Algebraic approximations of fibrations in abelian varieties over a curve

For every fibration f : X → B f : X \to B with X X a compact Kähler manifold, B B a smooth projective curve, and a general fiber of f f an abelian variety, we prove that f f has an algebraic approximation.

Fujita decomposition on families of abelian varieties

Let $$F:{\mathcal {V}}\rightarrow B$$ F : V → B be a smooth non-isotrivial $$1-$$ 1 - dimensional family of complex polarized abelian varieties and $$V_b=F^{-1}(b)$$ V b = F - 1 ( b ) be the general fiber. Let $${\mathcal {F}}^1\subset R^1F_*{\mathbb {C}}$$ F 1 ⊂ R 1 F ∗ C be the associated Hodge bundle filtration, $${\mathcal {F}}^1_b=H^{1.0}(V_b).$$ F b 1 = H 1.0 ( V b ) . Under the assumption that the Fujita decomposition for $${\mathcal {F}}^1$$ F 1 is non trivial, that is there is a non trivial flat sub-bundle $$0\ne {\mathbb {U}}\subset {\mathcal {F}}^1,$$ 0 ≠ U ⊂ F 1 , we show that $$V_b$$ V b has non-trivial endomorphism: $$End(V_b)\ne {\mathbb {Z}}.$$ E n d ( V b ) ≠ Z .

The Griffiths bundle is generated by groups

First the Griffiths line bundle of a $$\mathbf {Q}$$ Q -VHS $${\mathscr {V}}$$ V is generalized to a Griffiths character $${{\,\mathrm{grif}\,}}(\mathbf {G}, \mu ,r)$$ grif ( G , μ , r ) associated to any triple $$(\mathbf {G}, \mu , r)$$ ( G , μ , r ) , where $$\mathbf {G}$$ G is a connected reductive group over an arbitrary field F, $$\mu \in X_*(\mathbf {G})$$ μ ∈ X ∗ ( G ) is a cocharacter (over $$\overline{F}$$ F ¯ ) and $$r:\mathbf {G}\rightarrow GL(V)$$ r : G → G L ( V ) is an F-representation; the classical bundle studied by Griffiths is recovered by taking $$F=\mathbf {Q}$$ F = Q , $$\mathbf {G}$$ G the Mumford–Tate group of $${\mathscr {V}}$$ V , $$r:\mathbf {G}\rightarrow GL(V)$$ r : G → G L ( V ) the tautological representation afforded by a very general fiber and pulling back along the period map the line bundle associated to $${{\,\mathrm{grif}\,}}(\mathbf {G}, \mu , r)$$ grif ( G , μ , r ) . The more general setting also gives rise to the Griffiths bundle in the analogous situation in characteristic p given by a scheme mapping to a stack of $$\mathbf {G}$$ G -Zips. When $$\mathbf {G}$$ G is F-simple, we show that, up to positive multiples, the Griffiths character $${{\,\mathrm{grif}\,}}(\mathbf {G},\mu ,r)$$ grif ( G , μ , r ) (and thus also the Griffiths line bundle) is essentially independent of r with central kernel, and up to some identifications is given explicitly by $$-\mu $$ - μ . As an application, we show that the Griffiths line bundle of a projective $${{\mathbf {G}{\text{- }}{} \mathtt{Zip}}}^{\mu }$$ G - Zip μ -scheme is nef.

Lagrangian Constant Cycle Subvarieties in Lagrangian Fibrations

We show that the image of a dominant meromorphic map from an irreducible compact Calabi–Yau manifold X whose general fiber is of dimension strictly between 0 and $\dim X$ is rationally connected. Using this result, we construct for any hyper-Kähler manifold X admitting a Lagrangian fibration a Lagrangian constant cycle subvariety ΣH in X which depends on a divisor class H whose restriction to some smooth Lagrangian fiber is ample. If $\dim X = 4$, we also show that up to a scalar multiple, the class of a zero-cycle supported on ΣH in CH0(X) depend neither on H nor on the Lagrangian fibration (provided b2(X) ≥ 8).

Introduction

This chapter discusses some crucial notions to the interplay between cohomology and Chow groups, and also to the consequences, for the topology of a family of smooth projective varieties, of statements concerning Chow groups of the general or very general fiber. It surveys the main ideas and results presented throughout this volume. First, the chapter discusses the decomposition of the diagonal and spread. It then explains the generalized Bloch conjecture, the converse to the generalized decomposition of the diagonal. Next, the chapter turns to the decomposition of the small diagonal and its application to the topology of families. Finally, the chapter discusses integral coefficients and birational invariants before providing a brief overview of the following chapters.

Brick Manifolds and Toric Varieties of Brick Polytopes

Bott-Samelson varieties are a twisted product of $\mathbb{C}\mathbb{P}^1$'s with a map into $G/B$. These varieties are mostly studied in the case in which the map into $G/B$ is birational to the image; however in this paper we study a fiber of this map when it is not birational. We prove that in some cases the general fiber, which we christen a brick manifold, is a toric variety. In order to do so we use the moment map of a Bott-Samelson variety to translate this problem into one in terms of the "subword complexes" of Knutson and Miller. Pilaud and Stump realized certain subword complexes as the dual of the boundary of a polytope which generalizes the brick polytope defined by Pilaud and Santos. For a nice family of words, the brick polytope is the generalized associahedron realized by Hohlweg, Lange and Thomas. These stories connect in a nice way: we show that the moment polytope of the brick manifold is the brick polytope. In particular, we give a nice description of the toric variety of the associahedron. We give each brick manifold a stratification dual to the subword complex. In addition, we relate brick manifolds to Brion's resolutions of Richardon varieties.

Geometry of some twistor spaces of algebraic dimension one

It is shown that there exists a twistor space on the n-fold connected sum of complex projective planes nCP2, whose algebraic dimension is one and whose general fiber of the algebraic reduction is birational to an elliptic ruled surface or a K3 surface. The former kind of twistor spaces are constructed over nCP2 for any n ≥ 5, while the latter kind of example is constructed over 5CP2. Both of these seem to be the first such example on nCP2. The algebraic reduction in these examples is induced by the anti-canonical system of the twistor spaces. It is also shown that the former kind of twistor spaces contain a pair of non-normal Hopf surfaces.

The geometry of fourth-order differential operator

In the present paper, the equivalence problem for fourth-order differential operators with one variable under general fiber-preserving transformation using the Cartan method of equivalence is applied. Two versions of equivalence problems are considered. First, the direct equivalence problem and second equivalence problem is to determine the sufficient and necessary conditions on two fourth-order differential operators such that there exists a fiber-preserving transformation mapping one to the other according to gauge equivalence.

The diminished base locus is not always closed

We exhibit a pseudoeffective $\mathbb{R}$-divisor ${D}_{\lambda }$ on the blow-up of ${\mathbb{P}}^{3}$ at nine very general points which lies in the closed movable cone and has negative intersections with a set of curves whose union is Zariski dense. It follows that the diminished base locus ${\boldsymbol{B}}_{-}({D}_{\lambda })={\bigcup }_{A\,\text{ample}}\boldsymbol{B}({D}_{\lambda }+A)$ is not closed and that ${D}_{\lambda }$ does not admit a Zariski decomposition in even a very weak sense. By a similar method, we construct an $\mathbb{R}$-divisor on the family of blow-ups of ${\mathbb{P}}^{2}$ at ten distinct points, which is nef on a very general fiber but fails to be nef over countably many prime divisors in the base.