This is the special issue dedicated to Professor Yum-Tong Siu on the occasion of his 60th birthday.

We introduce a family of positivity conditions for Hermitian symmetric functions, establish basic properties, and connect the ideas with complex geometry. Let M be a complex manifold, and let M' denote its complex conjugate manifold. In this paper M will typically be either Cⁿ or the total space of a holomorphic line bundle over a compact complex manifold.

Let Bⁿ = {z ∆ Cn : |z| < 1} be the unit ball in C_n. The problem of classifying proper holomorphic mappings between Bⁿ and B^N has attracted considerable attention (see [Fo 1992] [DA 1988] [DA 1993] [W 1979] [H 1999][HJ 2001] for extensive references) since the work of Poincare [P 1907][T 1962] and Alexander [A 1977]. Let us denote by Prop(Bⁿ,B^N) the collection of proper holomorphic mappings from Bⁿ to B^N. It is known [A 1977] that any map F ∆ Prop(Bⁿ,Bⁿ) must be biholomorphic and must be equivalent to the identity map. Here we say that f, g ∆Prop(Bⁿ,B^N) are equivalent if there are automorphisms σ ∆ Aut(Bⁿ) and τ ∆ Aut(B^N)) such that f = τ ∘ g ∘ σ. For general N > n, the discovery of inner functions indicates that Prop(Bⁿ,B^N) is too complicated to be classified. Hence we may focus on Rat(Bⁿ,B^N), the collection of all rational proper holomorphic mappings from Bⁿ to B^N).

It is a fundamental problem in algebraic geometry to understand the behavior of a multiple linear system |nD| on a projective complex manifold X for large n. For example, the well-known Riemann-Roch problem is to compute the function n ↦ h⁰(O_X(nD)) := dim_C H⁰(X,O_X(nD)). In the introduction to his collected works [33], Zariski cited the Riemann-Roch problem as one of the four "difficult unsolved questions concerning projective varieties (even algebraic surfaces)". The other natural problems about |nD| are to find the fixed part and base points (see [32]), the very ampleness, the properties of the associated rational map and its image variety, the finite generation of the ring of sections.

For a genus g curve X, Riemann-Roch theorem gives good and effective solutions to these problems.

It was shown by J.N. Mather and S.S.-T. Yau that an isolated complex hypersurface singularity is completely determined by its moduli algebra. In this article it is shown, for the simple elliptic singularities, how to construct continuous invariants from the moduli algebras and, hence, associate invariants to the singularities themselves.

Projective manifolds X with nef anticanonical bundles (i.e. - K _X ∙ C = det T _X ∙ C ≥ 0 for all curves C ⊂ X ) can be regarded as an interpolation between Fano manifolds (ample anticanonical bundle) and Calabi-Yau manifolds resp. tori and symplectic manifolds (trivial canonical bundle). A differential-geometric analogue are varieties with semi-positive Ricci curvature although this class is strictly smaller -- to get the correct picture one has to consider sequences of metrics and make the negative part smaller and smaller. However we will work completely in the context of algebraic geometry. Our aim is twofold: classification and, as a consequence, boundedness in case of dimension 3. We shall not consider threefolds with trivial canonical bundles, the eventual boundedness of Calabi-Yau threefolds still being unknown. Fano threefolds have been classified a long time ago and threefolds with big and nef anticanonical bundle are very much related with Q-Fano threefolds; therefore we will concentrate here on projective threefolds X with - K_X nef and K³_X = 0, but K_X ≢ 0. The essential problem is to distinguish the positive and flat directions in X.

Let M be a strongly pseudoconvex hypersurface in Cⁿ⁺¹, i.e. the boundary of a domain Ω in Cⁿ⁺¹. The Szegö kernel K^S(x, y), x,y ∈ M, is smooth outside of the diagonal x = y. The singularity at (x, x) is determined by the local datum at x of M, even though K^S itself is a global object. Our problem is to write down the singularity at (x, x) in terms of the local equation of M in Cⁿ⁺¹. We fix a reference point, say p_*, in M and only consider the germ of M at p_*. Hence we we may shrink M near p_* without mentioning it. We use as the model structure the boundary of the Siegel upper half space.

The Torelli theorem (cf.[Tor13], [Andr58], [Weil-57]) states that two algebraic curves are isomorphic if and only if their Jacobian varieties are isomorphic as polarized abelian varieties.

André Weil ([Weil-CPII]) set up a program for doing arithmetics on K3 surfaces, based on a Torelli type theorem, which was later proven through the effort of several authors (for this long story and related references we refer to Chapter VIII of the book [BPV]). This result is crucial for answering questions about the existence of K3 surfaces or families thereof possessing certain curve configurations.

The general Torelli question, as set up by Griffiths ([Grif68], [Grif70], cf. also [Grif-Schmid], [Grif84]), is to associate to each projective variety X of general type its Hodge structure of weight n = dim(X), and ask whether the corresponding "period map" ψ _n is injective on the local moduli space (or Kuranishi space of X).

It was known since long time that, as soon as the dimension is at least two, there are families of varieties without Hodge structures, which are not rigid. Surfaces of general type with q = pg = 0 were constructed in the 30's by Campedelli and Godeaux ([Cam32], [God35]), and for instance, in the case of the Godeaux surfaces, the Kuranishi family has dimension 8.

A natural question which arises is: under which hypothesis on X is a local Torelli theorem valid for the Hodge structure of weight n = dim X?

In other words, when is the local period map ψ _n a local embedding?