Antonio Laface, Luca Ugaglia
In this note we consider the behavior of linear systems of P^3 through fat points under a cubo-cubic Cremona transformation. This allows us to produce a class of special systems which we conjecture to be the only ones.
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We introduce a method to construct special holomorphic tensors on orthogonal modular varieties from scalar-valued modular forms, and give applications to the Lang conjecture on the birational type of subvarieties of orthogonal modular varieties.
We use the language and tools available in model theory to redefine and clarify the rather involved notion of a {\em special subvariety} known from the theory of Shimura varieties (mixed and pure).
We prove two special cases of a conjecture of J. Fernández de Bobadilla for hypersurfaces with $1$-dimensional critical loci. We do this via a new numerical invariant for such hypersurfaces, called the beta invariant, first defined and explored by the second author in 2014. The beta invariant is an algebraically calculable invariant of the local ambient topological-type of the hypersurface, and the vanishing of the beta invariant is equivalent to the hypotheses of Bobadilla's conjecture.
Using orbifold metrics of the appropriately signed Ricci curvature on orbifolds with negative or numerically trivial canonical bundle and the two-dimensional Log Minimal Model Program, we prove that the fundamental group of special compact Kähler threefolds is almost abelian. This property was conjectured in all dimensions in [Cam04b], and also for orbifolds in [Cam07], where the notion of specialness was introduced. We briefly recall below the definition, basic properties, and the role of special manifolds in classification theory.
Let X be a Fano variety of dimension n, pseudoindex i_X and Picard number ρ_X. A generalization of a conjecture of Mukai says that ρ_X(i_X-1)\le n. We prove that the conjecture holds if: a) X has pseudoindex i_X \ge \frac{n+3}{3} and either has a fiber type extremal contraction or does not have small extremal contractions b) X has dimension five.
Informed by the Bloch-Beilinson conjectures, Voisin has made a conjecture about $0$-cycles on self-products of Calabi-Yau varieties. In this note, we consider variant versions of Voisin's conjecture for cubic fourfolds, and for hyperkähler varieties. We present examples for which these conjectures are verified, by considering certain very special cubic fourfolds and their Fano varieties of lines.
Symmetric products and puncturing Campana-special varieties
本文通过uniruled表面的对称积，推翻了Hassett-Tschinkel的打孔猜想，并提出修正猜想，同时验证了Campana关于潜在密度的猜想。
Motivated by the Beauville-Voisin conjecture about Chow rings of powers of $K3$ surfaces, we consider a similar conjecture for Chow rings of powers of EPW sextics. We prove part of this conjecture for the very special EPW sextic studied by Donten-Bury et alii. We also prove some other results concerning the Chow groups of this very special EPW sextic, and of certain related hyperkähler fourfolds.
The Segre-Gimigliano-Harbourne-Hirschowitz Conjecture can be naturally formulated for Hirzebruch surfaces F_n. We show that this Conjecture holds for imposed base points of equal multiplicity bounded by 8.
In this paper we prove the $Γ_{00}$ conjecture of van Geemen and van der Geer, under the additional assumption that the matrix of coefficients of the tangent has rank at most 2. This assumption is satisfied by Jacobians, and thus our result gives a characterization of the locus of Jacobians among all principally polarized abelian varieties. The proof is by reduction to the (stronger version of the) characterization of Jacobians by semidegenerate trisecants, i.e. by the existence of lines tangent to the Kummer variety at one point and intersecting it in another, proven by Krichever in the cours…
O'Grady conjectured that the Chow group of 0-cycles of the generic fiber of the universal family over the moduli space of polarized K3 surfaces of genus g is cyclic. This so-called generalized Franchetta conjecture has been solved only for low genera where there is a Mukai model (precisely, when g<11 and g=12, 13, 16, 18, 20), by the work of Pavic--Shen--Yin. In this paper, as a non-commutative analog, we study the Franchetta property for families of special cubic fourfolds (in the sense of Hassett) and relate it to O'Grady's conjecture for K3 surfaces. Most notably, by using special cubic fou…
We propose a conjecture on special values of $ L $-functions in a function field context with positive characteristic coefficients. For $ M $ a uniformizable $ t $-motive with everywhere good reduction we conjecture a relation between the value of the Goss $ L $-function $ L(M^\vee, s) $ at $ s = 0 $ and the uniformization of the abelian $ t $-module associated with $ M $. When $ M $ is a power of the Carlitz $ t $-motive the conjecture specializes to a theorem of Anderson and Thakur on Carlitz zeta values. Beyond this case we present numerical evidence.
For complete intersection Calabi-Yau manifolds in toric varieties, Gross and Haase-Zharkov have given a conjectural combinatorial description of the special Lagrangian torus fibrations whose existence was predicted by Strominger, Yau and Zaslow. We present a geometric version of this construction, generalizing an earlier conjecture of the first author.
We introduce the notion of dR-absolutely special subvarieties in motivic variations of Hodge structure as special subvarieties cut out by (de Rham-)absolute Hodge cycles and conjecture that all special subvarieties are dR-absolutely special. This is implied by Deligne's conjecture that all Hodge cycles are absolute Hodge cycles, but is a much weaker conjecture. We prove our conjecture for subvarieties satisfying a simple monodromy condition introduced by Klingler-Otwinowska-Urbanik. We study applications to typical respectively atypical intersections and $\bar{\mathbb{Q}}$-bialgebraic subvarie…
The Buchsbaum-Eisenbud-Horrocks rank conjecture proposes lower bounds for the Betti numbers of a graded module M based on the codimension of M. We prove a special case of this conjecture via Boij-Soederberg theory. More specifically, we show that the conjecture holds for graded modules where the regularity of M is small relative to the minimal degree of a first syzygy of M. Our approach also yields an asymptotic lower bound for the Betti numbers of powers of an ideal generated in a single degree.
The Weakly Special Conjecture contradicts orbifold Mordell, and hence the abc conjecture
该研究通过构造特定的弱特殊三维流形，证明了弱特殊猜想与orbifold莫尔德猜想矛盾，进而影响abc猜想，并展示了Enriques和K3表面的非可除退化情况。
For a fixed integer $d$, we study here the locus of degree $d$ hypersurfaces $X$ in $\mathbb{P}^{2n+1}$ such that $H^{2n}(X,\mathbb{Q}) \cap H^{n,n}(X,\mathbb{C}) \not= \mathbb{Q}$. We call this locus \textit{the Noether-Lefschetz locus}. Any irreducible component of this locus is locally a Hodge locus. So, we see that the study of this locus is very closely related to the variational Hodge conjecture. In this article we show that the Hodge locus corresponding to the cohomology class of a complete intersection subscheme in $\mathbb{P}^{2n+1}$ of codimension $n+1$ and degree less than $d$ satis…
Motivic integration on special rigid varieties and the motivic integral identity conjecture
本文通过发展特殊刚性簇上的等变动机积分新框架，证明了Kontsevich和Soibelman的动机积分恒等式猜想，并建立了具有Fubini型性质的动机体积概念。
We obtain results on the so-called Andre-Pink-Zannier conjecture which is a special case of a the Zilber-Pink conjecture on unlikely intersections in Shimura varieties. Our methods rely on an ergodic theorem of Richard-Zamojski and we are able to obtain stronger conclusions that those of the Andre-Pink-Zannier conjecture in the special case we consider. We work under the assumption of the S-Shafarevich conjecture and S-semisimplicity conjecture which hold for Shimura varieties of abelian type.
We prove in this paper the Ax-Lindemann-Weierstrass theorem for all mixed Shimura varieties and discuss the lower bounds for Galois orbits of special points of mixed Shimura varieties. In particular we reprove a result of Silverberg in a different approach. Then combining these results we prove the André-Oort conjecture for any mixed Shimura variety whose pure part is a subvariety of A_6^n.
We show the Alon-Tarsi conjecture on Latin squares is equivalent to a very special case of a conjecture made independently by Hadamard and Howe, and to the non-vanishing of some interesting integrals over SU(n). Our investigations were motivated by geometric complexity theory.
We define an analogue of the `Real' Deligne cohomology group at a prime of semi-stable or good reduction of a variety. We also define regulator maps to this group and formulate a conjecture about the image. This allows us to formulate a non-Archimedian version of Beilinson's Hodge-D-conjecture, S-integral and function field versions of Beilinson's global conjectures as well as a precise special value conjecture in the function field case. Finally we give a few examples where these conjectures are known to be true.
Equivalent conjectures on blowing-ups of $\mathbb P^2$
该研究刻画了代数曲面上nef且big除子的渐近特殊性，证明射影平面极一般位置点爆破上线性系统的SHGH猜想等价于nef类非特殊，并给出了$r < 2^n$时nef除子渐近非特殊的结论。
We formulate a general conjecture relating Chern classes of subbundles of Gauss-Manin bundles in Arakelov geometry to logarithmic derivatives of Artin L-functions of number fields. This conjecture may be viewed as a far-reaching generalisation of the (Lerch-)Chowla-Selberg formula computing logarithms of periods of elliptic curves in terms of special values of the $Γ$-function. We prove several special cases of this conjecture in the situation where the involved Artin characters are Dirichlet characters. This article contains the computations promised in the article {\it Conjectures sur les dé…
Beginning with the conjecture of Artin and Tate in 1966, there has been a series of successively more general conjectures expressing the special values of the zeta function of an algebraic variety over a finite field in terms of other invariants of the variety. In this article, we present the ultimate such conjecture, and provide evidence for it. In particular, we enhance Voevodsky's Z[1/p]-category of etale motivic complexes with a p-integral structure, and show that, for this category, our conjecture follows from the Tate and Beilinson conjectures. As the conjecture is stated in terms of mot…
Let $k$ be a base field and $G$ be an algebraic group over $k$. J.-P. Serre defined $G$ to be special if every $G$-torsor $T \to X$ is locally trivial in the Zariski topology for every reduced algebraic variety $X$ defined over $k$. In recent papers an a priori weaker condition is used: $G$ is called special if every $G$-torsor $T \to Spec(K)$ is split for every field $K$ containing $k$. We show that these two definitions are equivalent. We also generalize this fact and propose a strengthened version of the Grothendieck-Serre conjecture based on the notion of essential dimension.
We introduce $\ell$-Galois special subvarieties as an $\ell$-adic analog of the Hodge-theoretic notion of a special subvariety. The Mumford-Tate conjecture predicts that both notions are equivalent. We study some properties of these subvarieties and prove this equivalence for subvarieties satisfying a simple monodromy condition. As applications, we show that the $\ell$-Galois exceptional locus is a countable union of algebraic subvarieties and, if the derived group of the generic Mumford-Tate group of a family is simple, its part of positive period dimension coincides with the Hodge locus of p…
In this article first we prove that a special case of the order ideal conjecture, originating from the work of Evans and Griffith in equicharacteristic, implies the monomial conjecture due to M. Hochster. We derive a necessary and sufficient condition for the validity of this special case in terms certain syzygis of canonical modules of normal domains possessing free summands. We also prove some special cases of this observation.
We consider Calabi-Yau $n$-folds $X$ arising from certain hyperplane arrangements. Using Fu-Vial's theory of distinguished cycles for varieties with motive of abelian type, we show that the subring of the Chow ring of $X$ generated by divisors, Chern classes and intersections of subvarieties of positive codimension injects into cohomology. We also prove Voisin's conjecture for $X$, and Voevodsky's smash-nilpotence conjecture for odd-dimensional $X$.
On Special Subvarieties of the Universal Semi-abelian Scheme and Zilber--Pink Conjectures
本文分类普遍半阿贝尔概形的特殊子簇，证明Zilber--Pink猜想在混合Shimura簇与半阿贝尔簇间的蕴含关系，并修正相对Manin--Mumford猜想以涵盖Ribet型子簇。
In this note we give a counterexample to a conjecture proposed by Ciliberto about special linear systems of P^n through multiple base points.
We prove that the universal covering space of a complex projective manifold is holomorphically convex provided its fundamental group is linear.
This note contains some examples of hyperkähler varieties $X$ having a group $G$ of non-symplectic automorphisms, and such that the action of $G$ on certain Chow groups of $X$ is as predicted by Bloch's conjecture. The examples range in dimension from $6$ to $132$. For each example, the quotient $Y=X/G$ is a Calabi-Yau variety which has interesting Chow-theoretic properties; in particular, the variety $Y$ satisfies (part of) a strong version of the Beauville-Voisin conjecture.
We prove the isogeny property for special fibres of integral canonical models of compact Shimura varieties of $A_n$, $B_n$, $C_n$, and $D_n^{\dbR}$ type. The approach used also shows that many crystalline cycles on abelian varieties over finite fields which are specializations of Hodge cycles, are algebraic. These two results have many applications. First, we prove a variant of the conditional Langlands--Rapoport conjecture for these special fibres. Second, for certain isogeny sets we prove a variant of the unconditional Langlands--Rapoport conjecture (like for many basic loci). Third, we prov…
The André-Pink-Zannier conjecture concerns the intersection of subvarieties and the generalized Hecke orbit of a given point in mixed Shimura varieties. It is part of the Zilber-Pink conjecture. In this paper we focus on the universal family of principally polarized abelian varieties. We explain the moduli interpretation of the conjecture in this case and prove several different cases for this conjecture: its overlap with the André-Oort conjecture; when the subvariety is contained in an abelian scheme over a curve and the point is a torsion point on its fiber; when the subvariety is a curve.
Zeta function of F-gauges and special values
本文针对有限域上可对偶$F$-gauges，提出并证明Artin–Tate型Zeta函数特殊值猜想的无条件形式，并借助稳定Bockstein特征及栈式$F$-gauge理论使其无条件适用于光滑真簇。
We study special cycles on a Shimura variety of orthogonal type over a totally real field of degree $d$ associated with a quadratic form in $n+2$ variables whose signature is $(n,2)$ at $e$ real places and $(n+2,0)$ at the remaining $d-e$ real places for $1\leq e <d$. Recently, these cycles were constructed by Kudla and Rosu-Yott and they proved that the generating series of special cycles in the cohomology group is a Hilbert-Siegel modular form of half integral weight. We prove that, assuming the Beilinson-Bloch conjecture on the injectivity of the higher Abel-Jacobi map, the generating serie…
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…
In the moduli space of degree d polynomials, the special subvarieties are those cut out by critical orbit relations, and then the special points are the post-critically finite polynomials. It was conjectured that in the moduli space of degree d polynomials, the subvarieties containing a Zariski-dense set of special points are exactly these special subvarieties. In this article, we prove the first non-trivial case for this conjecture: the case of cubic polynomials.
The SYZ Conjecture explains Mirror Symmetry between mirror Calabi-Yau 3-folds M,M' in terms of special Lagrangian fibrations f : M --> B and f' : M' --> B over the same base B, whose fibres are dual 3-tori, except for singular fibres. One of the main problems in proving the SYZ Conjecture (or even in finding the right statement of it) is that the singularities of special Lagrangian 3-folds and fibrations are poorly understood. This paper studies the singularities of special Lagrangian fibrations. Our main rigorous results are the construction of examples of special Lagrangian fibrations on ope…
Consider a smooth, projective family of canonically polarized varieties over a smooth, quasi-projective base manifold Y, all defined over the complex numbers. It has been conjectured that the family is necessarily isotrivial if Y is special in the sense of Campana. We prove the conjecture when Y is a surface or threefold. The proof uses sheaves of symmetric differentials associated to fractional boundary divisors on log canonical spaces, as introduced by Campana in his theory of Orbifoldes Geometriques. We discuss a weak variant of the Harder-Narasimhan Filtration and prove a version of the Bo…
We explore an analogue of the André-Oort conjecture for subvarieties of Drinfeld modular varieties. The conjecture states that a subvariety $X$ of a Drinfeld modular variety contains a Zariski-dense set of complex multiplication (CM) points if and only if $X$ is a "special" subvariety (i.e. $X$ is defined by requiring additional endomorphisms). We prove this conjecture in two cases. Firstly when $X$ contains a Zariski-dense set of CM points with a certain behaviour above a fixed prime (which is the case if these CM points lie in one Hecke orbit), and secondly when $X$ is a curve containing inf…
The p-adic distance of special points to subvarieties
本文综述了关于半阿贝尔簇上挠点到子簇的p-adic距离一致上界的猜想及其类似情形下的研究进展。
The Bombieri-Dwork conjecture predicts that the differential equations satisfied by $G$-functions come from geometry. In this paper, we will look at special $G$-functions whose differential equations have a special singularity with maximally unipotent monodromy. We will formulate a Gamma conjecture about such $G$-functions, which has close connections with the mirror symmetry of Calabi-Yau threefolds and the Gamma conjecture in algebraic geometry. We will provide examples to support this conjecture, which involves numerical computations using Mathematica programs.
Modularity of special cycles on Shimura varieties: a survey
本文综述了Shimura变种上特殊循环生成函数的模性猜想，并提出相关猜想及更一般周期域商的猜想。
Let $\mathbb{V}$ be a polarized variation of integral Hodge structure on a smooth complex quasi-projective variety $S$. In this paper, we show that the union of the non-factor special subvarieties for $(S, \mathbb{V})$, which are of Shimura type with dominant period maps, is a finite union of special subvarieties of $S$. This generalizes previous results of Clozel and Ullmo arXiv:math/0404131, Ullmo \cite{Ullmo07} on the distribution of the non-factor (in particular, strongly) special subvarieties in a Shimura variety to the non-classical setting and also answers positively the geometric part …
A Conjecture of Flach and Morin
本文证明了Flach和Morin关于混合特征DVR上概型的p进Beilinson-Hyodo-Kato上同调中单值作用与特殊纤维上同调关系的猜想，在特殊纤维为正特征曲线上几何族纤维的情形下成立。
The paper gives a counter-example to the relative version of the Manin-Mumford conjecture.
Let $S$ be a Shimura variety and let $h$ be a Weil height function on $S$. We conjecture that the heights of special points in $S$ are discriminant negligible. Assuming this conjecture to be true, we prove that the sizes of the Galois orbits of special points grow as a fixed power of their discriminant (an invariant we will define in the text). In the case of Shimura varieties of abelian type, the height bound holds by the recently proved averaged Colmez formula, and our theorem gives a new proof of Tsimerman's Galois lower bound in this case. The main novelty is that our approach avoids the u…
We consider refined conjectures of Birch and Swinnerton-Dyer type for the Hasse-Weil-Artin L-series of abelian varieties over general number fields. We shall, in particular, formulate several new such conjectures and establish their precise relation to previous conjectures, including to the relevant special case of the equivariant Tamagawa number conjecture. We also derive a wide range of concrete interpretations and explicit consequences of these conjectures that, in general, involve a thoroughgoing mixture of difficult archimedean considerations related to refinements of the conjecture of De…
Let E be an elliptic curve over Q with complex multiplication by the ring of integers of an imaginary quadratic field K. In 1991, by studying a certain special value of the Katz two-variable p-adic L-function lying outside the range of $p$-adic interpolation, K. Rubin formulated a p-adic variant of the Birch and Swinnerton-Dyer conjecture when $E(K)$ is infinite, and he proved that his conjecture is true for E(K) of rank one. When E(K) is finite, however, the statement of Rubin's original conjecture no longer applies, and the relevant special value of the appropriate $p$-adic L-function is equ…
A solution to the Yau-Tian-Donaldson Conjecture through Special Fujita Approximations
通过证明光滑射影簇上大线丛存在特殊Fujita逼近，并利用Boucksom-Jonsson-Li的先前工作，解决了非阿基米德熵泛函的Boucksom-Jonsson正则化猜想，进而得到Yau-Tian-Donaldson猜想（一致版本）的解决：极化光滑射影簇(X,L)存在cscK度量当且仅当它是Aut°(X,L)-一致K-稳定的。
Lusztig's special pieces conjecture
研究简单代数群$G$的李代数$\mathfrak g$中特殊幂零轨道的特殊片，通过两种方法证明其是光滑$G$ - 簇被有限群作用的商，适用于经典$\mathfrak g$并给出新证，还涉及例外群及相关嵌入与定义。
We prove a $p$-adic analogue of the André-Oort conjecture for subvarieties of the universal abelian varieties containing a dense set of special points. Let $g$ and $n$ be integers with $n \geq 3$ and $p$ a prime number not dividing $n$. Let $R$ be a finite extension of $W[{\mathbb F}_p^{\mathrm alg}]$, the ring of Witt vectors of the algebraic closure of the field of $p$ elements. The moduli space $\cA = \cA_{g,1,n}$ of $g$-dimensional principally polarized abelian varieties with full level $n$-structure as well as the universal abelian variety $π:\cX \to \cA$ over $\cA$ may be defined over $R…
Weakly Special Manifolds with no rational curves
本文在假设丰盛猜想和终端卡拉比-约当簇上存在稠密有理曲线的前提下，证明了无有理曲线的复投影弱特殊流形是阿贝尔簇的 étale 商。若流形包含稠密整个曲线，则在假设朗猜想下同样成立，表明非超广复投影流形包含阿贝尔簇的像。
We make a conjecture about mean curvature flow of Lagrangian submanifolds of Calabi-Yau manifolds, expanding on \cite{Th}. We give new results about the stability condition, and propose a Jordan-Hölder-type decomposition of (special) Lagrangians. The main results are the uniqueness of special Lagrangians in hamiltonian deformation classes of Lagrangians, under mild conditions, and a proof of the conjecture in some cases with symmetry: mean curvature flow converging to Shapere-Vafa's examples of SLags.
In 1996, Strominger, Yau and Zaslow made a conjecture about the geometric relationship between two mirror Calabi-Yau manifolds. Roughly put, if X and Y are a mirror pair of such manifolds, then X should possess a special Lagrangian torus fibration $f:X\to B$ such that Y is obtained by dualizing the fibration f. This leaves a huge amount to be done to verify such conjectures. This paper takes a speculative point of view, in that it assumes that a special Lagrangian torus fibration exists on X. We address a number of questions of a topological nature: what is the relationship between the cohomol…
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