ON PROJECTIVE MANIFOLDS WITH PSEUDO-EFFECTIVE TANGENT BUNDLE

In this paper, we develop the theory of singular Hermitian metrics on vector bundles. As an application, we give a structure theorem of a projective manifold X with pseudo-effective tangent bundle; X admits a smooth fibration $X \to Y$ to a flat projective manifold Y such that its general fibre is rationally connected. Moreover, by applying this structure theorem, we classify all the minimal surfaces with pseudo-effective tangent bundle and study general nonminimal surfaces, which provide examples of (possibly singular) positively curved tangent bundles.

Rationally connected rational double covers of primitive Fano varieties

We show that for a Zariski general hypersurface $V$ of degree $M+1$ in ${\mathbb P}^{M+1}$ for $M\geqslant 5$ there are no Galois rational covers $X\dashrightarrow V$ of degree $d\geqslant 2$ with an abelian Galois group, where $X$ is a rationally connected variety. In particular, there are no rational maps $X\dashrightarrow V$ of degree 2 with $X$ rationally connected. This fact is true for many other families of primitive Fano varieties as well and motivates a conjecture on absolute rigidity of primitive Fano varieties. Comment: the final journal version

Maximal Chow constant and cohomologically constant fibrations

Motivated by the study of rationally connected fibrations, we study different notions of birationally simple fibrations. Our main result is the construction of maximal Chow constant and cohomologically constant fibrations. This paper is largely self-contained and we prove a number of basic properties of these fibrations. One application is to the classification of “rationalizations of singularities of cones.” We also consider consequences for the Chow groups of the generic fiber of a Chow constant fibration.

FANO HYPERSURFACES WITH ARBITRARILY LARGE DEGREES OF IRRATIONALITY

We show that complex Fano hypersurfaces can have arbitrarily large degrees of irrationality. More precisely, if we fix a Fano index $e$ , then the degree of irrationality of a very general complex Fano hypersurface of index $e$ and dimension n is bounded from below by a constant times $\sqrt{n}$ . To our knowledge, this gives the first examples of rationally connected varieties with degrees of irrationality greater than 3. The proof follows a degeneration to characteristic $p$ argument, which Kollár used to prove nonrationality of Fano hypersurfaces. Along the way, we show that in a family of varieties, the invariant ‘the minimal degree of a dominant rational map to a ruled variety’ can only drop on special fibers. As a consequence, we show that for certain low-dimensional families of varieties, the degree of irrationality also behaves well under specialization.

RC-positive metrics on rationally connected manifolds

In this paper, we prove that if a compact Kähler manifold X has a smooth Hermitian metric $\omega $ such that $(T_X,\omega )$ is uniformly RC-positive, then X is projective and rationally connected. Conversely, we show that, if a projective manifold X is rationally connected, then there exists a uniformly RC-positive complex Finsler metric on $T_X$ .

Instrumental Rationality and General Deterrence

The Supreme Court of Canada concluded in R. v. Nur that the use of general deterrence in sentencing is not “rationally connected”to its objective of lowering crime levels. Although this conclusion was drawn in the Charter section 1 context, its logic applies with equal force at the section 7 stage of analysis. As a law bearing no rational connection to its purpose is arbitrary, the author contends that judicial reliance on general deterrence in sentencing runs afoul of section 7 of the Canadian Charter of Rights and Freedoms. This conclusion is significant not only because it would forestall judicial use of general deterrence, but also for what it reveals about the relationship between the instrumental rationality principles. Commentators maintain that the Supreme Court’s “individualistic” approach to instrumental rationality resulted in the arbitrariness principle becoming subsumed by overbreadth. Yet, challenging the general deterrence provisions with overbreadth is not possible given the discretion given to judges to avoid its unnecessary application. The fact that a law can be arbitrary but not overbroad provides support for the Supreme Court’s insistence upon keeping the principles distinct. It also, however, requires that the Supreme Court adjust its position with respect to its method for proving arbitrariness.

Weak approximation for isotrivial families

We prove weak approximation for isotrivial families of rationally connected varieties defined over the function field of a smooth projective complex curve.