The continuous rank function for varieties of maximal Albanese dimension and its applications
Lidia Stoppino
发表信息：
Extended abstracts GEOMVAP 2019-geometry, topology, algebra, and applications; women in geometry and topology, 121-126, Trends Math. Res. Perspect. CRM
In this note, I review an aspect of some new techniques introduced recently in collaboration with Miguel Ángel Barja and Rita Pardini: the construction of the continuous rank function. I give a sketch of how to use this function to prove the Barja-Clifford-Pardini-Severi inequalities for varieties of maximal Albanese dimension and to obtain the classification of varieties satisfying the equalities.
For every noetherian smooth and separated scheme over an algebraically closed field, we define its derived Albanese in Voevodsky's triangulated category of effective Nisnevich motives. To justify our definition, we relate the derived Albanese with the Albanese scheme. We also prove that the derived Albanese satisfies the Nisnevich descent property.
We prove that every smooth projective variety with maximal Albanese dimension has a good minimal model. We also treat Ueno's problem on subvarieties of Abelian varieties.
Let X be a projective variety, possibly singular. A generalized Albanese variety of X was constructed by Esnault, Srinivas and Viehweg over algebraically closed base field as a universal regular quotient of the relative Chow group of 0-cycles by Levine-Weibel. In this paper, we obtain a functorial description of the Albanese of Esnault-Srinivas-Viehweg over a perfect base field, using duality theory of 1-motives with unipotent part.
We show that Albanese varieties of abelian covers of projective plane are isogenous to product of isogeny components of abelian varieties associated with singularities of the ramification locus. In particular Albanese varieties of abelian covers of projective plane ramified over arrangements of lines and uniformized by unit ball are isogenous to a product of Jacobians of Fermat curves. Periodicity of the sequence of (semi-abelian) Albanese varieties of unramified cyclic covers of complements to a plane singular curve is shown.
In the present paper, we study the (twisted) 3-canonical map of varieties of Albanese fiber dimension one. Based on a theorem about the regularity of direct image of canonical sheaves, we prove that the 3-canonical map is generically birational when the genus of a general fiber of the Albanese map is 2.
Let $X$ be a projective, normal, minimal and Gorenstein $n$-dimensional complex variety of general type. Suppose $X$ is of maximal Albanese dimension. We prove that $K^n_X \ge 2 n! χ(K_X)$
We show that if X is a nonsingular projective variety of general type over an algebraically closed field k of positive characteristic and X has maximal Albanese dimension and the Albanese map is separable, then |4K_X| induces a birational map.
We prove that the tetracanonical map of a variety $X$ of maximal Albanese dimension induces the Iitaka fibration. Moreover, if $X$ is of general type, then the tricanonical map is birational.
We study the birational geometry of varieties of maximal Albanese dimension. In particular we discuss criteria for a generically finite morphism of varieties of maximal Albanese dimension to be birational; we give a new characterization of Theta divisors; we study the Albanese map and refine some of the results of Kollár; finally we use these results to birationally classify varieties with $P_3(X)=2$ and $q(X)=dim (X)$. Our method combines the generic vanishing theorems of Green and Lazarsfeld, the theory of Fourier Mukai transforms and the results of Kollàr on higher direct images of dualizin…
We classify almost homogeneous normal varieties of Albanese codimension $1$, defined over an arbitrary field. We prove that such a variety has a unique normal equivariant completion. Over a perfect field, the group scheme of automorphisms of this completion is smooth, except in one case in characteristic $2$, and we determine its (reduced) neutral component.
In this paper we construct abelian varieties of large Mordell-Weil rank over function fields. We achieve this by using a generalization of the notion of Prym variety to higher dimensions and a structure theorem for the Mordell-Weil group of abelian varieties over function fields proven in our previous works. We consider abelian and dihedral covers of the projective space and apply the above results to the twists of their Albanese varieties.
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.
We show that any derived equivalent smooth, projective varieties of dimension 3 over a finite field $\mathbb{F}_q$ have equal zeta functions. This result is an application of the extension to smooth, projective varieties over any field of Popa and Schnell's proof that derived equivalent smooth, projective varieties over $\mathbb{C}$ have isogenous Albanese torsors; this result is proven in an appendix by Achter, Casalaina-Martin, Honigs and Vial.
Let X be a projective variety over an algebraically closed base field, possibly singular. The aim of this paper is to show that the generalized Albanese variety of Esnault-Srinivas-Viehweg can be computed from one general curve C in X, if the base field is of characteristic 0. We illustrate this by an example, which we also use to unravel some mysterious properties of the Albanese of Esnault-Srinivas-Viehweg.
Let $X$ be a smooth projective variety of Albanese fiber dimension 1 and of general type. We prove that the translates through 0 of all components of $V^0(ω_X)$ generate $\Pic^0(X)$. We then study the pluricanonical maps of $X$. We show that $|4K_X|$ induces a birational map.
Let $f: X\to Y$ be a surjective morphism of smooth $n$-dimensional projective varieties, with $Y$ of maximal Albanese dimension. Hacon and Pardini studied the structure of $f$ assuming $P_m(X)=P_m(Y)$ for some $m\geq 2$. We extend their result by showing that, under the above assumtions, $f$ is birationally equivalent to a quotient by a finite abelian group. We also study the pluricanonical map of varieties of maximal Albanese dimesnion. The main result is that the linear series $|5K_X|$ incuces the Iitaka model of $X$.
We study algebraic fiber spaces $f:X \longrightarrow Y$ where $Y$ is of maximal Albanese dimension. In particular we give an effective version a theorem of Kawamata: If $P_m(X)=1$ for some $m \ge 2$, then the Albanese map of $X$ is surjective. Combining this with \cite{CH} it follows that $X$ is birational to an abelian variety if and only if $P_2(X)=1$ and $q(X)= \text{\rm dim} (X)$.
Let k be an algebraically closed field and X a smooth projective k-variety. A famous theorem of A. A. Roitman states that the canonical map from the degree zero part of the Chow group of zero cycles on X to the group of k-points of its Albanese variety induces an isomorphism on torsion prime to the characteristic of k. In the present paper we prove a generalisation to quasi-projective varieties admitting a smooth compactification. As was first observed by Ramachandran, for such a generalisation one should replace the Chow group of zero cycles by Suslin's 0-th algebraic singular homology group …
For a smooth projective complex variety whose Albanese morphism is finite, we show that every Bridgeland stability condition on its bounded derived category of coherent sheaves is geometric, in the sense that all skyscraper sheaves are stable with the same phase. Furthermore, we describe the stability manifolds of irregular surfaces and abelian threefolds with Picard rank one, and show that they are connected and contractible.
In this thesis we looked into three different problems which share, as a common factor, the exstensive use of the Fourier--Mukai transform as research tool. In the first Part we investigated the syzygies of Kummer varieties (i.e. quotients of abelian varieties by the $\mathbb{Z}/2\Z$-action induced by the group operation), extending to higher syzygies results on projective normality and degree of equations of Sasaki, Kempf, and Khaled. The second Part of this Thesis (partially written in collaboration with Z.Jiang and M. Lahoz) is dedicated to the study of pluricanonical linear systems on vari…
Given a smooth complex projective variety X, a line bundle L of X an element v of H^1(O_X) and a section s in H^0(L) that deforms to first order in the direction v, we give a sufficient condition on v in terms of Koszul cohomology for this first order deformation to extend to an analytic deformation. We apply this result to improve known results on the paracanonical system of a variety of maximal Albanese dimension, due to Beauville in the case of surfaces and to Lazarsfeld-Popa in higher dimension. In particular, we prove the inequality p_g(X)>=χ(K_X)+q(X)-1 for a variety X of maximal Albanes…
We study and obtain Slope inequalities for fibred irregular varieties of non-maximal Albanese dimension. We give a comparison theorem between Clifford-Severi and Slope inequalities for this type of fibrations. We also obtain a set of Slope inequalities considering the geometry of the Albanese map and the associated eventual maps.
We prove the so-called Severi inequality, stating that the invariants of a minimal smooth complex projective surface of maximal Albanese dimension satisfy: K^2_S >= 4χ(S).
The main purpose of this paper is to prove the Iitaka conjecture $C_{n,m}$ on the assumption that the sufficiently general fibers have maximal Albanese dimension.
The paper studies a relation between fundamental group of the complement to a plane singular curve and the orbifold pencils containing it. The main tool is the use of Albanese varieties of cyclic covers ramified along such curves. Our results give sufficient conditions for a plane singular curve to belong to an orbifold pencil, i.e. a pencil of plane curves with multiple fibers inducing a map onto an orbifold curve whose orbifold fundamental group is non trivial. We construct an example of a cyclic cover of the projective plane which is an abelian surface isomorphic to the Jacobian of a curve …
We show the derived invariance of various geometric invariants of smooth complex projective varieties governed by the Albanese map, including the relative canonical ring and the class of the relative canonical model in a suitable variant of the Grothendieck ring of varieties. Then we derive some applications to the derived invariance of Hodge numbers.
Let $(X,B)$ be a log canonical pair over a normal variety $Z$ with maximal Albanese dimension. If $K_X+B$ is relatively abundant over $Z$ (for example, $K_X+B$ is relatively big over $Z$), then we prove that $K_X+B$ is abundant. In particular, the subadditvity of Kodaira dimensions $κ(K_X+B) \geq κ(K_F+B_F)+ κ(Z)$ holds, where $F$ is a general fiber, $K_F+B_F= (K_X+B)|_F$, and $κ(Z)$ means the Kodaira dimension of a smooth model of $Z$. We discuss several variants of this result in Section 4. We also give a remark on the log Iitaka conjecture for log canonical pairs in Section 5.
We continue our study on smooth complex projective varieties $X$ of maximal Albanese dimension and of general type satisfying $χ(X, ω_X)=0$. We formulate a conjectural characterization of such varieties and prove this conjecture when the Albanese variety has only three simple factors.
In [17], we proved a structure theorem on the Mordell-Weil group of abelian varieties over function fields that arise as the twists of abelian varieties by the cyclic covers of projective varieties in terms of the Prym varieties associated with covers. In this paper, we provide an explicit way to construct the abelian varieties with large ranks over the higher dimension function fields. To do so, we apply the above-mentioned theorem to the twists of Albanese varieties of the cyclic multiple planes.
We discuss $p$-adic unipotent Albanese maps for curves of positive genus, extending the theory of $p$-adic multiple polylogarithms. This construction is then used to relate linear Diophantine conjectures of `Birch and Swinnerton-Dyer type' to non-linear theorems of Faltings-Siegel type.
Let $X$ be a smooth complex projective algebraic variety of maximal Albanese dimension. We give a characterization of $κ(X)$ in terms of the set $V^0(X,ω_{X})$ $:=\{P\in {\text{\rm Pic}}^0(X)|h^0(X, ω_X \otimes P) \ne 0\}$. An immediate consequence of this is that the Kodaira dimension $κ(X)$ is invariant under smooth deformations. We then study the pluricanonical maps $ϕ_m:X -> \Bbb{P} (H^0(X,mK_X))$. We prove that if $X$ is of general type, $ϕ_m$ is generically finite for $m\geq 5$ and birational for $m\geq 5 \text{\rm dim} (X) +1$. More generally, we show that for $m\geq 6$ the image of $ϕ_…
We investigate properties of the Albanese map and the fundamental group of a complex projective variety with many rational points over some function field, and prove that every linear quotient of the fundamental group of such a variety is virtually abelian, as well as that its Albanese map is surjective, has connected fibres, and has no multiple fibres in codimension one.
Let $a_X:X\rightarrow \mathrm{Alb}\, X$ be the Albanese map of a smooth complex projective variety. Roughly speaking in this note we prove that for all $i \geq 0$ and $α\in \mathrm{Pic}^0\, X$, the cohomology ranks $h^i(\mathrm{Alb}\, X, \,{a_X}_* ω_X\otimes P_α)$ are derived invariants. In the case of varieties of maximal Albanese dimension this proves conjectures of Popa and Lombardi-Popa -including the derived invariance of the Hodge numbers $h^{0,j}$ -- and a weaker version of them for arbitrary varieties. Finally we provide an application to derived invariance of certain irregular fibrati…
We study smooth complex projective varieties $X$ of maximal Albanese dimension and of general type satisfying with vanishing holomorphic Euler characteristic. We prove that the Albanese variety of $X$ has at least three simple factors. Examples were constructed by Ein and Lazarsfeld, and we prove that in dimension 3, these examples are (up to abelian étale covers) the only ones. By results of Ueno, another source of examples is provided by varieties $X$ of maximal Albanese dimension and of general type with $p_g(X)=1$. Examples were constructed by Chen and Hacon, and again, we prove that in di…
Let $f: X \to B$ be a relatively minimal fibration of maximal Albanese dimension from a variety $X$ of dimension $n \ge 2$ to a curve $B$ defined over an algebraically closed field of characteristic zero. We prove that $K_{X/B}^n \ge 2n! χ_f$, which was conjectured by Barja in [2]. Via the strategy outlined in [5], it also leads to a new proof of the Severi inequality for varieties of maximal Albanese dimension. Moreover, when the equality holds and $χ_f > 0$, we prove that the general fiber $F$ of $f$ has to satisfy the Severi equality that $K_F^{n-1} = 2(n-1)! χ(F, ω_F)$. We also prove some …
Let X be a proper smooth variety over the complex numbers. We consider the generalized Albanese variety Alb(X,Y) of X of modulus Y, which is a higher dimensional analogue of the generalized Jacobian variety with modulus of Rosenlicht-Serre. Note that the divisor Y can have multiplicity, so the algebraic group Alb(X,Y) can have an additive part. The purpose of this paper is to give Hodge theoretic presentations of Alb(X,Y).
We relate the existence of some surfaces of general type and maximal Albanese dimension to the existence of some monodromy representations of the braid group $\mathsf{B}_2(C_2)$ in the symmetric group $\mathsf{S}_n$. Furthermore, we compute the number of such representations up to $n=9$, and we analyze the cases $n \in \{2, \, 3, \, 4\}$. For $n=2, \, 3$ we recover some surfaces with $p_g=q=2$ recently studied (with different methods) by the author and his collaborators, whereas for $n=4$ we obtain some conjecturally new examples.
In this paper we study the cohomology of smooth projective complex surfaces $S$ of general type with invariants $p_g = q = 2$ and surjective Albanese morphism. We show that on a Hodge-theoretic level, the cohomology is described by the cohomology of the Albanese variety and a K3 surface $X$ that we call the K3 partner of $S$. Furthermore, we show that in suitable cases we can geometrically construct the K3 partner $X$ and an algebraic correspondence in $S \times X$ that relates the cohomology of $S$ and $X$. Finally, we prove the Tate and Mumford-Tate conjectures for those surfaces $S$ that li…
Let $(X,Δ)$ be a pair. We study how the condition $κ(K_X + Δ)=0$ causes surjectivity or birationality of the Albanese map and the Albanese morphism of $X$ in both characteristic $0$ and characteristic $p > 0$. In particular in characteristic $0$ we generalize Kawamata's result to the cases of log canonial pairs, and in characteristic $p>0$ we generalize a result of Hacon-Patakfalvi to the cases of log pairs. Moreover we show that if $X$ is a normal projective threefold in characteristic $p>0$, the coefficients of the components of $Δ$ are $\le 1$ and $-(K_X+Δ)$ is semiample, then the Albanese …
We describe algebraically defined cohomological and homological Albanese and Picard 1-motives (or mixed motives) of any algebraic variety in characteristic zero, generalizing the classical Albanese and Picard varieties. We compute Hodge, l-adic and De Rham realizations proving Deligne's conjecture for the concerned mixed Hodge structures. We investigate functoriality, universality, homotopical invariance and invariance under formation of projective bundles. We compare our cohomological and homological 1-motives for normal schemes. For proper schemes, we obtain an Abel-Jacobi map from the (Levi…
Let X be a smooth proper variety over a perfect field k of arbitrary characteristic. Let D be an effective divisor on X with multiplicity. We introduce an Albanese variety Alb(X, D) of X of modulus D as a higher dimensional analogon of the generalized Jacobian of Rosenlicht-Serre with modulus for smooth proper curves. Basing on duality of 1-motives with unipotent part (which are introduced here), we obtain explicit and functorial descriptions of these generalized Albanese varieties and their dual functors. We define a relative Chow group of zero cycles w.r.t. the modulus D and show that Alb(X,…
We prove that any smooth complex projective variety with generic vanishing index bigger or equal than 2 has birational bicanonical map. Therefore, if X is a smooth complex projective variety X with maximal Albanese dimension and non-birational bicanonical map, then the Albanese image of X is fibered by subvarieties of codimension at most 1 of an abelian subvariety of Alb X.
From the point of view of uniform bounds for the birationality of pluricanonical maps, irregular varieties of general type and maximal Albanese dimension behave similarly to curves. In fact Chen-Hacon showed that, at least when their holomorphic Euler characteristic is positive, the tricanonical map of such varieties is always birational. In this paper we study the bicanonical map. We consider the natural subclass of varieties of maximal Albanese dimension formed by primitive varieties of Albanese general type. We prove that the only such varieties with non-birational bicanonical map are the n…
In this note we studied the primitive varieties of general type with $q(X) = dim(X)$ and non-birational bicanonical maps. Let $X$ be such a variety. We bounded the degree of its bicanonical map. If moreover the Albanese variety $Alb(X)$ is simple, we proved that the Euler number $χ(ω_X) = 1$, and $|2K_X|$ separates the points mapped to the same general point via the Albanese map.
In this short article we show that if $(X, B)$ is a compact Kähler klt pair of maximal Albanese dimension, then it has a good minimal model, i.e. there is a bimeromorphic contraction $ϕ:X\dashrightarrow X'$ such that $K_{X'}+B'$ is semi-ample.
