非阿贝尔拼接法简介

Multi-Harnack smoothings of real plane branches

We introduce a new method for the construction of smoothings of a real plane branch $(C, 0)$ by using Viro Patchworking method. Since real plane branches are Newton degenerated in general, we cannot apply Viro Patchworking method directly. Instead we apply the Patchworking method for certain Newton non degenerate curve singularities with several branches. These singularities appear as a result of iterating deformations of the strict transforms of the branch at certain infinitely near points of the toric embedded resolution of singularities of $(C,0)$. We characterize the $M$-smoothings obtaine…

A Non-Algebraic Patchwork

Itenberg and Shustin's pseudoholomorphic curve patchworking is in principle more flexible than Viro's original algebraic one. It was natural to wonder if the former method allows one to construct non-algebraic objects. In this paper we construct the first examples of patchworked real pseudoholomorphic curves in $Σ_n$ whose position with respect to the pencil of lines cannot be realised by any homologous real algebraic curve.

A real sextic surface with 45 handles

It follows from classical restrictions on the topology of real algebraic varieties that the first Betti number of the real part of a real nonsingular sextic in $\mathbb{CP}^3$ can not exceed $94$. We construct a real nonsingular sextic $X$ in $\mathbb{CP}^3$ satisfying $b_1(\mathbb{R}X)=90$, improving a result of F.Bihan. The construction uses Viro's patchworking and an equivariant version of a deformation due to E.Horikawa.

Real algebraic curves of bidegree (5,5) on the quadric ellipsoid

We complete the topological classification of real algebraic non-singular curves of bidegree $(5, 5)$ on the quadric ellipsoid. We show in particular that previously known restrictions form a complete system for this bidegree. Therefore, the main part of the paper concerns the construction of real algebraic curves. Our strategy is first to reduce to the construction of curves in the second Hirzebruch surface by degenerating the quadric ellipsoid to the quadratic cone. Next, we combine different classical construction methods on toric surfaces, such as Dessin d'enfants and and Viro's patchworki…

Complex conjugation and simplicial algebraic hypersurfaces

We call a real algebraic hypersurface in $(\mathbb{C}^*)^n$ simplicial if it is given by a real Laurent polynomial in $n$-variables that has exactly $n+1$ monomials with non-zero coefficients and such that the convex hull in $\mathbb{R}^n$ of the $n+1$ points of $\mathbb{Z} ^n$ corresponding to the exponents is a non-degenerate $n$-dimensional simplex. Such hypersurfaces are natural building blocks from which more complicated objects can be constructed, for example using O. Viro's Patchworking method. Drawing inspiration from related work by G. Kerr and I. Zharkov, we describe the action of th…

热带流形上的实相结构与高余维补丁构造

Real phase structures on tropical manifolds and patchworks in higher codimension

本文用热带几何将Viro组合补丁构造推广到任意余维，证明热带非奇异补丁为拓扑流形，并推广谱序列得到贝蒂数上界。

Real inflection points of real hyperelliptic curves

Given a real hyperelliptic algebraic curve $X$ with non-empty real part and a real effective divisor $\mc{D}$ arising via pullback from $\mathbb{P}^1$ under the hyperelliptic structure map, we study the real inflection points of the associated complete real linear series $|\mc{D}|$ on $X$. To do so we use Viro's patchworking of real plane curves, recast in the context of some Berkovich spaces studied by M. Jonsson. Our method gives a simpler and more explicit alternative to limit linear series on metrized complexes of curves, as developed by O. Amini and M. Baker, for curves embedded in toric …

Almost non-degenerate abelian fibrations

We provide evidence for the Principle : Non-uniruled abelian fibrations are almost non-degenerate.

Non-abelian groups acting on Severi-Brauer surfaces

We provide examples of finite non-abelian groups acting on non-trivial Severi-Brauer surfaces.

Degenerations of non-simple abelian surfaces

We study degenerations of non-simple principally polarized abelian surfaces to the boundary in the toroidal compactification of $\mathcal{A}_2$, and describe the degenerate abelian surfaces as well as the degenerate elliptic curves that live inside them.

Constructions of Non-Abelian Zeta Functions for Curves

In this paper, we introduce (local and) global non-abelian zeta functions for general curves. As an example, we compute the so-called rank two zeta functions for genus two curves by studying non-abelian Brill-Noether loci and their infinitesimal structures.

The Abelian/non-Abelian Correspondence and Gromov-Witten Invariants of Blow-ups

We prove the Abelian/non-Abelian Correspondence with bundles for target spaces that are partial flag bundles, combining and generalising results by Ciocan-Fontanine-Kim-Sabbah, Brown, and Oh. From this we deduce how genus-zero Gromov-Witten invariants change when a smooth projective variety X is blown up in a complete intersection defined by convex line bundles. In the case where the blow-up is Fano, our result gives closed-form expressions for certain genus-zero invariants of the blow-up in terms of invariants of X. We also give a reformulation of the Abelian/non-Abelian Correspondence in ter…

Non-Abelian L Function for Number Fields

This is an integrated part of our Geo-Arithmetic Program. In this paper we introduce and hence study non-abelian zeta functions and more generally non-abelian $L$-functions for number fields, based on geo-arithmetical cohomology, geo-arithmetical truncation and Langlands' theory of Eisenstein series.

Non-Abelian Class Field Theory for Riemann Surfaces

In this paper, using what we call a micro reciprocity law, we complete Weil's program for non-abelian class field theory of Riemann surfaces.

On Reid's recipe for non abelian groups

This paper presents two new explicit examples of Reid's recipe for non-abelian groups in $SL(3,\mathbb{C})$, namely the dihedral group $\mathbb{D}_{5,2}$ and a trihedral group of order 39.

非平凡阿贝尔簇的de Jong基本群是非阿贝尔的

The de Jong fundamental group of a non-trivial abelian variety is non-abelian

本文证明了非平凡阿贝尔簇的de Jong基本群是非阿贝尔的，并展示了该基本群依赖于基础域且具有大性质。

A Calabi-Yau threefold with non-Abelian fundamental group

The title is self-explanatory.

Big Actions with non abelian derived subgroup

For any $p>2$ we give an example of big action $(X,G)$ with non abelian derived subgroup. It is obtained as a covering of a curve related to the Ree curve.

Equivariant Bloch-Kato conjecture and non-abelian Iwasawa Main Conjecture

This is a contribution to the ICM 2002. We explain the relation between the (equivariant) Bloch-Kato conjecture for special values of L-functions and the Main Conjecture of (non-abelian) Iwasawa theory. On the way we will discuss briefly the case of Dirichlet characters in the abelian case. We will also discuss how "twisting" in the non-abelian case would allow to reduce the general conjecture to the case of number fields. This is one the main motivations for a non-abelian Main Conjecture.

On projective plane curves whose complements have finite non-abelian fundamental groups

We construct new examples of singular projective plane curves whose complements have finite and non-abelian fundamental groups, by generalizing the classical three cuspidal quartic curve discovered by Zariski.