Algebraic and complex geometry
36 results in OpenAI Math 2026.
Algebraic and complex geometry36
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The rational Hodge conjecture for CM abelian varieties
Proves the rational Hodge conjecture for every complex CM abelian variety, in every dimension and codimension. Through Milne's theorems, this also gives the Tate conjecture for all abelian varieties over finite fields and the Hodge standard conjecture for abelian varieties in every characteristic.
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Iitaka subadditivity, variation, and logarithmic additivity
Proves Campana's orbifold Iitaka subadditivity conjecture for smooth Fujiki-class- manifolds with rational simple-normal-crossing boundaries. For projective fibrations of smooth complex quasi-projective varieties with connected fibers, general fiber F, and , proves Popa's inequality , where variation measures the whole geometric generic fiber.
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Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity
Using logarithmic Iitaka subadditivity, proves log abundance in every dimension for normal compact Kähler log canonical pairs with effective rational boundary: an analytically nef ℚ-Cartier adjoint is semiample. It also proves projective log abundance over every algebraically closed field of characteristic zero and the effective Iitaka fibration conjecture for smooth projective varieties of nonnegative Kodaira dimension in that setting. A further result resolves the finite-rational-coefficient index conjecture for connected projective semi-log-canonical log Calabi–Yau pairs over such fields in each fixed dimension and for each fixed finite set of rational boundary coefficients, with a uniform index independent of the number of components.
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Log-canonical threefold abundance in numerical dimension one
Proves log abundance for projective log canonical threefold pairs over algebraically closed fields of characteristic p > 3 when the effective boundary is rational and the ℚ-Cartier adjoint is nef of numerical dimension one. The adjoint is semiample, without requiring the original variety to be terminal or ℚ-factorial.
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Numerical semiampleness and generalized minimal models
Proves numerical semiampleness for nef adjoints with pseudo-effective and M nef rational, for projective klt rational pairs over algebraically closed characteristic-zero fields and smooth compact Kähler rational klt simple-normal-crossing pairs, using Bott–Chern cohomology in the latter case. Separately, projective generalized log canonical rational pairs over such fields admit minimal models for pseudo-effective adjoints and Mori fiber spaces otherwise, with nef b-data fixed.
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The ordinary-double-point volume gap
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.
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Fujita’s freeness conjecture
Proves Fujita's freeness conjecture at its sharp bound in every dimension: for a smooth projective complex variety X of dimension n and an ample line bundle L, the adjoint is globally generated for every integer .
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Nagata’s conjecture and maximal Seshadri constants
Proves Nagata's strict inequality for every nonzero effective plane curve of degree d through r ≥ 10 very general complex points, with arbitrary multiplicities mi. It also proves maximal multipoint Seshadri constants for every smooth polarized projective variety of dimension n ≥ 2 and all sufficiently large r: at very general points over ℂ, and at the geometric generic tuple over any algebraically closed field of positive characteristic.
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Bloch’s conjecture for complex surfaces
Proves Bloch's conjecture: for every smooth connected projective complex surface S with , the Albanese map on integral degree-zero zero-cycles is an isomorphism. This combines the new theorem with the classical theorem of Bloch, Kas, and Lieberman.
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Hyperkähler SYZ and projective-space bases
Proves the strong hyperkähler SYZ conjecture: every holomorphic line bundle with nonzero nef isotropic first Chern class on a compact irreducible holomorphic symplectic Kähler manifold is semiample. It also proves that every projective Lagrangian fibration with normal projective base has projective space as its base, in every dimension and deformation type.
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Oka classification for minimal compact complex surfaces: Kodaira dimension zero and class VII
Proves that every complex K3 surface is Oka, including nonprojective surfaces. More generally, every connected minimal compact complex surface of Kodaira dimension zero is Oka; a connected minimal compact complex surface of class VII is Oka exactly when it is a Hopf or Enoki surface.
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P = W for fixed-determinant SLn moduli spaces
Proves on the full rational cohomology of smooth coprime fixed-determinant, trace-free Higgs moduli spaces and their character varieties for composite ranks over smooth projective complex curves of genus at least two. This includes variant cohomology and, together with the known prime-rank theorems, establishes the fixed-determinant P = W conjecture in every coprime rank.
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The equivariant cohomological Hikita conjecture
Proves the equivariant cohomological Hikita correspondence for every finite quiver, including loops and multiple arrows, with arbitrary dimension and framing vectors and commuting flavor torus. When every semistable point is stable and the gauge action is free, the equivariant cohomology of the Nakajima variety is canonically the coordinate ring of the scheme-theoretic cocharacter-fixed locus of its flavor-deformed Coulomb branch.
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Shafarevich counterexamples in dimension two and with large fundamental group
Constructs a smooth projective complex fourfold with large fundamental group whose universal cover contains no positive-dimensional compact analytic subvariety but is neither Stein nor holomorphically convex. A separate smooth projective complex surface already disproves unrestricted Shafarevich holomorphic convexity in dimension two.
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Zariski cancellation and affine fibrations over the complex numbers
Constructs an integral complex affine fourfold with , disproving affine-space cancellation over ℂ in dimension four. It also disproves the Dolgachev–Weisfeiler affine-fibration conjecture: smooth surjections and have every residue-field fiber isomorphic to affine three-space but are not Zariski-locally trivial.
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A characteristic-zero counterexample to Lipman–Zariski
Constructs a singular normal affine complex surface with free rank-two tangent sheaf, disproving the characteristic-zero Lipman–Zariski conjecture.
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A stable-coordinate counterexample in four variables
Constructs a polynomial in four complex variables that is not a coordinate but becomes one after adjoining a single variable, disproving the Stable Coordinate conjecture in four variables. Every fiber is affine three-space, yet none of its embeddings is rectifiable, also disproving the Abhyankar–Sathaye conjecture even when all fibers are affine spaces.
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A counterexample to Griffiths’ positivity conjecture
Constructs ample rank-two bundles on with no smooth Hermitian metric of strictly Griffiths-positive curvature, disproving Griffiths' positivity conjecture already on the quadric surface.
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Kobayashi’s canonical-ampleness conjecture
Every compact connected Kähler manifold of positive complex dimension with no nonconstant entire curve has ample canonical bundle and is therefore projective. This proves Kobayashi's canonical-ampleness conjecture in the smooth compact Kähler setting.
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Tangent splittings and product decompositions
A splitting of the tangent bundle of a compact Kähler manifold into two integrable holomorphic subbundles induces a compatible product decomposition of its universal cover, proving the two-summand form of Beauville's splitting conjecture. On smooth rationally connected projective manifolds, both summands are automatically integrable, establishing Höring's conjecture and the corresponding product decomposition.
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A counterexample to Pixton completeness in Chow
Constructs a tautological relation on a moduli space of stable pointed curves that vanishes in rational Chow, hence in rational cohomology, but lies outside Pixton's original relation span. This disproves the Chow and rational-cohomological forms of his original completeness conjecture.
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Irrational cubic fourfolds with Hodge-theoretic and categorical K3 associations
For every sufficiently large admissible Hassett discriminant, a very general smooth complex cubic fourfold is irrational despite having both an untwisted geometric K3 category and an integral Hodge-theoretic K3 association. This disproves Kuznetsov's rationality conjecture and the sufficiency of the associated-K3 criterion for rationality; the discriminant threshold is ineffective.
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Gepner symmetry and large-volume stability on threefolds
Proves Toda's Gepner conjecture for every smooth complex quintic threefold, constructing a numerical Bridgeland stability condition with the prescribed phase shift 2/5. Also constructs numerical Bridgeland stability conditions at every sufficiently large volume on all smooth projective complex threefolds with trivial canonical bundle, with the exact ordinary and square-root-Todd central charges.
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Termination of projective and Kähler fourfold minimal model programs
Proves termination of every existing generalized log canonical flip sequence on globally Weil ℚ-factorial compact Kähler fourfolds, with rational boundary, fixed rational analytically nef b-data, and projective small flip diagrams with the prescribed ample signs. Also proves termination of arbitrary permitted minimal model programs for projective log canonical fourfolds with rational boundary in characteristic zero.
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Fundamental groups of special complex varieties and root orbifolds
Proves Campana's abelianity conjecture: special compact Kähler manifolds have virtually abelian fundamental groups. Using this theorem, establishes the same conclusion for order-two root orbifolds of smooth projective complex fourfolds along one nonempty smooth connected divisor, when special in the stated differential-line sense. For smooth special complex quasi-projective varieties, proves that every finite-dimensional complex linear representation of the fundamental group has virtually nilpotent image of class at most two.
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Semialgebraic universal covers and bounded domains
Proves the Kollár–Pardon conjecture: the semialgebraic universal covers of connected normal projective complex varieties are exactly products , with D bounded symmetric and F simply connected, normal, and projective. A universal cover is quasi-projective exactly when the bounded symmetric factor is absent. In particular, a smooth projective variety covered by ℂn has a finite étale cover by an abelian variety.
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Counterexamples to Zariski’s multiplicity conjecture
Disproves Zariski's multiplicity conjecture by constructing reduced holomorphic hypersurface germs that are ambiently homeomorphic but have different multiplicities. The examples include hypersurfaces in ℂ4 with isolated critical points and multiplicities four and five.
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The Global Spherical Shell conjecture
Every connected minimal compact complex surface of class VII with contains a global spherical shell, proving the positive-b2 Global Spherical Shell conjecture. Such a shell is a holomorphically embedded neighborhood of the standard three-sphere in whose complement is connected.
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Projective contact classification and the LeBrun–Salamon conjecture
Proves the LeBrun–Salamon conjecture: every closed connected positive quaternionic-Kähler manifold of real dimension at least eight is homothetic to a compact symmetric Wolf space. It also proves contact-Fano homogeneity and classifies smooth connected complex projective contact manifolds of complex dimension at least three: those with are adjoint varieties with their canonical contact structures, while those with have underlying manifold for a smooth projective variety Z.
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The generalized Mukai conjecture
Proves the generalized Mukai conjecture: every positive-dimensional smooth complex projective Fano manifold of dimension n, Picard number ρ and pseudoindex ι satisfies , with equality exactly for . Here the pseudoindex is the least anticanonical degree of a rational curve.
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Topological triviality of μ-constant surface singularities
Proves topological right-triviality for every holomorphic one-parameter family of isolated hypersurface singularities in ℂ3 with constant Milnor number, resolving the surface case of the μ-constant problem. After shrinking the parameter disk and representatives, ambient homeomorphisms vary jointly continuously, fix the origin section and preserve the defining functions.
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Virasoro constraints for complete intersections and projective-bundle towers
Proves the full ordinary unreduced descendant Virasoro conjecture for smooth complete intersections in complex projective space, in every genus and curve class with arbitrary cohomology insertions. The constraints also pass from any smooth projective complex base satisfying them to the projectivization of every algebraic vector bundle of rank at least two, and hence to projective-bundle towers.
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Bounded klt complements for Fano contractions
Proves the finite-rational-coefficient form of Shokurov’s bounded-klt-complement conjecture for ϵ-lc complex Fano-type pairs with nef anti-log-canonical divisor. For ϵ-lc Fano contractions over any algebraically closed characteristic-zero field, it gives klt complements near every base point, with index bounded only by dimension and positive rational ϵ.
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The Campana–Peternell conjecture in dimension six
Proves the Campana–Peternell conjecture in complex dimension six: every smooth connected complex projective Fano sixfold with nef tangent bundle is rational homogeneous.
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Anticanonical nonvanishing in every dimension
If X is a smooth connected complex projective variety and admits a smooth Hermitian metric with nonnegative curvature, then for some m > 0. Thus smooth semipositivity forces a nonzero section of a positive tensor power of the anticanonical bundle in every dimension.
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Global quantum geometric Langlands at irrational level
Proves the unramified de Rham quantum geometric Langlands equivalence for every connected simple complex algebraic group on every smooth projective connected complex curve, at every shifted level . It identifies the full derived categories of twisted D-modules for the group and its Langlands dual, retaining all global forms and connected components.
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