Picard–Lefschetz theory

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,...,x_n in P¹. Pick any other point x in P¹. The fundamental group π₁(P¹ – {x₁, ..., x_n}, x) is generated by loops w_i going around the points x_i, and to each point x_i there is a vanishing cycle 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 2k. The monodromy action of π₁(P¹ – {x₁, ..., x_n}, x) 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 ${\displaystyle \gamma }$ ∈ H_k(Y_x) is given by

${\displaystyle w_{i}(\gamma )=\gamma +(-1)^{(k+1)(k+2)/2}\langle \gamma ,\delta _{i}\rangle \delta _{i}}$

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 Picard & Simart (1897, p.95). Lefschetz (1924, chapters II, V) gave the explicit formula in all dimensions.

Consider the projective family of hyperelliptic curves of genus ${\displaystyle g}$ 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 ${\displaystyle g}$ tori, the intersection form on ${\displaystyle H_{1}}$ 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 ${\displaystyle 1}$-cycles from the ${\displaystyle j}$-th torus. Then, the Picard-Lefschetz formula reads

${\displaystyle w_{j}(\gamma )=\gamma -\delta }$

if the ${\displaystyle j}$-th torus contains the vanishing cycle. Otherwise it is the identity map.

• 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