Related papers: Simpson's construction of varieties with many loca…

We give the first examples of $\mathcal{O}$-acyclic smooth projective geometrically connected varieties over the function field of a complex curve, whose index is not equal to one. More precisely, we construct a family of Enriques surfaces…

This paper is a survey of recent work on the asymptotic behavior of the syzygies of a smooth complex projective variety as the positivity of the embedding line bundle grows. After a quick overview of results from the 1980s and 1990s…

Using the theory of cohomology support locus, we give a necessary condition for the Albanese map of a smooth projective surface being a submersion. More precisely, assuming the cohomology support locus of any finite abelian cover of a…

Two high-level "pictures" of probability theory have emerged: one that takes as central the notion of random variable, and one that focuses on distributions and probability channels (Markov kernels). While the channel-based picture has been…

In this paper, we explain a simple and uniform construction of a smooth integral model associated to a quadratic, (anti)-hermitian, and (anti)-quaternionic hermitian lattice defined over an arbitrary local field. As one major application,…

We present constructions of countable two-dimensional subshifts of finite type (SFTs) with interesting properties. Our main focus is on properties of the topological derivatives and subpattern posets of these objects. We present a countable…

We show that birational smooth complex projective varieties with numerically effective canonical bundles along the exceptional loci have the same Betti numbers. In particular, birational smooth minimal models share the same Betti numbers.…

We consider minimal compact complex surfaces S with Betti numbers b_1=1 and n=b_2>0. A theorem of Donaldson gives n exceptional line bundles. We prove that if in a deformation, these line bundles have sections, S is a degeneration of…

We study smooth projective varieties with small dual variety using methods from symplectic topology. We prove the affine parts of such varieties are subcritical, and that the hyperplane class is invertible in their quantum cohomology. We…

We describe smooth rational projective algebraic surfaces over an algebraically closed field of characteristic different from 2 which contain $n \ge \b_2-2$ disjoint smooth rational curves with self-intersection -2, where $\b_2$ is the…

We extend the notion of absolute subsets of Betti moduli spaces of smooth algebraic varieties to the case of normal varieties. As a consequence we prove that twisted cohomology jump loci in rank one over a normal variety are a finite union…

A birational map from a projective space onto a not too much singular projective variety with a single irreducible non-singular base locus scheme (special birational transformation) is a rare enough phenomenon to allow meaningful and…

We prove that each irreducible component of the cohomology jump loci of rank one local systems over a compact K\"ahler manifold contains at least one torsion point. This generalizes a theorem of Simpson for smooth complex projective…

We study simplicial complexes with a given number of vertices whose Stanley-Reisner ring has the minimal possible Betti numbers. We find that these simplicial complexes have very special combinatorial and topological structures. For…

We prove vanishing of the higher direct images of the structure (and the canonical) sheaf for a proper birational morphism with source a smooth variety and target the quotient of a smooth variety by a finite group of order prime to the…

We partially resolve conjectures of Deligne and Simpson concerning $\mathbb{Z}$-local systems on quasi-projective varieties that underlie a polarized variation of Hodge structure. For local systems with $\mathbb{Q}$-anisotropic monodromy,…

We explore the relation between the positive dimensional irreducible components of the characteristic varieties of rank one local systems on a smooth surface and the associated (rational or irrational) pencils. Our study, which may viewed…

We present scheme theoretic methods that apply to the study of secant varieties. This mainly concerns finite schemes and their smoothability. The theory generalises to the base fields of any characteristic, and even to non-algebraically…

We investigate the collapsibility of systolic finite simplicial complexes of arbitrary dimension. The main tool we use in the proof is discrete Morse theory. We shall consider a convex subcomplex of the complex and project any simplex of…

We discuss properties of complex algebraic orbifold groups, their characteristic varieties, and their abelian covers. In particular, we deal with the question of (quasi)-projectivity of orbifold groups. We also prove a structure theorem for…