arXiv subjects

Enlin Yang

Publications and source records attributed to Enlin Yang.

11 recordsLinked to original sources

arXiv2026

Microlocalization and singular supports of constructible \'etale sheaves

Let $X$ be a smooth scheme over a perfect field of characteristic $p>0$, and let $F$ be a constructible complex of finite Tor-dimension with finite coefficients of characteristic prime to $p$. We prove \[ {\rm SS}_\mu(F)= {\rm SS}(F), \] where ${\rm SS}_\mu(F)$ is Saito's microlocal singular support and ${\rm SS}(F)$ is Beilinson's singular support. This answers a question of Saito. As applications, we resolve another question of Saito regarding support estimates for microlocalization along a smooth closed subscheme, and we prove Saito's conjecture on characteristic classes (\emph{Invent. Math. 207: 597-695, 2017}), showing that the cohomological characteristic classes of Abbes and Saito are the cycle classes associated with the corresponding characteristic cycles.

math.AG↗

arXiv2022

The quadratic Artin conductor of a motivic spectrum

Given a motivic spectrum $K$ over a smooth proper scheme which is dualizable over an open subscheme, we define its quadratic Artin conductor under some assumptions, and prove a formula relating the quadratic Euler characteristic of $K$, the rank of $K$ and the quadratic Artin conductor. As a consequence, we obtain a quadratic refinement of the classical Grothendieck-Ogg-Shafarevich formula.

math.AG↗

arXiv2022

Cohomological Milnor formula and Saito's conjecture on characteristic classes

We confirm the quasi-projective case of Saito's conjecture, namely that the cohomological characteristic classes defined by Abbes and Saito can be computed in terms of the characteristic cycles. We construct a cohomological characteristic class supported on the non-acyclicity locus of a separated morphism relatively to a constructible sheaf. As applications of the functorial properties of this class, we prove cohomological analogs of the Milnor formula and the conductor formula for constructible sheaves on (not necessarily smooth) varieties.

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arXiv2022

The limit and boundary characteristic classes in Borel-Moore motivic homology

We show that the zero-dimensional part of the pro-Chern-Schwarz-MacPherson class defined by Aluffi can be lifted to the zeroth Suslin homology. The proof uses the pro-characteristic class in the limit Borel-Moore motivic homology, which has a quadratic refinement in the limit Borel-Moore Milnor-Witt homology. In characteristic zero, this construction factors through the group of constructible functions, in a way compatible with the covariant functoriality; in positive characteristic this property fails, and we show that the failure can be measured by the boundary characteristic class in the boundary Borel-Moore motivic homology. We prove a push-forward formula for the boundary characteristic class, and conjecture it to agree with the Swan class defined by Kato-Saito.

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arXiv2021

Some results on the motivic nearby cycle

We extend Ayoub's formalism of motivic nearby cycle functor to the $\infty$-categorical level, and prove some desired cohomological properties by relating the motivic nearby cycle functor to the notion of local acyclicity in motivic homotopy.

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arXiv2018

K\"unneth formulas for motives and additivity of traces

We prove several K\"unneth formulas in motivic homotopy categories and deduce a Verdier pairing in these categories following SGA5, which leads to the characteristic class of a constructible motive, an invariant closely related to the Euler-Poincar\'e characteristic. We prove an additivity property of the Verdier pairing using the language of derivators, following the approach of May and Groth-Ponto-Shulman; using such a result we show that in the presence of a Chow weight structure, the characteristic class for all constructible motives is uniquely characterized by proper covariance, additivity along distinguished triangles, refined Gysin morphisms and Euler classes. In the relative setting, we prove the relative K\"unneth formulas under some transversality conditions, and define the relative characteristic class.

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arXiv2018

On the relative twist formula of $\ell$-adic sheaves

We propose a conjecture on the relative twist formula of $\ell$-adic sheaves, which can be viewed as a generalization of Kato-Saito's conjecture. We verify this conjecture under some transversal assumptions. We also define a relative cohomological characteristic class and prove that its formation is compatible with proper push-forward. A conjectural relation is also given between the relative twist formula and the relative cohomological characteristic class.

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arXiv2017

Relative singular support and the semi-continuity of characteristic cycles for \'etale sheaves

Recently, the singular support and the characteristic cycle of an \'etale sheaf on a smooth variety over a perfect field are constructed by Beilinson and Saito, respectively. In this article, we extend the singular support to a relative situation. As an application, we prove the generic constancy for singular supports and characteristic cycles of \'etale sheaves on a smooth fibration. Meanwhile, we show the failure of the lower semi-continuity of characteristic cycles in a higher relative dimension case, which is different from Deligne and Laumon's result in the relative curve case.

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arXiv2017

Characteristic class and the epsilon factor of an \'etale sheaf

We prove a twist formula for the epsilon factor of a constructible sheaf on a projective smooth variety over a finite field in terms of characteristic class of the sheaf. This formula is a modified version of the formula conjectured by Kato and Saito in [Ann. Math., 168 (2008):33-96, Conjecture 4.3.11]. We give two applications of the twist formula. Firstly, we prove that the characteristic classes of constructible \'etale sheaves on projective smooth varieties over a finite field are compatible with proper push-forward. Secondly, we show that the two Swan classes in the literature are the same on proper smooth surfaces over a finite field.

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arXiv2015

Semi-continuity for total dimension divisors of \'etale sheaves

In this article, we extend a pull-back inequality for total dimension divisors of \'etale sheavs due to Saito. Using this formula, we generalize Deligne and Laumon's lower semi-continuous property for Swan conductors of \'etale sheaves on relative curves to higher relative dimensions in a geometric situation.

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arXiv2012

Derivations of Siegel Modular Forms from Connections

We introduce a method in differential geometry to study the derivative operators of Siegel modular forms. By determining the coefficients of the invariant Levi-Civita connection on a Siegel upper half plane, and further by calculating the expressions of the differential forms under this connection, we get a non-holomorphic derivative operator of the Siegel modular forms. In order to get a holomorphic derivative operator, we introduce a weaker notion, called modular connection, on the Siegel upper half plane than a connection in differential geometry. Then we show that on a Siegel upper half plane there exists at most one holomorphic modular connection in some sense, and get a possible holomorphic derivative operator of Siegel modular forms.

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