Hyperbolicity for Log Smooth Families with Maximal Variation

We prove that the base space of a log smooth family of log canonical pairs of log general type is of log general type as well as algebraically degenerate, when the family admits a relative good minimal model over a Zariski open subset of the base and the relative log canonical model is of maximal variation.

ALGEBRAIC FIBER SPACES AND CURVATURE OF HIGHER DIRECT IMAGES

Let $p:X\rightarrow Y$ be an algebraic fiber space, and let $L$ be a line bundle on $X$ . In this article, we obtain a curvature formula for the higher direct images of $\unicode[STIX]{x1D6FA}_{X/Y}^{i}\otimes L$ restricted to a suitable Zariski open subset of $X$ . Our results are particularly meaningful if $L$ is semi-negatively curved on $X$ and strictly negative or trivial on smooth fibers of $p$ . Several applications are obtained, including a new proof of a result by Viehweg–Zuo in the context of a canonically polarized family of maximal variation and its version for Calabi–Yau families. The main feature of our approach is that the general curvature formulas we obtain allow us to bypass the use of ramified covers – and the complications that are induced by them.

A Bochner principle and its applications to Fujiki class 𝒞 manifolds with vanishing first Chern class

We prove a Bochner-type vanishing theorem for compact complex manifolds [Formula: see text] in Fujiki class [Formula: see text], with vanishing first Chern class, that admit a cohomology class [Formula: see text] which is numerically effective (nef) and has positive self-intersection (meaning [Formula: see text], where [Formula: see text]). Using it, we prove that all holomorphic geometric structures of affine type on such a manifold [Formula: see text] are locally homogeneous on a non-empty Zariski open subset. Consequently, if the geometric structure is rigid in the sense of Gromov, then the fundamental group of [Formula: see text] must be infinite. In the particular case where the geometric structure is a holomorphic Riemannian metric, we show that the manifold [Formula: see text] admits a finite unramified cover by a complex torus with the property that the pulled back holomorphic Riemannian metric on the torus is translation invariant.

Continuity of homotopies

This chapter includes some additional material on homotopies. In particular, for a smooth variety V, there exists an “inflation” homotopy, taking a simple point to the generic type of a small neighborhood of that point. This homotopy has an image that is properly a subset of unit vector V, and cannot be understood directly in terms of definable subsets of V. The image of this homotopy retraction has the merit of being contained in unit vector U for any dense Zariski open subset U of V. The chapter also proves the continuity of functions and homotopies using continuity criteria and constructs inflation homotopies before proving GAGA type results for connectedness. Additional results regarding the Zariski topology are given.

On stabilizers of algebraic function fields of one variable

Let $\tilde K$ be a fixed algebraic closure of an infinite field K. We consider an absolutely integral curve Γ in $\mathbb{P}_{K}^{n}$ with n ≥ 2. The curve $\it\Gamma_{\tilde{K}}$ should have only finitely many inflection points, finitely many double tangents, and there exists no point in $\mathbb{P}_{\tilde{K}}^{n}$ through which infinitely many tangents to $\it\Gamma_{\tilde{K}}$ go. In addition there exists a prime number q such that $\it\Gamma_{\tilde{K}}$ has a cusp of multiplicity q and the multiplicities of all other points of $\it\Gamma_{\tilde{K}}$ are at most q. Under these assumptions, we construct a non-empty Zariski-open subset O of $\mathbb{P}_{\tilde{K}}^{n}$ such that if n ≥ 3, the projection from each point o ∈ O(K) birationally maps Γ onto an absolutely integral curve Γ′ in $\mathbb{P}_{K}^{n-1}$ with the same properties as Γ (keeping q unchanged). If n = 2, then the projection from each o ∈ O(K) maps Γ onto $\mathbb{P}_{K}^{1}$ and leads to a stabilizing element t of the function field F of Γ over K. The latter means that F/K(t) is a finite separable extension whose Galois closure ${\hat F}$ is regular over K.

Lowest weights in cohomology of variations of Hodge structure

LetXbe an irreducible complex analytic space withj:U ↪ Xan immersion of a smooth Zariski-open subset, and let 𝕍 be a variation of Hodge structure of weightnoverU. Assume thatXis compact Kähler. Then, provided that the local monodromy operators at infinity are quasi-unipotent,IHk(X, 𝕍) is known to carry a pure Hodge structure of weightk+n, whileHk(U, 𝕍) carries a mixed Hodge structure of weight at leastk+n. In this note it is shown that the image of the natural mapIHk(X, 𝕍) →Hk(U, 𝕍) is the lowest-weight part of this mixed Hodge structure. In the algebraic case this easily follows from the formalism of mixed sheaves, but the analytic case is rather complicated, in particular when the complementX — Uis not a hypersurface.

Lowest weights in cohomology of variations of Hodge structure

LetXbe an irreducible complex analytic space withj:U ↪ Xan immersion of a smooth Zariski-open subset, and let 𝕍 be a variation of Hodge structure of weightnoverU. Assume thatXis compact Kähler. Then, provided that the local monodromy operators at infinity are quasi-unipotent,IHk(X, 𝕍) is known to carry a pure Hodge structure of weightk+n, whileHk(U, 𝕍) carries a mixed Hodge structure of weight at leastk+n. In this note it is shown that the image of the natural mapIHk(X, 𝕍) →Hk(U, 𝕍) is the lowest-weight part of this mixed Hodge structure. In the algebraic case this easily follows from the formalism of mixed sheaves, but the analytic case is rather complicated, in particular when the complementX — Uis not a hypersurface.

Jacobians of singular spectral curves and completely integrable systems

Consider an isospectral manifold formed by matrices M ∈ glr(ℂ)[x] with a fixed leading term. The description of such a manifold is well known in the case of a diagonal leading term with different eigenvalues. On the other hand, there are many important systems where this term has multiple eigenvalues. One approach is to impose conditions in the sub-leading term. The result is that the isospectral set is a smooth manifold, bi-holomorphic to a Zariski open subset of the generalized Jacobian of a singular curve.

VARIATION OF THE GIESEKER AND UHLENBECK COMPACTIFICATIONS

In this article, we study the variation of the Gieseker and Uhlenbeck compactifications of the moduli spaces of Mumford-Takemoto stable vector bundles of rank 2 by changing polarizations. Some canonical rational morphisms among the Gieseker compactifications are shown to exist. In particular, we proved that when the second Chern class is sufficiently large, these morphisms are genuine rational maps. Moreover, as a consequence of studying the morphisms from the Gieseker compactifications to the Uhlenbeck compactifications, we show that there is an everywhere-defined canonical algebraic map between two adjacent Uhlenbeck compactifications which restricts to the identity on some Zariski open subset.