Related papers: Non-genericity of infinitesimal variations of Hodg…
We compute the Hodge numbers of the variation of Hodge structure of the middle cohomology (with compact support) of the Landau-Ginzburg model dual to a weighted projective space. We state a conjectural formula for the Hodge numbers of…
Categories of models of algebraic theories have good categorical properties except for gluing. Building upon insights and examples from Synthetic Differential Geometry, we introduce a generalisation of models of algebraic theories to…
Random shapes arise naturally in many contexts. The topological and geometric structure of such objects is interesting for its own sake, and also for applications. In physics, for example, such objects arise naturally in quantum gravity, in…
Motivated by a question of Hansen and Li, we show that a smooth and proper rigid analytic space $X$ with projective reduction satisfies Hodge symmetry in the following situations: (1) the base non-archimedean field $K$ is of residue…
This paper investigates the density of hypersurfaces in a projective normal simplicial toric variety over a finite field having a quasismooth intersection with a given quasismooth subscheme. The result generalizes the formula found by B.…
Spinorial methods have proven to be a powerful tool to study geometric properties of spin manifolds. Our aim is to continue the spinorial study of manifolds that are not necessarily spin. We introduce and study the notion of $G$-invariance…
This paper reproduces the text of a part of the Author's DPhil thesis. It gives a proof of the classification of non-trivial, finite homogeneous geometries of sufficiently high dimension which does not depend on the classification of the…
The Green--Griffiths--Lang and Kobayashi hyperbolicity conjectures for generic hypersurfaces of polynomial degree are proved using intersection theory for non-reductive geometric invariant theoretic quotients and recent work of Riedl and…
Let $G/\Gamma$ be the quotient of a semisimple Lie group by an arithmetic lattice. We show that for reductive subgroups $H$ of $G$ that is large enough, the orbits of $H$ on $G/\Gamma$ intersect nontrivially with a fixed compact set. As a…
We construct and study the moduli of hypersurfaces in toric orbifolds. Let $X$ be a projective toric orbifold and $\alpha \in Cl(X)$ an ample class. The moduli space is constructed as a quotient of the linear system $|\alpha|$ by $G =…
We prove the non-existence of special generic maps on $3$-dimensional complex projective space as our new result and a corollary by several methods. Special generic maps are generalizations of Morse functions with exactly two singular…
The main goal of this article is to relate asymptotic geometric properties on a tower of coverings of a non-compact K\"ahler manifold of finite volume with reasonable geometric assumptions to its universal covering. Applicable examples…
We find sharp upper bounds on the order of the automorphism group of a hypersurface in complex projective space in every dimension and degree. In each case, we prove that the hypersurface realizing the upper bound is unique up to…
We prove the existence of various families of irreducible homaloidal hypersurfaces in projective space $\mathbb P^ r$, for all $r\geq 3$. Some of these are families of homaloidal hypersurfaces whose degrees are arbitrarily large as compared…
We introduce a new approach to the geometric Bombieri--Lang conjecture for hyperbolic varieties in characteristic 0. The main idea is to construct an entire curve on a special fiber of a variety over a complex function field from an…
We define and construct mixed Hodge structures on real schematic homotopy types of complex quasi-projective varieties, giving mixed Hodge structures on their homotopy groups and pro-algebraic fundamental groups. We also show that these…
We study higher-degree generalizations of symplectic groupoids, referred to as {\em multisymplectic groupoids}. Recalling that Poisson structures may be viewed as infinitesimal counterparts of symplectic groupoids, we describe "higher''…
We give a geometric proof that Hasse principle holds for the following varieties defined over global function fields: smooth quadric hypersurfaces in odd characteristic, smooth cubic hypersurfaces of dimension at least $4$ in characteristic…
We define a categorical birational invariant for minimal geometrically rational surfaces with a conic bundle structure over a perfect field via components of a natural semiorthogonal decomposition. Together with the similar known result on…
We (1) characterize the Schubert varieties that arise as variations of Hodge structure (VHS); (2) show that the isotropy orbits of the infinitesimal Schubert VHS `span' the space of all infinitesimal VHS; and (3) show that the cohomology…