We classify these threefolds, which are the ones such that their universal cover is not compact and not covered by positive-dimensional compact analytic subsets. We show that these threefolds have nonnegative Kodaira dimension, and that their Iitaka-Moishezon fibrations are, after a suitable finite etale cover, bimeromorphic to a submersion with fibres complex tori, and base of general type and $π_1$-general type.
We give a complex polarized variation of Hodge structure over a compact K"ahler manifold $M$ which controls all finite-dimensional complex polarized variations of Hodge structure over $M$ and their tensor relations. As a corollary, we obtain the cohomology algebra with values in a local system admitting multiplicative Hodge structures.
We prove that a compact contact threefold which is bimeromorphically equivalent to a Kaehler manifold and not rationally connected is the projectivised tangent bundle of a Kaehler surface.
Let $X$ be a compact K{ä}hler manifold of dimension three. We prove that there exists a projective manifold $Y$ such that $π\_1(X)\simeq π\_1(Y)$. We also prove the bimeromorphic existence of algebraic approximations for compact K{ä}hler manifolds of algebraic dimension $\dim(X)-1$. Together with the work of Graf and the third author, this settles in particular the bimeromorphic Kodaira problem for compact K{ä}hler threefolds.
Extending well-known results on surfaces, we give bounds on the cohomological action of automorphisms of compact Kähler threefolds. More precisely, if the action is virtually unipotent we prove that the norm of $(f^n)^*$ grows at most as $cn^4$; in the general case, we give a description of the spectrum of $f^*$, and bounds on the possible conjugates over $\mathbb Q$ of the dynamical degrees $λ_1(f),λ_2(f)$. Examples on complex tori show the optimality of the results.
We study compact Kähler threefolds X with infinite fundamental group whose universal cover can be compactified. Combining techniques from $L^2$ -theory, Campana's geometric orbifolds and the minimal model program we show that this condition imposes strong restrictions on the geometry of X. In particular we prove that if a projective threefold with infinite fundamental group has a quasi-projective universal cover, the latter is then isomorphic to the product of an affine space with a simply connected manifold.
We prove that every compact Kähler threefold $X$ of Kodaira dimension $κ= 0$ or $1$ has a $\mathbf{Q}$-factorial bimeromorphic model $X'$ with at worst terminal singularities such that for each curve $C \subset X'$, the pair $(X',C)$ admits a locally trivial algebraic approximation such that the restriction of the deformation of $X'$ to some neighborhood of $C$ is a trivial deformation. As an application, we prove that every compact Kähler threefold with $κ= 0$ or $1$ has an algebraic approximation.
Let $X$ be a non-uniruled compact Kähler space of dimension 3. We show that the group of bimeromorphic automorphisms of $X$ is Jordan. More generally, the same result holds for any compact Kähler space admitting a quasi-minimal model.
In this note, we study the normal compact Kähler (possibly singular) threefold $X$ admitting the action of a free abelian group $G$ of maximal rank, all the non-trivial elements of which are of positive entropy. If such $X$ is further assumed to have only terminal singularities, then we prove that it is either a rationally connected projective threefold or bimeromorphic to a quasi-étale quotient of a complex $3$-torus.
Let X be a compact Kaehler threefold such that the base of the MRC-fibration has dimension two. We prove that X is bimeromorphic to a Mori fibre space. Together with our earlier result arXiv:1304.4013 this completes the MMP for compact Kaehler threefolds: let X be a non-projective compact Kaehler threefold. Then X has a minimal model or X is bimeromorphic to a Mori fibre space over a non-projective Kaehler surface.
The moduli space of multiply-connected Calabi-Yau threefolds is shown to contain codimension-one loci on which the corresponding variety develops a certain type of hyperquotient singularity. These have local descriptions as discrete quotients of the conifold, and are referred to here as hyperconifolds. In many (or possibly all) cases such a singularity can be resolved to yield a distinct compact Calabi-Yau manifold. These considerations therefore provide a method for embedding an interesting class of singularities in compact Calabi-Yau varieties, and for constructing new Calabi-Yau manifolds. …
In this paper we study \emph{threefolds isogenous to a product of mixed type} i.e. quotients of a product of three compact Riemann surfaces $C_i$ of genus at least two by the action of a finite group $G$, which is free, but not diagonal. In particular, we are interested in the systematic construction and classification of these varieties. Our main result is the full classification of threefolds isogenous to a product of mixed type with $χ(\mathcal O_X)=-1$ assuming that any automorphism in $G$, which restricts to the trivial element in $Aut(C_i)$ for some $C_i$, is the identity on the product.…
We consider the connected-sum method of constructing compact Riemannian 7-manifolds with holonomy G_2 developed in math.DG/0012189. The method requires pairs of projective complex threefolds endowed with anticanonical K3 divisors, the latter `matching' via a certain non-holomorphic map. Suitable examples of threefolds were previously obtained in math.DG/0012189 by blowing up curves in Fano threefolds. In this paper, we give further suitable algebraic threefolds using theory of K3 surfaces with non-symplectic involution due to Nikulin. These threefolds are not obtainable from Fano threefolds, a…
The present work deals with the canonical map of smooth, compact complex surfaces of general type in a polarization of type $(1,2,2)$ on an abelian threefold. A natural and classical question is whether the canonical system of such surfaces is very ample in the general case. In this work, we provide a positive answer to this question. First, we describe the structure of the canonical map of those smooth ample surfaces of type $(1,2,2)$ in an abelian threefold which are bidouble cover of principal polarizations. Then, we study the general behavior of the canonical map of general ample surfaces …
The purpose of this paper is to study the bimeromorphic invariants of compact complex manifolds in terms of Bott-Chern cohomology. We prove a blow-up formula for Bott-Chern cohomology. As an application, we show that for compact complex threefolds the non-Kählerness degrees, introduced by Angella-Tomassini [Invent. Math. 192, (2013), 71-81], are bimeromorphic invariants. Consequently, the $\partial\bar{\partial}$-Lemma on threefolds admits the bimeromorphic invariance.
We calculate the D-brane superpotentials for two Calabi-Yau manifolds with three deformations by the generalized hypergeometric GKZ systems, which give rise to the flux superpotentials $\mathcal{W}_{GVW}$ of the dual F-theory compactification on the relevant Calabi-Yau fourfolds in the weak decoupling limit. We also compute the Ooguri-Vafa invariants from A-model expansion with mirror symmetry, which are related to the open Gromov-Witten invariants.
In the paper [FSM] we described some Siegel modular threefolds which admit a Calabi-Yau model. Using a different method we give in this paper an enlarged list of such varieties that admits a Calabi-Yau model in the following weak sense: there exists a desingularization in the category of complex spaces of the Satake compactification which admits a holomorphic three-form without zeros and whose first Betti number vanishes Basic for our method is the paper [GN] of van Geemen and Nygaard.
In this paper, we extend the GKZ-system method to the more general case: compact Complete Intersection Calabi-Yau manifolds (CICY). For several one-deformation modulus compact CICYs with D-branes, the on-shell superpotentials in this paper from the extended GKZ-system method are exactly consistent with published results obtained from other methods. We further compute the off-shell superpotentials of these models. Then we obtain both the on-shell and off-shell superpotentials for sev- eral two-deformation moduli compact CICYs with D-branes by using the extended GKZ-system method. The discrete s…
The string landscape satisfies interesting finiteness properties imposed by supersymmetry and string-theoretical consistency conditions. We study N=1 supersymmetric compactifications of Type IIB string theory on smooth elliptically fibered Calabi-Yau threefolds at large volume with magnetized D9-branes and D5-branes. We prove that supersymmetry and tadpole cancellation conditions imply that there is a finite number of such configurations. In particular, we derive an explicitly computable bound on the number of magnetic flux quanta, as well as the number of D5-branes, which is independent of th…
A numerically flat rank-two bundle without a holomorphic connection on a $\partial\bar\partial$-threefold
School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）；The Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong（香港中文大学数学科学研究所和数学系）
We present the first examples of smooth elliptic Calabi-Yau threefolds with Mordell-Weil rank 10, the highest currently known value. They are given by the Schoen threefolds introduced by Namikawa; there are six isolated fibers of Kodaira Type IV. We explicitly compute the Shioda homomorphism for the generators of the Mordell-Weil group and their induced height pairing. Compactification of F-theory on these threefolds gives an effective theory in six dimensions which contains ten abelian gauge group factors. We compute the massless matter spectrum. In particular, we show that the charged single…
We give a differential-geometric construction and examples of Calabi-Yau threefolds, at least one of which is {\it{new}}. Ingredients in our construction are {\it admissible pairs}, which were dealt with by Kovalev in \cite{K03} and further studied by Kovalev and Lee in \cite{KL11}. An admissible pair $(\overline{X},D)$ consists of a three-dimensional compact Kähler manifold $\overline{X}$ and a smooth anticanonical $K3$ divisor $D$ on $\overline{X}$. If two admissible pairs $(\overline{X}_1,D_1)$ and $(\overline{X}_2,D_2)$ satisfy the {\it gluing condition}, we can glue $\overline{X}_1\setmin…
Let $X$ be a normal compact Kähler space of dimension $n$. A surjective endomorphism $f$ of such $X$ is int-amplified if $f^*ξ-ξ=η$ for some Kähler classes $ξ$ and $η$. First, we show that this definition generalizes the notion in the projective setting. Second, we prove that for the cases of $X$ being smooth, a surface or a threefold with mild singularities, if $X$ admits an int-amplified endomorphism with pseudo-effective canonical divisor, then it is a $Q$-torus. Finally, we consider a normal compact Kähler threefold $Y$ with only terminal singularities and show that, replacing $f$ by a pos…
