An explicit Q-algebra automorphism of symmetric functions, given by q-deformations of power sums, converts the S_n-equivariant Euler characteristics of stable map moduli spaces to those of ε-stable quasimap moduli spaces over Grassmannians.
The Logarithmic Quot space: foundations and tropicalisation
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abstract
We construct a logarithmic version of the Hilbert scheme, and more generally the Quot scheme, of a simple normal crossings pair. The logarithmic Quot space admits a natural tropicalisation called the space of tropical supports, which is a functor on the category of cone complexes. The fibers of the map to the space of tropical supports are algebraic. The space of tropical supports is representable by ``piecewise linear spaces'', which are introduced here to generalise fans and cone complexes to allow non--convex geometries. The space of tropical supports can be seen as a polyhedral analogue of the Hilbert scheme. The logarithmic Quot space parameterises quotient sheaves on logarithmic modifications that satisfy a natural transversality condition. We prove that our moduli space is a separated and universally closed logarithmic algebraic space. The logarithmic Hilbert space parameterizes families of proper monomorphisms, and in this way is exactly analogous to the classical Hilbert scheme. The new complexity of the space can then be viewed as stemming from the complexity of proper monomorphisms in logarithmic geometry. Our construction generalises the logarithmic Donaldson--Thomas space studied by Maulik--Ranganathan to arbitrary rank and dimension, and the good degenerations of Quot schemes of Li--Wu to simple normal crossings geometries.
fields
math.AG 5years
2026 5representative citing papers
A Weil-pairing refinement of the DDR cycle is introduced and proved to satisfy a multiple-cover formula, yielding refined log-GW invariants of toric surfaces that also satisfy the formula.
Logarithmic Hochschild homology is functorial for strong log Fourier-Mukai transforms on smooth proper log pairs, yielding a dg bicategory of logarithmic correspondences with compatible Chern characters and Euler pairings.
Logarithmic Hilbert schemes of points on smooth pointed curves are iterated weighted blow-ups of symmetric products, from which their integral Chow rings are computed using recent formulas for weighted blow-ups.
The decomposition theorem for logarithmic Hochschild homology extends from firm to general logarithmic orbifolds, enabling computations for symmetric products and proving invariance under root stack operations.
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A correlated refinement of the double double ramification cycle
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The decomposition theorem for logarithmic Hochschild homology extends from firm to general logarithmic orbifolds, enabling computations for symmetric products and proving invariance under root stack operations.