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Wenchuan Hu

Publications and source records attributed to Wenchuan Hu.

At least 19 recordsLinked to original sources
arXiv2026

Lawson homology groups of Chow varieties

Let $C_{p,d}(\mathbb{P}^n)$ denote the Chow variety of effective algebraic $p$-cycles of degree $d$ in complex projective space $\mathbb{P}^n$. In this paper, we compute the rational Lawson homology groups $L_qH_k(C_{p,d}(\mathbb{P}^n))_\mathbb{Q}$ for $0 \leq 2q\leq k \leq 2d$. Additionally, we prove that the rational Lawson homology groups of a natural completion of the Chow monoid of algebraic $p$-cycles in projective spaces are isomorphic to the corresponding rational singular homology groups. We also establish the stability of Lawson homology groups of Chow varieties under natural embeddings and algebraic suspension maps within a specified range.

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arXiv2026

Chow groups of Chow varieties

Let $C_{p,d}(\mathbb{P}^n)$ be the Chow variety of effective algebraic $p$-cycles of degree $d$ in complex projective $n$-space $\mathbb{P}^n$. In this paper, we compute the rational Chow groups $\mathrm{Ch}_q(C_{p,d}(\mathbb{P}^n))_\mathbb{Q}$ for $0 \le q \le d$. We show that these Chow groups are isomorphic to the corresponding rational singular homology groups $H_{2q}(C_{p,d}(\mathbb{P}^n), \mathbb{Q})$ in this range, a result that was previously known. Furthermore, we prove that the rational Chow groups of a natural completion of the Chow monoid of algebraic $p$-cycles on projective spaces coincide with the corresponding rational singular homology groups. We also establish the stability of Chow groups of Chow varieties under natural embeddings and algebraic suspension maps within a certain range. Finally, we determine the Chow groups, up to a certain level, for the space of algebraic cycles of fixed degree.

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arXiv2025

Tilings of the sphere by congruent pentagons V: Edge combination $a^{4}b$ with rational angles

We classify edge-to-edge tilings of the sphere by congruent pentagons with the edge combination $a^4b$ and with rational angles in degree: they are a one-parameter family of symmetric $a^4b$-pentagonal subdivisions of the tetrahedron with $12$ tiles; a sequence of unique symmetric $a^4b$-pentagons admitting a symmetric $3$-layer earth map tiling by $4m$ tiles for any $m\ge4$, among which each odd $m$ case admits two standard flip modifications; and a unique non-symmetric and degenerate $a^4b$-pentagon admitting a non-symmetric $3$-layer earth map tiling and its standard flip modification with $20$ tiles. The full classification from this series and all induced non-edge-to-edge quadrilateral tilings from degenerate pentagons are summarized with their 3D pictures.

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arXiv2025

The $L_{p}$ Gaussian Minkowski problem for $C$-pseudo-cones

The $L_{p}$ Gaussian Minkowski problem for $C$-pseudo-cones is studied in this paper, and the existence and uniqueness results are established. This extends our previous work on the Minkowski problem for $C$-pseudo-cones with respect to the Gaussian surface area measure ($p=1$) and the Gaussian cone measure ($p=0$).

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arXiv2025

The Gaussian Minkowski-type problems for $C$-pseudo-cones

The Gaussian surface area measure and the Gaussian cone measure for $C$-pseudo-cones are introduced and their corresponding Gaussian Minkowski problem and Gaussian log-Minkowski problem are proposed, respectively. The existence and uniqueness of solutions to these problems for $C$-pseudo-cones are established.

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arXiv2024

Extended Shuffle Product for Multiple Zeta Values

The shuffle algebra on positive integers encodes the usual multiple zeta values (MZVs) (with positive arguments) thanks to the representations of MZVs by iterated Chen integrals of Kontsevich. Together with the quasi-shuffle (stuffle) algebra, it provides the algebraic framework to study relations among MZVs. This paper enlarges the shuffle algebra uniquely to what we call the extended shuffle algebra that encodes convergent multiple zeta series with arbitrary integer arguments, not just the positive ones in the usual case. To achieved this goal, we first replace the Rota-Baxter operator of weight zero (the integral operator) that characterizes the shuffle product by the differential operator which extends the shuffle product to the larger space. We then show that the subspace corresponding to the convergent MZVs with integer arguments becomes a subalgebra under this extended shuffle product. Furthermore, by lifting the extended shuffle algebra to the locality algebra of Chen symbols, we prove that taking summations of fractions from Chen symbols defines an algebra homomorphism from the above subalgebra to the subalgebra of real numbers spanned by convergent multiple zeta series.

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arXiv2023

Hopf algebras for the shuffle algebra and fractions from multiple zeta values

The algebra of multiple zeta values (MZVs) is encoded as a stuffle (quasi-shuffle) algebra and a shuffle algebra. The MZV stuffle algebra has a natural Hopf algebra structure. This paper equips a Hopf algebra structure to the MZV shuffle algebra. The needed coproduct is defined by a recursion through a family of weight-increasing linear operators. To verify the Hopf algebra axioms, we make use of a family of fractions, called Chen fractions, that have been used to study MZVs and also serve as the function model for the MZV shuffle algebra. Applying natural derivations on functions and working in the context of locality, a locality Hopf algebra structure is established on the linear span of Chen fractions. This locality Hopf algebra is then shown to descend to a Hopf algebra on the MZV shuffle algebra, whose coproduct satisfies the same recursion as the first-defined coproduct. Thus the two coproducts coincide, establishing the needed Hopf algebra axioms on the MZV shuffle algebra.

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arXiv2021

Topological and Geometric filtration for products

We show that the Friedlander-Mazur conjecture holds for the product of an elliptic curve with some smooth projective variety of dimension 3. Moreover, we show that the Friedlander-Mazur conjecture is stable under a surjective map. As applications, we show that the Friedlander-Mazur conjecture holds uniruled threefolds and unirational varieties up to certain range.

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arXiv2021

The equivalence of Friedlander-Mazur and standard conjectures for threefolds

We show that the Friedlander-Mazur conjecture holds for a complex smooth projective variety X of dimension three implies the standard conjectures hold for X. This together with a result of Friedlander yields the equivalence of the two conjectures in dimension three. From this we provide some new examples whose standard conjectures hold.

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arXiv2020

Holomorphic vector fields and rationality

We show that a nonsingular complex projective variety admitting a holomorphic vector field with nonempty isolated zeroes, is rational using a key technique by Harvey-Lawson on finite volume flows. This statement was conjectured by J. Carrell. By the same technique, we obtain a uniform upper bound of Betti numbers of nonsingular complex projective variety admitting a holomorphic vector field with exact one zero point. Such an upper bound depends only on the dimension of the variety, which is a stronger version of a result of Akyildiz and Carrell.

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arXiv2019

Holomorphic vector fields and Chow groups

We show that the chow group of $p$-cycles with rational coefficients are isomorphic to the corresponding rational homology groups for smooth complex projective varieties carrying a holomorphic vector field with an isolated zero locus. As applications, we obtain Chow groups and Lawson homology groups with rational coefficients and verify the Friedlander-Mazur conjecture and the Generalized Hodge conjecture for those varieties.

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arXiv2019

The multiplicative group action on singular varieties and Chow varieties

We answer two questions of Carrell on a singular complex projective variety admitting the multiplicative group action, one positively and the other negatively. The results are applied to Chow varieties and we obtain Chow groups of 0-cycles and Lawson homology groups of 1-cycles for Chow varieties. A short survey on the structure of the Chow varieties is included for comparison and completeness. Moreover, we give counterexamples to Shafarevich's question on the rationality of the irreducible components of Chow varieties.

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arXiv2011

Lawson Homology for Abelian Varieties

In this paper we introduce the Fourier-Mukai transform for Lawson homology of abelian varieties and prove an inversion theorem for the Lawson homology as well as the morphic cohomology of abelian varieties. As applications, we obtain the direct sum decomposition of the Lawson homology and the morphic cohomology groups with rational coefficients, inspired by Beauville's works on the Chow theory. An analogue of the Beauville conjecture for Chow groups is proposed and is shown to be equivalent to the (weak) Suslin conjecture for Lawson homology. A filtration on Lawson homology is proposed and conjecturally it coincides to the filtration given by the direct sum decomposition of Lawson homology for abelian varieties. Moreover, a refined Friedlander-Lawson duality theorem is obtained for abelian varieties. We summarize several related conjectures in Lawson homology theory in the appendix for convenience.

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arXiv2010

On Additive invariants of actions of additive and multiplicative groups

The additive invariants of an algebraic variety is calculated in terms of those of the fixed point set under the action of additive and multiplicative groups, by using Bialynicki-Birula's fixed point formula for a projective algebraicset with a G_m-action or G_a-action. The method is also generalized to calculate certain additive invariants for Chow varieties. As applications, we obtain the Hodge polynomial of Chow varieties in characteristic zero and the number of points for Chow varieties over finite fields. As applications, we obtain the l-adic Euler-Poincare characteristic for the Chow varieties of certain projective varieties over an algebraically closed field of arbitrary characteristic. Moreover, we show that the virtual Hodge (p,0) and (0,q)-numbers of the Chow varieties and affine group varieties are zero for all p,q positive.

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arXiv2009

Lawson Homology for projective varieties with C^*-action

The Lawson homology of a smooth projective variety with a $\C^*$-action is given in terms of that of the fixed point set of this action. We also consider such a decomposition for the Lawson homology of certain singular projective varieties with a $\C^*$-action. As applications, we calculate the Lawson homology and higher Chow groups for several examples.

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arXiv2009

A map from Lawson homology to Deligne Cohomology

A natural map from Lawson homology to Deligne cohomology groups for smooth complex projective varieties is constructed by using the Harvey-Lawson spark complexes. We also compare this to Abel-Jacobi type constructions by others.

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