Monodromy Filtrations and the Topology of Tropical Varieties

Abstract

We find restrictions on the topology of tropical varieties that arise from a certain natural class of varieties. We develop a theory of tropical degenerations that is a nonconstant coefficient analogue of Tevelev's theory of tropical compactifications, and use it to construct normal crossings degenerations of a subvariety X of a torus, under mild hypotheses on X. These degenerations allow us to construct a natural, "multiplicity-free" parameterization of Trop(X) by a topological space Γ_X. We give a geometric interpretation of the cohomology of Γ_X in terms of the action of a monodromy operator on the cohomology of X. This gives bounds on the Betti numbers of ${\Gamma }_X$ in terms of the Betti numbers of $X$. When $X$ is a sufficiently general complete intersection, this allows us to show that the cohomology of Trop(X) vanishes in degree less than dim(X). In addition, we give a description for the top power of the monodromy operator acting on middle cohomology in terms of the volume pairing on ${\Gamma }_X$.

22 pages. Canadian Journal of Mathematics, to appear

References in corpus (5)

• Stable pair, tropical, and log canonical compact moduli of del Pezzo surfaces
• Uniformizing Tropical Curves I: Genus Zero and One
• Mirror Symmetry via Logarithmic Degeneration Data I
• Analytification is the limit of all tropicalizations
• Homology of tropical varieties

Cited by in corpus (12)

• Uniformizing Tropical Curves I: Genus Zero and One
• Faithful realizability of tropical curves
• Moduli of stable maps in genus one and logarithmic geometry II
• The motivic zeta functions of a matroid
• Boundary complexes and weight filtrations
• Integration over Tropical Plane Curves and Ultradiscretization
• Analytification is the limit of all tropicalizations
• Euler characteristic and signature of real semi-stable degenerations
• When do the recession cones of a polyhedral complex form a fan?
• Excluded homeomorphism types for dual complexes of surfaces
• Initial degenerations of flag varieties
• Cohomologically tropical varieties