Over a perfect field, we determine the sheaf of $\mathbb{A}^1$-connected components of a class of threefolds given by the Blow-up of a variety admitting a $\mathbb{P}^1$-fibration over either an $\mathbb{A}^1$-rigid or a non-uniruled surface, along a smooth curve. As a consequence, we verify that the sheaf of $\mathbb{A}^1$-connected components for such varieties is $\mathbb{A}^1$-invariant.
We consider certain $K3$-fibered Calabi--Yau threefolds. One class of such Calabi--Yau threefolds are constructed by Hunt and Schimmrigk using twist maps. They are realized in weighted projective spaces as orbifolds of hypersurfaces. Our main goal of this paper is to investigate arithmetic properties of these Calabi--Yau threefolds. We also consider deformations of our Calabi--Yau threefolds, and we study the variation of the zeta-functions using $p$-adic rigid cohomology theory.
The Hodge numbers of generic elliptically fibered Calabi-Yau threefolds over toric base surfaces fill out the "shield" structure previously identified by Kreuzer and Skarke. The connectivity structure of these spaces and bounds on the Hodge numbers are illuminated by considerations from F-theory and the minimal model program. In particular, there is a rigorous bound on the Hodge number h_{21} <= 491 for any elliptically fibered Calabi-Yau threefold. The threefolds with the largest known Hodge numbers are associated with a sequence of blow-ups of toric bases beginning with the Hirzebruch surfac…
This paper classifies a class of holomorphic D-branes, closely related to framed torsion-free sheaves, on threefolds fibered in resolved ADE surfaces over a general curve C, in terms of representations with relations of a twisted Kronheimer--Nakajima-type quiver in the category Coh(C) of coherent sheaves on C. For the local Calabi--Yau case C\cong\A^1 and special choice of framing, one recovers the N=1 ADE quiver studied by Cachazo--Katz--Vafa.
In part I of this paper we constructed certain fibered Calabi-Yaus by a quotient construction in the context of weighted hypersurfaces. In this paper look at the case of K3 fibrations more closely and study the singular fibers which occur. This differs from previous work since the fibrations we discuss have constant modulus, and the singular fibers have torsion monodromy.
We study each of the 16 types of complete intersection Calabi-Yau threefolds in projective n-space times the projective line, for various n, and prove existence of isolated rational curves of bidegree (d,0) for all positive integers d on a general threefold of each of these types. As a corollary we find isolated rational curves in all degrees on a general CICY of each of the 5 possible types in usual projective n-space.
In this paper we construct noncommutative resolutions of a certain class of Calabi-Yau threefolds studied by F. Cachazo, S. Katz and C. Vafa. The threefolds under consideration are fibered over a complex plane with the fibers being deformed Kleinian singularities. The construction is in terms of a noncommutative algebra introduced by V. Ginzburg, which we call the "N=1 ADE quiver algebra".
Using the techniques introduced by Corvaja and Zannier we solve the non-split case of the geometric Lang-Vojta Conjecture for affine surfaces isomorphic to the complement of a conic and two lines in the projective plane. In this situation we deal with sections of an affine threefold fibered over a curve, whose boundary, in the natural projective completion, is a quartic bundle over the base whose fibers have three irreducible components. We prove that the image of each section has bounded degree in terms of the Euler Characteristic of the base curve.
We study sections of a Calabi-Yau threefold fibered over a curve by K3 surfaces. We show that there exist infinitely many isolated sections on certain K3 fibered Calabi-Yau threefolds and the subgroup of the Néron-Severi group generated by these sections is not finitely generated. This also gives examples of K3 surfaces over the function field F of a complex curve with Zariski dense F-rational points, whose geometric model is Calabi-Yau.
Let $X$ be a Gorenstein minimal projective 3-fold with at worst locally factorial terminal singularities. Suppose the canonical map is of fiber type. Denote by $F$ a smooth model of a generic irreducible component in fibers of the canonical map of $X$ and so $F$ is a smooth curve or a smooth surface. The main result of the paper is that there is a computable constant $K$ (independent of $X$) such that $g(F)\leq 647$ or $p_g(F)\leq 38$ whenever $p_g(X)\geq K$. The method heavily relies on both a Noether type of inequality and a Miyaoka-Yau inequality and that is the reason we only treat a Goren…
Elliptic and genus one fibered Calabi-Yau spaces play a prominent role in string theory and mathematics. In this article we discuss a class of genus one fibered Calabi-Yau threefolds with 5-sections from various perspectives. In algebraic geometry, such Calabi-Yaus can be constructed as complete intersections in Grassmannian fibrations and as Pfaffian varieties. These constructions naturally fit into the framework of homological projective duality and lead to dual pairs of Calabi-Yaus. From a physics perspective, these spaces can be realised as low-energy configurations ("phases") of gauged li…
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.
Revisiting the Quantum Geometry of Torus-fibered Calabi-Yau Threefolds
Laboratoire de Physique Théorique et Hautes Energies, CNRS and Sorbonne Université（高能理论物理实验室，法国国家科学研究中心与索邦大学）；Institute for Theoretical Physics & Department of Mathematics, Utrecht University（乌得勒支大学理论物理研究所与数学系）
To every elliptic Calabi-Yau threefold with a section $X$ there can be associated a Lie group $G$ and a representation $ρ$ of that group. The group is determined from the Weierstrass model, which has singularities that are generically rational double points; these double points lead to local factors of $G$ which are either the corresponding A-D-E groups or some associated non-simply laced groups. The representation $ρ$ is a sum of representations coming from the local factors of $G$, and of other representations which can be associated to the points at which the singularities are worse than ge…
We construct stable bundle extensions on elliptically fibered Calabi-Yau threefolds. We show that these bundles can solve the topological anomaly constraint in heterotic string theory without the need of invoking background fivebranes.
We carry out a systematic analysis of Calabi-Yau threefolds that are elliptically fibered with section ("EFS") and have a large Hodge number h^{2, 1}. EFS Calabi-Yau threefolds live in a single connected space, with regions of moduli space associated with different topologies connected through transitions that can be understood in terms of singular Weierstrass models. We determine the complete set of such threefolds that have h^{2, 1} >= 350 by tuning coefficients in Weierstrass models over Hirzebruch surfaces. The resulting set of Hodge numbers includes those of all known Calabi-Yau threefold…
This article constructs a smooth weak Fano threefold of Picard number two with small anti-canonical morphism that arises as a blowup of a smooth curve of genus 5 and degree 8 in $\mathbb{P}^3$. While the existence of this weak Fano was known as a numerical possibility in \cite{CM13} and constructed in BL12, this paper removes the dependencies on the results in \cite{JPR11} needed in the construction of BL12 and constructs the link in the style of ACM17.
We characterise smooth curves in P^3 whose blow-up produces a threefold with anticanonical divisor big and nef. These are curves C of degree d and genus g lying on a smooth quartic, such that (i) $4d-30 \le g\le 14$ or $(g,d) = (19,12)$, (ii) there is no 5-secant line, 9-secant conic, nor 13-secant twisted cubic to C. This generalises the classical similar situation for the blow-up of points in P^2. We describe then Sarkisov links constructed from these blow-ups, and are able to prove the existence of Sarkisov links which were previously only known as numerical possibilities.
We give further counterexamples to the conjectural construction of Bridgeland stability on threefolds due to Bayer, Macrì, and Toda. This includes smooth projective threefolds containing a divisor that contracts to a point, and Weierstraß elliptic Calabi-Yau threefolds. Furthermore, we show that if the original conjecture, or a minor modification of it, holds on a smooth projective threefold, then the space of stability conditions is non-empty on the blow up at an arbitrary point. More precisely, there are stability conditions on the blow up for which all skyscraper sheaves are semistable.
We study links between algebraic cycles on threefolds and finite-dimensionality of their motives with coefficients in Q. We decompose the motive of a non-singular projective threefold X with representable algebraic part of CH_0(X) into Lefschetz motives and the Picard motive of a certain abelian variety, isogenous to the corresponding intermediate Jacobian J^2(X) when the ground field is C. In particular, it implies motivic finite-dimensionality of Fano threefolds over a field. We also prove representability of zero-cycles on several classes of threefolds fibered by surfaces with algebraic H^2…
We study local, global and local-to-global properties of threefolds with certain singularities. We prove criteria for these threefolds to be rational homology manifolds and conditions for threefolds to satisfy rational Poincaré duality. We relate the topological Euler characteristic of elliptic Calabi-Yau threefolds with $\mathbb Q$-factorial terminal singularities to dimensions of Lie algebras and certain representations, Milnor and Tyurina numbers and other birational invariants of an elliptic fibration. We give an interpretation in terms of complex deformations. We state a conjecture on the…
We give a proof of the existence of $G=SU(5)$, stable holomorphic vector bundles on elliptically fibered Calabi--Yau threefolds with fundamental group $\bbz_2$. The bundles we construct have Euler characteristic 3 and an anomaly that can be absorbed by M-theory five-branes. Such bundles provide the basis for constructing the standard model in heterotic M-theory. They are also applicable to vacua of the weakly coupled heterotic string. We explicitly present a class of three family models with gauge group $SU(3)_C\times SU(2)_L\times U(1)_Y$.
We propose a general definition of mathematical instanton bundle with given charge on any Fano threefold extending the classical definitions on $\mathbb P^3$ and on Fano threefold with cyclic Picard group. Then we deal with the case of the blow up of $\mathbb P^3$ at a point, giving an explicit construction of instanton bundles satisfying some important extra properties: moreover, we also show that they correspond to smooth points of a component of the moduli space.
We find all K-polystable smooth Fano threefolds that can be obtained as blowup of projective space along the disjoint union of a twisted cubic curve and a line.
We show that the Ueno-Campana's threefold cannot be obtained as the blow-up of any smooth threefold along a smooth centre, answering negatively a question raised by Oguiso and Truong.
It is known that the smooth rational threefolds of P^5 having a rational non-special surface of P^4 as general hyperplane section have degree d=3,... ,7. We study such threefolds X from the point of view of linear systems of surfaces in P^3, looking in each case fosr an explicit description of a birational map from P^3 to X. For d=3,..., 6 we prove that there exists a line L on X such that the projection map of X centered at L is birational; we completely describe the base loci B of the linear systems found in this way and give a description of any such threefold X as a suitable blowing-down o…
We develop a combinatorial approach to the construction of general smooth compact base surfaces that support elliptic Calabi-Yau threefolds. This extends previous analyses that have relied on toric or semi-toric structure. The resulting algorithm is used to construct all classes of such base surfaces $S$ with $h^{1, 1} (S) < 8$ and all base surfaces over which there is an elliptically fibered Calabi-Yau threefold $X$ with Hodge number $h^{2, 1} (X) \geq 150$. These two sets can be used todescribe all 6D F-theory models that have fewer than seven tensor multiplets or more than 150 neutral scala…
In this paper, we consider the existence problem of rank one and two stable Ulrich bundles on imprimitive Fano 3-folds obtained by blowing-up one of $\mathbb{P}^{3}$, $Q$ (smooth quadric in $\mathbb{P}^{4}$), $V_{3}$ (smooth cubic in $\mathbb{P}^{4}$) or $V_{4}$ (complete intersection of two quadrics in $\mathbb{P}^{5}$) along a smooth irreducible curve. We prove that the only class which admits Ulrich line bundles is the one obtained by blowing up a genus 3, degree 6 curve in $\mathbb{P}^{3}$. Also, we prove that there exist stable rank two Ulrich bundles with $c_{1}=3H$ on a generic member o…
We study the syzygetic structure of projections of del Pezzo surfaces in order to construct singular Calabi-Yau threefolds. By smoothing those threefolds, we obtain new examples of Calabi-Yau threefolds with Picard group of rank 1. We also give an example of type II primitive contraction whose exceptional divisor is the blow-up of the projective plane at a point.
In this paper we study compactifications of the ${\cal N}=2$ heterotic $E_8\times E_8$ string on $(K3\times T^2)/\mathbb{Z}_3$ with various gauge backgrounds and calculate the topological couplings in the effective supergravity action that arise from one-loop amplitudes. We then identify candidates for dual type IIA compactifications on Calabi-Yau threefolds and compare the heterotic results with the corresponding topological string amplitudes. We find that the dual Calabi-Yau geometries are $K3$ fibrations that are also genus one fibered with three-sections. Moreover, we show that the interse…
Let X be the blow-up of the three dimensional complex projective space along r general points of a smooth elliptic quartic curve B of P^3 and let L be any line bundle of X. The aim of this paper is to provide an explicit algorithm for determining the dimension of H^0(X,L).
We prove that for every compact, connected, differentiable 3--manifold $M$ there is a compact complex manifold $X$ which can be obtained from projective 3--space by a sequence of smooth, real blow ups and downs such that $M$ is diffeomorphic to the set of real points of $X$. By earlier results, such an $X$ can almost never be projective.
We determine the Cox rings of the minimal resolutions of cubic surfaces with at most rational double points, of blow ups of the projective plane at non-general configurations of six points and of three dimensional smooth Fano varieties of Picard numbers one and two.
We deal with instanton bundles on the product ${\mathbb P}^1\times{\mathbb P}^2$ and the blow up of ${\mathbb P}^3$ along a line. We give an explicit construction leading to instanton bundles. Moreover, we also show that they correspond to smooth points of a unique irreducible component of their moduli space.
In this article, a sequel to "Global Frobenius Liftability I" (math:1708:03777v2), we continue the development of a comprehensive theory of Frobenius liftings modulo $p^2$. We study compatibility of divisors and closed subschemes with Frobenius liftings, Frobenius liftings of blow-ups, descent under quotients by some group actions, stability under base change, and the properties of associated F-splittings. Consequently, we characterise Frobenius liftable surfaces and Fano threefolds, confirming the conjecture stated in our previous paper.
We give new proofs of the K-polystability of two smooth Fano threefolds. One of them is a~smooth divisor in $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^2$ of degree $(1,1,1)$, which is unique up to isomorphism. Another one is the~blow up of the complete intersection $$ \Big\{x_0x_3+x_1x_4+x_2x_5=x_0^2+ωx_1^2+ω^2x_2^2+\big(x_3^2+ωx_4^2+ω^2x_5^2\big)+\big(x_0x_3+ωx_1x_4+ω^2x_2x_5\big)\Big\}\subset\mathbb{P}^5 $$ in the conic cut out by $x_0=x_1=x_2=0$, where $ω$ is a~primitive cube root of unity.
We construct a family of fibered threefolds $X_m \to (S , Δ)$ such that $X_m$ has no étale cover that dominates a variety of general type but it dominates the orbifold $(S,Δ)$ of general type. Following Campana, the threefolds $X_m$ are called \emph{weakly special} but not \emph{special}. The Weak Specialness Conjecture predicts that a weakly special variety defined over a number field has a potentially dense set of rational points. We prove that if $m$ is big enough the threefolds $X_m$ present behaviours that contradict the function field and analytic analogue of the Weak Specialness Conject…
