In mathematics, Picard–Lefschetz theory studies the topology of a
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...
by looking at the
critical points of a
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
on the manifold. It was introduced by
Émile Picard
Charles Émile Picard (; 24 July 1856 – 11 December 1941) was a French mathematician. He was elected the fifteenth member to occupy seat 1 of the Académie française in 1924.
Life
He was born in Paris on 24 July 1856 and educated there at th ...
for complex surfaces in his book , and extended to higher dimensions by . It is a complex analog of
Morse theory
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differenti ...
that studies the topology of a real
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
by looking at the critical points of a real function. extended Picard–Lefschetz theory to varieties over more general fields, and Deligne used this generalization in his proof of 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 ...
.
Picard–Lefschetz formula
The Picard–Lefschetz formula describes the
monodromy
In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name implies, the fundamental meaning of ''mono ...
at a critical point.
Suppose that ''f'' is a holomorphic map from an -dimensional projective complex manifold to the projective line P
1. Also suppose that all critical points are non-degenerate and lie in different fibers, and have images ''x''
1,...,''x''
''n'' in P
1. Pick any other point ''x'' in P
1. The
fundamental group
In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...
is generated by loops ''w''
''i'' going around the points ''x''
''i'', and to each point ''x''
''i'' there is a
vanishing cycle In mathematics, vanishing cycles are studied in singularity theory and other parts of algebraic geometry. They are those homology (mathematics), homology cycles of a smooth fiber in a family which vanish in the singular fiber.
For example, in a map ...
in the homology ''H''
''k''(''Y''
''x'') of the fiber at ''x''. Note that this is the middle homology since the fibre has complex dimension ''k'', hence real dimension 2''k''.
The monodromy action of on ''H''
''k''(''Y''
''x'') is described as follows by the Picard–Lefschetz formula. (The action of monodromy on other homology groups is trivial.) The monodromy action of a generator ''w''
''i'' of the fundamental group on is given by
:
where δ
''i'' is the vanishing cycle of ''x''
''i''. This formula appears implicitly for ''k'' = 2 (without the explicit coefficients of the vanishing cycles δ
''i'') in . gave the explicit formula in all dimensions.
Example
Consider the projective family of hyperelliptic curves of genus
defined by
:
where
is the parameter and
. Then, this family has double-point degenerations whenever
. Since the curve is a connected sum of
tori, the intersection form on
of a generic curve is the matrix
:
we can easily compute the Picard-Lefschetz formula around a degeneration on
. Suppose that
are the
-cycles from the
-th torus. Then, the Picard-Lefschetz formula reads
:
if the
-th torus contains the vanishing cycle. Otherwise it is the identity map.
See also
*
Lefschetz pencil In mathematics, a Lefschetz pencil is a construction in algebraic geometry considered by Solomon Lefschetz, used to analyse the algebraic topology of an algebraic variety V.
Description
A ''pencil'' is a particular kind of linear system of div ...
References
*
*
*
*
*
*
{{DEFAULTSORT:Picard-Lefschetz theory
Algebraic geometry