# Algebraic and complex geometry

> 36 AI-made results in algebraic and complex geometry, each explained in plain English with its trust label, Lean status and sources.

Canonical: https://closedform.nishachay.tech/science/mathematics/algebraic-and-complex-geometry/
Last updated: 2026-10-14

36 results in OpenAI Math 2026.

36 AI-made results in algebraic and complex geometry, from 86 papers in the OpenAI math collection, each explained in plain English. 8 have a formal Lean proof listed; 28 are claimed in a paper and not yet machine-checked. The oldest question here dates back to 1950: [The rational Hodge conjecture for CM abelian varieties](https://closedform.nishachay.tech/e/openai-math-2026-032/). Results rated Landmark include [Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity](https://closedform.nishachay.tech/e/openai-math-2026-034/) and [Nagata’s conjecture and maximal Seshadri constants](https://closedform.nishachay.tech/e/openai-math-2026-039/).

## Results (36)

- [The rational Hodge conjecture for CM abelian varieties](https://closedform.nishachay.tech/e/openai-math-2026-032/) - Entry 032 · The rational Hodge conjecture is claimed for every abelian variety with complex multiplication, in every dimension. (Claimed, not checked)
- [Iitaka subadditivity, variation, and logarithmic additivity](https://closedform.nishachay.tech/e/openai-math-2026-033/) - Entry 033 · Twisting a family of shapes never destroys the forms its fibers and base contribute. (Formal proof listed)
- [Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity](https://closedform.nishachay.tech/e/openai-math-2026-034/) - Entry 034 · A family of 14 papers claims log abundance in characteristic zero and the effective Iitaka fibration conjecture. (Claimed, not checked)
- [Log-canonical threefold abundance in numerical dimension one](https://closedform.nishachay.tech/e/openai-math-2026-035/) - Entry 035 · In characteristic above 3, a nef log canonical threefold divisor of numerical dimension one is claimed to give a map. (Claimed, not checked)
- [Numerical semiampleness and generalized minimal models](https://closedform.nishachay.tech/e/openai-math-2026-036/) - Entry 036 · Nef adjoint divisors with an extra nef part are claimed to be numerically semiample; minimal models are claimed too. (Claimed, not checked)
- [The ordinary-double-point volume gap](https://closedform.nishachay.tech/e/openai-math-2026-037/) - Entry 037 · Every singular point of a mild kind is at most as roomy as a double cone, in any dimension. (Claimed, not checked)
- [Fujita’s freeness conjecture](https://closedform.nishachay.tech/e/openai-math-2026-038/) - Entry 038 · On every smooth shape, n + 1 copies of a positive bundle are always enough for Fujita's freeness. (Claimed, not checked)
- [Nagata’s conjecture and maximal Seshadri constants](https://closedform.nishachay.tech/e/openai-math-2026-039/) - Entry 039 · Nagata's 1959 rule for curves through random points holds for every count from 10 points up. (Formal proof listed)
- [Bloch’s conjecture for complex surfaces](https://closedform.nishachay.tech/e/openai-math-2026-040/) - Entry 040 · On a complex surface with no holomorphic two-forms, zero-cycles carry only their degree and Albanese data. (Claimed, not checked)
- [Hyperkähler SYZ and projective-space bases](https://closedform.nishachay.tech/e/openai-math-2026-041/) - Entry 041 · Every nonzero nef isotropic line bundle on a compact hyperkähler manifold is claimed to give a torus fibration. (Claimed, not checked)
- [Oka classification for minimal compact complex surfaces: Kodaira dimension zero and class VII](https://closedform.nishachay.tech/e/openai-math-2026-042/) - Entry 042 · A claimed proof that every complex K3 surface is Oka: local holomorphic maps into it can be copied by entire maps. (Claimed, not checked)
- [P = W for fixed-determinant SL n moduli spaces](https://closedform.nishachay.tech/e/openai-math-2026-043/) - Entry 043 · Two ways of layering the cohomology of SL_n Higgs moduli are claimed to agree in every composite coprime rank. (Claimed, not checked)
- [The equivariant cohomological Hikita conjecture](https://closedform.nishachay.tech/e/openai-math-2026-044/) - Entry 044 · For every finite quiver, the cohomology of a Nakajima variety is claimed to equal functions on a Coulomb fixed scheme. (Claimed, not checked)
- [Shafarevich counterexamples in dimension two and with large fundamental group](https://closedform.nishachay.tech/e/openai-math-2026-046/) - Entry 046 · Two claimed counterexamples to Shafarevich's conjecture: a surface and a fourfold whose universal covers fail convexity. (Claimed, not checked)
- [Zariski cancellation and affine fibrations over the complex numbers](https://closedform.nishachay.tech/e/openai-math-2026-047/) - Entry 047 · A four-dimensional shape that is not ordinary space, until you add one line. (Formal proof listed)
- [A characteristic-zero counterexample to Lipman–Zariski](https://closedform.nishachay.tech/e/openai-math-2026-048/) - Entry 048 · A singular complex surface whose vector fields still form a free basis, against the Lipman–Zariski conjecture. (Claimed, not checked)
- [A stable-coordinate counterexample in four variables](https://closedform.nishachay.tech/e/openai-math-2026-049/) - Entry 049 · A polynomial that becomes a coordinate with one extra variable, but not without it, disproves two conjectures. (Formal proof listed)
- [A counterexample to Griffiths’ positivity conjecture](https://closedform.nishachay.tech/e/openai-math-2026-050/) - Entry 050 · Algebraic positivity need not come with positive curvature, even on the simple surface built from two spheres. (Formal proof listed)
- [Kobayashi’s canonical-ampleness conjecture](https://closedform.nishachay.tech/e/openai-math-2026-051/) - Entry 051 · Compact Kähler manifolds with no entire curves have ample canonical bundle, as Kobayashi conjectured. (Claimed, not checked)
- [Tangent splittings and product decompositions](https://closedform.nishachay.tech/e/openai-math-2026-052/) - Entry 052 · Two integrable families of directions on a compact Kähler manifold make its universal cover a product. (Formal proof listed)
- [A counterexample to Pixton completeness in Chow](https://closedform.nishachay.tech/e/openai-math-2026-053/) - Entry 053 · A tautological relation in genus 10^60 that Pixton’s original relations miss, in Chow and in cohomology. (Claimed, not checked)
- [Irrational cubic fourfolds with Hodge-theoretic and categorical K3 associations](https://closedform.nishachay.tech/e/openai-math-2026-054/) - Entry 054 · Very general cubic fourfolds of large admissible discriminant are irrational despite a K3 derived category. (Claimed, not checked)
- [Gepner symmetry and large-volume stability on threefolds](https://closedform.nishachay.tech/e/openai-math-2026-055/) - Entry 055 · A stability condition on every smooth quintic threefold with Toda’s exact 2/5 Gepner symmetry. (Claimed, not checked)
- [Termination of projective and Kähler fourfold minimal model programs](https://closedform.nishachay.tech/e/openai-math-2026-056/) - Entry 056 · Every permitted minimal model program on a projective log canonical fourfold claims to stop after finitely many steps. (Claimed, not checked)
- [Fundamental groups of special complex varieties and root orbifolds](https://closedform.nishachay.tech/e/openai-math-2026-057/) - Entry 057 · Special compact Kähler manifolds claim virtually abelian fundamental groups in every dimension, as Campana predicted. (Claimed, not checked)
- [Semialgebraic universal covers and bounded domains](https://closedform.nishachay.tech/e/openai-math-2026-058/) - Entry 058 · Unrolled shapes with a polynomial description split into just three kinds of pieces. (Formal proof listed)
- [Counterexamples to Zariski’s multiplicity conjecture](https://closedform.nishachay.tech/e/openai-math-2026-059/) - Entry 059 · Two hypersurface germs with the same embedded topology but different multiplicities, against Zariski’s 1971 question. (Claimed, not checked)
- [The Global Spherical Shell conjecture](https://closedform.nishachay.tech/e/openai-math-2026-060/) - Entry 060 · Every minimal class VII surface with positive second Betti number contains a global spherical shell. (Claimed, not checked)
- [Projective contact classification and the LeBrun–Salamon conjecture](https://closedform.nishachay.tech/e/openai-math-2026-062/) - Entry 062 · Contact Fano manifolds are homogeneous, which settles the LeBrun–Salamon conjecture on quaternionic-Kähler spaces. (Claimed, not checked)
- [The generalized Mukai conjecture](https://closedform.nishachay.tech/e/openai-math-2026-063/) - Entry 063 · A claimed proof that rational curves on a Fano manifold cap its Picard number, with equality only for product spaces. (Claimed, not checked)
- [Topological triviality of μ -constant surface singularities](https://closedform.nishachay.tech/e/openai-math-2026-064/) - Entry 064 · A claimed proof that surface singularities with a constant Milnor number keep the same topology along a family. (Claimed, not checked)
- [Virasoro constraints for complete intersections and projective-bundle towers](https://closedform.nishachay.tech/e/openai-math-2026-065/) - Entry 065 · Claimed proofs that curve counts on complete intersections and on projective bundles obey the Virasoro equations. (Claimed, not checked)
- [Bounded klt complements for Fano contractions](https://closedform.nishachay.tech/e/openai-math-2026-066/) - Entry 066 · One complement index, fixed by dimension and singularity bound, works for every mildly singular Fano contraction. (Formal proof listed)
- [The Campana–Peternell conjecture in dimension six](https://closedform.nishachay.tech/e/openai-math-2026-067/) - Entry 067 · A claimed proof that every six-dimensional Fano manifold with nef tangent bundle is a symmetric, homogeneous space. (Claimed, not checked)
- [Anticanonical nonvanishing in every dimension](https://closedform.nishachay.tech/e/openai-math-2026-068/) - Entry 068 · A claimed proof that a smooth nonnegatively curved anticanonical bundle always has a section of some power. (Claimed, not checked)
- [Global quantum geometric Langlands at irrational level](https://closedform.nishachay.tech/e/openai-math-2026-069/) - Entry 069 · A claimed proof of quantum geometric Langlands duality on any smooth complex curve at every irrational level. (Claimed, not checked)

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