This is a note on MacPherson’s local Euler obstruction, which plays an important role recently in the Donaldson–Thomas theory by the work of Behrend.

We introduce MacPherson’s original definition and prove that it is equivalent to the algebraic definition used by Behrend, following the method of González-Sprinberg. We also give a formula of the local Euler obstruction in terms of Lagrangian intersections. As an application, we consider a scheme or DM stack $X$ admitting a symmetric obstruction theory. Furthermore, we assume that there is a ${\mathbb{C}}^{\ast }$ action on $X$ that makes the obstruction theory ${\mathbb{C}}^{\ast }$-equivariant. The ${\mathbb{C}}^{\ast }$-action on the obstruction theory naturally gives rise to a cosection map in the Kiem–Li sense. We prove that Behrend’s weighted Euler characteristic of $X$ is the same as the Kiem–Li localized invariant of $X$ by the ${\mathbb{C}}^{\ast }$-action.

We study closed nonpositively curved Riemannian manifolds $M$ that admit “fat $k$-flats”; that is, the universal cover $\tilde{M}$ contains a positive-radius neighborhood of a $k$-flat on which the sectional curvatures are identically zero. We investigate how the fat $k$-flats affect the cardinality of the collection of closed geodesics. Our first main result is to construct rank $1$ nonpositively curved manifolds with a fat $1$-flat that corresponds to a twisted cylindrical neighborhood of a geodesic on $M$. As a result, $M$ contains an embedded closed geodesic with a flat neighborhood, but $M$ nevertheless has only countably many closed geodesics. Such metrics can be constructed on finite covers of arbitrary odd-dimensional finite volume hyperbolic manifolds. Our second main result is a proof of a closing theorem for fat flats, which implies that a manifold $M$ with a fat $k$-flat contains an immersed, totally geodesic $k$-dimensional flat closed submanifold. This guarantees the existence of uncountably many closed geodesics when $k\ge 2$. Finally, we collect results on thermodynamic formalism for the class of manifolds considered in this paper.

We establish some results on the structure of the geometric unipotent radicals of pseudo-reductive $k$-groups. In particular, our main theorem gives bounds on the nilpotency class of geometric unipotent radicals of standard pseudo-reductive groups, which are sharp in many cases. A major part of the proof rests upon consideration of the following situation: let $k\text{'}$ be a purely inseparable field extension of $k$ of degree $p^e$, and let $G$ denote the Weil restriction of scalars ${\mathrm{R}}_{k\text{'}/k}(G\text{'})$ of a reductive $k\text{'}$-group $G\text{'}$. When $G={\mathrm{R}}_{k\text{'}/k}(G\text{'})$, we also provide some results on the orders of elements of the unipotent radical ${\mathcal{R}}_u(G_{\overline{k}})$ of the extension of scalars of $G$ to the algebraic closure $\overline{k}$ of $k$.

In view of Mori theory, rational homogenous manifolds satisfy a recursive condition: every elementary contraction is a rational homogeneous fibration, and the image of any elementary contraction also satisfies the same property. In this paper, we show that a smooth Fano $n$-fold with the same condition and Picard number greater than $n-6$ is either a rational homogeneous manifold or the product of $n-7$ copies of ${\mathbb{P}}^1$ and a Fano $7$-fold $X_0$ constructed by G. Ottaviani. We also clarify that $X_0$ has a non-nef tangent bundle and in particular is not rational homogeneous.

For $n\ge 4$, we show that generic closed Riemannian $n$-manifolds have no nontrivial totally geodesic submanifolds, answering a question of Spivak. Although the result is widely believed to be true, we are not aware of any proof in the literature.

In this paper, we compute the Artin part of a relative cohomological motive, introduced by Ayoub and Zucker, as a “weight zero part” in two challenging contexts. For this, we first introduce, in a very natural way, the part of punctual weight $⩽0$ of any complex of mixed Hodge modules and verify that the Hodge realization of the Artin part of smooth cohomological motives coincide with the part of punctual weight $⩽0$ of its realization. Second, we compute the Artin part of the motivic nearby sheaf, introduced by Ayoub, and relate it to the Betti cohomology of Berkovich spaces defined by tubes in non-Archimedean geometry. In particular, the former result provides a motivic interpretation of the Betti cohomology of the analytic Milnor fiber (seen as a Berkovich space).

Let $U$ be the complement of a smooth anticanonical divisor in a del Pezzo surface of degree at most 7 over a number field $k$. We show that there is an effective uniform bound for the size of the Brauer group of $U$ in terms of the degree of $k$.

In this paper, we study multiplicative dependence of values of polynomials or rational functions over a number field. As an application, we obtain new results on multiplicative dependence in the orbits of a univariate polynomial dynamical system. We also obtain a generalization of the Northcott theorem replacing the finiteness of preperiodic points from a given number field by the finiteness of algebraic integers having two multiplicatively dependent elements in their orbits.

In this paper, we study finite generation of symbolic Rees rings of the defining ideal $\mathfrak{p}$ of the space monomial curve $(t^a,t^b,t^c)$ for pairwise coprime integers $a$, $b$, $c$. Suppose that the base field is of characteristic $0$, and the ideal $\mathfrak{p}$ is minimally generated by three polynomials. In Theorem 1.1, under the assumption that the homogeneous element $\xi$ of the minimal degree in $\mathfrak{p}$ is a negative curve, we determine the minimal degree of an element $\eta$ such that the pair $\{\xi ,\eta \}$ satisfies Huneke’s criterion in the case where the symbolic Rees ring is Noetherian. By this result we can decide whether the symbolic Rees ring ${\mathcal{R}}_s(\mathfrak{p})$ is Notherian using computers. We give a necessary and sufficient condition for finite generation of the symbolic Rees ring of $\mathfrak{p}$ in Proposition 4.10 under some assumptions. We give an example of an infinitely generated symbolic Rees ring of $\mathfrak{p}$ in which the homogeneous element of the minimal degree in ${\mathfrak{p}}^{(2)}$ is a negative curve in Example 5.7. We give a simple proof to (generalized) Huneke’s criterion.