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Differential geometry, especially the use of curvature, plays a central role in modern Hodge theory. The vector bundles that occur in the theory (Hodge bundles) have metrics given by the polarizations of the Hodge structures, and the sign and singularity properties of the resulting curvatures have far reaching implications in the geometry of families of algebraic varieties. A special property of the curvatures is that they are 1 st order invariants expressed in terms of the norms of algebro-geometric bundle mappings. This partly expository paper will explain some of the positivity and singularity properties of the curvature invariants that arise in the Hodge theory with special emphasis on the norm property. [ABSTRACT FROM AUTHOR]

We discuss the Hard Lefschetz Condition (HLC) on various cohomology groups and verify them for the Nakamura manifold of completely solvable type and the Kodaira-Thurston manifold. A general Demailly-Griffiths-Kähler identity is also given. [ABSTRACT FROM AUTHOR]

We propose a Hodge theory for the spaces featuring at the second step either in the Frölicher spectral sequence of an arbitrary compact complex manifold or in the spectral sequence associated with a pair of complementary regular holomorphic foliations on such a manifold. The main idea is to introduce a Laplace-type operator associated with a given Hermitian metric on whose kernel in every bidegree is isomorphic to in either of the two situations discussed. The surprising aspect is that this operator is not a differential operator since it involves a harmonic projection, although it depends on certain differential operators. We then use this Hodge isomorphism for to give sufficient conditions for the degeneration at of the spectral sequence considered in each of the two cases in terms of the existence of certain metrics on . For example, in the Frölicher case, we prove degeneration at if there exists an SKT metric (i.e. such that ) whose torsion is small compared to the spectral gap of the elliptic operator defined by . In the foliated case, we obtain degeneration at under a hypothesis involving the Laplacians and associated with the splitting induced by the foliated structure. [ABSTRACT FROM AUTHOR]

We provide a construction of associating a de Rham subbundle to a Higgs subbundle in characteristic p in the geometric case. As applications, we obtain a Higgs semistability result and a W2-unliftable result. [ABSTRACT FROM AUTHOR]

We establish a relation between the Hodge structure of the double cover of a nonsingular cubic surface branched along its Hessian and the Hodge structure of the triple cover of P3 branched along the cubic surface. Then we introduce a method to study the infinitesimal variations of Hodge structure of the double cover of the cubic surface. Using these results, we compute the Néron-Severi lattices for the double cover of a generic cubic surface and the Fermat cubic surface. [ABSTRACT FROM AUTHOR]

Let G be a reductive affine algebraic group and let X be an affine algebraic G-variety. We establish a (poly)stability criterion for points x ∈ X in terms of intrinsically defined closed subgroups Hx of G and relate it with the numerical criterion of Mumford and with Richardson and Bate-Martin-Röhrle criteria, in the case X = GN. Our criterion builds on a close analogue of a theorem of Mundet and Schmitt on polystability and allows the generalization to the algebraic group setting of results of Johnson-Millson and Sikora about complex representation varieties of finitely presented groups. By well established results, it also provides a restatement of the non-abelian Hodge theorem in terms of stability notions. [ABSTRACT FROM AUTHOR]

Let X be a smooth projective curve of genus g ≥ 2 over ℂ. Fix n ≥ 2, d ∈ ℤ. A pair (E, ϕ) over X consists of an algebraic vector bundle E of rank n and degree d over X and a section ϕ ∈ H0(E). There is a concept of stability for pairs which depends on a real parameter τ. Let ${\mathfrak M}_\tau(n, d)$ be the moduli space of τ-semistable pairs of rank n and degree d over X. Here we prove that the cohomology groups of ${\mathfrak M}_\tau(n, d)$ are Hodge structures isomorphic to direct summands of tensor products of the Hodge structure H1(X). This implies a similar result for the moduli spaces of stable vector bundles over X. [ABSTRACT FROM AUTHOR]

We investigate a method of construction of Calabi–Yau manifolds, that is, by smoothing normal crossing varieties. We develop theories for calculating the Picard groups of the Calabi–Yau manifolds obtained in this method. As an application, we show that many examples of Kawamata and Namikawa are different Calabi–Yau three-folds although they have same Hodge numbers. [ABSTRACT FROM AUTHOR]

We investigate Calabi–Yau three folds which are small resolutions of fiber products of elliptic surfaces with section admitting reduced fibers. We start by the classification of all fibers that can appear on such varieties. Then, we find formulas to compute the Hodge numbers of obtained three folds in terms of the types of singular fibers of the elliptic surfaces. Next, we study Kummer fibrations associated to these fiber products. [ABSTRACT FROM AUTHOR]

In this paper, we continue the investigation of coherent systems of type (n, d, k) on the projective line which are stable with respect to some value of a parameter α. We consider the case k = 1 and study the variation of the moduli spaces with α. We determine inductively the first and last moduli spaces and the flip loci, and give an explicit description for ranks 2 and 3. We also determine the Hodge polynomials explicitly for ranks 2 and 3 and in certain cases for arbitrary rank. [ABSTRACT FROM AUTHOR]

Let X be a smooth projective curve of genus g ≥ 2 over the complex numbers. A holomorphic pair on X is a couple (E, ϕ), where E is a holomorphic bundle over X of rank n and degree d, and ϕ ∈ H0(E) is a holomorphic section. In this paper, we determine the Hodge polynomials of the moduli spaces of rank 2 pairs, using the theory of mixed Hodge structures. We also deal with the case in which E has fixed determinant. [ABSTRACT FROM AUTHOR]