Picard–Lefschetz theory

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,...,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.

Example

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.

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. An International Journal of Mathematics, 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