[Paper Review] Monodromies at infinity of non-tame polynomials

Kiyoshi Takeuchi, Mihai Tibăr|arXiv (Cornell University)|Aug 22, 2012

Advanced Differential Equations and Dynamical Systems22 references3 citations

TL;DR

This paper extends the theory of monodromy at infinity for non-tame polynomials by introducing a notion of 'atypical' eigenvalues and proving cohomological concentration for generalized eigenspaces corresponding to non-atypical eigenvalues. Using motivic Milnor fibers and Newton polyhedra, it generalizes prior results on Jordan normal forms of monodromy operators to non-tame, non-degenerate polynomials with non-isolated singularities.

ABSTRACT

We consider the monodromy at infinity and the monodromies around the bifurcation points of polynomial functions $f : \CC^n \longrightarrow \CC$ which are not tame and might have non-isolated singularities. Our description of their Jordan blocks in terms of the Newton polyhedra and the motivic Milnor fibers relies on two new issues: the non-atypical eigenvalues of the monodromies and the corresponding concentration results for their generalized eigenspaces.

Motivation & Objective

• To overcome the limitations of prior methods that fail for non-tame polynomials due to lack of cohomological concentration.
• To define a finite set of 'bad' or atypical eigenvalues $ A_f $ that obstruct concentration in the cohomology of generic fibers.
• To generalize the computation of Jordan normal forms of monodromy operators at infinity to non-tame polynomials with non-isolated singularities.
• To establish a framework using motivic Milnor fibers and Newton polyhedra to describe the monodromy action beyond the tame case.

Proposed method

• Introduce the concept of atypical eigenvalues $ A_f $ via the Newton polyhedron at infinity $ \Gamma_\infty(f) $, excluding eigenvalues that disrupt cohomological concentration.
• Prove cohomological concentration for generalized eigenspaces $ H^j(f^{-1}(R);\mathbb{C})_\lambda $ when $ \lambda \notin A_f $, showing vanishing for $ j \neq n-1 $.
• Construct a compactification $ \widetilde{X_\Sigma} $ of $ \mathbb{C}^n $ to analyze horizontal divisors at infinity and refine Sabbah's theorem in this context.
• Use motivic Milnor fibers $ \SS_f^b \in \mathcal{M}_\mathbb{C}^{\hat{\mu}} $ to encode monodromy data, combining contributions from local singularities and atypical fibers.
• Express the monodromy action in terms of motivic invariants via the Hodge realization $ \chi_h(\SS_f^b) $ in $ K_0(\mathrm{HS}^{\mathrm{mon}}) $.
• Derive explicit formulas for the $ \lambda $-part of the Jordan normal form of $ \Phi_{n-1}^b $ using compact faces $ \gamma \prec \Gamma_+(f_i) $ and associated Laurent polynomials $ g_\gamma $.

Experimental results

Research questions

• RQ1 How can monodromy at infinity be described for non-tame polynomials with non-isolated singularities?
• RQ2 What conditions on eigenvalues prevent cohomological concentration in the fibers of non-tame polynomials?
• RQ3 Can the Jordan normal form of the monodromy operator $ \Phi_{n-1}^\infty $ be computed beyond the tame case using motivic and Newton-theoretic tools?
• RQ4 How do motivic Milnor fibers and Newton polyhedra encode the monodromy action for non-atypical eigenvalues?
• RQ5 What is the role of compact faces of $ \Gamma_+(f_i) $ in determining the size and structure of Jordan blocks for $ \Phi_{n-1}^b $?

Key findings

• For any non-atypical eigenvalue $ \lambda \notin A_f $, the generalized eigenspace $ H^j(f^{-1}(R);\mathbb{C})_\lambda $ vanishes for $ j \neq n-1 $, establishing cohomological concentration.
• The $ \lambda $-part of the monodromy $ \Phi_{n-1}^\infty $ is completely determined by the motivic Milnor fiber $ \SS_f^b $ for $ \lambda \notin A_f $, via $ [H_f^b]_\lambda = \chi_h(\SS_f^b)_\lambda $.
• The number of Jordan blocks of size $ \geq k $ for eigenvalue $ \lambda \notin A_f $ is given by a signed sum over compact faces $ \gamma \prec \Gamma_+(f_i) $, involving Hodge numbers $ e^{p,q} $ of $ (1-\mathbb{L})^{m_\gamma} \cdot [Z_{\Delta_\gamma}^*] $.
• The formula for the Jordan block structure generalizes [17, Theorems 5.9, 5.14, 5.16] to non-tame polynomials by excluding contributions from faces in $ \Gamma_+^\circ(f_i) $ for $ i > l $.
• The motivic Milnor fiber $ \SS_f^b $ is decomposed into contributions from isolated singularities $ p_i $ and proper transforms of divisors, with $ \SS_f^b = [Z_{f,b}] + \sum_{i=1}^l \SS_{f_i,p_i} + \sum_{i=l+1}^{l+r} \SS_{f_i,p_i}^\circ $.
• The construction of the Laurent polynomial $ g_\gamma $ on $ T_{\Delta_\gamma} \simeq (\mathbb{C}^*)^{\dim \gamma + 1} $ ensures non-degeneracy and invariance under translation by $ \tau_\gamma $, enabling motivic integration.