Normal functions v provide a method for studying algebraic cycles Z(t) subset of X-t varying in a family of smooth projective varieties. Associated with v is an infinitesimal invariant 8v that reflects the first-order variation of pairs (X-t, Z(t)). Over the years, 8v has been widely used in the study of various geometric questions. We note that whereas v is a transcendental invariant, like periods of algebraic integrals, delta(v) has a natural filtration whose associated graded gives algebraic sections of coherent sheaves. In a number of interesting cases, these sections have had geometric interpretations. In this paper, we will discuss an identification between the singularities v and delta(v). The formal proof of this result will be given in a separate work.

Normal functions v provide a method for studying algebraic cycles Z(t) subset of X-t varying in a family of smooth projective varieties. Associated with v is an infinitesimal invariant 8v that reflects the first-order variation of pairs (X-t, Z(t)). Over the years, 8v has been widely used in the study of various geometric questions. We note that whereas v is a transcendental invariant, like periods of algebraic integrals, delta(v) has a natural filtration whose associated graded gives algebraic sections of coherent sheaves. In a number of interesting cases, these sections have had geometric interpretations. In this paper, we will discuss an identification between the singularities v and delta(v). The formal proof of this result will be given in a separate work.

We propose an analog of the Satake–Baily–Borel compactification and Borel’s extension theorem for arbitrary period maps. The proposed analog is constructed as a proper topological completion of the period map. It is conjectured that the construction is projective algebraic, and the conjecture is reduced to a certain extension problem.

Let B be a smooth projective varieity, and Z ⊂B a simple normal crossing divisor. Assume that B = B - Z admits a variation of pure, polarized Hodge structure. The divisor Z is naturally stratified, and Schmid's nilpotent orbit theorem defines a family/variation of nilpotent orbits along each strata. We study the rich geometric structure encoded by this family, its relationship to the induced (quotient) variation of pure Hodge structure on the strata, and establish a relationship between the extension data in the nilpotent orbits and the normal bundles of the smooth irreducible components of Z.

Using the general framework due to Donagi-Markman and Markushevich we shall derive an expression for the differential of Abel-Jacobi mappings on Fano threefolds. This formula involves information normal to the Lagrangian submanifolds constructed in and . It may be applied to give new proofs of a number of classical results about these varieties.

We introduce the notion of infinitesimal variations of mixed Hodge structures and invariants associated to them. We describe these invariants in the case of a pair $(X,Y)$ with $X$ a Fano 3-fold and $Y$ a smooth anticanonical K3 surface and in more detail in the case when $X$ is a cubic threefold. In this last setting, we obtain a generic global Torelli theorem for pairs.

We shall show that a smooth, quasi-projective variety X has a holomorphically convex universal covering (X) over tilde when (i) pi(1)(X) is residually nilpotent and (ii) there is an admissible variation of Hodge structure over X whose monodromy representation has a finite kernel, and where in each case a corresponding period mapping is assumed to be proper. We shall also briefly discuss the mixed case which will more fully be addressed elsewhere.

The global behavior of period mappings defined on generally non-complete algebraic varieties B as well as their local lbehavior around points in the boundary Z = (B) over bar \B of smooth completions of B have been extensively investigated. In this paper we shall study the global behavior of period mappings in neighborhoods of the entire boundary Z when dimB = 2. One method will be to decompose the dual graph of the boundary into basic building blocks of cycles and trees and analyze these separately. A main tool will be a global version of the classical nilpotent orbit theorem of Schmid.

The study of the variation of Hodge structure in a family of algebraic varieties is an important topic in algebraic ge-ometry. Of special interest is analysis of the Hodge structure in a family of varieties X-t, t is an element of delta = unit disc, where X-t is smooth for t &NOTEQUexpressionL; 0 while X-0 may be singular. It is known that if the monodromy is finite, then the Hodge structure fills in at t = 0. When the mon-odromy is infinite there is a well-developed understanding of how the Hodge structure degenerates. In this paper, we shall define and study properties of the first order variation of the Hodge structure at t = 0, both when the monodromy is finite, e.g., for a family of smooth surfaces acquiring a normal non-Gorenstein singulariy, and for the case when the monodromy is infinite. In the latter case, we shall give a Torelli type result that may be used to infer Torelli properties in the interior of a moduli space from these properties on the boundary.

Looijenga–Lunts and Verbitsky showed that the cohomology of a compact hyper-Kähler manifold X admits a natural action by the Lie algebra $$\mathfrak {so} (4, b_2(X)-2)$$ , generalizing the Hard Lefschetz decomposition for compact Kähler manifolds. In this paper, we determine the Looijenga–Lunts–Verbitsky (LLV) decomposition for all known examples of compact hyper-Kähler manifolds, and propose a general conjecture on the weights occurring in the LLV decomposition, which in particular determines strong bounds on the second Betti number $$b_2(X)$$ of hyper-Kähler manifolds (see Kim and Laza in Bull Soc Math Fr 148(3):467–480, 2020). Specifically, in the $$K3^{[n]}$$ and $$\mathrm {Kum}_n$$ cases, we give generating series for the formal characters of the associated LLV representations, which generalize the well-known Göttsche formulas for the Euler numbers, Betti numbers, and Hodge numbers for these series of hyper-Kähler manifolds. For the two exceptional cases of O’Grady (OG6 and OG10) we refine the known results on their cohomology. In particular, we note that the LLV decomposition leads to a simple proof for the Hodge numbers of hyper-Kähler manifolds of $$\mathrm {OG}10$$ type. In a different direction, for all known examples of hyper-Kähler manifolds, we establish the so-called Nagai’s conjecture on the monodromy of degenerations of hyper-Kähler manifolds. More consequentially, we note that Nagai’s conjecture is a first step towards a more general and more natural conjecture, that we state here. Finally, we prove that this new conjecture is satisfied by the known types of hyper-Kähler manifolds.

We give an informal, expository account of a project to construct completions of period maps.

It is well known that positivity properties of the curvature of a vector bundle have implications on the algebro-geometric properties of the bundle, such as numerical positivity, vanishing of higher cohomology leading to existence of global sections etc. It is also well known that bundles arising in Hodge theory tend to have positivity properties. From these considerations several issues arise: (i) In general for bundles that are semi-positive but not strictly positive; what further natural conditions lead to the existence of sections of its symmetric powers? (ii) In Hodge theory the Hodge metrics generally have singularities; what can be said about these and their curvatures, Chern forms etc.? (iii) What are some algebro-geometric applications of positivity of Hodge bundles? The purpose of these partly expository notes is fourfold. One is to summarize some of the general measures and types of positivity that have arisen in the literature. A second is to introduce and give some applications of norm positivity. This is a concept that implies the di_erent notions of metric semi-positivity that are present in many of the standard examples and one that has an algebro-geometric interpretation in these examples. A third purpose is to discuss and compare some of the types of metric singularities that arise in algebraic geometry and in Hodge theory. Finally we shall present some applications of the theory from both the classical and recent literature.

We give conditions under which natural lines bundles associated with completions of period mappings are semi-ample and ample.

We discuss progress towards a conjectural Hodge theoretic completion of a period map. The completion is defined, and we conjecture that it admits the structure of a compact complex analytic variety. The conjecture is proved when the image of the period map has dimension 1,2. Assuming the conjecture holds, we then prove that the augmented Hodge line bundle extends to an ample line bundle on the completion. In particular, the completion is a projective algebraic variety that compactifies the image, analogous to the Satake-Baily-Borel compactification.

The motivation behind this work is to construct a “Hodge theoretically maximal” completion of a period map. This is done up to finite data (we work with the Stein factorization of the period map). The image of the extension is a Moishezon variety that compactifies a finite cover of the image of the period map.

This work is part of a project to construct completions of period mappings. A proper topological SBB-esque completion is constructed. The fibres of are projective varieties, and the image is a union of quasi-projective varieties; one wants to endow the topological completion with a compatible algebraic structure. This raises questions about: (i) the global geometry of the fibres; and (ii) the existence of period matrix representations on neighborhoods of such fibres over which the restricted extension is still proper. The purpose of this paper is to investigate these questions.

Looijenga--Lunts and Verbitsky showed that the cohomology of a compact hyper-K\"ahler manifold $X$ admits a natural action by the Lie algebra $\mathfrak{so} (4, b_2(X)-2)$, generalizing the Hard Lefschetz decomposition for compact K\"ahler manifolds. In this paper, we determine the Looijenga--Lunts--Verbitsky (LLV) decomposition for all known examples of compact hyper-K\"ahler manifolds, and propose a general conjecture on the weights occurring in the LLV decomposition, which in particular determines strong bounds on the second Betti number $b_2(X)$ of hyper-K\"ahler manifolds. Specifically, in the $K3^{[n]}$ and $\mathrm{Kum}_n$ cases, we give generating series for the formal characters of the associated LLV representations, which generalize the well-known G\"ottsche formulas for the Euler numbers, Betti numbers, and Hodge numbers for these series of hyper-K\"ahler manifolds. For the two exceptional cases of O'Grady we refine the known results on their cohomology. In particular, we note that the LLV decomposition leads to a simple proof for the Hodge numbers of hyper-K\"ahler manifolds of O'Grady 10 type. In a different direction, for all known examples of hyper-K\"ahler manifolds, we establish the so-called Nagai's conjecture on the monodromy of degenerations of hyper-K\"ahler manifolds. More consequentially, we note that Nagai's conjecture is a first step towards a more general and more natural conjecture, that we state here. Finally, we prove that this new conjecture is satisfied by the known types of hyper-K\"ahler manifolds.

Talk given at the Algebraic Geometry in Mexico conference taking place at Puerta Escondido, December 2 - 7, 2018. Partly based on work in progress with Mark Green, Radu Laza and Colleen Robles.

Let P be the image of a period map. We discuss progress towards a conjectural Hodge theoretic completion P, an analogue of the Satake-Baily-Borel compactification in the classical case. The set P is defined and given the structure of a compact Hausdorff topological space. We conjecture that it admits the structure of a compact complex analytic variety. We verify this conjecture when dim P ≤ 2. In general, P admits a finite cover S (also a compact Hausdorff space, and constructed from Stein factorizations of period maps). Assuming that S is a compact complex analytic variety, we show that a lift of the augmented Hodge line bundle Λ extends to an ample line bundle, giving P the structure of a projective normal variety. Our arguments rely on refined positivity properties of Chern forms associated to various Hodge bundles; properties that might be of independent interest.

We discuss progress towards a conjectural Hodge theoretic completion of a period map. The completion is defined, and we conjecture that it admits the structure of a compact complex analytic variety. The conjecture is proved when the image of the period map has dimension 1,2. Assuming the conjecture holds, we then prove that the augmented Hodge line bundle extends to an ample line bundle on the completion. In particular, the completion is a projective algebraic variety that compactifies the image, analogous to the Satake-Baily-Borel compactification.