cone example in which an lc center fails \(S_2\) and is not seminormal, demonstrating that the characteristic restrictions are substantive [2302.07329]. For isolated lc centers, Chou gives a depth criterion in terms of the reduced discrepancy-\((-1)\) divisor \(E\) on a log resolution. If \(P\in X\) is an isolated lc center of dimension \(n\), then for every \(3\le t\le n\), \[ \operatorname{depth}_P \mathcal O_X\ge t \quad\Longleftrightarrow\quad H^{i-1}(E,\mathcal O_E)=0 \text{ for all } 10\), a Cohen ring \(C_k\subset R\), and a module-finite map \[ A=C_k[[x_1,\dots,x_d]]\to R. \] After perfectoid base change and perfectoidization one obtains \(R_\infty^A\), and for \(\phi\in\operatorname{Hom}_R(R_\infty^A,R)\) an ideal \(a\subset R\) is called \(\phi\)-compatible if \(\phi(a^\infty)\subset a\). A prime \(\mathfrak p\subset R\) is a center of perfectoid purity if \(R\) is perfectoid-pure and \(\mathfrak p\) is uniformly perfectoid-compatible; the finite set of all such primes is denoted \(CP(R)\) [2504.15569]. The relation to classical center theories is explicit. If \(R\) is a quasi-Gorenstein normal local \(\mathbb C\)-algebra with log-canonical divisor \(\Delta\), then the log-canonical centers of \((R,\Delta)\) coincide with the centers of perfectoid purity after reduction mod \(p\) and passage to \(R_\infty^A\) [2504.15569]. If \(R\) has characteristic \(p>0\), is \(F\)-finite and \(F\)-pure, then these primes are exactly Schwede’s centers of \(F\)-purity [2504.15569]. In this sense, centers of perfectoid purity interpolate between lc centers and \(F\)-pure centers. Several structural theorems parallel the older theories. If \(R\) is perfectoid-pure, then there are only finitely many uniformly perfectoid-compatible ideals, hence \(CP(R)\) is finite [2504.15569]. The conductor \(c\) of the normalization \(R\to R^N\) is uniformly perfectoid-compatible, so the absence of nontrivial compatible ideals forces normality. There is also a largest uniformly perfectoid-compatible ideal \[ \beta(R)=\bigcap_{\phi\in\operatorname{Hom}_R(R_\infty^A,R)} \beta(R,\phi), \] which satisfies \(\beta(R)=R\) if and only if \(R\) is perfectoid-pure; when \(R\) is perfectoid-pure, \(\beta(R)\in CP(R)\) is the unique maximal center and \(R/\beta(R)\) is a normal domain [2504.15569]. Fayolle also proves étale stability: \[ CP(S)\cap R = CP(R),\qquad CP(R)\cdot S = CP(S) \] for finite-étale maps and localizations of étale maps [2504.15569]. The basic mixed-characteristic example is \[ R=W(k)[[x,y]]/(xy-p). \] Here \(R\) is perfectoid-pure, the two height-one primes \((x,p)\) and \((y,p)\) are uniformly perfectoid-compatible, and \[ CP(R)=\{(x,p),(y,p)\}. \] Moreover \(\beta(R)=(p)\), and \[ R/\beta(R)\cong k[[x,y]]/(xy) \] is the nodal curve in characteristic \(p\) [2504.15569]. This example exhibits nontrivial mixed-characteristic “canonical” centers in exact analogy with the node’s lc and \(F\)-pure center pictures. Source: https://www.emergentmind.com/topics/canonical-centers