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Higher rational and higher Du Bois singularities have recently been introduced as natural generalizations of the standard definitions of rational and Du Bois singularities. In this note, we discuss these properties for isolated…
We prove that the higher direct images $R^qf_*\Omega^p_{\mathcal Y/S}$ of the sheaves of relative K\"ahler differentials are locally free and compatible with arbitrary base change for flat proper families whose fibers have $k$-Du Bois local…
The goal of this paper is to generalize results concerning the deformation theory of Calabi-Yau and Fano threefolds with isolated hypersurface singularites, due to the first author, Namikawa and Steenbrink. In particular, under the…
A bielliptic surface (or hyperelliptic surface) is a smooth surface with a numerically trivial canonical divisor such that the Albanese morphism is an elliptic fibration. In the first part of this paper, we study the structure of bielliptic…
We present a complete reimplementation of the LinearSystem package of Magma, with substantial improvements in design and performance. The resulting efficiency enables computations that were previously out of reach. We briefly describe the…
We present the construction of Inoue surfaces of type $S^{(+)}/S^{(-)}$ in terms of data arising from real quadratic number fields. We then describe the automorphism group of such surfaces in terms of this data.
A well known conjecture asserts that a cubic fourfold X is rational if it has a cohomologically associated K3 surface. G.Ouchi proved that if X admits a finite group G of symplectic automorphisms, whose order is different from 2, then X has…
The Galois group of a family of cubic surfaces is the monodromy group of the 27 lines of its generic fibre. We describe a method to compute this group for linear systems of cubic surfaces using certified numerical computations. Applying…
Recently Engel et al. (2025) have shown that the integral Hodge conjecture fails for very general abelian varieties. Using Deligne's theory of absolute Hodge cycles, we deduce a similar statement for the integral Tate conjecture.
Let $\Omega$ ' $\subset$ R^d , d = 1, 2, . . . be an open bounded smooth domain, and $\Omega = \Omega'\times (0,H)\subset \mathbb{R}^d \times \mathbb{R}_+.$ The coordinates in $\Omega$ are designated as x = (x ' , y) $\in$ $\Omega$ ' x (0,…
Using the general framework due to Donagi-Markman \cite{DM} and Markushevich \cite{M} we shall derive an expression for the differential of Abel-Jacobi mappings on Fano threefolds. This formula involves information normal to the Lagrangian…
Given a hypersurface singularity $(X,0) \subset (\mathbb{C}^{n+1},0)$ defined by a holomorphic function $f:(\mathbb{C}^{n+1},0) \to (\mathbb{C},0)$, we introduce an alternating version of Teissier's Jacobian Newton polygon, associated with…
We give an explicit formula of the associated graded map to the nonabelian Gauss-Manin connection with respect to the nonabelian Hodge filtration.
In this paper, we prove that a very general cubic threefold does not admit a universal codimension-two cycle and hence is stably irrational.
In this paper, we construct higher Chow cycles of type $(2, 1)$ on a certain family of surfaces, which are constructed by a product of certain hypergeometric curves of degree $N$. We prove that for a very general member, these cycles are…
This paper gives computations of all the $G$-theory groups of several classes of simplicial toric varieties, including all affine toric surfaces when the base field is algebraically closed and has characteristic zero, all weighted…
It is well known that there are totally 130 deformation families of quasi-smooth terminal weighted hypersurface Fano threefolds and all members belonging to 95 families of Fano indices one are birationally rigid. Among remaining $35$…
We propose a concept of truncation for arbitrary smooth projective toric varieties and construct explicit cellular resolutions for nef truncations of their total coordinate rings. We show that these resolutions agree with the short…
In this paper, we show that there is a finite SAGBI basis of the coordinate ring of a Kronecker quiver moduli space, indexed by primitive semi-standard tableaux pairs. This induces a toric degeneration of the Kronecker moduli space to a…
Let $X = G/H$ be an affine homogeneous spherical variety with abelian regular centralizer and no type N roots. In this paper, we formulate a relative geometric Langlands conjecture in the Dolbeault setting for $M = T^*X$. More concretely,…