In mathematics, Picard–Lefschetz theory studies the topology of a complex manifold by looking at the critical points of a holomorphic function on the manifold. It was introduced by Émile Picard for complex surfaces in his book Picard & Simart (1897), and extended to higher dimensions by Solomon Lefschetz (1924). It is a complex analog of Morse theory that studies the topology of a real manifold by looking at the critical points of a real function. Pierre Deligne and Nicholas Katz (1973) extended Picard–Lefschetz theory to varieties over more general fields, and Deligne used this generalization in his proof of the Weil conjectures.
Picard–Lefschetz formula
The Picard–Lefschetz formula describes the monodromy at a critical point.
Suppose that f is a holomorphic map from an (k+1)-dimensional projective complex manifold to the projective line P1. Also suppose that all critical points are non-degenerate and lie in different fibers, and have images x1,...,xn in P1. Pick any other point x in P1. The fundamental group π1(P1 – {x1, ..., xn}, x) is generated by loops wi going around the points xi, and to each point xi there is a vanishing cycle in the homology Hk(Yx) of the fiber at x. Note that this is the middle homology since the fibre has complex dimension k, hence real dimension 2k. The monodromy action of π1(P1 – {x1, ..., xn}, x) on Hk(Yx) 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 wi of the fundamental group on \( \gamma \) ∈ Hk(Yx) is given by
\( w_{i}(\gamma )=\gamma +(-1)^{{(k+1)(k+2)/2}}\langle \gamma ,\delta _{i}\rangle \delta _{i} \)
where δi is the vanishing cycle of xi. This formula appears implicitly for k = 2 (without the explicit coefficients of the vanishing cycles δi) in Picard & Simart (1897, p.95). Lefschetz (1924, chapters II, V) gave the explicit formula in all dimensions.
Example
Consider the projective family of hyperelliptic curves of genus g {\displaystyle g} g defined by
\( {\displaystyle y^{2}=(x-t)(x-a_{1})\cdots (x-a_{k})} \)
where \( {\displaystyle t\in \mathbb {A} ^{1}} \) is the parameter and \( {\displaystyle k=2g+1} \). Then, this family has double-point degenerations whenever \( {\displaystyle t=a_{i}} \). Since the curve is a connected sum of g {\displaystyle g} g tori, the intersection form on \( H_{1} \) of a generic curve is the matrix
\( {\displaystyle {\begin{bmatrix}0&1\\1&0\end{bmatrix}}^{\oplus g}={\begin{bmatrix}0&1&0&0&\cdots &0&0\\1&0&0&0&\cdots &0&0\\0&0&0&1&\cdots &0&0\\0&0&1&0&\cdots &0&0\\\vdots &\vdots &\vdots &\vdots &\cdots &\vdots &\vdots \\0&0&0&0&\cdots &0&1\\0&0&0&0&\cdots &1&0\end{bmatrix}}} \)
we can easily compute the Picard-Lefschetz formula around a degeneration on \( {\displaystyle \mathbb {A} _{t}^{1}} \). Suppose that \( {\displaystyle \gamma ,\delta } \) are the 1-cycles from the j j-th torus. Then, the Picard-Lefschetz formula reads
\( {\displaystyle w_{j}(\gamma )=\gamma -\delta } \)
if the j-th torus contains the vanishing cycle. Otherwise it is the identity map.
References
Deligne, Pierre; Katz, Nicholas (1973), Groupes de monodromie en géométrie algébrique. II, Lecture Notes in Mathematics, 340, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0060505, ISBN 978-3-540-06433-6, MR 0354657
Lamotke, Klaus (1981), "The topology of complex projective varieties after S. Lefschetz", Topology, 20 (1): 15–51, doi:10.1016/0040-9383(81)90013-6, ISSN 0040-9383, MR 0592569
Lefschetz, S. (1924), L'analysis situs et la géométrie algébrique, Gauthier-Villars, MR 0033557
Lefschetz, Solomon (1975), Applications of algebraic topology. Graphs and networks, the Picard-Lefschetz theory and Feynman integrals, Applied Mathematical Sciences, 16, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90137-4, MR 0494126
Picard, É.; Simart, G. (1897), Théorie des fonctions algébriques de deux variables indépendantes. Tome I (in French), Paris: Gauthier-Villars et Fils.
Vassiliev, V. A. (2002), Applied Picard–Lefschetz theory, Mathematical Surveys and Monographs, 97, Providence, R.I.: American Mathematical Society, doi:10.1090/surv/097, ISBN 978-0-8218-2948-6, MR 1930577
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