Coherent sheaf cohomologyIn mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties. Many geometric questions can be formulated as questions about the existence of sections of line bundles or of more general coherent sheaves; such sections can be viewed as generalized functions. Cohomology provides computable tools for producing sections, or explaining why they do not exist. It also provides invariants to distinguish one algebraic variety from another.
Tautological bundleIn mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of -dimensional subspaces of , given a point in the Grassmannian corresponding to a -dimensional vector subspace , the fiber over is the subspace itself. In the case of projective space the tautological bundle is known as the tautological line bundle. The tautological bundle is also called the universal bundle since any vector bundle (over a compact space) is a pullback of the tautological bundle; this is to say a Grassmannian is a classifying space for vector bundles.
Rational pointIn number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field of real numbers, a rational point is more commonly called a real point. Understanding rational points is a central goal of number theory and Diophantine geometry. For example, Fermat's Last Theorem may be restated as: for n > 2, the Fermat curve of equation has no other rational points than (1, 0), (0, 1), and, if n is even, (–1, 0) and (0, –1).
Kähler manifoldIn mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnoldus Schouten and David van Dantzig in 1930, and then introduced by Erich Kähler in 1933. The terminology has been fixed by André Weil.
Abelian varietyIn mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory. An abelian variety can be defined by equations having coefficients in any field; the variety is then said to be defined over that field.
Moduli spaceIn mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spaces frequently arise as solutions to classification problems: If one can show that a collection of interesting objects (e.g., the smooth algebraic curves of a fixed genus) can be given the structure of a geometric space, then one can parametrize such objects by introducing coordinates on the resulting space.
CodimensionIn mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective algebraic varieties, the codimension equals the height of the defining ideal. For this reason, the height of an ideal is often called its codimension. The dual concept is relative dimension. Codimension is a relative concept: it is only defined for one object inside another.
Euler sequenceIn mathematics, the Euler sequence is a particular exact sequence of sheaves on n-dimensional projective space over a ring. It shows that the sheaf of relative differentials is stably isomorphic to an -fold sum of the dual of the Serre twisting sheaf. The Euler sequence generalizes to that of a projective bundle as well as a Grassmann bundle (see the latter article for this generalization.) Let be the n-dimensional projective space over a commutative ring A. Let be the sheaf of 1-differentials on this space, and so on.
Adequate equivalence relationIn algebraic geometry, a branch of mathematics, an adequate equivalence relation is an equivalence relation on algebraic cycles of smooth projective varieties used to obtain a well-working theory of such cycles, and in particular, well-defined intersection products. Pierre Samuel formalized the concept of an adequate equivalence relation in 1958. Since then it has become central to theory of motives. For every adequate equivalence relation, one may define the of pure motives with respect to that relation.
Elliptic surfaceIn mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper morphism with connected fibers to an algebraic curve such that almost all fibers are smooth curves of genus 1. (Over an algebraically closed field such as the complex numbers, these fibers are elliptic curves, perhaps without a chosen origin.) This is equivalent to the generic fiber being a smooth curve of genus one. This follows from proper base change.
André WeilAndré Weil ('veɪ; ɑ̃dʁe vɛj; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was one of the most influential mathematicians of the twentieth century. His influence is due both to his original contributions to a remarkably broad spectrum of mathematical theories, and to the mark he left on mathematical practice and style, through some of his own works as well as through the Bourbaki group, of which he was one of the principal founders.
Chern classIn mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches of mathematics and physics, such as string theory, Chern–Simons theory, knot theory, Gromov-Witten invariants. Chern classes were introduced by . Chern classes are characteristic classes. They are topological invariants associated with vector bundles on a smooth manifold.
