Monthly Lecture Series - On the Geometry of the Cotangent Bundle and Hyperbolicity

Behrouz Taji, UNSW, Sydney: Boundedness results for families of non-canonically polarized varieties

Thanks to Faltings, Arakelov and Parshin’s solution to Mordell’s conjecture we know that smooth complex projective curves of genus at least equal to 2 have finite number of rational points. A key input in the proof of this fundamental result is the boundedness of families of smooth projective curves of a fixed genus (greater than 1) over a fixed base scheme. The latter was generalized by the combined spectacular results of Kovács-Lieblich and Bedulev-Viehweg to higher dimensional analogues of such curves; the so-called canonically polarized projective manifolds. In this talk I will discuss our recent extension of this boundedness result to the case of families of varieties with semiample canonical bundle (for example Calabi-Yaus). This is based on joint work with Kenneth Ascher (UC Irvine).

For function theory the hyperbolicity problem seeks conditions for a compact complex manifold to admit no nonconstant holomorphic map from C. The corresponding problem in number theory is for the set of rational points to be finite or contained in some proper subvariety.

Nonexistence of nonconstant entire curves comes from sufficiently independent holomorphic jet differentials vanishing on some ample divisor. For a complex submanifold of an abelian variety such jet differentials are constructed from the position-forgetting map of the jet space of the submanifold to conclude hyperbolicity unless the submanifold is invariant under some linear translation. For complex hypersurfaces of the complex projective space, hyperbolicity for a generic hypersurface of sufficiently high degree follows from the method of vertical jet differentials on the universal complex hypersurface.

In this talk we will discuss the reduction of the lower bound for the degree of a generic n-dimensional hypersurface to be hyperbolic from the best known bound of the order of (n log n)n to polynomial order in n, possibly even to quadratic in n, by combining techniques from the abelian variety setting and the complex projective space setting. The analogy between hyperbolicity in function theory and in number theory will be considered. For abelian varieties differentiation in jet differentials in function theory is replaced by the difference map in number theory, but similar implementation for the hypersurface setting is not yet clear.

In this talk, I will survey joint work with Eric Riedl on algebraic hyperbolicity of very general hypersurfaces and the description of Lang-type loci in hypersurfaces.

If $X$ is a submanifold of an Abelian variety, the Ueno fibration $p:X\to Z$ turns $X$ into a bundle with fibres $B$, an Abelian subvariety of $A$, and $Z\subset A/B$, of general type. Let $d_X$ be the Kobayashi pseudodistance on $X$. Then $d_X=p^*(d_Z)$, and $d_Z$ is a metric generically on $Z$ by Lang's conjecture (solved here by K. Yamanoi).

The goal is to give a similar description for arbitrary projective $X$, using its `Core map' $c:X\to Z$, which has `special' fibres $X_z$, and `orbifold base' $(Z,D_c)$ of general type, so that its naturally defined Kobayashi pseudodistance should be generically a metric on $Z . $D_c$ is a divisor on $Z$, nonzero in general, which encodes the multiple fibres of $c$.

$X_z$ `Special' means that $\Omega^p_{X_z}$ has no rank-one subsheaf of maximal Kodaira dimension $p$. They are conjecturally exactly the ones with $d_X\equiv 0$.

The talk will give the relevant definitions and illustrate the conjecture by examples.

The concept of a pair has become standard in algebraic geometry. One of its key motivations arises from litaka's program in the 1970s, which aimed to classify open varieties $U$.The approach involves compactifying $U$ into a projective variety $X$ by adding a boundary divisor $0$, and then studying the geometry of $U$ through the pair $(X,D)$.
Over time, the theory of pairs has evolved significantly and has become a cornerstone of birational geometry. After reviewing some historical developments in the theory of pairs (including the concept of hyperbolicity for pairs), we turn our attention to Calabi-Yau pairs.
In particular, we present a framework, developed in collaboration with Alessio Corti and Alex Massarenti, which allows one to explicitly describe the birational geometry of Calabi-Yau pairs.

The Kobayashi pseudometric $d_M$ on a complex manifold $M$ is the maximal pseudometric such that any holomorphic map from the Poincare disk to $M$ is distance-decreasing. Kobayashi conjectured that this pseudometric vanishes on Calabi-Yau manifolds, and in particular, Calabi-Yau manifolds have "entire curves". Using ergodicity of complex structures, together with S. Lu and M. Verbitsky we proved Kobayashi's conjecture for all K3 surfaces and for many classes of hyperkaehler manifolds. Together with Christian Lehn we gave a generalization of these results to primitive symplectic varieties.

A projective manifold is algebraically hyperbolic if the degree of any curve is bounded from above by its genus times a constant, which is independent from the curve. This is a property which follows from Kobayashi hyperbolicity. Together with Misha Verbitsky we proved that hyperkaehler manifolds are not algebraically hyperbolic when the Picard rank is at least 3, or if the Picard rank is 2 and the SYZ conjecture on existence of Lagrangian fibrations holds. We also prove that if the automorphism group of a hyperkahler manifold is infinite, then it is algebraically non-hyperbolic.