[Paper Review] On the isotriviality of families of projective manifolds over curves

Eckart Viehweg, Kang Zuo|ArXiv.org|Feb 24, 2000

Algebraic Geometry and Number Theory14 references95 citations

TL;DR

This paper establishes sharp lower bounds on the number of singular fibers in non-isotrivial families of projective manifolds over curves, proving that such families over $\mathbb{P}^1$ have at least three singular fibers and over elliptic curves have at least one, under mild positivity conditions on the canonical bundle of the fiber. The key technique combines Hodge theory, the negativity of the Kodaira-Spencer map, and ampleness criteria for direct images of pluricanonical bundles.

ABSTRACT

Let Y be a projective non-singular curve of genus g, X a projective manifold, both defined over the field of complex numbers, and let f:X ---> Y be a surjective morphism with general fibre F. If the Kodaira dimension of X is non-negative, and if Y is the projective line we show that f has at least 3 singular fibres. In general, for non-isotrivial morphisms f, one expects that the number of singular fibres is at least 3, if g=0, or at least 1, if g=1. Using the strong additivity of the Kodaira dimension, this is verified, if either F is of general type, or if F has a minimal model with a semi-ample canonical divisor. The corresponding result has been obtained by Migliorini and Kovacs, for families of surfaces of general type and for families of canonically polarized manifolds, and by Oguiso-Viehweg for families of elliptic surfaces. As a byproduct we obtain explicit bounds for the degree of the direct image of powers of the dualizing sheaf, generalizing those obtained by Bedulev-Viehweg for families of surfaces of general type.

Motivation & Objective

• To establish lower bounds on the number of singular fibers in families of projective manifolds over curves when the family is not birationally isotrivial.
• To generalize classical results of Arakelov, Parshin, and others on families of curves to higher-dimensional fibers with positive canonical bundle or minimal model structure.
• To provide effective bounds on the degree of the determinant of the direct image of pluricanonical bundles in terms of the genus of the base, number of singular fibers, and fiber invariants.
• To prove that manifolds of general type cannot dominate an elliptic curve or $\mathbb{P}^1$ via a smooth morphism, unless the base has sufficient singular fiber count.

Proposed method

• Uses the negativity of the curvature of the Hodge metric on subbundles of the Hodge bundle, derived from the Kodaira-Spencer map and its kernel.
• Applies the ampleness criterion for $f_*\omega_X^\nu$ via the condition that $\kappa(F) = \dim(F)$ or $K_{F'}$ semi-ample, which ensures $\det(f_*\omega_X^\nu)$ is ample for large $\nu$.
• Employs cyclic coverings and vanishing theorems to control the behavior of direct images, replacing global vanishing with local negativity of the Kodaira-Spencer map.
• Uses the Stein factorization and pullback via finite covers to reduce to the case of semistable families over $\mathbb{P}^1$ or elliptic curves.
• Applies the ampleness of $\det(f_*\omega_X^\nu)$ to derive contradiction when the number of singular fibers is too small.
• Constructs a uniform bound $e$ on the volume of canonical systems via moduli spaces of polarized manifolds with fixed Hilbert polynomial $h(t)$, ensuring uniformity in the bound.

Experimental results

Research questions

• RQ1 What is the minimal number of singular fibers in a non-isotrivial family of projective manifolds over $\mathbb{P}^1$ with $\kappa(F) = \dim(F)$?
• RQ2 Can a manifold of general type admit a smooth morphism to an elliptic curve?
• RQ3 Under what conditions is the determinant of $f_*\omega_X^\nu$ ample, and how does this relate to isotriviality?
• RQ4 Can effective bounds on the degree of $\det(f_*\omega_X^\nu)$ be established in terms of the base curve and fiber geometry?
• RQ5 How does the semi-ampleness of $\omega_F$ ensure uniformity in the constants of the bound for $\deg(f_*\omega_X^\nu)$?

Key findings

• If $f: X \to \mathbb{P}^1$ is a surjective morphism from a projective manifold of non-negative Kodaira dimension, then $f$ has at least three singular fibers.
• If $f: X \to E$ is a morphism to an elliptic curve with $X$ of general type, then $f$ must have at least one singular fiber.
• For a non-birationally isotrivial family with $\omega_F$ semi-ample and fixed Hilbert polynomial $h(t)$, the degree of $\det(f_*\omega_X^\nu)$ is bounded by $ (n(2g-2+s) + \delta) \cdot \nu \cdot e \cdot r $, where $n = \dim(F)$, $g$ is the genus of the base, $s = \deg(S)$, $\delta$ counts non-semistable fibers, $r = \text{rank}(f_*\omega_X^\nu)$, and $e$ depends only on $h(t)$.
• The bound becomes $n(2g-2+s)\nu e r$ when the family is semistable, showing a clean dependence on the base curve's geometry.
• The ampleness of $\det(f_*\omega_X^\nu)$ for large $\nu$ implies isotriviality, and this is guaranteed under the assumptions $\kappa(F) = \dim(F)$ or $K_{F'}$ semi-ample.
• The proof yields explicit bounds generalizing those of Arakelov, Parshin, and Bedulev, and extends them to higher-dimensional fibers with canonical or minimal models.