In this note, we study the problem of localizing the signature of a fibered surface, i.e., a compact complex surface equipped with the structure of a fiber space over a compact Riemann surface. As applications, we give an estimate for the number of critical points and a formula for the Horikawa index for a certain class of fibered surfaces.

The Horikawa index and the local signature are introduced for relatively minimal, Eisenbud–Harris general fibred algebraic surfaces of genus 4.

The Fatou–Julia decomposition of complex codimension-one foliations is given by Ghys, Gomez-Mont and Saludes. The Julia sets are expected to play a role of minimal sets of real codimension-one foliations. For example, it is known that the Godbillon–Vey class is trivial if the Julia set is empty. In this paper, we propose another decomposition obtained in a slightly different way and announce some results.

The paper presents some recent results obtained by singularity researchers in the area of characteristic classes, in fact a promenade along the Japanese singularity road. Main subjects concern characteristic classes of singular varieties, equivariant Chern classes, bivariant Chern classes and motivic characteristic classes.

We calculate the fundamental groups $\pi = \pi_1 (\mathbb{P}^2 \smallsetminus B)$ for all irreducible plane sextics $B \subset \mathbb{P}^2$ with simple singularities for which $\pi$ is known to admit a dihedral quotient $\mathbb{D}_{10}$. All groups found are shown to be finite, two of them being of large order: 960 and 21600.

We analyze irreducible plane sextics whose fundamental group factors to $\mathbb{D}_{14}$. We produce explicit equations for all curves and show that, in the simplest case of the set of singularities $3\mathbf{A}_6$, the group is $\mathbb{D}_{14} \times \mathbb{Z}_3$.

A $\mathbb{D}_{10}$-sextic is an irreducible sextic $C \subset \mathbb{CP}^2$ with simple singularities such that the fundamental group $\pi_1 (\mathbb{CP}^2 \setminus C)$ factors to the dihedral group $\mathbb{D}_{10}$. A $\mathbb{D}_{10}$-sextic is not of torus type. In this paper, we show that if $C$ is a $\mathbb{D}_{10}$-sextic with the set of singularities $4\mathbf{A}_4$ or $4\mathbf{A}_4 \oplus \mathbf{A}_1$, then $\pi_1 (\mathbb{CP}^2 \setminus C)$ is isomorphic to $\mathbb{D}_{10} \times \mathbb{Z}/3\mathbb{Z}$. This positively answers a conjecture by Degtyarev.

A short survey on the Nash blow-up of singular varieties, applications and examples in particular for orbifolds, followed by some new results for threefolds.

We will survey basic results and recent progress in the existence, classification and moduli problems of complex and Kähler structures on compact homogeneous manifolds. We also state and discuss some related conjectures for further study in this field.

Let $B$ be a reduced sextic curve in $\mathbb{P}^2$. In the case when singularities of $B$ are only six cusps, Zariski proved that there exists a non-Galois triple cover branched at $B$ if and only if $B$ is given by an equation of the form $G_2^3 + G_3^2$, where $G_i$ denotes a homogeneous polynomial of degree $i$. In this article, we generalize Zariski's statement to any reduced sextic curve with at worst simple singularities. To this purpose, we give formulae for numerical invariants of non-Galois triple covers by using Tan's canonical resolution.

This paper shows some criteria for a scheme of finite type over an algebraically closed field to be non-singular in terms of jet schemes. For the base field of characteristic zero, the scheme is non-singular if and only if one of the truncation morphisms of its jet schemes is flat. For the positive characteristic case, we obtain a similar characterization under the reducedness condition on the scheme. We also obtain by a simple discussion that the scheme is non-singular if and only if one of its jet schemes is non-singular.

We study the geometry of a linear system of plane curves $C(\tau) (\tau \in \mathbb{C})$ spanned by two irreducible curves $C$, $C'$ of degree $d$ such that $\pi_1 (\mathbb{P}^2 - C \cup C')$ is abelian. We will show that the fundamental group $\pi_1 (\mathbb{C}^2 - C(\vec{\tau}))$ is isomorphic to $\mathbb{Z} \times F(r- 1)$ for a generic $\vec{\tau}$ where $\vec{\tau} = (\tau_1, \dots, \tau_r)$ and $C(\vec{\tau}) = C(\tau_1) \cup \dots \cup C(\tau_r)$.

On exactly one typewritten page, Ernst Witt sketched in the 60th of the last century a proof of the Prime Number Theorem, even in a generalized abstract situation. A closer inspection reveals a deep correlation to several other publications before and around this time. We elaborate his outline in more detail (without trying to find the "historical truth" of priority) and demonstrate that only some part of his approach can be combined with the quite recent very elegant method of Newman and Korevaar to yield a really simple analytic proof of a version of the abstract Prime Number Theorem that is close to Beurling's original one.

We present a method to produce examples of non-homeomorphic conjugate complex varieties based on the genus theory of lattices. As an application, we give examples of arithmetic Zariski pairs.

We study holomorphic germs $f : (\mathbb{C}^{n+1}, 0) \to (\mathbb{C}, 0)$ with the following properties:

(i) the critical set $H$ of the germ $f$ is a hyperplane $H = \{x = 0\}$;

(ii) the transversal singularity of the germ $f$ in points of the set $H \setminus \{0\}$ has type $A_{k-1}$.

We will investigate the topological structure of the Milnor fibre for $f$ and give explicit formula for the middle Betti number of the Milnor fibre and for the quasihomogeneous case we express it in terms of weights and degrees.

Zero-dimensional ideals in the formal power series and the associated vector space consisting of algebraic local cohomology classes are considered in the context of Grothendieck local duality. An algorithmic strategy for computing relative Čech cohomology representations of the algebraic local cohomology classes are described. A new algorithmic method for computing standard bases of a given zero-dimensional ideal is derived by using algebraic local cohomology and the Grothendieck local duality.

This is a survey on a universal bivariant theory $\mathbb{M}_{\mathcal{S}}^{\mathcal{C}} (X \to Y)$, which is a prototype of a bivariant analogue of Levine–Morel's algebraic cobordism, and its application to constructing a bivariant theory $F\Omega (X \to Y)$ of cobordism groups. Before giving such a survey, we recall the genus such as signature, which is the main important invariant defined on the cobordism group, i.e, a ring homomorphism from the cobordism group to a commutative ring with a unit. We capture the Euler–Poincaré characteristic and genera as a drastic generalization of the very natural counting of finite sets.