[Paper Review] Monodromy at Infinity and the Weights of Cohomology

Alexandru Dimca, Morihiko Saito|ArXiv.org|Feb 25, 2000

Algebraic Geometry and Number Theory12 references4 citations

TL;DR

This paper establishes bounds on the size of Jordan blocks for eigenvalue 1 in the monodromy at infinity of polynomial maps, showing they are constrained by the maximal multiplicity of the divisor at infinity in a good compactification. It links these blocks to global invariant cycles in the weight filtration and proves that larger Jordan blocks in graded pieces arise from even larger blocks in the total cohomology, with applications to period integrals and rationality of asymptotic expansions.

ABSTRACT

We show that for a polynomial map, the size of the Jordan blocks for the eigenvalue 1 of the monodromy at infinity is bounded by the multiplicity of the reduced divisor at infinity of a good compactification of a general fiber. The existence of such Jordan blocks is related to global invariant cycles of the graded pieces of the weight filtration. These imply some applications to period integrals. We also show that such a Jordan block of size greater than 1 for the graded pieces of the weight filtration is the restriction of a strictly larger Jordan block for the total cohomology group. If there are no singularities at infinity, we have a more precise statement on the monodromy.

Motivation & Objective

• To determine the maximal size of Jordan blocks for eigenvalue 1 in the monodromy at infinity of a polynomial map.
• To clarify the relationship between such Jordan blocks and the global invariant cycles in the graded pieces of the weight filtration.
• To establish that nontrivial Jordan blocks for eigenvalue 1 in the graded weight pieces arise from strictly larger blocks in the total cohomology.
• To apply these results to period integrals, particularly the rationality and asymptotic behavior of period integrals at infinity.

Proposed method

• Use of good compactifications of general fibers to define the divisor at infinity and its multiplicity $m_s$.
• Analysis of the monodromy action on the cohomology of the general fiber, focusing on the eigenvalue 1 and its Jordan blocks.
• Application of the weight filtration and its compatibility with the monodromy action via the mixed Hodge structure.
• Use of the nearby cycle functor and the theory of variations of mixed Hodge structures on a Zariski-open subset of $\mathbb{C}$.
• Reduction to the normal crossing case via embedded resolution and sheaf-theoretic calculations of nearby cycles.
• Duality arguments and the use of the dual filtration on homology to relate invariants to period integrals.

Experimental results

Research questions

• RQ1 What bounds exist on the size of Jordan blocks for eigenvalue 1 in the monodromy at infinity of a polynomial map?
• RQ2 How are these Jordan blocks related to the global invariant cycles in the graded pieces of the weight filtration?
• RQ3 Can a Jordan block for eigenvalue 1 in a graded piece of the weight filtration be extended to a strictly larger block in the total cohomology?
• RQ4 Under what conditions is the period integral of a differential form a rational function of the parameter at infinity?
• RQ5 How does the asymptotic expansion of period integrals at infinity relate to the multiplicity of the divisor at infinity?

Key findings

• The size of Jordan blocks for eigenvalue 1 in the monodromy at infinity is bounded by $m_s$, the maximal multiplicity of the divisor at infinity, and by $j$ for $j > 0$.
• If the monodromy on $\mathrm{Gr}^W_i H^j(X_s,\mathbb{Q})$ has a Jordan block of size $r > 0$ for eigenvalue 1, then $\mathrm{Gr}^W_{i'} H^j(X_s,\mathbb{Q})$ has nonzero global invariants with $i' = i + r + 1 \leq j + m'_s$.
• Such a Jordan block of size $r > 1$ for eigenvalue 1 in a graded piece is the restriction of a strictly larger Jordan block in the total cohomology group.
• For cohomologically tame polynomials, the graded pieces $\mathrm{Gr}^W_i H^n(X_s,\mathbb{Q})$ are constant for $i \neq n$, and the monodromy data satisfy explicit formulas involving $m(\infty,1,r)$ and $m'(s,\lambda,r)$.
• The period integral $\int_{\gamma_t} \omega$ is a rational function of $t$ if the de Rham class of $\omega$ has weight $\leq i'$, where $i'$ is determined by the invariant cycle condition.
• For rational asymptotic expansions at infinity, the order of logarithmic terms $r(\alpha)$ is bounded by $m'_s - 1$ when $\alpha \in \mathbb{Z}$, improving the general monodromy theorem bound of $j$.