Threefolds with big and nef anticanonical bundles II

In a follow-up to our paper [Threefolds with big and nef anticanonical bundles I, Math. Ann., 2005, 333(3), 569–631], we classify smooth complex projective threefolds Xwith −K _X big and nef but not ample, Picard number γ(X) = 2, and whose anticanonical map is small. We assume also that the Mori contraction of X and of its flop X ⁺ are not both birational.     

Generalization of a criterion for semistable vector bundles

It is known that a vector bundle E on a smooth projective curve Y defined over an algebraically closed field is semistable if and only if there is a vector bundle F on Y such that both H⁰(X,EF) and H¹(X,EF) vanishes. We extend this criterion for semistability to vector bundles on curves defined over perfect fields. Let X be a geometrically irreducible smooth projective curve defined over a perfect field k, and let E be a vector bundle on X. We prove that E is semistable if and only if there is a vector bundle F on X such that Hⁱ(X,EF)=0 for all i. We also give an explicit bound for the rank of F.     

A criterion for ample vector bundles over a curve in positive characteristic

Let X be a smooth projective curve defined over an algebraically closed field of positive characteristic. We give a necessary and sufficient condition for a vector bundle over X to be ample. This generalizes a criterion given by Lange in [Math. Ann. 238 (1978) 193-202] for a rank two vector bundle over X to be ample.     

Vector bundles as direct images of line bundles

LetX be a smooth irreducible projective variety over an algebraically closed fieldK andE a vector bundle onX. We prove that, if dimX ≥ 1, there exist a smooth irreducible projective varietyZ overK, a surjective separable morphismf:Z →X which is finite outside an algebraic subset of codimension ≥ 3 inX and a line bundleL onX such that the direct image ofL byf is isomorphic toE. WhenX is a curve, we show thatZ, f, L can be so chosen thatf is finite and the canonical mapH ¹(Z, O) →H ¹(X, EndE) is surjective. Dedicated to the memory of Professor K G Ramanathan     

A criterion for the strongly semistable principal bundles over a curve in positive characteristic

Let X be an irreducible smooth projective curve over an algebraically closed field k of positive characteristic and G a simple linear algebraic group over k. Fix a proper parabolic subgroup P of G and a nontrivial anti-dominant character λ of P. Given a principal G-bundle E_G over X, let E_G(λ) be the line bundle over E_G/P associated to the principal P-bundle E_G→E_G/P for the character λ. We prove that E_G is strongly semistable if and only if the line bundle E_G(λ) is numerically effective. For any connected reductive algebraic group H over k, a similar criterion is proved for strongly semistable H-bundles.     

Fano 5-folds with nef tangent bundles and Picard numbers greater than one

We prove that smooth Fano 5-folds with nef tangent bundles and Picard numbers greater than one are rational homogeneous manifolds.     

Isospectral connections on line bundles

Dedicated to Professor Shingo Murakami for his 60th Birthday     

A stability condition for line bundles on reducible varieties

Over a family of varieties with singular special fiber, the relative Picard functor (i.e. the moduli space of line bundles) may fail to be compact. We propose a stability condition for line bundles on reducible varieties that is aimed at compactifying it. This stability condition generalizes the notion of ‘balanced multidegree’ used by Caporaso in compactifying the relative Picard functor over families of curves. Unlike the latter, it is defined ‘asymptotically’; an important theme of this paper is that although line bundles on higher-dimensional varieties are more complicated than those on curves, their behavior in terms of stability asymptotically approaches that of line bundles on curves.Using this definition of stability, we prove that over a one-parameter family of varieties having smooth total space, any line bundle on the generic fiber can be extended to a unique semistable line bundle on the (possibly reducible) special fiber, provided the special fiber is not too complicated in a combinatorial sense.     

A generalization of the existence theorem of special line bundles

LetX be a generic smooth irreducible complex projective curve of genusg withg4. In this paper, we generalize the existence theorem of special divisors to high dimensional indecomposable vector bundles. We give a necessary and sufficient condition on the existence ofn-dimensional indecomposable vector bundlesE onX with det(E)=d, dimH ⁰(X,E)h. We also determine under what condition the set of all such vector bundles will be finite and how many elements it contains.Project partly supported by the National Natural Science Foundation of China.     

Direct image of parabolic line bundles

Given a vector bundle E, on an irreducible projective variety X, we give a necessary and sufficient criterion for E to be a direct image of a line bundle under a surjective étale morphism. The criterion in question is the existence of a Cartan subalgebra bundle of the endomorphism bundle $\text{End}(E)$. As a corollary, a criterion is obtained for E to be the direct image of the structure sheaf under an étale morphism. The direct image of a parabolic line bundle under any ramified covering map has a natural parabolic structure. Given a parabolic vector bundle, we give a similar criterion for it to be the direct image of a parabolic line bundle under a ramified covering map.