Related papers: A remark on the B-model Comparison of Hodge struct…

We classify the homogeneous finite-dimensional permutation structures, i.e., homogeneous structures in a language of finitely many linear orders, giving a nearly complete answer to a question of Cameron, and confirming the classification…

We formulate and prove a non-abelian analog of Deligne's Fixed Part theorem on Hodge classes, revisiting previous work of Jost--Zuo, Katzarkov--Pantev and Landesman--Litt. To this aim we study algebraically isomonodromic extensions of local…

We prove that the cyclic homology of a saturated $A_\infty$ category admits the structure of a `polarized variation of Hodge structures', building heavily on the work of many authors: the main point of the paper is to present complete…

We develop the homotopy theory of semisimplicial sets constructively and without reference to point-set topology to obtain a constructive model for $\omega$-groupoids. Most of the development is folklore, but for a few results the author is…

In recent work, comonads and associated structures have been used to analyse a range of important notions in finite model theory, descriptive complexity and combinatorics. We extend this analysis to Hybrid logic, a widely-studied extension…

Based on some analogies with the Hodge theory of isolated hypersurface singularities, we define Hodge-type numerical invariants (called H-numbers) of any, not necessarily algebraic, link in $S^3$. They contain the same information as the…

We present a new structure theorem for finite fields of odd order that relates multiplicative and additive structure in an interesting way. This theorem has several applications, including an improved understanding of Dickson and Chebyshev…

A relative derived category for the category of modules over a presheaf of algebras is constructed to identify the relative Yoneda and Hochschild cohomologies with its homomorphism groups. The properties of a functor between this category…

Let H be a quasi-Hopf algebra, a weak Hopf algebra or a braided Hopf algebra. Let B be an H-bicomodule algebra such that there exists a morphism of H-bicomodule algebras v:H\rightarrow B. Then we can define an object B^{co(H)} which is a…

We study some semi-infinite invariants associated to Landau-Ginzburg models. These specialize classically to the usual twisted de Rham package and in the case of vanishing potential to the chiral de Rham complex of Malikov, Schechtman and…

We give a new proof of the finiteness of B-representations. As a consequence of the finiteness of B-representations and Koll\'ar's gluing theory on lc centers, we prove that the (relative) abundance conjecture for slc pairs is implied by…

We generalize the finiteness theorem for the locus of Hodge classes with fixed self-intersection number, due to Cattani, Deligne, and Kaplan, from Hodge classes to self-dual classes. The proof uses the definability of period mappings in 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 prove in this paper the Ax-Schanuel conjecture for all admissible variations of mixed Hodge structures.

Functorial semi-norms are semi-normed refinements of functors such as singular (co)homology. We investigate how different types of representability affect the (non-)triviality of finite functorial semi-norms on certain functors or classes.…

We study the homotopy theory of a certain type of diagram categories whose vertices are in variable categories with a functorial path, leading to a good calculation of the homotopy category in terms of cofibrant objects. The theory is…

Given a variation of Hodge structures on a quasi-projective base $S$, whose generic Mumford-Tate group is non-product, we prove that the (countable) union of positive components of the Hodge locus is either an algebraic subvariety of $S$,…

We study the infinitesimal variation of Hodge structure for families of algebraic curves and extend the classical theory from smooth curves to singular and non--planar settings. Using the deformation space $\mathrm{Ext}^1(\Omega_X,\mathcal…

We establish some cohomological bounds in D-module theory that are known in the holonomic case and folklore in general. The method rests on a generalization of the b-function lemma for non-holonomic D-modules.

Hodge theory associates to a smooth projective variety over $\mathbb{C}$ a piece of linear algebra information, called a $\mathbb{Q}$-Hodge structure. Conversely, it is a natural question which abstract $\mathbb{Q}$-Hodge structures arise…