Albanese Morphism Research Articles

Paracanonical base locus, Albanese morphism, and semi-orthogonal indecomposability of derived categories

Paracanonical base locus, Albanese morphism, and semi-orthogonal indecomposability of derived categories

Generic vanishing, 1-forms, and topology of Albanese maps

Given a bounded constructible complex of sheaves F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {F}$$\\end{document} on a complex Abelian variety, we prove an equality relating the cohomology jump loci of F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {F}$$\\end{document} and its singular support. As an application, we identify two subsets of the set of holomorphic 1-forms with zeros on a complex smooth projective irregular variety X; one from Green-Lazarsfeld’s cohomology jump loci and one from the Kashiwara’s estimates for singular supports. This result is related to Kotschick’s conjecture about the equivalence between the existence of nowhere vanishing global holomorphic 1-forms and the existence of a fibre bundle structure over the circle. Our results give a conjecturally equivalent formulation using singular support, which is equivalent to a criterion involving cohomology jump loci proposed by Schreieder. As another application, we reprove a recent result proved by Schreieder and Yang; namely if X has simple Albanese variety and admits a fibre bundle structure over the circle, then the Albanese morphism cohomologically behaves like a smooth morphism with respect to integer coefficients. In a related direction, we address the question whether the set of 1-forms that vanish somewhere is a finite union of linear subspaces of H0(X,ΩX1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^0(X,\\Omega _X^1)$$\\end{document}. We show that this is indeed the case for forms admitting zero locus of codimension 1.

Groups of symplectic involutions on symplectic varieties of Kummer type and their fixed loci

AbstractWe describe the Galois action on the middle$\ell $-adic cohomology of smooth, projective fourfolds$K_A(v)$that occur as a fiber of the Albanese morphism on moduli spaces of sheaves on an abelian surfaceAwith Mukai vectorv. We show this action is determined by the action on$H^2_{\mathrm {\acute{e}t}}(A_{\bar {k}},{\mathbb Q}_{\ell }(1))$and on a subgroup$G_A(v) \leqslant (A\times \hat {A})[3]$, which depends onv. This generalizes the analysis carried out by Hassett and Tschinkel over${\mathbb C}$[21]. As a consequence, over number fields, we give a condition under which$K_2(A)$and$K_2(\hat {A})$are not derived equivalent.The points of$G_A(v)$correspond to involutions of$K_A(v)$. Over${\mathbb C}$, they are known to be symplectic and contained in the kernel of the map$\operatorname {\mathrm {Aut}}(K_A(v))\to \mathrm {O}(H^2(K_A(v),{\mathbb Z}))$. We describe this kernel for all varieties$K_A(v)$of dimension at least$4$.When$K_A(v)$is a fourfold over a field of characteristic 0, the fixed-point loci of the involutions contain K3 surfaces whose cycle classes span a large portion of the middle cohomology. We examine the fixed-point locus on fourfolds$K_A(0,l,s)$over${\mathbb C}$whereAis$(1,3)$-polarized, finding the K3 surface to be elliptically fibered under a Lagrangian fibration of$K_A(0,l,s)$.

On the characterization of abelian varieties for log pairs in zero and positive characteristic

Let (X,Δ) be a pair. We study how the values of the log Kodaira dimension and log plurigenera relates to surjectivity and birationality of the Albanese map and the Albanese morphism of X in both characteristic 0 and characteristic p>0. In particular, we generalize some well known results for smooth varieties in both zero and positive characteristic to varieties with various types of singularities. Moreover, we show that if X is a normal projective threefold in characteristic p>0, the coefficients of the components of Δ are ≤1 and -(K X +Δ) is semiample, then the Albanese morphism of X is surjective under certain assumptions on p and the singularities of the general fibers of the Albanese morphism. This is a positive characteristic analogue in dimension 3 of a result of Zhang on a conjecture of Demailly–Peternell–Schneider.

On the abundance theorem for numerically trivial canonical divisors in positive characteristic

Abstract In this paper, we prove the abundance theorem for numerically trivial canonical divisors on strongly F-regular varieties, assuming that the geometric generic fibers of the Albanese morphisms are strongly F-regular.

Erratum to “When is the Albanese morphism an algebraic fiber space in positive characteristic?”

Erratum to “When is the Albanese morphism an algebraic fiber space in positive characteristic?”

Slope inequalities and a Miyaoka–Yau type inequality

For a minimal smooth projective surface S of general type over a field of characteristic p>0 , we prove that K^2_S\le 32\chi(\mathcal{O}_S). Moreover, if 18\chi(\mathcal{O}_S)<K^2_S\le 32\chi(\mathcal{O}_S) , the Albanese morphism of S must induce a genus 2 fibration. A classification of surfaces with K^2_S=32\chi(\mathcal{O}_S) is also given. The inequality also implies \chi(\mathcal{O}_S)>0 , which answers completely a question of Shepherd-Barron.

On Severi type inequalities

We study Severi type inequalities for big line bundles on irregular varieties via cohomological rank functions. We show that these Severi type inequalities are related to some naturally defined birational invariants of the general fibers of the Albanese morphisms. As an application, we show that the volume of an irregular threefold of general type is at least $$\frac{3}{8}$$ . We also show that the volume of a smooth projective variety X of general type and of maximal Albanese dimension is at least $$2(\dim X)!$$ . Moreover, if $${{\,\mathrm{vol}\,}}(X)=2(\dim X)!$$ , the canonical model of X is a double cover of a principally polarized abelian variety $$(A, \Theta )$$ branched over some divisor $$D\in |2\Theta |$$ .

Duality for Commutative Group Stacks

Abstract We study in this article the dual of a (strictly) commutative group stack $G$ and give some applications. Using the Picard functor and the Picard stack of $G$, we first give some sufficient conditions for $G$ to be dualizable. Then, for an algebraic stack $X$ with suitable assumptions, we define an Albanese morphism $a_X: X\longrightarrow A^1(X)$ where $A^1(X)$ is a torsor under the dual commutative group stack $A^0(X)$ of $\textrm{Pic}_{X/S}$. We prove that $a_X$ satisfies a natural universal property. We give two applications of our Albanese morphism. On the one hand, we give a geometric description of the elementary obstruction and of universal torsors (standard tools in the study of rational varieties over number fields). On the other hand, we give some examples of algebraic stacks that satisfy Grothendieck’s section conjecture.

Quotients of the square of a curve by a mixed action, further quotients and Albanese morphisms

We study mixed surfaces, the minimal resolution S of the singularities of a quotient $$(C \times C)/G$$ of the square of a curve by a finite group of automorphisms that contains elements not preserving the factors. We study them through the further quotients $$(C \times C)/G'$$ where $$G' \supset G$$ . As a first application we prove that if the irregularity is at least 3, then S is also minimal. The result is sharp. The main result is a complete description of the Albanese morphism of S through a well determined further quotient $$(C \times C)/G'$$ that is an etale cover of the symmetric square of a curve. In particular, if the irregularity of S is at least 2, then S has maximal Albanese dimension. We apply our result to all the semi-isogenous mixed surfaces of maximal Albanese dimension constructed by Cancian and Frapporti, relating them with the other constructions appearing in the literature of surfaces of general type having the same invariants.

Polarized endomorphisms of normal projective threefolds in arbitrary characteristic

Let $X$ be a projective variety over an algebraically closed field $k$ of arbitrary characteristic $p \ge 0$. A surjective endomorphism $f$ of $X$ is $q$-polarized if $f^\ast H \sim qH$ for some ample Cartier divisor $H$ and integer $q > 1$. Suppose $f$ is separable and $X$ is $\mathbb{Q}$-Gorenstein and normal. We show that the anti-canonical divisor $-K_X$ is numerically equivalent to an effective $\mathbb{Q}$-Cartier divisor, strengthening slightly the conclusion of Boucksom, de Fernex and Favre (Theorem C) and also covering singular varieties over an algebraically closed field of arbitrary characteristic. Suppose $f$ is separable and $X$ is normal. We show that the Albanese morphism of $X$ is an algebraic fibre space and $f$ induces polarized endomorphisms on the Albanese and also the Picard variety of $X$, and $K_X$ being pseudo-effective and $\mathbb{Q}$-Cartier means being a torsion $\mathbb{Q}$-divisor. Let $f^{Gal}:\overline{X}\to X$ be the Galois closure of $f$. We show that if $p>5$ and co-prime to $deg\, f^{Gal}$ then one can run the minimal model program (MMP) $f$-equivariantly, after replacing $f$ by a positive power, for a mildly singular threefold $X$ and reach a variety $Y$ with torsion canonical divisor (and also with $Y$ being a quasi-\'etale quotient of an abelian variety when $\dim(Y)\le 2$). Along the way, we show that a power of $f$ acts as a scalar multiplication on the Neron-Severi group of $X$ (modulo torsion) when $X$ is a smooth and rationally chain connected projective variety of dimension at most three.

When is the Albanese morphism an algebraic fiber space in positive characteristic?

In this paper, we study the Albanese morphisms in positive characteristic. We prove that the Albanese morphism of a variety with nef anti-canonical divisor is an algebraic fiber space, under the assumption that the general fiber is $F$-pure. Furthermore, we consider a notion of $F$-splitting for morphisms, and investigate it of the Albanese morphisms. We show that an $F$-split variety has $F$-split Albanese morphism, and that the $F$-split Albanese morphism is an algebraic fiber space. As an application, we provide a new characterization of abelian varieties.

Birational characterization of Abelian varieties and ordinary Abelian varieties in characteristic p>0

Let $k$ be an algebraically closed field of characteristic $p>0$. We give a birational characterization of ordinary abelian varieties over $k$: a smooth projective variety $X$ is birational to an ordinary abelian variety if and only if $\kappa_S(X)=0$ and $b_1(X)=2 \dim X$. We also give a similar characterization of abelian varieties as well: a smooth projective variety $X$ is birational to an abelian variety if and only if $\kappa(X)=0$, and the Albanese morphism $a: X \to A$ is generically finite. Along the way, we also show that if $\kappa _S (X)=0$ (or if $\kappa(X)=0$ and $a$ is generically finite) then the Albanese morphism $a:X\to A$ is surjective and in particular $\dim A\leq \dim X$.

Rational cohomology tori

We study normal compact K\"ahler spaces whose rational cohomology ring is isomorphic to that of a complex torus. We call them rational cohomology tori. We classify, up to dimension three, those with rational singularities. We then give constraints on the degree of the Albanese morphism and the number of simple factors of the Albanese variety for rational cohomology tori of general type (hence projective) with rational singularities. Their properties are related to the birational geometry of smooth projective varieties of general type, maximal Albanese dimension, and with vanishing holomorphic Euler characteristic. We finish with the construction of series of examples.

Abelianization of the F-divided fundamental group scheme

Let (X , x0) be a pointed smooth proper variety defined over an algebraically closed field. The Albanese morphism for (X , x0) 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.

Some (big) irreducible components of the moduli space of minimal surfaces of general type with $p_g= q = 1$ and $K^2 = 4$

This paper is devoted to the irregular surfaces of general type having the smallest invariants, p_g = q = 1 . We consider the still unexplored case K^2 = 4 , classifying those whose Albanese morphism has general fibre of genus 2 and such that the direct image of the bicanonical sheaf under the Albanese morphism is a direct sum of line bundles. We find 8 unirational families, and we prove that all are irreducible components of the moduli space of minimal surfaces of general type. This is unexpected because the assumption on the direct image bicanonical sheaf is a priori only a closed condition. One more unexpected property is that all these components have dimension strictly bigger than the expected one.

Aspherical Kähler manifolds with solvable fundamental group

We survey recent developments which led to the proof of the Benson-Gordon conjecture on Kähler quotients of solvable Lie groups. In addition, we prove that the Albanese morphism of a Kähler manifold which is a homotopy torus is a biholomorphic map. The latter result then implies the classification of compact aspherical Kähler manifolds with (virtually) solvable fundamental group up to biholomorphic equivalence. They are all biholomorphic to complex manifolds which are obtained as a quotient of $$\mathbb{C}^{n}$$ by a discrete group of complex isometries.

Generalized Albanese morphisms

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button. We define nef line bundles ${\mathcal L}_r$ on a projective variety X with the property that, for a curve $C \subset X$, the intersection ${\mathcal L}_r.C$ is zero, if and only if the restriction morphism ${\rm Hom}(\pi_1(X),{\rm U}(r)) \to {\rm Hom}(\pi_1(C),{\rm U}(r))$ has finite image up to conjugation. This yields a rational morphism $\smash[b]{\xymatrix{X \ar@{-->}[r]^-{{\rm alb}_r} & {\rm Alb}_r(X)}}$ contracting those curves C with ${\mathcal L}_r.C=0$. For $r=1$ this is the Stein factorization of the Albanese morphism.

Generalized hyperelliptic surfaces

This article presents some results on the surfaces of general type whose Albanese morphism is a holomorphic fibre bundle.

Variations of the Albanese morphisms

Variations of the Albanese morphisms