A remark on amoebas in higher codimensions

Amoeba finite basis does not exist in general

We show that the amoeba of a generic complex algebraic variety of codimension $1<r<n$ do not have a finite basis. In other words, it is not the intersection of finitely many hypersurface amoebas. Moreover we give a geometric characterization of the topological boundary of hypersurface amoebas refining an earlier result of F. Schroeter and T. de Wolff \cite{SW-13}.

Generalized logarithmic Gauss map and its relation to (co)amoebas

We define the generalized logarithmic Gauss map for algebraic varieties of the complex algebraic torus of any codimension. Moreover, we describe the set of critical points of the logarithmic mapping restricted to our variety, and we show an analogous of Mikhalkin's result on the critical points of the logarithmic map restricted to a hypersurfaces.

A Nullstellensatz for amoebas

The amoeba of an affine algebraic variety V in (C^*)^r is the image of V under the map (z_1, ..., z_r) -> (log|z_1|, ..., log|z_r|). We give a characterisation of the amoeba based on the triangle inequality, which we call testing for lopsidedness. We show that if a point is outside the amoeba of V, there is an element of the defining ideal which witnesses this fact by being lopsided. This condition is necessary and sufficient for amoebas of arbitrary codimension, as well as for compactifications of amoebas inside any toric variety. Our approach naturally leads to methods for approximating hype…

Remark on nefness in higher codimension

In this work, following the fundamental work of Boucksom we construct the nef cone of a compact complex manifold in higher codimension and give explicit examples where these cones are different. In the third section, we give two versions of Kawamata-Viehweg vanishing theorems in terms of nefness in higher codimension and numerical dimensions. We also show by examples the optimality of the divisoral Zariski decomposition defined by Boucksom.

A remark on the codimension of the Green-Griffiths locus of generic projective hypersurfaces of high degree

We show that for every smooth generic projective hypersurface $X\subset\mathbb P^{n+1}$, there exists a proper subvariety $Y\subsetneq X$ such that $\operatorname{codim}_X Y\ge 2$ and for every non constant holomorphic entire map $f\colon\mathbb C\to X$ one has $f(\mathbb C)\subset Y$, provided $°X\ge 2^{n^5}$. In particular, we obtain an effective confirmation of the Kobayashi conjecture for threefolds in $\mathbb P^4$.

关于叶状结构的第二陈类的注记

Remarks on the second Chern class of a foliation

本文研究了叶状结构的第二陈类及其奇点方案的度数，提出对于射影空间上的叶状结构，其奇点方案的度数下界为d+1，并探讨了等号成立的条件。