Shafarevich's hyperbolicity conjecture asserts that a family of curves over a quasi-projective 1-dimensional base is isotrivial unless the logarithmic Kodaira dimension of the base is positive. More generally it has been conjectured by Viehweg that the base of a smooth family of canonically polarized varieties is of log general type if the family is of maximal variation. In this paper, we relate the variation of a family to the logarithmic Kodaira dimension of the base and give an affirmative answer to Viehweg's conjecture for families parametrized by surfaces.
We answer an open question concerning the boundedness of canonical fiber spaces in high dimensions and prove the following: for any set of integers $n\geq 3$, $0<d<n$ and $N>0$, there exists a nonsingular projective $n$-fold $X$ of general type so that $X$ is canonically fibred by $d$-dimensional varieties $F$ with $p_g(F)\geq N$. This disproves the desired boundedness parallel to Beauville's boundedness theorem in the surface case.
In this paper we prove that a smooth family of canonically polarized manifolds parametrized by a special (in the sense of Campana) quasi-projective variety is isotrivial.
Effective bounds for the finite number of surjective holomorphic maps between canonically polarized compact complex manifolds of any dimension with fixed domain are proven. Both the case of a fixed target and the case of varying targets are treated. In the case of varying targets, bounds on the complexity of Chow varieties are used.
A Finsler metric, whose holomorphic curvature is bounded from above by a negative constant, is constructed on the moduli stack of canonically polarized manifolds including singularities. Demailly's version of Ahlfors' lemma yields hyperbolicity of the moduli stack.
Generalizing the well-known Shafarevich hyperbolicity conjecture, it has been conjectured by Viehweg that a quasi-projective manifold that admits a generically finite morphism to the moduli stack of canonically polarized varieties is necessarily of log general type. Given a quasi-projective surface that maps to the moduli stack, we employ extension properties of logarithmic pluri-forms to establish a strong relationship between the moduli map and the minimal model program of the surface. As a result, we can describe the fibration induced by the moduli map quite explicitly. A refined affirmativ…
This paper investigates the geometry of smooth canonically polarized surfaces defined over a field of positive characteristic which have a nontrivial global vector field, and the implications that the existence of such surfaces has in the moduli problem of canonically polarized surfaces. In particular, an explicit real valued function f(x) is obtained such that if $X$ is a smooth canonically polarized surface defined over an algebraically closed field of characteristic p>0 such that $K_X^2 <f(p)$, then $X$ is unirational and the order of its algebraic fundamental group is at most two. As a con…
Determining the number of singular fibers in a family of varieties over a curve is a generalization of Shafarevich's Conjecture and has implications for the types of subvarieties that can appear in the corresponding moduli stack. We consider families of log canonically polarized varieties over $¶^1$, i.e. families $g:(Y,D)\to ¶^1$ where $D$ is an effective snc divisor and the sheaf $ω_{Y/¶^1}(D)$ is $g$-ample. After first defining what it means for fibers of such a family to be singular, we show that with the addition of certain mild hypotheses (the fibers have finite automorphism group, $\sO_…
We study canonically polarized Gorenstein $3$-folds with at most terminal singularities and satisfying $K_X^3=\frac 43p_g(X)-\frac {10}3$ and $p_g(X) \ge 7$. We characterize the canonical maps of such $3$-folds, describe a structure theorem for the locally factorial ones and completely classify the smooth ones. New examples of canonically polarized smooth $3$-folds with $K_X^3=\frac 43p_g(X)-\frac {10}3$ and $p_g(X) \ge 7$ are constructed. These examples are natural extensions of those constructed by M.~Kobayashi.
We show that the number of deformation types of canonically polarized manifolds over an arbitrary variety with proper singular locus is finite, and that this number is uniformly bounded in any finite type family of base varieties. As a corollary we show that a direct generalization of the geometric version of Shafarevich's original conjecture holds for infinitesimally rigid families of canonically polarized varieties.
1) We give a 3-dimensional analogue of M. Noether's inequality for canonically polarized threefolds: $K^3\ge 2(2p_g-5)/3$. This inequality is sharp by known examples of M. Kobayashi. 2) Given a minimal 3-fold $X$ of general type with canonical singularities, if $K^3< (3p_g-5)/2$, we show that $X$ must be birationally fibred by curves of genus 2.
This paper studies the existence of non trivial $μ_p$ actions on a canonically polarized surface X defined over an algebraically closed field of characteristic p>0. In particular, an explicit function $f(K_X^2)$ is obtained such that if $p>f(K_X^2)$, then there does not exist a non trivial $μ_p$-action on X. This implies that the connected component of the automorphism scheme of X containing the identity is either smooth or is obtained by successive extensions by $α_p$.
This paper investigates the structure of the automorphism scheme of a smooth canonically polarized surface $X$ defined over an algebraically closed field of characteristic 2. In particular it is investigated when Aut(X) is not smooth. This is a situation that appears only in positive characteristic and it is closely related to the structure of the moduli stack of canonically polarized surfaces. Restrictions on certain numerical invariants of X are obtained in order for Aut(X) to be smooth or not and information is provided about the structure of the component of Aut(X) containing the identity.…
This paper investigates the geometry of a smooth canonically polarized surface $X$ defined over an algebraically closed field of characteristic $p>0$ in the case when the automorphism scheme of $X$ is not smooth. This is a situation that appears only in positive characteristic and it is closely related to the structure of the moduli stack of canonically polarized surfaces. Restrictions on certain numerical invariants of $X$ are obtained in order for Aut(X) to be smooth or not and information is provided about the structure of the component of Aut(X) containing the identity. In particular, it i…
In this article we establish a version of Y. Miyaoka generic semi-positivity theorem in the context of log-canonical orbifold pairs. As an application, we show that the canonical bundle associated to a lc pair is big as soon as there exists a generically injective morphism from an ample line bundle to some symmetric power of the cotangent bundle associated to the orbifold pair.
This paper investigates the geometry of canonically polarized surfaces defined over a field of positive characteristic which have a nontrivial global vector field, and the implications that the existence of such surfaces has in the moduli problem of canonically polarized surfaces. In particular, an explicit integer valued function f(x) is obtained with the following properties. If X is a canonically polarized surface with canonical singularities defined over an algebraically closed field of characteristic p>0 such that p>f(K_X^2) and X has a nontrivial global vector field, then X is unirationa…
Given an effectively parameterized family of canonically polarized manifolds the Kaehler-Einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle. We use a global elliptic equation to show that this metric is strictly positive. For degenerating families we obtain a singular hermitian metric. Main application is an analytic proof of the quasi-projectivity of the moduli space of canonically polarized manifolds. Further applications in arXiv:1002.4858v2.
Consider the moduli functor of canonically polarized complex manifolds with Hilbert polynomial h, and let M_h be the corresponding coarse quasi-projective moduli scheme. We show that M_h is Brody hyperbolic in the following sense: Assume that for some quasi-projective variety U there exists a morphism U --> M_h, quasi-finite over its image, which is induced by a family f: V --> U, belonging to the moduli problem. Then all holomorphic maps from the complex plane to U are constant.
Let (X,D) be a dlt pair, where X is a normal projective variety. Let S denote the support of the rounddown of D, and K the canonical divisor of X. We show that any smooth family of canonically polarized varieties over X§is isotrivial if the divisor -(K+D) is ample. This result extends results of Viehweg-Zuo and Kebekus-Kovacs. To prove this result we show that any extremal ray of the moving cone is generated by a family of curves, and these curves are contracted after a certain run of the minimal model program. In the log Fano case, this generalizes a theorem by Araujo from the klt to the dlt …
The main result of this note is an effective uniform bound for the number of deformation types of certain nonisotrivial families of canonically polarized manifolds. It extends the author's earlier such bound for the classical Shafarevich Conjecture over function fields to the case of higher dimensional fibers, but without the disadvantageous iterated use of Chow or Hilbert varieties that was the core of the proofs in the earlier approaches.
Generalizing the well-known Shafarevich hyperbolicity conjecture, it has been conjectured by Viehweg that a quasi-projective manifold that admits a generically finite morphism to the moduli stack of canonically polarized varieties is necessarily of log general type. Given a quasi-projective threefold Y that admits a non-constant map to the moduli stack, we employ extension properties of logarithmic pluri-forms to establish a strong relationship between the moduli map and the minimal model program of Y: in all relevant cases the minimal model program leads to a fiber space whose fibration facto…
We study the classification problem for polarized varieties with high nefvalue. We give a complete list of isomorphism classes for normal polarized varieties with high nefvalue. This generalizes classical work on the smooth case by Fujita, Beltrametti and Sommese. As a consequence we obtain that polarized varieties with slc singularities and high nefvalue, are birationally equivalent to projective bundles over nodal curves.
We classify the polarized symplectic automorphisms of Fano varieties of smooth cubic fourfolds (equipped with the Plücker polarization) and study the fixed loci.
We extend some of the results obtained for subvarieties of the moduli stack of canonically polarized manifolds in "Base spaces of non-isotrivial families of smooth minimal models" (math.AG/0103122) to moduli of polarized minimal models of Kodaira dimension zero. To this aim we first show that for those manifolds there are no non isotrivial families of minimal models, which are birationally isotrivial. In the last chapter, we discuss the rigidity of curves in the moduli stack of canonically polarized manifolds and of polarized minimal models of Kodaira dimension zero.
We describe polarized complexity-one T-varieties combinatorially in terms of so-called divisorial polytopes, and show how geometric properties of such a variety can be read off the corresponding divisorial polytope. We compare our description with other possible descriptions of polarized complexity-one T-varieties. We also describe how to explicitly find generators of affine complexity-one T-varieties.
In this note it is shown that associating to a principally polarized abelian variety its de Rham cohomology defines a faithfully flat morphism of the moduli space of principally polarized abelian varieties in positive characteristics to the moduli space of symplectic $F$-Zips as defined in \cite{MW}.
In this paper, we study principally polarized abelian varieties $X$ of dimension $g$ that contain a curve $ν:C\to X$ such that the class of $C$ is $m$ times the minimal class. Welters introduced the formalism of stable pairs to handle this problem in the case $m=2$. We generalize the results of Welters and construct families of principally polarized abelian varieties for any $m$ and compute the dimension of the locus of these abelian varieties.
Let $\mathcal{A}_g$ denote the moduli space of principally polarized abelian varieties of dimension $g \ge 3.$ In this paper we prove the connectedness of the singular sublocus of $\mathcal{A}_g$ consisting of those abelian varieties which possess an involution different from $-id$.
We give an explicit characterization of all principally polarized abelian varieties $(A,Θ)$ such that there is a finite subgroup of automorphisms $G$ of $A$ that preserve the numerical class of $Θ$, and such that the quotient variety $A/G$ is smooth. We also give a complete classification of smooth quotients of Jacobians of curves.
We study invariant divisors on the total spaces of the homogeneous deformations of rational complexity-one T-varieties constructed by Ilten and Vollmert. In particular, we identify a natural subgroup of the Picard group for any general fiber of such a deformation, which is canonically isomorphic to the Picard group of the special fiber. This isomorphism preserves Euler characteristic, intersection numbers, and the canonical class.
We prove that the embedding of a $GV$-subscheme in a principally polarized abelian variety does not factor through any nontrivial isogeny. As an application, we present a new proof of a theorem of Clemens--Griffiths identifying the intermediate Jacobian of a smooth cubic threefold in $\mathbf{P}^4$ to the Albanese variety of its Fano surface of lines.
In a previous paper we showed that for every polarization on an abelian variety there is a dual polarization on the dual abelian variety. In this note we extend this notion of duality to families of polarized abelian varieties. As a main consequence we obtain an involution on the set of moduli spaces of polarized abelian varieties of dimension $g$. In particular, the moduli spaces $\mathcal A_{(d_1,...,d_g)}$ and $\mathcal A_{(d_1,\frac{d_1d_g}{d_{g-1}},...,\frac{d_1d_g}{d_2},d_g)}$ are canonically isomorphic.
The article has two parts. The first part is devoted to proving a singular version of the logarithmic Kodaira-Akizuki-Nakano vanishing theorem of Esnault and Viehweg. This is then used to prove other vanishing theorems. In the second part these vanishing theorems are used to prove an Arakelov-Parshin type boundedness result for families of canonically polarized varieties with rational Gorenstein singularities.
We study the geometry of some varieties of sums of powers related to the Klein quartic. This allows us to describe the birational geometry of certain moduli spaces of abelian surfaces. In particular we show that the moduli space $\mathcal{A}_2(1,7)^{-}_{sym}$ of $(1,7)$-polarized abelian surfaces with a symmetric theta structure and an odd theta characteristic is unirational by showing that it admits a dominant morphism from a unirational conic bundle.
Let $(X,D)$ be a smooth log pair over $\mathbb{C}$ such that the complement $U := X \setminus D$ carries a maximally varied family of polarized manifolds. We prove a version of second main theorem on $(X,D)$ by using the Viehweg-Zuo construction of the family and McQuillan's tautological inequality. As an application, we generalize a classical result of Nadel about the distribution of entire curves in the (compactified) base space of polarized families.
In this paper, we show that a general polarized abelian variety $(X,L)$ of type $(1,\dots,1,d)$ and dimension $g$ satisfies property $(N_p)$ if $ d \geq \sum_{i=0}^{g} (p+2)^i$. In particular, the case $p=0$ affirmatively solves a conjecture by L. Fuentes Garc\'ıa on projective normality.
Let f: X -> Y be a smooth family of canonically polarized complex varieties over a smooth base. Generalizing the classical Shafarevich hyperbolicity conjecture, Viehweg conjectured that Y is necessarily of log general type if the family has maximal variation. A somewhat stronger and more precise version of Viehweg's conjecture was shown by the authors in arXiv:math/0511378 in the case where Y is a quasi-projective surface. Assuming that the minimal model program holds, this very short paper proves the same result for projective base manifolds Y of arbitrary dimension.
To every abelian subvariety of a principally polarized abelian variety $(A, \mathcal{L})$ we canonically associate a numerical class in the Néron-Severi group of $A$. We prove that these classes are characterized by their intersection numbers with $ \mathcal{L}$; moreover, the cycle class induced by an abelian subvariety in the Chow ring of $A$ modulo algebraic equivalence can be described in terms of its numerical divisor class. Over the field of complex numbers, this correspondence gives way to an explicit description of the (coarse) moduli space that parametrizes non-simple principally pola…
Semi-log canonical varieties are a higher-dimensional analogue of stable curves. They are the varieties appearing as the boundary $Δ$ of a log canonical pair $(X,Δ)$, and also appear as limits of canonically polarized varieties in moduli theory. For certain three-fold pairs $(X,Δ)$ we show how to compute the PL homeomorphism type of the dual complex of a dlt minimal model directly from the normalization data of $Δ$.
The formula of expanding the Abel variety theta function restricted to Abel subvariety into theta functions of this subvariety is obtained. With the help of this formula the solution of differential equations with Jacobi theta functions, restricted on a nonprincipally polarized Abel subvariety and their translations are rewritten in terms of the theta functions of these subvarieties. This is exemplified by the CKP equations, the Bogoyavlenskij system, and the Toda $g_2^{(1)}$-chain.
For a polarized abelian variety $(X,L)$, Z. Jiang and G. Pareschi introduce an invariant $β(X,L)$, called the basepoint-freeness threshold. Using this invariant, we show that a general polarized abelian variety $(X,L)$ of dimension $g$ is projectively normal if $χ(L) \geq 2^{2g-1}$ and the type of $L$ is not $(2,4,\dots,4)$. This bound is sharp since it is known that any polarized abelian variety of type $(2,4,\dots,4)$ is not projectively normal. We also give an application of $β(X,L)$ to the Infinitesimal Torelli Theorem for $Y \in |L|$.
We apply Angehrn-Siu-Helmke's method to estimate basepoint freeness thresholds of higher dimensional polarized abelian varieties. We showed that a conjecture of Caucci holds for very general polarized abelian varieties in the moduli spaces $\mathcal A_{g, l}$ with only finitely many possible exceptions of polarization types $l$ in each dimension $g$. We improved the bound of basepoint freeness thresholds of any polarized ableian $4$-folds and simple abelian $5$-folds.
In this note we derive from deep results due to Clozel-Ullmo the density of Noether-Lefschetz loci inside the moduli space of marked (polarized) irreducible holomorphic symplectic (IHS) varieties. In particular we obtain the density of Hilbert schemes of points on projective $K3$ surfaces and of projective generalized Kummer varieties in their moduli spaces. We present applications to the existence of rational curves on projective deformations of such varieties, to the study of relevant cones of divisors, and a refinement of Hassett's result on cubic fourfolds whose Fano variety of lines is is…
We obtain, by a direct computation, explicit descriptions of all principally polarized semi-abelic varieties of torus rank up to 3. We describe the geometry of their symmetric theta divisors and obtain explicit formulas for the involution and its fixed points. These results allow us to give a new proof of the statements about the dimensions, for small genus, of the loci of ppav with theta divisor containing two-torsion points of multiplicity three. We also prove a result about the closure of this set. Our computations used in our work arXiv:1103.1857 to compute the class of the closure of the …
We apply Schmidt's Subspace Theorem to establish Arithmetic General Theorems for projective varieties over number and function fields. Our first result extends an analogous result of M. Ru and P. Vojta. One aspect to its proof makes use of a filtration construction which appears in work of Autissier. Further, we consider work of M. Ru and J. T.-Y. Wang, which pertains to an extension of K. F. Roth's theorem for projective varieties in the sense of D. McKinnon and M. Roth. Motivated by these works, we establish our second Arithmetic General Theorem, namely a form of Roth's theorem for exception…
In this paper, we discuss a generalization of log canonical singularities in the non-$\mathbb{Q}$-Gorenstein setting. We prove that if a normal complex projective variety has a non-invertible polarized endomorphism, then it has log canonical singularities in our sense. As a corollary, we give an affirmative answer to a conjecture of Broustet and Höring.
Consider a polarized abelian variety $(A,L)$ over the field of complex numbers. Following Demailly, one can associate to $(A,L)$ a real number $ε(A,L)$, its {\em Seshadri constant}, which in effect measures how much of the positivity of $L$ can be concentrated at any given point of $A$. There has been considerable recent interest in finding bounds on the Seshadri constants of abelian varieties and on smooth projective varieties in general. In the present paper we first generalize an approach of Buser and Sarnak to give a lower bound on the minimal period length of $(A,L)$ in terms of the type …
We construct two pencils of bielliptic curves of genus three and genus five. The first pencil is associated with a general abelian surface with a polarization of type $(1,2)$. The second pencil is related to the first by an unramified double cover, the Prym variety of which is canonically isomorphic to the Jacobian of a very general curve of genus two. Our results are obtained by analyzing suitable elliptic fibrations on the associated Kummer surfaces and rational double covers among them.
In this article we study the deformations of hyperelliptic polarized varieties $(X,L)$ of dimension $m$ and sectional genus $g$ such that the image $Y$ of the morphism $φ$ induced by $|L|$ is smooth. If $L^m < 2g-2$, it is known that, by adjunction and the Clifford's theorem, any deformation of $(X,L)$ is hyperelliptic. Thus, we focus on when $L^m=2g-2$ or $L^m=2g$. We prove that, if $(X,L)$ is Fano-K3, then, except when $Y$ is a hyperquadric, all deformations of $(X,L)$ are again hyperelliptic (if $Y$ is a hyperquadric, the general deformation of $φ$ is an embedding). This contrasts with the …
We present a structure theorem for the moduli space R_7 of Prym curves of genus 7 as a projective bundle over the moduli space of 7-nodal rational curves. The existence of this parametrization implies the unirationality of R_7 and that of the moduli space of Nikulin surfaces of genus 7, as well as the rationality of the moduli space of Nikulin surfaces of genus 7 with a distinguished line. Using the results in genus 7, we then establish that R_8 is uniruled.
Let G be a finite group. For each integral representation $ρ$ of G we consider $ρ-$decomposable principally polarized abelian varieties; that is, principally polarized abelian varieties (X,H) with $ρ(G)-$action, of dimension equal to the degree of $ρ$, which admit a decomposition of the lattice for X into two G-invariant sublattices isotropic with respect to $\Im H$, with one of the sublattices $\mathbb{Z}G-$isomorphic to $ρ$. We give a construction for $ρ-$decomposable principally polarized abelian varieties, and show that each of them is isomorphic to a product of elliptic curves. Conversely…
In this paper, we prove the Kobayashi hyperbolicity of the coarse moduli spaces of canonically polarized or polarized Calabi-Yau manifolds in the sense of complex $V$-spaces (a generalization of complex $V$-manifolds in the sense of Satake). As an application, we prove the following hyperbolic version of Campana's isotriviality conjecture: for the smooth family of canonically polarized or polarized Calabi-Yau manifolds, when the Kobayashi pseudo-distance of the base vanishes identically, the family must be isotrivial, that is, any two fibers are isomorphic. We also prove that for the smooth pr…
