In algebraic geometry, Serre duality is a duality present on non-singular projective algebraic varieties V of dimension n (and in greater generality) for vector bundles and the more general coherent sheaves. It shows that a cohomology group Hⁱ is the dual space of another one, Hⁿ⁻ⁱ. If the variety is defined over the complex numbers, this is therefore quite distinct from Poincaré duality, which relates Hⁱ to H²ⁿ⁻ⁱ because as a manifold V has dimension 2n. Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In mathematics, duality has numerous meanings. ... In mathematics, a singular point of an algebraic variety V is a point P that is special (so, singular), in the geometric sense that V is not locally flat there. ... In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In mathematics, especially in algebraic geometry and the theory of complex manifolds, a coherent sheaf F on a locally ringed space X is a sheaf isomorphic with the cokernel of a morphism of OX_modules OXm → OXn. ... In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ... In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ... In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit , satisfying . ... In mathematics, the Poincaré duality theorem is a basic result on the structure of the homology and cohomology groups of manifolds. ...

The case of algebraic curves was already implicit in the Riemann-Roch theorem. For a curve C the coherent groups Hⁱ vanish for i > 1; but H¹ does enter implicitly. In fact the basic relation of the theorem involves L(D) and L(K−D), where D is a divisor and K a divisor of the canonical class. After Serre we recognise l(K−D) as the dimension of H¹(D), where now D means the line bundle determined by the divisor D. That is, Serre duality in this case relates groups H⁰(D) and H¹(KD*), and we are reading off dimensions (notation: K is the canonical line bundle, D* is the dual line bundle, and juxtaposition is tensor product of line bundles). In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ... In mathematics, specifically in complex analysis and algebraic geometry, the Riemann-Roch theorem is an important tool in the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... In mathematics, the canonical bundle of a non-singular algebraic variety of dimension is the line bundle which is the th exterior power of the cotangent bundle on . ... In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ...

In this formulation the theorem can be rearranged to read as a calculation of the Euler characteristic of a sheaf In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F. This is the main step, in numerous areas, from sheaf theory as a description of a geometric...

h⁰(D) − h¹(D),

in terms of the genus of the curve, which is In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ...

h¹(C,O_C),

and the degree of D. It is this expression that can be generalised to higher dimensions.

Serre duality of curves is therefore something very classical; but it has interesting light to cast. For example, in Riemann surface theory, the deformation theory of complex structures is studied classically by means of quadratic differentials (namely sections of L(K²)). The deformation theory of Kunihiko Kodaira and D. C. Spencer identifies deformations via H¹(T), where T is the tangent bundle sheaf K*. The duality shows why these approaches coincide. In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. ... In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution P of a problem to slightly different solutions Pε, where ε is a small number, or vector of small quantities. ... In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution P of a problem to slightly different solutions Pε, where ε is a small number, or vector of small quantities. ... Kunihiko Kodaira (小平 邦彦 Kodaira Kunihiko, 16 March 1915 – 26 July 1997) was a Japanese mathematician known for distinguished work in algebraic geometry and the theory of complex manifolds; and as the founder of the Japanese school of algebraic geometers. ... Donald C. Spencer (April 25, 1912 - December 23, 2001) was an American mathematician, known for major work on deformation theory of structures arising in differential geometry, and on several complex variables from the point of view of partial differential equations. ... In mathematics, the tangent bundle of a manifold is a vector bundle which as a set is the disjoint union of all the tangent spaces at every point in the manifold with natural topology and smooth structure. ...

The origin of the theory lay in Serre's earlier work on several complex variables. In the generalisation of Alexandre Grothendieck, Serre duality becomes a part of coherent duality in a much broader setting. While the role of K above in general Serre duality is played by the determinant line bundle of the cotangent bundle, when V is a manifold, in full generality K cannot just be a single sheaf in the absence of some hypothesis of non-singularity on V. The formulation in full generality uses a derived category and Ext functors, to allow for the fact that K is now represented by a chain complex of sheaves. The statement of the theorem is recognisably Serre's, however. The theory of functions of several complex variables is the branch of mathematics dealing with functions f(z1, z2, ... , zn) on the space Cn of n-tuples of complex numbers. ... Alexander Grothendieck (born March 28, 1928, Berlin) is one of the greatest mathematicians of the 20th century, with major contributions to algebraic geometry, homological algebra, and functional analysis. ... Coherent duality in mathematics refers to a number of generalisations of Serre duality, applying to coherent sheaves, in algebraic geometry and complex manifold theory, as well as some aspects of commutative algebra that are part of the local theory. ... In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. ... In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. ... In mathematics, a singular point of an algebraic variety V is a point P that is special (so, singular), in the geometric sense that V is not locally flat there. ... In mathematics, the derived category D(C) of a category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C (which therefore should already be an abelian category). ... In mathematics, the Ext functors of homological algebra are derived functors of functors. ... In homological algebra, a chain complex is a sequence of abelian groups or modules A0, A1, A2. ...