Related papers: The global derived period map

We generalize Griffiths' theorem on the Hodge filtration of the primitive cohomology of a smooth projective hypersurface, using the local Bernstein-Sato polynomials, the V-filtration of Kashiwara and Malgrange along the hypersurface and the…

In order to develop the foundations of logarithmic derived geometry, we introduce a model category of logarithmic simplicial rings and a notion of derived log \'etale maps and use this to define derived log stacks.

We continue our investigation of Discrete Riemann Surfaces with the discussion of the discrete analogs of period matrices, Riemann's bilinear relations, exponential of constant argument, series and electrical moves. We show that given a…

Given a smooth proper family $\phi:X\rightarrow S$, we study the (quasi)-periods of the fibers of $\phi$ as (germs of) functions on $S$. We show that they field they generate has the same algebraic closure as that given by the flag variety…

We give two graph theoretical characterizations of tope graphs of (complexes of) oriented matroids. The first is in terms of excluded partial cube minors, the second is that all antipodal subgraphs are gated. A direct consequence is a third…

We introduce a natural generalization of twisted maps, called \emph{warped maps}. While twisted maps play an important role in the study of Deligne--Mumford stacks, warped maps are better suited for studying Artin stacks. Heuristically,…

We develop a general framework for Abel maps associated with a family $X/S$ of integral curves using derived algebraic geometry. For compactified Picard schemes, our approach yields relative quasi-smooth derived enhancements of the Quot…

We prove a mixed version of a conjecture of Griffiths: that the closure of the image of any admissible mixed period map is quasiprojective, with a natural ample bundle. Specifically, we consider the map from the image of the mixed period…

In this paper we prove a conjecture of Griffiths about vanishing of the zeroth cohomology groups of locally homogeneous vector bundles on compact quotients of non-classical period domains, and construct a new $G_\R$-invariant complex…

In this paper, we consider time-varying real analytic vector fields as curves on the space of real analytic vector fields. Using a suitable topology on the space of real analytic vector fields, we study and characterize different properties…

We study varieties associated to hypergraphs from the point of view of projective geometry and matroid theory. We describe their decompositions into matroid varieties, which may be reducible and can have arbitrary singularities by the…

In this series we introduce and investigate the concept of connectoids, which captures the connectivity structure of various discrete objects like undirected graphs, directed graphs, bidirected graphs, hypergraphs or finitary matroids. In…

This is the second part of a series of papers devoted to develop Homotopical Algebraic Geometry. We start by defining and studying generalizations of standard notions of linear and commutative algebra in an abstract monoidal model category,…

For a linear algebraic group $G$ over $\bf Q$, we consider the period domains $D$ classifying $G$-mixed Hodge structures, and construct the extended period domains $D_{\mathrm{BS}}$, $D_{\mathrm{SL}(2)}$, and $\Gamma \backslash D_{\Sigma}$.…

A generalized Tate curve is a universal family of curves with fixed genus and degeneration data which becomes Schottky uniformized Riemann surfaces and Mumford curves by specializing moduli and deformation parameters. By considering each…

We consider stacks of filtered phi-modules over rigid analytic spaces and adic spaces. We show that these modules parametrize p-adic Galois representations of the absolute Galois group of a p-adic field with varying coefficients over an…

We show how derived categories build bridges across the current mathematical mainstream, linking geometric and algebraic, commutative and noncommutative, local and global banks. Arches in these bridges are pieces of semiorthogonal…

In this paper, we expand the foundations of derived complex analytic geometry introduced in [DAG-IX] by J. Lurie. We start by studying the analytification functor and its properties. In particular, we prove that for a derived complex scheme…

Around 1970 Griffiths introduced the moduli of polarized Hodge structures/the period domain $D$ and described a dream to enlarge $D$ to a moduli space of degenerating polarized Hodge structures. Since in general $D$ is not a Hermitian…

We study a class of complex polynomial equations on a finite graph with a view to understanding how holistic phenomena emerge from combinatorial structure. Particular solutions arise from orthogonal projections of regular polytopes,…