代数几何

We prove the claim in the title by showing that a definable Grauert Direct Image Theorem in o-minimal geometry would imply a weak representability-like property of the definable Picard functor. However, this weak representability cannot…

In this paper, we study how global index formulas arise in the theory of one-dimensional holomorphic foliation from the microlocal point of view. We give short proofs and generalizations to a few exisiting index formulas concerning…

Let $Y$ be a compact Gorenstein analytic space with only isolated singularities and trivial dualizing sheaf. A recent paper of Imagi studies the deformation theory of $Y$ in case the singularities of $Y$ are weighted homogeneous and…

We construct motivic power operations on the mod-$p$ motivic cohomology of $\Fb_p$-schemes using a motivic refinement of Nizio{\l}'s theorem. The key input is a purity theorem for motivic cohomology established by Levine. Our operations…

We study homotopy properties of regular mappings from spheres into a real retract rational variety $Y$. We show that the homotopy classes which are represented by such mappings form subgroups of the homotopy groups of $Y$, and that the…

Racinet's double shuffle Lie algebra $\mathfrak{dmr}_0$ is a Lie subalgebra of the Lie algebra $\mathfrak{tder}$ of tangential derivations of the free Lie algebra with generators $x_0,x_1$, i.e. of derivations such that $x_1\mapsto 0$ and…

For a Fano threefold admitting a full exceptional collection of vector bundles of length four we show that all full exceptional collections consist of shifted vector bundles. We prove this via a detailed study of the group generated by…

Twistor spaces are certain compact complex threefolds with an additional real fibre bundle structure. We focus here on twistor spaces over $3\mathbb{C}\mathbb{P}^2$. Such spaces are either small resolutions of double solids or they can be…

We present an explicit expression of the cohomology complex of a constructible sheaf of abelian groups on the small \'etale site of an irreducible curve over an algebraically closed field, when the torsion of the sheaf is invertible in the…

Non-commutative crepant resolutions (NCCRs) are non-commutative analogues of the usual crepant resolutions that appear in algebraic geometry. In this paper we survey some results around NCCRs.

Let $A$ be an integral matrix and let $P$ be the convex hull of its columns. By a result of Gelfand, Kapranov and Zelevinski, the so-called principal $A$-determinant locus is equal to the union of the closures of the discriminant loci of…

This paper studies the class of spherical objects over any Kodaira $n$-cycle of projective lines and provides a parametrization of their isomorphism classes in terms of closed curves on the $n$-punctured torus without self-intersections.…

The canonical projections of the unit spheres are generalized to special generic maps and round fold maps, for example. They are generalizations from the viewpoint of singularity theory of differentiable maps and these maps restrict the…

The derived category of coherent systems is an interesting triangulated category associated with a smooth, projective curve $C$. These categories admit Bridgeland stability conditions, as recently shown by Feyzbakhsh and Novik. Their…

For a smooth algebraic curve defined over a number field, one can associate a bipartite graph known as a dessin d'enfant. In this paper, we investigate the regularity and automorphism groups of dessins d'enfants with uniform passports, that…

We show that the space of Belyi maps admits a natural parametrization by an infinite-dimensional sphere arising from Voiculescu's theory of noncommutative probability spaces. We show that this sphere decomposes into sectors, each of which…

A rational elliptic surface with section is a smooth, rational, complex, projective surface $\mathcal{X}$ that admits a relatively minimal fibration $f: \mathcal{X}\longrightarrow \bbP^1$ such that its general fibre is a smooth irreducible…

Given a smooth holomorphic symplectic variety $X$ with a Hamiltonian $G$-action, $G$-invariant Lagrangians $C's$ induce Lagrangians in the symplectic quotient $X// G$. Given clean intersections $B=C_1\cap C_2$ whose conormal sequence…

The excursion algebra associated to a scheme X over a finite field and a reductive group G is the algebra of global functions on the stack of arithmetic G-local systems on X. When X is a curve, this algebra acts on the space of automorphic…

Determinantal methods for bounding the rank and border rank of tensors or polynomials are subject to a major barrier. For instance, it is known that using determinantal methods one cannot prove a lower bound for the border rank of a 3-way…