On semistability of Albanese maps

In this note, we show that the Albanese map for a smooth projective variety X with numerically effective anticanonical bundle is surjective, equi-dimensional and semistable.     

Pluricanonical maps of varieties of maximal Albanese dimension

Let X be a smooth complex projective algebraic variety of maximal Albanese dimension. We give a characterization of in terms of the set . An immediate consequence of this is that the Kodaira dimension is invariant under smooth deformations. We then study the pluricanonical maps . We prove that if X is of general type, is generically finite for and birational for . More generally, we show that for the image of is of dimension equal to and for , is the stable canonical map. Received July 7, 2000 / Published online April 12, 2001     

Seminégativité des courbes compactes des revêtements non compacts des surfaces projectives complexes

We show that the intersection form of a compact complex curveof any non compact cover ƒ:→ S of a smooth projective complex surface S is seminegative. Ifis not exceptional in , then C ≔ ƒ(C) is a fibre of the Albanese map α: S → B, whose image is a positive genus curve B (after possibly replacing S by a suitable finite cover), provided C is smooth and its normal bundle in S is torsion.These results agree with a question raised by I. Shafarevich in [1].     

Abelianization of the <Emphasis Type="Italic">F</Emphasis>-divided fundamental group scheme

Let (X , x ₀) be a pointed smooth proper variety defined over an algebraically closed field. The Albanese morphism for (X , x ₀) produces a homomorphism from the abelianization of the F-divided fundamental group scheme of X to the F-divided fundamental group of the Albanese variety of X. We prove that this homomorphism is surjective with finite kernel. The kernel is also described.     

Pluricanonical maps of varieties of Albanese fiber dimension two

In this paper we prove that for any smooth projective variety of Albanese fiber dimension two and of general type, the \(6\) -canonical map is birational. And we also show that the \(5\) -canonical map is birational for any such variety with some geometric restrictions.     

On Ueno’s conjecture K

We show that if X is a smooth complex projective variety with Kodaira dimension 0 then the Kodaira dimension of a general fiber of its Albanese map is at most . J. A. Chen was partially supported by NCTS, TIMS, and NSC of Taiwan. C. D. Hacon was partially supported by NSF research grant no: 0456363 and an AMS Centennial Scholarship. We would like to thank J. Kollár, R. Lazarsfeld, C.-H. Liu, M. Popa, P. Roberts, and A. Singh for many useful comments on the contents of this paper.     

On varieties of maximal Albanese dimension

We study the birational geometry of varieties of maximal Albanese dimension in this article. Given a non-birational, generically finite, and surjective morphism f: X → Y between varieties of maximal Albanese dimension, we show that the plurigenera P _m (X) and P _m (Y) for some m?≥ 2 could be equal only in very restrictive situations. We also prove that the 5-th pluricanonical map of a variety of maximal Albanese dimension always induces the Iitaka model.     

Unramified Brauer group of the moduli spaces of PGL r (ℂ)-bundles over curves

Let X be an irreducible smooth complex projective curve of genus g, with g ≥ 2. Let N be a connected component of the moduli space of semistable principal PGL _r (?)-bundles over X; it is a normal unirational complex projective variety. We prove that the Brauer group of a desingularization of N is trivial.     

On the bicanonical map of a surface section of a threefold

Let (ℳ, ℒ) be a 3-fold of log-general type polarized by a very ample line bundle ℒ. We study the pairs (ℳ, ℒ) in the case when there exists at least one smooth surface Ŝ ∈ |ℒ| such that the bicanonical map associated to |2K_Ŝ| is not birational. As one consequence of our classification we obtain the result:if a smooth projective threefold has non- negative Kodaira dimension, then given any smooth very ample divisor Ŝon the threefold, the bicanonical map associated to |2K_Ŝ|is birational.     

The separability of the Gauss map and the reflexivity for a projective surface

It is known that if a projective variety X in P ^N is reflexive with respect to the projective dual, then the Gauss map of X defined by embedded tangent spaces is separable, and moreover that the converse is not true in general. We prove that the converse holds under the assumption that X is of dimension two. Explaining the subtleness of the problem, we present an example of smooth projective surfaces in arbitrary positive characteristic, which gives a negative answer to a question raised by S. Kleiman and R. Piene on the inseparability of the Gauss map.      

Primitive cohomology and the tube mapping

Let X be a smooth complex projective variety of dimension d. We show that its primitive cohomology in degree d is generated by certain “tube classes,” constructed from the monodromy in the family of all hyperplane sections of X. The proof makes use of a result about the group cohomology of certain representations that may be of independent interest.     

Existence of a non‐reflexive embedding with birational Gauss map for a projective variety

We study the relationship between the generic smoothness of the Gauss map and the reflexivity (with respect to the projective dual) for a projective variety defined over an algebraically closed field. The problem we discuss here is whether it is possible for a projective variety X in ℙ^N to re‐embed into some projective space ℙ^M so as to be non‐reflexive with generically smooth Gauss map. Our result is that the answer is affirmative under the assumption that X has dimension at least 3 and the differential of the Gauss map of X in ℙ^N is identically zero; hence the projective varietyX re‐embedded in ℙ^M yields a negative answer to Kleiman–Piene's question: Does the generic smoothness of the Gauss map imply reflexivity for a projective variety? A Fermat hypersurface in ℙ^N with suitable degree in positive characteristic is known to satisfy the assumption above. We give some new, other examples of X in ℙ^N satisfying the assumption. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Base manifolds for fibrations of projective irreducible symplectic manifolds

Given a projective irreducible symplectic manifold M of dimension 2n, a projective manifold X and a surjective holomorphic map f:M→X with connected fibers of positive dimension, we prove that X is biholomorphic to the projective space of dimension n. The proof is obtained by exploiting two geometric structures at general points of X: the affine structure arising from the action variables of the Lagrangian fibration f and the structure defined by the variety of minimal rational tangents on the Fano manifold X.     

Quasi-projective reduction of toric varieties

We define a quasi–projective reduction of a complex algebraic variety X to be a regular map from X to a quasi–projective variety that is universal with respect to regular maps from X to quasi–projective varieties. A toric quasi–projective reduction is the analogous notion in the category of toric varieties. For a given toric variety X we first construct a toric quasi–projective reduction. Then we show that X has a quasi–projective reduction if and only if its toric quasi–projective reduction is surjective. We apply this result to characterize when the action of a subtorus on a quasi–projective toric variety admits a categorical quotient in the category of quasi–projective varieties. Received October 29, 1998; in final form December 28, 1998     

Compact Kähler quotients of algebraic varieties and Geometric Invariant Theory

Given an action of a complex reductive Lie group G on a normal variety X, we show that every analytically Zariski-open subset of X admitting an analytic Hilbert quotient with projective quotient space is given as the set of semistable points with respect to some G-linearised Weil divisor on X. Applying this result to Hamiltonian actions on algebraic varieties, we prove that semistability with respect to a momentum map is equivalent to GIT-semistability in the sense of Mumford and Hausen. It follows that the number of compact momentum map quotients of a given algebraic Hamiltonian G-variety is finite. As further corollary we derive a projectivity criterion for varieties with compact Kähler quotient.     

On the elementary obstruction to the existence of rational points

The differentials of a certain spectral sequence converging to the Brauer-Grothendieck group of an algebraic variety X over an arbitrary field are interpreted as the ∪-product with the class of the so-called “elementary obstruction.” This class is closely related to the cohomology class of the first-degree Albanese variety of X. If X is a homogeneous space of an algebraic group, then the elementary obstruction can be described explicitly in terms of natural cohomological invariants of X. This reduces the calculation of the Brauer-Grothendieck group to the computation of a certain pairing in the Galois cohomology.     

Surfaces with p g = q = 1, K 2 = 6 and Non-birational Bicanonical Maps

Let S be a smooth minimal projective surface of general type with p_g(S) = q(S) = 1,K_S~2= 6. We prove that the degree of the bicanonical map of S is 1 or 2. So if S has non-birational bicanonical map, then it is a double cover over either a rational surface or a K3 surface.     

Towards the classification of weak Fano threefolds with ρ = 2

In this paper, examples of type II Sarkisov links between smooth complex projective Fano threefolds with Picard number one are provided. To show examples of these links, we study smooth weak Fano threefolds X with Picard number two and with a divisorial extremal ray. We assume that the pluri-anticanonical morphism of X contracts only a finite number of curves. The numerical classification of these particular smooth weak Fano threefolds is completed and the geometric existence of some numerical cases is proven.     

Unitary representations of the fundamental group of orbifolds

Let X be a smooth complex projective variety of dimension n and \(\mathcal {L}\) an ample line bundle on it. There is a well known bijective correspondence between the isomorphism classes of polystable vector bundles E on X with \(c_{1}(E) = 0 = c_{2} (E) \cdot c_{1} (\mathcal {L})^{n-2}\) and the equivalence classes of unitary representations of π₁(X). We show that this bijective correspondence extends to smooth orbifolds.