Donaldson-Thomas invariant

Donaldson-Thomas invariant (plural Donaldson-Thomas invariants)

Introduced in 1998 by Simon Donaldson and Richard Thomas.

Calabi-Yau threefold
Calabi-Yau manifold#English

(algebraic geometry) Given a compact moduli space of sheaves on a Calabi-Yau threefold, its Donaldson-Thomas invariant is the virtual number of its points, i.e., the integral of the cohomology class 1 against the virtual fundamental class.


Calabi-Yau manifold

en-noun

Calabi-Yau manifold (plural Calabi-Yau manifolds)

The name was coined by Candelas et al. (1985), after Eugenio Calabi (1954, 1957) who first conjectured that such surfaces might exist, and Shing-Tung Yau (1978), who proved the Calabi conjecture.

(algebraic geometry) A kind of manifold having properties, such as Ricci flatness, that yield applications in theoretical physics.


橢圓曲線

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тоюань цюйсянь

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超橢圓曲線

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чаотоюань цюйсянь

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(algebraic geometry) hyperelliptic curve


faithfully flat

(algebraic geometry, of a morphism of schemes) Flat and surjective.


GAGA

(algebraic geometry) acronym of Géometrie Algébrique et Géométrie Analytique (a landmark paper by Jean-Pierre Serre)


distinguished open set

distinguished open set (plural distinguished open sets)

(algebraic geometry) A particularly basic kind of set, generalizing the notion of the complement of a hypersurface, originating in the study of algebraic varieties but in modern mathematics also extending to the setting of affine schemes. Formally: (in the context of affine or projective varieties) the compliment of the zero locus of a polynomial in affine or projective space; (in scheme theory) the subset of prime ideals of a commutative ring which do not contain some element of the ring.


zero locus

(algebraic geometry, of a polynomial, or, more generally, a function) The set of points (usually in affine or projective space, but more generally in the domain of the function) where the given polynomial (or function) takes the value zero.


vanishing locus

(algebraic geometry) Synonym of zero locus.


affine scheme

locally ringed space
locally ringed space

(algebraic geometry) An abstract, geometric object in mathematics which generalizes the notion of an affine variety qua domain of a commutative ring of polynomial functions: a space which is the result of a particular construction beginning with an arbitrary commutative ring whereby elements of the ring can be thought of as functions on the space; a space which can be constructed in such a way. Formally: the spectrum of a commutative ring; a locally ringed space isomorphic to the spectrum of a commutative ring.


structure sheaf

structure sheaf (plural structure sheaves)

(sheaf theory, algebraic geometry) The sheaf associated to a ringed space.


B-sheaf

(sheaf theory, algebraic geometry) A kind of partial, simpler to describe sheaf which can be extended into a sheaf (and which is, in a technical sense, equivalent to that sheaf). For a formal definition, see B-sheaf on Wikipedia.Wikipedia.


quasi-projective

From quasi- + projective, the sense being that projective varieties are closed subsets of projective space while quasi-projective varieties are only required to be locally closed, a strictly weaker condition. The second, scheme-theoretic sense generalizes the first.

(algebraic geometry, of a variety) Locally closed with respect to the Zariski topology on projective n-space; isomorphic to such a set.

(algebraic geometry, of a scheme) Comprising an open subscheme of a projective scheme


Fourier-Mukai transform

Fourier-Mukai transform (plural Fourier-Mukai transforms)

(algebraic geometry) A functor Φ_K between derived category of coherent sheaves D(X) → D(Y) for schemes X and Y, which is, in a sense, an integral transform along a kernel object K ∈ D(X×Y).


Cox ring

Named after David A. Cox, whose earlier construction led to these.

(algebraic geometry) A sort of universal homogeneous coordinate ring for a projective variety, and (roughly speaking) a direct sum of the spaces of sections of all isomorphism classes of line bundles.