[Paper Review] Monodromy map for tropical Dolbeault cohomology

Yifeng Liu|arXiv (Cornell University)|Apr 23, 2017

Algebraic Geometry and Number Theory5 references3 citations

TL;DR

This paper introduces a monodromy map for tropical Dolbeault cohomology on algebraic varieties over non-Archimedean fields, constructing a functorial map $\mathrm{N}_X: H^{p,q}_{\mathrm{trop}}(X) \to H^{p-1,q+1}_{\mathrm{trop}}(X)$ via sheaf-level superform operations. It conjectures that iterated monodromy maps induce Hodge isomorphisms $H^{p,0}_{\mathrm{trop}}(X) \xrightarrow{\sim} H^{0,p}_{\mathrm{trop}}(X)$, providing evidence through injectivity and isomorphism results in specific cases using weight spectral sequences and arithmetic geometry.

ABSTRACT

We define monodromy maps for tropical Dolbeault cohomology of algebraic varieties over non-Archimedean fields. We propose a conjecture of Hodge isomorphisms via monodromy maps, and provide some evidence.

Motivation & Objective

• To define a monodromy map for tropical Dolbeault cohomology on varieties over non-Archimedean fields, generalizing complex geometric notions to the tropical setting.
• To propose a conjectural Hodge isomorphism for tropical Dolbeault cohomology via iterated monodromy maps, replacing the classical Hodge symmetry $H^{p,q} \simeq H^{q,p}$.
• To provide arithmetic-geometric evidence for the conjecture using weight spectral sequences and monodromy-weight conjectures.
• To explore the analytic and cohomological properties of tropical Dolbeault cohomology, particularly in the context of semistable models and non-Archimedean geometry.

Proposed method

• Constructs a canonical sheaf map $\mathscr{A}^{p,q}_{X^{\mathrm{an}}} \to \mathscr{A}^{p-1,q+1}_{X^{\mathrm{an}}}$ using superforms and star-shaped integration on simplices.
• Defines the monodromy map $\mathrm{N}_X$ as a functorial map on tropical Dolbeault cohomology groups via sheaf-level operations.
• Applies the weight spectral sequence and monodromy-weight conjecture to establish injectivity and isomorphism results in specific cases.
• Uses arithmetic geometry techniques, including base change to algebraic closures and completion, to analyze cohomology groups over non-Archimedean fields.
• Relies on explicit computations of contraction maps $\mathcal{C}_Q$ and star-shaped integrals $\mathcal{I}_Q'$ on $\mathbf{R}^{n+1}$ to derive cohomological identities.
• Compares the monodromy map with Mikhalkin-Zharkov's $\phi \cap$ map in tropical homology, suggesting equivalence up to scalar factors.

Experimental results

Research questions

• RQ1 Does the monodromy map $\mathrm{N}_X: H^{p,q}_{\mathrm{trop}}(X) \to H^{p-1,q+1}_{\mathrm{trop}}(X)$ exist as a functorial map in tropical Dolbeault cohomology?
• RQ2 Can the iterated monodromy map $\mathrm{N}_X^{p-q}: H^{p,q}_{\mathrm{trop}}(X) \to H^{q,p}_{\mathrm{trop}}(X)$ induce an isomorphism for $p \geq q$?
• RQ3 Is the monodromy map $\mathrm{N}_X: H^{1,0}_{\mathrm{trop}}(X) \to H^{0,1}_{\mathrm{trop}}(X)$ an isomorphism when $X$ is a proper smooth variety with a semistable model?
• RQ4 How does the monodromy map relate to the Hodge isomorphism conjecture in tropical geometry, and how does it differ from complex geometry?
• RQ5 Can the monodromy map be realized analytically, independent of arithmetic geometry, especially in cases like $K = \widehat{K_0^\mathrm{a}}$?

Key findings

• The monodromy map $\mathrm{N}_X^p: H^{p,0}_{\mathrm{trop}}(X) \to H^{0,p}_{\mathrm{trop}}(X)$ is injective for all $p \geq 0$ when $K_0 \cong k((t))$ for $k$ a finite field or characteristic zero field.
• For $K_0$ a finite extension of $\mathbf{Q}_p$ or $k((t))$ with $k$ finite, and $X_0$ admitting a strictly semistable model, $\mathrm{N}_X: H^{1,0}_{\mathrm{trop}}(X) \to H^{0,1}_{\mathrm{trop}}(X)$ is an isomorphism.
• The map $\mathrm{N}_X^p: H^{p,0}_{\mathrm{trop}}(X) \to H^{0,p}_{\mathrm{trop}}(X)$ is isomorphic to the composition of the flipping map $\mathrm{J}$ and multiplication by $p!$, as shown in Lemma 3.2(1).
• The monodromy map induces a numerical Hodge diamond inequality: $h^{p,0}_{\mathrm{trop}}(X) \leq h^{p-1,1}_{\mathrm{trop}}(X) \leq \cdots \geq h^{0,p}_{\mathrm{trop}}(X)$, which is new compared to complex geometry.
• The construction of the monodromy map is compatible with Mikhalkin-Zharkov’s $\phi \cap$ map on tropical homology, suggesting equivalence up to scalar factors under compactness assumptions.
• The monodromy map fails to be an isomorphism in general for $K = \widehat{\mathbf{C}((t))^\mathrm{a}}$, indicating that the conjectural Hodge isomorphism is not universally valid.