mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, vanishing cycles are studied in

singularity theory In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it ...

and other parts of

algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

. They are those

homology Homology may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor * Sequence homology, biological homology between DNA, RNA, or protein sequences *Homologous chrom ...

cycles of a smooth fiber in a family which vanish in the singular fiber. For example, in a map from a connected complex surface to the complex projective line, a generic fiber is a smooth

Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...

of some fixed genus g and, generically, there will be isolated points in the target whose preimages are nodal curves. If one considers an isolated critical value and a small loop around it, in each fiber, one can find a smooth loop such that the singular fiber can be obtained by pinching that loop to a point. The loop in the smooth fibers gives an element of the first homology group of a surface, and the monodromy of the critical value is defined to be the monodromy of the first homology of the fibers as the loop is traversed, i.e. an invertible map of the first homology of a (real) surface of genus g. A classical result is the Picard–Lefschetz formula,Given i

for Morse functions. detailing how the monodromy round the singular fiber acts on the vanishing cycles, by a

shear mapping In plane geometry, a shear mapping is a linear map that displaces each point in a fixed direction, by an amount proportional to its signed distance from the line that is parallel to that direction and goes through the origin. This type of mappi ...

. The classical, geometric theory of

Solomon Lefschetz Solomon Lefschetz (russian: Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear o ...

was recast in purely algebraic terms, in SGA7. This was for the requirements of its application in the context of

l-adic cohomology In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension ...

; and eventual application to the

Weil conjectures In mathematics, the Weil conjectures were highly influential proposals by . They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory. Th ...

. There the definition uses

derived categories In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction proce ...

, and looks very different. It involves a functor, the ''nearby cycle functor'', with a definition by means of the

higher direct image In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology and algebraic geometry. Given a sheaf ''F'' defined on a topo ...

and pullbacks. The ''vanishing cycle functor'' then sits in a

distinguished triangle In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy cate ...

with the nearby cycle functor and a more elementary functor. This formulation has been of continuing influence, in particular in D-module theory.

References

*Dimca, Alexandru; Singularities and Topology of Hypersurfaces. *Section 3 of Peters, C.A.M. and J.H.M. Steenbrink: ''Infinitesimal variations of Hodge structure and the generic Torelli problem for projective hypersurfaces'', in : ''Classification of Algebraic Manifolds'', K. Ueno ed., Progress inMath. 39, Birkhauser 1983. *For the

étale cohomology In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectur ...

version, see the chapter on monodromy in * , see especially Pierre Deligne, ''Le formalisme des cycles évanescents'', SGA7 XIII and XIV. *

External links

Vanishing Cycle
in the Encyclopedia of Mathematics Algebraic topology Topological methods of algebraic geometry {{topology-stub

See also

References

External links