Algebraic geometry
Provisional editorial order for Top Open Problems: Millennium Prize Problems first, followed by assessed difficulty tiers; the unassessed remainder keeps its previous catalogue order. About rankings
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- Math-4 Hodge Conjecture
- Math-6 Birch and Swinnerton-Dyer Conjecture
- Math-10 Tate Conjecture for Algebraic Cycles
- Math-11 Generalized Hodge Conjecture
- Math-16 Grothendieck Period Conjecture
- Math-19 Grothendieck's Standard Conjecture D
- Math-25 Abundance Conjecture for the minimal model program
- Math-26 Termination of Flips Conjecture in the Minimal Model Program
- Math-27 The Standard Conjectures on Algebraic Cycles
- Math-28 Grothendieck Standard Conjecture of Hodge Type
- Math-29 Deligne's Absolute Hodge Conjecture
- Math-30 Grothendieck's Variational Hodge Conjecture
- Math-34 Grothendiecks Section Conjecture in Anabelian Geometry
- Math-80 Weight–Monodromy Conjecture
- Math-82 Green-Griffiths-Lang Conjecture on entire curves in varieties of general type
- Math-135 Campana's Conjecture Characterizing Special Varieties by Potential Density of Rational Points
- Math-147 Iitaka Subadditivity Conjecture for Kodaira Dimension
- Math-167 Murre's Chow-Kunneth Conjectures
- Math-187 Voevodsky's Smash-Nilpotence Conjecture
- Math-191 Beilinson-Soule Vanishing Conjecture for Rational Motivic Cohomology
- Math-195 Kimura–O'Sullivan Finite-Dimensionality Conjecture
- Math-198 Nonvanishing conjecture for projective log canonical pairs
- Math-226 Parshin's conjecture
- Math-238 Kawamata's K-Equivalence Implies D-Equivalence Conjecture
- Math-241 Virasoro conjecture
- Math-244 Morrison’s Cone Conjecture for Calabi–Yau Pairs
- Math-258 Holomorphic Convexity of Universal Covers
- Math-271 Rational Cubic Fourfolds and K3 Categories
- Math-295 Griffiths Positivity Conjecture for Ample Vector Bundles
- Math-302 Zariski's Multiplicity Conjecture
- Math-316 Bass–Quillen conjecture
- Math-320 Original Dubrovin conjecture for Fano manifolds
- Math-326 Hartshorne Conjecture on complete intersections in codimension two
- Math-367 Fujita Freeness Conjecture
- Math-368 Kobayashi Conjecture: Hyperbolic Compact Kähler Manifolds Have Ample Canonical Bundle
- Math-369 Nagata Conjecture on Plane Curve Linear Systems
- Math-371 Igusa’s Standard p-Adic Monodromy Conjecture
- Math-373 In spherical or hyperbolic geometry, must polyhedra with the same volume and Dehn invariant be scissors-congruent
- Math-376 Hyperkähler SYZ (Lagrangian Fibration) Conjecture
- Math-389 Green’s Canonical Syzygy Conjecture for Arbitrary Curves
- Math-403 Clemens Conjecture for Rational Curves on a General Quintic Threefold
- Math-407 Coleman-Oort conjecture: eventual special-subvariety exclusion
- Math-423 Zariski Cancellation Problem (characteristic zero, dimension ≥ 3)
- Math-425 Boundedness of Compact Hyperkahler Manifolds: Finiteness of Deformation Types in Each Dimension
- Math-457 Unit propagation and a trace-Bockstein reduction for Greek-letter redshift
- Math-470 Fujita Very Ampleness Conjecture
- Math-472 Campana–Peternell Conjecture for Fano Manifolds
- Math-497 SHGH Conjecture for Linear Systems on Blown-Up Planes
- Math-503 F-Conjecture for the Nef Cone of Mbar_(0,n)
- Math-2189 The Abundance Conjecture