Hironaka's theorem

Hironaka's theorem
Name	Hironaka's theorem
Field	Algebraic geometry
Named after	Heisuke Hironaka
Year	1964
Statement	Resolution of singularities for algebraic varieties over fields of characteristic zero

Hironaka's theorem is a foundational result in algebraic geometry proving that any algebraic variety over a field of characteristic zero admits a resolution of singularities by a sequence of blowups along smooth centers, establishing desingularization in the category of varieties. The theorem influenced work in complex geometry, commutative algebra, and arithmetical algebraic geometry, and it shaped subsequent advances by researchers affiliated with institutions such as the Institute for Advanced Study and universities like Harvard University and Princeton University.

Statement of the theorem

Hironaka's original formulation asserts that for every reduced, separated, finite-type scheme over a field of characteristic zero, there exists a proper birational morphism from a smooth scheme obtained by a finite sequence of blowups along nonsingular centers that is an isomorphism over the nonsingular locus. This statement connects objects studied by mathematicians such as Oscar Zariski, Alexander Grothendieck, Jean-Pierre Serre, David Mumford, and Oscar Zariski with techniques from the work of Kunihiko Kodaira and Heisuke Hironaka himself, and it is compatible with concepts developed at places like the Institute Henri Poincaré and the Collège de France.

Historical context and motivation

The problem of resolving singularities dates to classical work by Bernhard Riemann and concrete classifications by Max Noether and Federigo Enriques, with algebraic formalizations by Oscar Zariski and conceptual reformulations by Alexander Grothendieck in the mid-20th century. Interest from scholars at institutions such as University of Tokyo, Massachusetts Institute of Technology, and University of California, Berkeley grew alongside contributions from researchers including Zariski, Grothendieck, Jean-Pierre Serre, André Weil, and John Tate, motivated by applications in the Weil conjectures, the development of étale cohomology by Grothendieck and collaborators, and arithmetic questions considered by Gerd Faltings and Jean-Pierre Serre.

Outline of the proof

Hironaka introduced intricate inductive invariants and a combinatorial strategy involving sequences of permissible blowups, employing resolution functions and principalization techniques inspired by methods used by Zariski and later refined by researchers at Harvard University and the Institute for Advanced Study. The proof uses embedded resolution, monomialization, and careful control of multiplicity along centers, building on ideas related to work by David Mumford, Oscar Zariski, Heisuke Hironaka, and subsequent simplifications by mathematicians in seminars at Centre National de la Recherche Scientifique and École Normale Supérieure. Later expositions by scholars affiliated with Princeton University, University of Paris, Nagoya University, and Rutgers University clarified the inductive structure and made the algorithmic content more explicit.

Applications and consequences

The theorem enabled progress on questions posed by André Weil and informed proofs of results in Hodge theory influenced by Phillip Griffiths and Pierre Deligne, while supporting techniques used by Gerd Faltings and Fedor Bogomolov in arithmetic geometry. It underpins simplifications in studies of moduli spaces by researchers at Institut des Hautes Études Scientifiques and guides work on birational geometry pursued by mathematicians such as Shigefumi Mori, Vladimir Voevodsky, and Yujiro Kawamata. The resolution result also affected developments in D-module theory associated with Alexander Beilinson and Joseph Bernstein and in analytic settings connected to Kähler manifolds studied by Shing-Tung Yau.

Examples and counterexamples

Concrete instances treated by the theorem include resolution of surface singularities originally classified by Federigo Enriques and Max Noether, and higher-dimensional examples motivated by constructions of Heisuke Hironaka and counterexamples in positive characteristic investigated by researchers at University of California, Berkeley and University of Tokyo such as Abhyankar and Raynaud. Specific pathological behaviours in characteristic p were exhibited in work by Shreeram Abhyankar and later by mathematicians associated with RIMS and École Polytechnique, demonstrating limits of the characteristic-zero hypothesis emphasized by Hironaka and later authors.

Generalizations and related results

Subsequent generalizations and alternative approaches include algorithmic resolution procedures by groups at Max Planck Institute for Mathematics and proofs for formal schemes, complex analytic spaces, and arithmetic schemes pursued by researchers at University of Cambridge, Université de Bordeaux, and Tohoku University. Related milestones include the work of Michael Temkin on non-Archimedean analytic spaces, contributions by Mark Spivakovsky and Herwig Hauser on simplifications of the original strategy, and connections to the minimal model program developed by Shigefumi Mori, Vyacheslav Shokurov, and others at institutions like Steklov Institute of Mathematics.

Category:Algebraic geometry