# Monodromy transformation

A transformation of the fibres (or of their homotopy invariants) of a fibre space corresponding to a path in the base. More precisely, let $p : E \rightarrow B$ be a locally trivial fibre space and let $\gamma : [ 0 , 1 ] \rightarrow B$ be a path in $B$ with initial point $a = \gamma ( 0)$ and end-point $b = \gamma ( 1)$. A trivialization of the fibration $\gamma ^ {*} ( E)$ defines a homeomorphism $T _ \gamma$ of the fibre $p ^ {-} 1 ( a)$ onto the fibre $p ^ {-} 1 ( b)$, $T _ \gamma : p ^ {-} 1 ( a) \rightarrow p ^ {-} 1 ( b)$. If the trivialization of $\gamma ^ {*} ( E)$ is modified, then $T _ \gamma$ changes into a homotopically-equivalent homeomorphism; this also happens if $\gamma$ is changed to a homotopic path. The homotopy type of $T _ \gamma$ is called the monodromy transformation corresponding to a path $\gamma$. When $a = b$, that is, when $\gamma$ is a loop, the monodromy transformation $T _ \gamma$ is a homeomorphism of $F = p ^ {-} 1 ( a)$ into itself (defined, yet again, up to a homotopy). This mapping, and also the homomorphisms induced by it on the homology and cohomology spaces of $F$, is also called a monodromy transformation. The correspondence of $\gamma$ with $T _ \gamma$ gives a representation of the fundamental group $\pi _ {1} ( B , a )$ on $H ^ {*} ( F )$.

The idea of a monodromy transformation arose in the study of multi-valued functions (see Monodromy theorem). If $S \rightarrow P ^ {1} ( \mathbf C )$ is the Riemann surface of such a function, then by eliminating the singular points of the function from the Riemann sphere $P ^ {1} ( \mathbf C )$, an unbranched covering is obtained. The monodromy transformation in this case is also called a covering or deck transformation.

The monodromy transformation arises most frequently in the following situation. Let $D = \{ {z \in \mathbf C } : {| z | < 1 } \}$ be the unit disc in the complex plane, let $X$ be an analytic space, let $f : X \rightarrow D$ be a proper holomorphic mapping (cf. Proper morphism), let $X _ {t}$ be the fibre $f ^ { - 1 } ( t)$, $t \in D$, $D ^ {*} = D \setminus \{ 0 \}$, and let $X ^ {*} = f ^ { - 1 } ( D ^ {*} )$. Diminishing, if necessary, the radius of $D$, the fibre space $f : X ^ {*} \rightarrow D ^ {*}$ can be made locally trivial. The monodromy transformation $T$ associated with a circuit around $0$ in $D$ is called the monodromy of the family $f : X \rightarrow D$ at $0 \in D$, it acts on the (co)homology spaces of the fibre $X _ {t}$, where $t \in D ^ {*}$. The most studied case is when $X$ and the fibres $X _ {t}$, $t \neq 0$, are smooth. The action of $T$ on $H ^ {*} ( X _ {t} , \mathbf Q )$, in this case, is quasi-unipotent [4], that is, there are positive integers $k$ and $N$ such that $( T ^ {k} - 1 ) ^ {N} = 0$. The properties of the monodromy display many characteristic features of the degeneracy of the family $f : X \rightarrow D$. The monodromy of the family $f : X \rightarrow D$ is closely related to mixed Hodge structures (cf. Hodge structure) on the cohomology spaces $H ^ {*} ( X _ {0} )$ and $H ^ {*} ( X _ {t} )$( see [5][7]).

When the singularities of $f : X \rightarrow D$ are isolated, the monodromy transformation can be localized. Let $x$ be a singular point of $f$( or, equivalently, of $X _ {0}$) and let $B$ be a sphere of sufficiently small radius in $X$ with centre at $x$. Diminishing, if necessary, the radius of $D$, a local trivialization of the fibre space $B \cap f ^ { - 1 } ( D ^ {*} )$ can be defined. It is compatible with the trivialization of the fibre space $\partial B \cap f ^ { - 1 } ( D) \rightarrow D$ on the boundary. This gives a diffeomorphism $T$ of the manifold of "vanishing cycles" $V _ {t} = B \cap X _ {t}$ into itself which is the identity on $\partial V _ {t}$, and which is called the local monodromy of $f$ at $x$. The action of the monodromy transformation on the cohomology spaces $H ^ {*} ( V _ {t} )$ reflects the singularity of $f$ at $x$( see [1], [2], [7]). It is known that the manifold $V _ {t}$ is homotopically equivalent to a bouquet of $\mu$ $n$- dimensional spheres, where $n + 1 = \mathop{\rm dim} X$ and $\mu$ is the Milnor number of the germ of $f$ at $x$.

The simplest case is that of a Morse singularity when, in a neighbourhood of $x$, $f$ reduces to the form $f = z _ {0} ^ {2} + \dots + z _ {n} ^ {2}$( cf. Morse lemma). In this case $\mu = 1$, and the interior $V _ {t} ^ {0}$ of $V _ {t}$ is diffeomorphic to the tangent bundle of the $n$- dimensional sphere $S ^ {n}$. A vanishing cycle $\delta$ is a generator of the cohomology group with compact support $H _ {c} ^ {n} ( V _ {t} ^ {0} , \mathbf Z ) \cong \mathbf Z$, defined up to sign. In general, if $f : X \rightarrow D$ is a proper holomorphic mapping (as above, having a unique Morse singularity at $x$), then a cycle $\delta _ {x}$ vanishing at $x$ is the image of a cycle $\delta \in H _ {c} ^ {n} ( V _ {t} ^ {0} )$ under the natural mapping $H _ {c} ^ {n} ( V _ {t} ^ {0} ) \rightarrow H ^ {n} ( X _ {t} )$. In this case the specialization homomorphism $r _ {t} ^ {*} : H ^ {i} ( X _ {0} ) \rightarrow H ^ {i} ( X _ {t} )$ is an isomorphism for $i \neq n , n + 1$, and the sequence

$$0 \rightarrow H ^ {n} ( X _ {0} ) \rightarrow H ^ {n} ( X _ {t} ) \mathop \rightarrow \limits ^ { {( , \delta _ {x} ) }} \mathbf Z \rightarrow$$

$$\rightarrow \ H ^ {n+} 1 ( X _ {0} ) \rightarrow H ^ {n+} 1 ( X _ {t} ) \rightarrow 0$$

is exact. The monodromy transformation $T$ acts trivially on $H ^ {i} ( X _ {t} )$ for $i \neq n$ and its action on $H ^ {n} ( X _ {t} )$ is given by the Picard–Lefschetz formula: For $z \in H ^ {n} ( X _ {t} )$,

$$T _ {z} = z \pm ( z , \delta _ {x} ) \delta _ {x} .$$

The sign in this formula and the values of $( \delta _ {x} , \delta _ {x} )$

are collected in the table.

<tbody> </tbody>
 $n \mathop{\rm mod} 4$ $0$ $1$ $2$ $3$ $\pm$ $-$ $-$ $+$ $+$ $( \delta _ {x} , \delta _ {x} )$ $2$ $0$ $- 2$ $0$

A monodromy transformation preserves the intersection form on $H ^ {n} ( X _ {t} )$.

Vanishing cycles and monodromy transformations are used in the Picard–Lefschetz theory, associating the cohomology space of a projective complex manifold and its hyperplane sections. Let $X \subset P ^ {N}$ be a smooth manifold of dimension $n + 1$, and let $\{ X _ {t} \}$, $t \in P ^ {1}$, be a pencil of hyperplane sections of $X$ with basic set (axis of the pencil) $Y \subset X$; let the following conditions be satisfied: a) $Y$ is a smooth submanifold in $X$; b) there is a finite set $S \subset P ^ {1}$ such that $X _ {t}$ is smooth for $t \in P ^ {1} \setminus S$; and c) for $s \in S$ the manifold $X _ {s}$ has a unique non-degenerate quadratic singular point $x _ {s}$, where $x _ {s} \in Y$. Pencils with these properties (Lefschetz pencils) always exist. Let $\sigma : \overline{X}\; \rightarrow X$ be a monoidal transformation with centre on the axis $Y$ of the pencil, and let $f : \overline{X}\; \rightarrow P ^ {1}$ be the morphism defined by the pencil $\{ X _ {t} \}$; here $f ^ { - 1 } ( t) \cong X _ {t}$ for all $t \in P ^ {1}$. Let a point $0 \in P ^ {1} \setminus S$ be fixed; then the monodromy transformation gives an action of $\pi _ {1} ( P ^ {1} \setminus S , 0 )$ on $H ^ {i} ( X _ {0} )$( non-trivial only for $i = n$). To describe the action of the monodromy on $H ^ {n} ( X _ {0} )$ one chooses points $s ^ \prime$, situated near $s \in S$, and paths $\gamma _ {s} ^ \prime$ leading from $0$ to $s ^ \prime$. Let $\gamma _ {s} \in \pi _ {1} ( p ^ {1} \setminus S , 0 )$ be the loop constructed as follows: first go along $\gamma _ {s} ^ \prime$, then once round $s$ and, finally, return along $\gamma _ {s} ^ \prime$ to $0$. In addition, let $\delta _ {s}$ be a cycle vanishing at $x _ {s}$( more precisely, take a vanishing cycle in $H ^ {n} ( X _ {s ^ \prime } )$ and transfer it to $H ^ {n} ( X _ {0} )$ by means of the monodromy transformation corresponding to the path $\gamma _ {s} ^ \prime$). Finally, let $E \subset H ^ {n} ( X _ {0} , \mathbf Q )$ be the subspace generated by the vanishing cycles $\delta _ {s}$, $s \in S$( the vanishing cohomology space). Then the following hold.

1) $\pi _ {1} ( P ^ {1} \setminus S , 0 )$ is generated by the elements $\gamma _ {s}$, $s \in S$;

2) the action of $\gamma _ {s}$ is given by the formula

$$\gamma _ {s} ( z ) = z \pm ( z , \delta _ {s} ) \delta _ {s} ;$$

3) the space $E \subset H ^ {n} ( X _ {0} )$ is invariant under the action of the monodromy group $\pi _ {1} ( P ^ {1} \setminus S , 0 )$;

4) the space of elements in $H ^ {n} ( X _ {0} )$ that are invariant relative to monodromy coincides with the orthogonal complement of $E$ relative to the intersection form on $H ^ {n} ( X _ {0} )$, and also with the images of the natural homomorphisms $H _ {n} ( \overline{X}\; ) \rightarrow H _ {n} ( X _ {0} )$ and $H ^ {n} ( X ) \rightarrow H ^ {n} ( X _ {0} )$;

5) the vanishing cycles $\pm \delta _ {s}$ are conjugate (up to sign) under the action of $\pi _ {1} ( P ^ {1} \setminus S , 0 )$;

6) the action of $\pi _ {1} ( P ^ {1} \setminus S , 0 )$ on $E$ is absolutely irreducible.

The formalism of vanishing cycles, monodromy transformations and the Picard–Lefschetz theory has also been constructed for $l$- adic cohomology spaces of algebraic varieties over any field (see [3]).

#### References

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How to Cite This Entry:
Monodromy transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Monodromy_transformation&oldid=47885
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article