Completing the pseudonorm program for birational classification of varieties of general type: unrestricted uniform Torelli theorem, realization, computation, and canonical-metric links

The pseudonorm $\langle\langle\eta\rangle\rangle=\int_M(\eta\wedge\bar\eta)^{1/m}$ on the $m$-th pluricanonical space $H^0(M,mK_M)$ of a smooth complex projective variety is a birational invariant. Chi and Yau (2008) proved a Torelli-type theorem — an isometry between such pseudonormed spaces of varieties of general type is induced by a unique birational map — under a base-point-freeness hypothesis and a 'concentrating singularities' condition (CS), and posed the four principal questions of what Yau later called the pseudonorm program. The open problem is: (1) a uniform birational Torelli theorem with bound depending only on the dimension for arbitrary varieties of general type, using the full pluricanonical spaces, with no base-point-freeness, canonical-polarity, index, or smooth-minimal-model hypotheses, extended to mildly singular (klt) varieties and positive characteristic; (2) a geometric and algebraic characterization of the pseudonormed spaces that arise as pluricanonical spaces; (3) effective computation of the pseudonorms; (4) the precise relation between the induced pluricanonical pseudo-metrics and Kähler–Einstein metrics.

Background

Birational geometry studies complex projective varieties up to birational equivalence: equivalence by rational maps that are isomorphisms between dense open subsets. For a smooth projective variety $M$ of dimension $n$, the canonical line bundle $K_M$ is the determinant of the cotangent bundle (the bundle of holomorphic $n$-forms), and $M$ is of general type when $K_M$ is big, i.e. when its pluricanonical sections grow at the maximal rate. The birational classification of such varieties is governed by the pluricanonical spaces $H^0(M,mK_M)$, $m\geq 1$: the plurigenera $P_m=\dim H^0(M,mK_M)$ are birational invariants, and by results of Hacon–McKernan and Takayama there is a constant $m_n$ depending only on $n$ such that the $m$-th pluricanonical rational map $\varphi_{|mK_M|}:M\dashrightarrow\mathbb{P}H^0(M,mK_M)^*$, defined away from the base locus (the common zero set of all pluricanonical sections), restricts to a birational map onto its image for all $m\geq m_n$.

Chi and Yau (2008) observed that each pluricanonical space carries a metric structure that is itself a birational invariant. A section $\eta$ of $mK_M$ is locally $f(z)(dz_1\wedge\cdots\wedge dz_n)^{\otimes m}$, and the density $\langle\eta\rangle_m=|f|^{2/m}dx_1dy_1\cdots dx_ndy_n$ patches to a globally defined nonnegative $(n,n)$-form; for $M$ compact one sets $\langle\langle\eta\rangle\rangle=\int_M\langle\eta\rangle_m$. The functional $\eta\mapsto\langle\langle\eta\rangle\rangle$ is positive and satisfies the triangle inequality (because $|a+b|^{\alpha}\leq|a|^{\alpha}+|b|^{\alpha}$ for $0<\alpha<1$) but is homogeneous of degree $2/m$ rather than $1$; following the source paper it is called a pseudonorm, and $(H^0(M,mK_M),\langle\langle\cdot\rangle\rangle)$ a pseudonormed space. Every birational map $M'\dashrightarrow M$ of general-type varieties induces a linear isometry of the corresponding pseudonormed spaces. For $n=1$, $m=2$ the pseudonorm is the genuine $L^1$-norm on holomorphic quadratic differentials (sections of $K_C^{\otimes 2}$ on a curve $C$), and Royden's theorem (1971) — a key ingredient in his description of the isometry group of Teichmüller space, the moduli space of compact Riemann surfaces of a fixed genus — states that an isometry between the $L^1$-normed spaces of quadratic differentials of compact Riemann surfaces of genus at least $2$ is induced, up to a unit scalar, by a unique biholomorphism. A Torelli-type theorem, named after the classical result that periods determine a curve, asserts that specified data determine a variety; Chi and Yau asked for such a theorem with birational equivalence as the equivalence relation and pseudonormed pluricanonical spaces as the data.

Their 2008 paper proved it under two auxiliary hypotheses: (i) the relevant pluricanonical system is base-point-free and defines a birational map onto its image, and (ii) a 'concentrating singularities' condition (CS): for a general hyperplane $H$ in the image of the pluricanonical map there must exist a section $\eta_0$ with $2\operatorname{lct}(D_{\eta_0})+2/m<1$ — here $D_{\eta_0}$ is the divisor of zeros of $\eta_0$ and $\operatorname{lct}(D,x)$, the log canonical threshold, is the supremum of $c>0$ for which the pair $(M,cD)$ has log canonical (mild) singularities at $x$, smaller values meaning worse singularities — whose characteristic indicatrix $C(D_{\eta_0})$, the locus of points where $D_{\eta_0}$ is most singular, is contained, off the fixed base locus of $|mK_M|$, in the zero locus of the sections defining $H$. (The associated characteristic index of $D$ at $x$ is the pair $(\operatorname{lct}(D,x),\mu(D,x))$, where the log canonical multiplicity $\mu(D,x)$ counts how many divisors of a log resolution attain the threshold simultaneously; it refines the log canonical threshold.) The mechanism is an asymptotic expansion of $\langle\langle\eta_0+t\eta\rangle\rangle$ as $t\to 0$ whose leading coefficient detects $C(D_{\eta_0})$; this allows hyperplanes of the pluricanonical image to be recognized from metric data alone, from which the birational map is reconstructed. Under these hypotheses they obtained: for surfaces of general type, every linear isometry between the $m$-pluricanonical pseudonormed spaces with $m\geq 75$ is induced, up to a unit scalar, by a unique birational map; in all dimensions, the same conclusion for $m$ of the form $2r(n+2)!$ with $r\geq r_0(n)$, provided the varieties admit smooth minimal models (minimal models — birational models with nef canonical bundle, i.e. a canonical bundle of nonnegative degree on every algebraic curve, and mildest singularities — exist for varieties of general type by Birkar–Cascini–Hacon–McKernan, but are smooth only in low dimension); and, with proofs deferred to companion works, general-type statements either with a bound depending also on the index of the minimal model (the smallest $j$ making $jK$ Cartier) or for the auxiliary subspaces $V(M,m,r)$, the image of $\operatorname{Sym}^r H^0(M,mK_M)$ in $H^0(M,rmK_M)$ with $r\geq 2n+1$ and $m\geq m_n$, which are uniform in $n$ but are not the full pluricanonical spaces.

The same paper posed the program's four principal questions: (1) the Torelli-type theorem itself in full generality; (2) Existence — characterize which pseudonormed spaces arise as pluricanonical spaces, with relations to Geometric Invariant Theory (GIT, the algebraic theory of quotient varieties and moduli spaces under reductive group actions); (3) Computation — compute the pseudonormed spaces effectively, as periods are computed in classical Torelli theory; (4) Relations with other invariants — connect the new invariant pseudo-metrics on $M$ induced by the pluricanonical series (including a hermitian metric on the section space for which the pseudonorm serves as a Kähler-type potential) with canonical metrics of Kähler geometry, notably Kähler–Einstein metrics (the unique constant-Ricci-curvature metrics carried by canonically polarized manifolds).

Subsequent work settled substantial parts of axis (1) in restricted regimes: Chi's complete treatment (2016) removed the (CS) condition via detailed asymptotic expansions, defined a resolution-free singular index refining the log canonical threshold, and proved for canonically polarized manifolds (those with $K_M$ ample) in every dimension $n$ an explicit constant $c_n$, depending only on $n$, such that an isometry of $m$-pluricanonical pseudonormed spaces for suitable $m\geq c_n$ forces an isomorphism; an alternative route through Rudin's equimeasurability theorem for $L^p$-isometries — which asserts that a linear isometry between $L^p$-spaces, for $p>0$ not an even integer, sends characteristic functions of measurable sets to characteristic functions, so that the underlying measures are equimeasurable — was identified (recorded in Chi's 2022 paper, per an observation of S. Antonakoudis); the p-adic analogue was proved by Chi (2022), with closely related results independently obtained by Lee (preprint 2022, published 2025); the linear-isometry-rigidity philosophy was extended to bounded hyperconvex domains (bounded domains carrying a bounded plurisubharmonic exhaustion function) and $L^p$-Bergman theory (the study of the Bergman kernel and projection on spaces of $L^p$-integrable holomorphic functions) by Deng–Wang–Zhang–Zhou, and to Stein manifolds (complex manifolds holomorphically embeddable as closed submanifolds of some $\mathbb{C}^N$, the analytic analogue of affine varieties), Stein morphisms, and direct images of pluricanonical bundles by Inayama; and Finski proved that submultiplicative norms on section rings are asymptotically equivalent to sup-norms coming from polarizing metrics. However, the unrestricted uniform statement for full pluricanonical spaces of arbitrary general-type varieties in dimension at least $3$ remains open, and no work addresses the realization or computation questions or proves the Kähler–Einstein connection.

Problem Statement

Complete the pseudonorm theory of birational classification of varieties of general type along the axes left open by Chi and Yau (2008) and not closed by the subsequent literature. The common objective: establish that the pseudonormed pluricanonical spaces form a complete, effectively computable, and geometrically anchored system of birational invariants of varieties of general type. Concretely, resolve the following four linked questions.

(1) Uniform, unrestricted birational Torelli theorem. For every $n\geq 3$, prove or disprove: there exists $m_0(n)$, depending only on $n$, such that for any two smooth complex projective varieties $M$ and $M'$ of general type of dimension $n$ — with no base-point-freeness, canonical-polarity, index, or smooth-minimal-model hypothesis — and any $m\geq m_0(n)$, every linear isometry $\iota:(H^0(M,mK_M),\langle\langle\cdot\rangle\rangle)\to(H^0(M',mK_{M'}),\langle\langle\cdot\rangle\rangle)$ is of the form $c\cdot\psi^*$ for a unique birational map $\psi:M'\dashrightarrow M$ and a constant $c$ with $|c|=1$. The full-generality target includes the extension to mildly singular projective varieties of general type with Kawamata log terminal singularities (klt, the standard class of mild singularities in the minimal model program) — the class for which the source paper states its fundamental question — and to ground fields of positive characteristic, where no analytic integral defines the pseudonorm and an algebraic replacement must be constructed and related to the complex theory.

(2) Realization (Existence). Characterize geometrically and algebraically the pseudonormed vector spaces $(V,\langle\langle\cdot\rangle\rangle)$ that are isometric to $(H^0(M,mK_M),\langle\langle\cdot\rangle\rangle)$ for some projective variety $M$ of general type and some $m\geq 2$; the characterization must be verifiable from the pseudonormed space alone, and should ideally be accompanied by an effective procedure that constructs a birational model of $M$ from such data. The source paper also asks for the relations of this question with Geometric Invariant Theory.

(3) Computation. Develop effective methods to compute the pseudonorm $\langle\langle\eta\rangle\rangle$ and the invariants derived from the asymptotic expansions (for example, characteristic-index data of divisors of sections, i.e. the refined log-canonical-threshold singularity data described in the background) for varieties given explicitly, e.g. by defining equations, with certified precision or exactness guarantees — the analogue, for pseudonormed pluricanonical spaces, of the classical effective computation of periods.

(4) Canonical-metric connection. Determine the precise relationship between the pseudo-metrics on $M$ and the hermitian/Finsler geometry on the pluricanonical spaces induced by the pseudonorms (for which the pseudonorm serves as a Kähler-type potential; these induced objects are those of the source paper's fourth question, Chi and Yau (2008), surveyed in Yau (2015)) and the canonical metrics of Kähler geometry — in particular whether, for canonically polarized $M$, these converge to, are uniformly comparable with, or otherwise determine the Kähler–Einstein metric as $m\to\infty$.

These four questions are the source paper's own program and are linked by the common objective above.

References

1. Chen-Yu Chi and Shing-Tung Yau, A geometric approach to problems in birational geometry, Proceedings of the National Academy of Sciences USA 105 (2008), no. 48, 18696-18701, DOI 10.1073/pnas.0809030105, https://doi.org/10.1073/pnas.0809030105 (full text also at arXiv:0811.2965, https://arxiv.org/abs/0811.2965, where the preprint version carries the slightly different title 'A new geometric approach to problems in birational geometry').
2. Chen-Yu Chi, Pseudonorms and theorems of Torelli type, Journal of Differential Geometry 104 (2016), no. 2, 239-273, DOI 10.4310/jdg/1476367057, https://doi.org/10.4310/jdg/1476367057.
3. Chen-Yu Chi, Pseudonorms and p-adic birational Torelli theorem, arXiv:2211.09335 (2022), https://arxiv.org/abs/2211.09335.
4. Shing-Tung Yau, On the Pseudonorm Project of Birational Classification of Algebraic Varieties, in: Geometry and Analysis on Manifolds, Progress in Mathematics 308, Birkhäuser/Springer, Cham, 2015, pp. 327-339, DOI 10.1007/978-3-319-11523-8_11, https://doi.org/10.1007/978-3-319-11523-8_11.
5. Takahiro Inayama, Pseudonorms on direct images of pluricanonical bundles, Journal of Functional Analysis 284 (2023), no. 12, 109916, DOI 10.1016/j.jfa.2023.109916, https://doi.org/10.1016/j.jfa.2023.109916 (preprint arXiv:1910.05771, https://arxiv.org/abs/1910.05771).
6. Fusheng Deng, Zhiwei Wang, Liyou Zhang, Xiangyu Zhou, Linear invariants of complex manifolds and their plurisubharmonic variations, Journal of Functional Analysis 279 (2020), no. 1, 108514, DOI 10.1016/j.jfa.2020.108514, https://doi.org/10.1016/j.jfa.2020.108514 (preprint arXiv:1901.08920, https://arxiv.org/abs/1901.08920).
7. Fusheng Deng, Zhiwei Wang, Liyou Zhang, Xiangyu Zhou, Holomorphic Invariants of Bounded Domains, The Journal of Geometric Analysis 30 (2020), no. 2, 1204-1217, DOI 10.1007/s12220-019-00229-9, https://doi.org/10.1007/s12220-019-00229-9.
8. Fusheng Deng, Jiafu Ning, Zhiwei Wang, Xiangyu Zhou, Linear isometric invariants of bounded domains, Izvestiya: Mathematics 88 (2024), no. 4, 626-638, DOI 10.4213/im9542e, https://doi.org/10.4213/im9542e (Russian original: Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya 88 (2024), no. 4, 31-43, DOI 10.4213/im9542; preprint arXiv:2209.03510, https://arxiv.org/abs/2209.03510).
9. Siarhei Finski, Submultiplicative norms and filtrations on section rings, Proceedings of the London Mathematical Society 131 (2025), no. 2, article e70077, DOI 10.1112/plms.70077, https://doi.org/10.1112/plms.70077 (preprint arXiv:2210.03039, https://arxiv.org/abs/2210.03039).
10. Shuang-Yen Lee, Global Igusa zeta functions and $K$-equivalence, Asian Journal of Mathematics 29 (2025), no. 2, 97-114, DOI 10.4310/ajm.250717204543, https://doi.org/10.4310/ajm.250717204543 (preprint arXiv:2210.06767, https://arxiv.org/abs/2210.06767, titled 'Global Igusa zeta function and $K$-equivalence').
11. H. L. Royden, Automorphisms and isometries of Teichmüller space, in: Advances in the Theory of Riemann Surfaces, Annals of Mathematics Studies 66, Princeton University Press, Princeton, 1971, pp. 369-384, DOI 10.1515/9781400822492-026, https://doi.org/10.1515/9781400822492-026.
12. Christopher D. Hacon and James McKernan, Boundedness of pluricanonical maps of varieties of general type, Inventiones Mathematicae 166 (2006), no. 1, 1-25, DOI 10.1007/s00222-006-0504-1, https://doi.org/10.1007/s00222-006-0504-1.
13. Shigeharu Takayama, Pluricanonical systems on algebraic varieties of general type, Inventiones Mathematicae 165 (2006), no. 3, 551-587, DOI 10.1007/s00222-006-0503-2, https://doi.org/10.1007/s00222-006-0503-2.
14. Walter Rudin, $L^p$-isometries and equimeasurability, Indiana University Mathematics Journal 25 (1976), no. 3, 215-228, DOI 10.1512/iumj.1976.25.25018, https://doi.org/10.1512/iumj.1976.25.25018.
15. Caucher Birkar, Paolo Cascini, Christopher D. Hacon, and James McKernan, Existence of minimal models for varieties of log general type, Journal of the American Mathematical Society 23 (2010), no. 2, 405-468, DOI 10.1090/S0894-0347-09-00649-3, https://doi.org/10.1090/S0894-0347-09-00649-3 (preprint arXiv:math/0610203, https://arxiv.org/abs/math/0610203).