Related papers: Birational smooth minimal models have equal Hodge …

The moduli space of (1,3)-polarized abelian surfaces with full level-2 structure is birational to a double cover of the Barth-Nieto quintic. Barth and Nieto have shown that these varieties have Calabi-Yau models Z and Y, respectively. In…

Hodge classes on the moduli space of admissible covers with monodromy group G are associated to irreducible representations of G. We evaluate all linear Hodge integrals over moduli spaces of admissible covers with abelian monodromy in terms…

This paper contains some applications of Bridgeland-Douglas stability conditions on triangulated categories, and Joyce's work on counting invariants of semistable objects, to the study of birational geometry. We introduce the notion of…

We prove that the uniformizing map of any arithmetic quotient, as well as the period map associated to any pure polarized $\mathbb{Z}$-variation of Hodge structure $\mathbb{V}$ on a smooth complex quasi-projective variety $S$, are…

In the wild nonabelian Hodge correspondence on curves, filtered Stokes G-local systems are regarded as the objects on the Betti side. In this paper, we demonstrate a construction of the moduli space of them, called the Betti moduli space,…

In this note we initiate a program to obtain global descriptions of Calabi-Yau moduli spaces, to calculate their Picard group, and to identify within that group the Hodge line bundle, and the closely-related Bagger-Witten line bundle. We do…

The purpose of this paper is to establish several new results about the Hodge theory of Lagrangian fibrations on (not necessarily compact) holomorphic symplectic manifolds. Let $M$ be a holomorphic symplectic manifold of dimension $2n$ that…

We consider a simple and natural coboundary operator, on the Lie algebra valued differential forms on a manifold, which in the abelian case reduces to usual exterior derivative of such forms. Using the corresponding de Rham cohomology Lie…

The Hodge decomposition provides a very powerful mathematical method for the analysis of 2D and 3D vector fields. It states roughly that any vector field can be $L^2$-orthogonally decomposed into a curl-free, divergence-free, and a harmonic…

We associate a family of ideal sheaves to any Q-effective divisor on a complex manifold, called higher multiplier ideals, using the theory of mixed Hodge modules and V-filtrations. This family is indexed by two parameters, an integer…

Minimal BCOV theory is a classical field theory which describes a subclass of deformations of the category of perfect complexes on a Calabi-Yau variety. We compute minimal models for $L_\infty$-algebras describing minimal BCOV theory and…

By normalizing the space of commuting pairs of elements in a reductive Lie group G, and the corresponding space for the Langlands dual group, we construct pairs of hyperkahler orbifolds which satisfy the conditions to be mirror partners in…

We study one parameter degenerations of complex projective manifolds by introducing certain type of Hodge metrics coming from the pluricanonical forms. We show that degenerations with at most canonical singularities are all in the finite…

We prove the existence of a smoothing for a toroidal crossing space under mild assumptions. By linking log structures with infinitesimal deformations, the result receives a very compact form for normal crossing spaces. The main approach is…

We prove several results about p-divisible groups and Rapoport-Zink spaces. Our main goal is to prove that Rapoport-Zink spaces at infinite level are naturally perfectoid spaces, and to give a description of these spaces purely in terms of…

We consider a class of special Lagrangian subspaces of Calabi-Yau manifolds and identify their mirrors, using the recent derivation of mirror symmetry, as certain holomorphic varieties of the mirror geometry. This transforms the counting of…

Let $\mathbb{V}$ be an admissible and graded-polarized integral variation of mixed Hodge structures over a smooth and irreducible complex algebraic variety $S$. We show that if the typical Hodge locus…

We discuss the Hodge theory of algebraic non-commutative spaces and analyze how this theory interacts with the Calabi-Yau condition and with mirror symmetry. We develop an abstract theory of non-commutative Hodge structures, investigate…

Let $k$ be a field of positive characteristic. We prove that the only linear relations between the Hodge numbers $h^{i,j}(X) = \dim H^j(X,\Omega_X^i)$ that hold for every smooth proper variety $X$ over $k$ are the ones given by Serre…

In this paper we prove the following results: $1)$ We show that any arithmetic quotient of a homogeneous space admits a natural real semi-algebraic structure for which its Hecke correspondences are semi-algebraic. A particularly important…