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Zongbin Chen

Publications and source records attributed to Zongbin Chen.

12 recordsLinked to original sources
arXiv2026

On the cohomological purity of the affine Springer fibers

We address questions posed by G\'erard Laumon and Jean-Loup Waldspurger concerning the cohomological purity of affine Springer fibers. More precisely, we show that an affine Springer fiber is cohomologically pure if and only if its $\xi$-stable quotient is cohomologically pure, and that this is further equivalent to the cohomological purity of a certain sequence of truncated affine Springer fibers. We deduce from this a sheaf-theoretic reformulation of cohomological purity for affine Springer fibers. We then compare this new criterion with a previously known one via a microlocal analysis of the relevant intersection complexes. As a corollary, we show that both the primitive part of the cohomology of an affine Springer fiber and the cohomology of its $\xi$-stable quotient depend only on the root valuation datum of the defining element.

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arXiv2024

On the dependence of the affine Springer fibers on their root valuation datum

For the group $\mathrm{GL}_{d}$, we confirm a conjecture of Goresky, Kottwitz and MacPherson, which states that the cohomology of the affine Springer fibers depend only on the root valuation datum of their defining elements. The proof relies on a microlocal analysis of the intermediate extensions appearing in Ngô's support theorem, and a Whitney regularity property of the union of the strict $δ$-strata over the equisingular strata for the miniversal deformation of plane curve singularities.

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arXiv2024

A decomposition theorem for the affine Springer fibers

According to Laumon, an affine Springer fiber is homeomorphic to the universal abelian covering of the compactified Jacobian of a spectral curve. We construct equivariant deformations $f_{n}:\overline{\mathcal{P}}_{n}\to \mathcal{B}_{n}$ of the finite abelian coverings of this compactified Jacobian, and decompose the complex $Rf_{n,*}\mathbf{Q}_{\ell}$ as direct sum of intersection complexes. Pass to the limit, we obtain a similar expression for the homology of the affine Springer fibers. A quite surprising consequence is that we can reduce the homology to its $Λ^{0}$-invariant subspace. As an application, we get a sheaf-theoretic reformulation of the purity hypothesis of Goresky, Kottwitz and MacPherson. In an attempt to solve it, we propose a conjecture about the punctural weight of the intermediate extension of a smooth $\ell$-adic sheaf of pure weight.

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arXiv2021

Truncated affine Springer fibers and Arthur's weighted orbital integrals

We explain an algorithm to calculate Arthur's weighted orbital integral in terms of the number of rational points on the fundamental domain of the associated affine Springer fiber. The strategy is to count the number of rational points of the truncated affine Springer fibers in two ways: by the Arthur-Kottwitz reduction and by the Harder-Narasimhan reduction. A comparison of results obtained from these two approaches gives us recurrence relations between the number of rational points on the fundamental domains of the affine Springer fibers and Arthur's weighted orbital integrals. As an example, we calculate Arthur's weighted orbital integrals for the group $\mathrm{GL}_{2}$ and $\mathrm{GL}_{3}$.

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arXiv2021

On the local behavior of weighted orbital integrals and the affine Springer fibers

The paper deals with two problems that seem to be of different nature: the local behavior of the weighted orbital integrals and the dependence of the affine Springer fibers on the root valuation datum of their defining elements. The intersection happens when we try to count the number of points on the fundamental domains of the affine Springer fibers: the number of points can be expressed in terms of certain weighted orbital integrals and vice versa. For the first problem, we prove the homogeneity of the Arthur-Shalika germ expansion of the weighted orbital integrals, and establish its parabolic descent. This is then applied to our conjecture about the dependence of the Poincar\'e polynomials of the fundamental domains on the root valuation datum. We show that the generating series of the number of points on the fundamental domains with respect to the root valuation datum of the defining elements is a rational fraction, and we bound its denominator with datum associated to the nilpotent orbits.

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arXiv2016

On the fundamental domain of affine Springer fibers

For $G$ a connected reductive group, $γ\in \kg(F)$ semisimple regular integral, we introduce a fundamental domain $F_γ$ for the affine Springer fibers $\xx_γ$. There is a beautiful way to reduce the purity conjecture of $\xx_γ$ to that of $F_γ$, we call it the Arthur-Kottwitz reduction. When restricted to the unramified case, it turns out that these fundamental domains behave well in family. We formulate a rationality conjecture about a generating series of their Poincaré polynomials. We then study them in detail for the group $\gl_{3}$. In particular, we pave them in affine spaces and we prove the rationality conjecture.

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arXiv2015

The $ξ$-stability on the affine grassmannian

We introduce a notion of $ξ$-stability on the affine grassmannian $\xx$ for the classical groups, this is the local version of the $ξ$-stability on the moduli space of Higgs bundles on a curve introduced by Chaudouard and Laumon. We prove that the quotient $\xx^ξ/T$ of the stable part $\xx^ξ$ by the maximal torus $T$ exists as an ind-$k$-scheme, and we introduce a reduction process analogous to the Harder-Narasimhan reduction for vector bundles. For the group $\mathrm{SL}_{d}$, we calculate the Poincaré series of the quotient $\xx^ξ/T$.

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arXiv2015

Geometric construction of generators of CoHA of doubled quiver

Let $Q$ be the doubled of a quiver. According to Efimov, Kontsevich and Soibelman, the cohomological Hall algebra (CoHA) associated to $Q$ is a free super-commutative algebra. In this short note, we confirm a conjecture of Hausel which gives a geometric realisation of the generators of the CoHA.

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arXiv2014

Truncated affine grassmannians and truncated affine Springer fibers for $\mathrm{GL}_{3}$

We state a conjecture on how to construct affine pavings for cohomologically pure projective algebraic varieties, which admit an action of torus such that the fixed points and $1$-dimensional orbits are finite. Experiments on the affine grassmannian for $\mathrm{GL}_{3}$ under the guideline of this conjecture, together with the work of Berenstein-Fomin-Zelevinsky and Kamnitzer, have led to the conjecture that the truncated affine grassmannians for $\mathrm{GL}_{r+1}$ admit affine pavings. For the group $\mathrm{GL}_{3}$, we construct affine pavings for the truncated affine grassmannians, and we use it to study the affine Springer fibers. In particular, we find a family of truncated affine Springer fibers which are cohomologically pure in the sense of Deligne.

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