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SEMISTABLE REDUCTION-THEOREM

Semistable reduction theorem

Mathematical theory in the field of algebraic geometry

In algebraic geometry, semistable reduction theorems state that, given a proper flat morphism of schemes X → S {\displaystyle X\to S} , there exists a

Semistable reduction theorem

Semistable_reduction_theorem

Semistable abelian variety

{\displaystyle A} is semistable if it has good or semistable reduction at all primes. The fundamental semistable reduction theorem of Alexander Grothendieck

Semistable abelian variety

Semistable_abelian_variety

Fermat's Last Theorem

17th-century conjecture proved by Andrew Wiles in 1994

modularity theorem—if proved for semi-stable elliptic curves—would mean that all semistable elliptic curves must be modular; Ribet's theorem showed that

Fermat's Last Theorem

Fermat's_Last_Theorem

Glossary of arithmetic and diophantine geometry

Néron model, potential good reduction, Tate curve, semistable abelian variety, semistable elliptic curve, Serre–Tate theorem. Grothendieck–Katz conjecture

Glossary of arithmetic and diophantine geometry

Glossary_of_arithmetic_and_diophantine_geometry

Langlands–Tunnell theorem

Basic Introduction

inputs in Wiles's proof of the modularity of semistable elliptic curves, and hence of Fermat's Last Theorem. In the dihedral case, the relevant modular

Langlands–Tunnell theorem

Langlands–Tunnell_theorem

Resolution of singularities

Concept in algebraic geometry

1090/S0273-0979-10-01301-7 Abramovich, D; de Jong, A. J. (1997), "Smoothness, semistability, and toroidal geometry", Journal of Algebraic Geometry, 6 (4): 789–801

Resolution of singularities

Resolution_of_singularities

Linear algebraic group

Subgroup of the group of invertible n×n matrices

subtleties when a reductive group G acts on a projective variety X. In particular, the theory defines open subsets of "stable" and "semistable" points in X

Linear algebraic group

Linear_algebraic_group

Richard Taylor (mathematician)

British-American mathematician (born 1962)

partial proof of the Sato–Tate conjecture uses Wiles's theorem about modularity of semistable elliptic curves. He received the Whitehead Prize in 1990

Richard Taylor (mathematician)

Richard_Taylor_(mathematician)

Complete algebraic curve

parametrized by the base scheme S.[citation needed] See also Semistable reduction theorem. This generalizes the classical construction due to Tate (cf

Complete algebraic curve

Complete_algebraic_curve

GIT quotient

set of semistable points.) A GIT quotient is a categorical quotient of the locus of semistable points; i.e., "the" quotient of the semistable locus. Since

GIT quotient

GIT_quotient

Dan Abramovich

Israeli-American mathematician

ISBN 978-3-0348-9550-7. S2CID 15820921. Abramovich, D.; Karu, K. (2000). "Weak semistable reduction in characteristic 0". Inventiones Mathematicae. 139 (2). Springer

Dan Abramovich

Dan_Abramovich

List of conjectures

quotes as of August 2026[update]. The conjecture terminology may persist: theorems often enough may still be referred to as conjectures, using the anachronistic

List of conjectures

List_of_conjectures

P-adic Hodge theory

Mathematical theory

next. In order, these are the categories of crystalline representations, semistable representations, de Rham representations, Hodge–Tate representations,

P-adic Hodge theory

P-adic_Hodge_theory

Nigel Hitchin

British mathematician

the Hitchin system, which has an interpretation as a moduli space of semistable Higgs bundles over a compact Riemann surface or algebraic curve. This

Nigel Hitchin

Nigel_Hitchin

Galois representation

Mathematical terminology

sub-representation. Semistable representations. These are two dimensional representations related to the representations coming from semistable elliptic curves

Galois representation

Galois_representation

K-stability of Fano varieties

Li independently. Theorem: (Fujita–Li, Blum–Xu)—A Q {\displaystyle \mathbb {Q} } -Fano variety X {\displaystyle X} is K-semistable if and only if β X

K-stability of Fano varieties

K-stability_of_Fano_varieties

Nonabelian Hodge correspondence

Correspondsnce between Higgs bundles and fundamental group representations

or a compact Kähler manifold. The theorem can be considered a vast generalisation of the Narasimhan–Seshadri theorem which defines a correspondence between

Nonabelian Hodge correspondence

Nonabelian_Hodge_correspondence

K-stability

Algebro-geometric stability condition

Asymptotically Hilbert semistable ⟹ {\displaystyle \implies } Asymptotically Chow semistable ⟹ {\displaystyle \implies } K-semistable It is however not know

K-stability

Kobayashi–Hitchin correspondence

Vector bundles theorem

theory, the Kobayashi–Hitchin correspondence (or Donaldson–Uhlenbeck–Yau theorem) relates stable vector bundles over a complex manifold to Einstein–Hermitian

Kobayashi–Hitchin correspondence

Kobayashi–Hitchin_correspondence

Tate curve

is non-integral and the Tate curve has semistable reduction of multiplicative type. Conversely, every semistable elliptic curve over a local field is isomorphic

Tate curve

Tate_curve

Gauge theory (mathematics)

Study of vector bundles, principal bundles, and fibre bundles

arbitrary Kähler manifold is known as the nonabelian Hodge theorem. The dimensional reduction of the Yang–Mills equations to three dimensions by imposing

Gauge theory (mathematics)

Gauge_theory_(mathematics)

List of algebraic geometry topics

Algebraic torus Reductive group Borel subgroup Radical of an algebraic group Unipotent radical Lie–Kolchin theorem Haboush's theorem (also known as the

List of algebraic geometry topics

List_of_algebraic_geometry_topics

Hitchin system

Type of integrable system

{\displaystyle \chi _{L}} above. Note that this definition is not relevant to semistability. To obtain the Hitchin fibration mentioned above, we need to take L

Hitchin system

Hitchin_system

Langlands program

Conjectures connecting number theory and geometry

Langlands conjectures for finite fields. Andrew Wiles' proof of modularity of semistable elliptic curves over rationals can be viewed as an instance of the Langlands

Langlands program

Langlands_program

Wassim Michael Haddad

Lebanese-Greek-American mathematician

stability, finite-time stability, semistability, stability of sets, stability of periodic orbits, and stability theorems via vector Lyapunov functions. In

Wassim Michael Haddad

Wassim_Michael_Haddad

Christopher Deninger

German mathematician (born 1958)

bundles, including those of degree 0 and having potentially strongly semistable reduction. In another paper of 2005, they related the resulting representations

Christopher Deninger