Jorge Vitorio Pereira
We will use flat divisors, and canonically associated singular holomorphic foliations, to investigate some of the geometry of compact complex manifolds. The paper is mainly concerned with three distinct problems: the existence of fibrations, the topology of smooth hypersurfaces and the topological closure of transcendental leaves of foliations.
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We study positivity properties of the moduli (b-)divisor associated to a relative log pair $(X,B)/Y$ with relatively trivial log canonical class.
We study the Iitaka-Kodaira dimension of nef relative anti-canonical divisors. As a consequence, we prove that given a complex projective variety with klt singularities, if the anti-canonical divisor is nef, then the dimension of a general fibre of the maximal rationally connected fibration is at least the Iitaka-Kodaira dimension of the anti-canonical divisor.
We study lc pairs polarized by a nef and log big divisor. After proving the minimal model theory for projective lc pairs polarized by a nef and log big divisor, we prove the effectivity of the Iitaka fibrations and some boundedness results for dlt pairs polarized by a nef and log big divisor.
We give a simple geometric proof of the decomposition theorem in terms of Thom-Whitney stratifications by reduction to fibrations by normal crossings divisors over the strata and explain the relation with the local purity theorem an unpublished result of Deligne and Gabber.
Roughly speaking, by using the semi-stable minimal model program, we prove that the moduli part of an lc-trivial fibration coincides with that of a klt-trivial fibration induced by adjunction after taking a suitable generically finite cover. As an application, we obtain that the moduli part of an lc-trivial fibration is b-nef and abundant by Ambro's result on klt-trivial fibrations.
We prove that the moduli b-divisor of an lc-trivial fibration from a log canonical pair is log abundant. The result follows from a theorem on the restriction of the moduli b-divisor, based on a theory of lc-trivial morphisms, which allows us to treat $\mathbb{R}$-divisors and proper morphisms possibly with disconnected fibres. We also prove a theorem on extending a finite cover over a closed subvariety to that over a variety in arbitrary characteristic.
The aim of this paper is to study the Okounkov bodies associated to abundant divisors. As a main result, we prove that the valuative Okounkov bodies of an abundant divisor encode all the numerical properties. We apply this result to recover the asymptotic base loci of an abundant divisor from the valuative Okounkov bodies. We also give a criterion of when the valuative and limiting Okounkov bodies of an abundant divisor coincide by comparing their Euclidean volumes. To obtain these results, we prove some variants of Fujita's approximations for Okounkov bodies using Iitaka fibrations.
Transcendental b-divisors I -- Correspondence with currents
研究紧Kähler流形上的超越b-除子，建立闭正当前与 nef b-除子的对应关系，并通过应用建立nef b-除子的交理论，回答了Dang-Favre的问题。
In this paper we investigate non-rationality of divisors on 3-fold log Fano fibrations $(X,B)\to Z$ under mild conditions. We show that if $D$ is a component of $B$ with coefficient $\ge t>0$ which is contracted to a point on $Z$, then $D$ is birational to $\mathbb{P}^1\times C$ where $C$ is a smooth projective curve with gonality bounded depending only on $t$. Moreover, if $t>\frac{1}{2}$, then genus of $C$ is bounded depending only on $t$.
Using Gauss-Manin derivatives of normal functions, we arrive at some remarkable results on the non-triviality of the transcendental regulator for $K_m$ of a very general projective algebraic manifold. Our strongest results are for the transcendental regulator for $K_1$ of a very general $K3$ surface. We also construct an explicit family of $K_1$ cycles on $H \oplus E_8 \oplus E_8$-polarized $K3$ surfaces, and show they are indecomposable by a direct evaluation of the real regulator. Critical use is made of natural elliptic fibrations, hypersurface normal forms, and an explicit parametrization …
Transcendental Brauer elements are notoriously difficult to compute. Work of Wittenberg, and later, Ieronymou, gives a method for computing 2-torsion transcendental classes on surfaces that have a genus 1 fibration with rational 2-torsion in the Jacobian fibration. We use ideas from a descent paper of Poonen and Schaefer to remove this assumption on the rational 2-torsion.
Automorphism groups and Homogeneous fibrations on projective varieties with nef anticanonical divisors
本文研究具有 nef 反典范除子的代数簇的 Albanese 态射与极大有理链连通纤维化的齐次结构，构造自然群同态诱导 Aut^0 的 Chevalley 分解（相差等源），并类比 Nishi-Matsumura 定理，用现代语言重证之。
On singular fibers of parabolic fibrations
本文描述了模除子数值平凡且判别除子为零的抛物纤维化的奇异纤维结构。
Transcendental b-divisors II -- Monotonicity theorem
本文发展了 nef b-除子的一般交理论，扩展了可移动交理论，定义了 b-除子的限制体积，并证明了交积的单调性定量版本，从而得出新的当前体和上同调类体积不等式。
Minimal degree fibrations in curves and the asymptotic degree of irrationality of divisors
本文研究复射影簇中高次数超曲面的非有理度，证明相关映射经由有理纤维化分解，并引入极小纤维化次数给出紧界，推出不平衡高次完全交的非有理度渐近为次数乘积。
Transcendental Epsilon Multiplicity via Divisor Volumes
本文证明ε重数可以取超越数，通过一个单理想公式将渐近余长度不变量转化为除子体积的几何与算术，并利用移位射影丛构造和Baker定理得到超越值。
Irregular threefolds with numerically trivial canonical divisor
本研究对正特征下具有数值平凡canonical除数的不规则三维流形进行分类，揭示其双纤维结构及显式描述。
On the Iitaka conjecture for anticanonical divisors in positive characteristic
本文研究正特征下纤维化中的Iitaka型不等式，证明在三重或曲线目标时，若一般纤维正则且诱导对强F-正则，则成立。同时给出特征2和3下非正常纤维的反例。
In this paper we are interested in the isotopy classes of symplectic log Calabi-Yau divisors in a fixed symplectic rational surface. We give several equivalent definitions and prove the stability, finiteness and rigidity results. Motivated by the problem of counting toric actions, we obtain a general counting formula of symplectic log Calabi-Yau divisors in a restrictive region of $c_1$-nef cone. A detailed count in the case of 2- and 3-point blow-ups of complex projective space for all symplectic forms is also given. In our framework the complexity of the combinatorics of analyzing Delzant po…
We classify, up to automorphisms, the elliptic fibrations on the singular K3 surface $X$ whose transcendental lattice is isometric to $\langle 6\rangle\oplus \langle 2\rangle$.
Stein degree on non-Fano type fibrations
研究非Fano型log Calabi-Yau纤维上垂直除子的Stein度无界性。
Let X be a smooth, complex Fano variety. For every prime divisor D in X, we set c(D):=dim ker(r:H^2(X,R)->H^2(D,R)), where r is the natural restriction map, and we define an invariant of X as c_X:=max{c(D)|D is a prime divisor in X}. In a previous paper we showed that c_X<9, and that if c_X>2, then either X is a product, or X has a flat fibration in Del Pezzo surfaces. In this paper we study the case c_X=2. We show that up to a birational modification given by a sequence of flips, X has a conic bundle structure, or an equidimensional fibration in Del Pezzo surfaces. We also show a weaker prope…
Stein degree over fields of characteristic zero
该研究针对特征零域，证明广义对数 Calabi-Yau 纤维化的水平除子或 Fano 型纤维化的垂直除子的斯坦因次数，可被与基域无关的常数界定。
Given a complex projective manifold $X$ and a divisor $D$ with normal crossings, we say that the logarithmic tangent bundle $T_X(-\log D)$ is R-flat if its pull-back to the normalization of any rational curve contained in $X$ is the trivial vector bundle. If moreover $-(K_X+D)$ is nef, then the log canonical divisor $K_X+D$ is torsion and the maximally rationally chain connected fibration turns out to be a smooth locally trivial fibration with typical fiber $F$ being a toric variety with boundary divisor $D_{|F}$.
Given a generic $K3$ surface $Y_k$ of the Apéry-Fermi pencil, we use the Kneser-Nishiyama technique to determine all its non isomorphic elliptic fibrations. These computations lead to determine those fibrations with 2-torsion sections T. We classify the fibrations such that the translation by T gives a Shioda-Inose structure. The other fibrations correspond to a K3 surface identified by it transcendental lattice. The same problem is solved for a singular member $Y_2$ of the family showing the differences with the generic case. In conclusion we put our results in the context of relations betwee…
We study linear functions on fibrations whose central fibre is a linear free divisor. We analyse the Gauss-Manin system associated to these functions, and prove the existence of a primitive and homogenous form. As a consequence, we show that the base space of the semi-universal unfolding of such a function carries a Frobenius manifold structure.
Let $f \colon (X,Δ) \to Y$ be a fibration such that $K_X + Δ$ is torsion along the fibres of $f$. Assume that $Y$ has dimension 2, or that $Y$ has dimension 3 and the fibres have dimension at most 3. Then the restriction of the moduli part to its augmented base locus is semiample.
Let $X$ be a general divisor in $|3M+nL|$ on the rational scroll $\mathrm{Proj}(\oplus_{i=1}^{4}\mathcal{O}_{\mathbb{P}^{1}}(d_{i}))$, where $d_{i}$ and $n$ are integers, $M$ is the tautological line bundle, $L$ is a fibre of the natural projection to $\mathbb{P}^{1}$, and $d_{1}\geqslant...\geqslant d_{4}=0$. We prove that $X$ is rational $\iff$ $d_{1}=0$ and $n=1$.
Fundamental properties of basic slc-trivial fibrations, II
本文证明了基本slc-平凡纤维化中模$\mathbb Q$-b-除子的b-数值平凡蕴含$\mathbb Q$-b-线性平凡，并得出基空间为曲线时模部分半丰富的推论。
In this paper we show that a smooth toric variety $X$ of Picard number $r\leq 3$ always admits a nef primitive collection supported on a hyperplane admitting non-trivial intersection with the cone $\Nef(X)$ of numerically effective divisors and cutting a facet of the pseudo-effective cone $\Eff(X)$, that is $\Nef(X)\cap\partial\overline{\Eff}(X)\neq\{0\}$. In particular this means that $X$ admits non-trivial and non-big numerically effective divisors. Geometrically this guarantees the existence of a fiber type contraction morphism over a smooth toric variety of dimension and Picard number lowe…
A known conjecture of Grinenko in birational geometry asserts that a Mori fibre space with the structure of del Pezzo fibration of low degree is birationally rigid if and only if its anticanonical class is an interior point in the cone of mobile divisors. The conjecture is proved to be true for smooth models (with a generality assumption for degree 3). It is speculated that the conjecture holds for, at least, Gorenstein models in degree 1 and 2. In this article, I present a (Gorenstein) counterexample in degree 2 to this conjecture.
Optimal bound for singularities on Fano type fibrations of relative dimension one
本文针对相对维数为1的Fano型纤维化，在Shokurov猜想的基础上，给出了任意$0<ε≤1$时$(Z,B_Z+M_Z)$为$δ$-lc的最优$δ$值，完善了该领域的奇点最优界研究。
We use the theory of foliations to study the relative canonical divisor of a normalized inseparable base-change. Our main technical theorem states that it is linearly equivalent to a divisor with positive integer coefficients divisible by $p-1$. We deduce many consequences about the fibrations of the minimal model program: for example the general fibers of terminal $3$-fold Mori fiber spaces are normal in characteristic $p\geq 5$ and smooth in characteristic $p\geq 11$.
A criterion for associated bundles and the weak Beauville--Bogomolov decomposition
本文给出纤维丛成为伴随纤维丛的判别准则，证明满足特定条件的满射纤维丛均为伴随纤维丛，进而推出丰反典范除子代数簇的弱博韦尔-博戈莫洛夫分解。
A contraction theorem for divisors fibering over a curve
本文针对纤维化到曲线上的 Q-Cartier 除子，给出了双亚缅收缩存在的充分条件，并作为推论恢复了紧 Kähler 三维簇的收缩结果。
In this article we study the structure of klt projective varieties with nef anticanonical divisor (and more generally, varieties of semi-Fano type), especially the canonical fibrations associated to them. We show that: 1. the Albanese map for such variety is a locally constant fibration (that is, an analytic fibre bundle with connected fibres that $X$ is equal to the product of the universal cover of the Albanese torus by the fibre of the Albanese map quotient by a diagonal action of the fundamenatl group of the Albanese torus); 2. if the smooth locus is simply connected, the MRC fibration of …
On the Chow ring of very general abelian varieties and a question of Pirola
研究维数≥4的非常一般阿贝尔簇及亏格4的非常一般雅可比簇中满足特定条件的除子性质，利用相关结论证明皮罗拉关于库默纤维化有理截面的猜想。
Let $f: \mathbb{C} \to X$ be a transcendental holomorphic curve into a complex projective manifold $X$. Let $L$ be a very ample line bundle on $X$. Let $s$ be a very generic holomorphic section of $L$ and $D$ the zero divisor given by $s$. We prove that the \emph{geometric} defect of $D$ (defect of truncation $1$) with respect to $f$ is zero. We also prove that $f$ almost misses general enough analytic subsets on $X$ of codimension $2$.
We study the geometry of elliptic fibrations satisfying the conditions of Step 8 of Tate's algorithm. We call such geometries F$_4$-models, as the dual graph of their special fiber is the twisted affine Dynkin diagram $\widetilde{\text{F}}_4^t$. These geometries are used in string theory to model gauge theories with the exceptional Lie group F$_4$ on a smooth divisor $S$ of the base. Starting with a singular Weierstrass model of an F$_4$-model, we present a crepant resolution of its singularities. We study the fiber structure of this smooth elliptic fibration and identify the fibral divisors u…
Isoresidual fibrations and the birational geometry of moduli of pointed stable curves
本研究针对带点稳定曲线模空间$\u0001overline{M}_{0,n}$的双有理几何问题，通过亚纯微分分层映射构造非多面极端射线，证明n≥8时其除子伪有效锥非多面，且这类空间不是Mori Dream Space。
Isogenies between $K3$ surfaces of the Apéry-Fermi pencil
该研究针对Apéry-Fermi铅笔中的K3曲面Y₁₀和Y₂，分析其带2阶、3阶挠截面的椭圆纤维化的商曲面Y/τ的结构，明确了不同情形下的商曲面类型，并建立了3-同源与基变换的关联。
Algebraic fibre spaces with strictly nef relative anti-log canonical divisor
本文证明具有严格 nef 相对反 log 典范除子的纤维化必为纤维有理连通的局部常值纤维化，底空间为典范极化双曲流形，并在三维情形下证实了 Campana-Peternell 猜想的奇异版本。
Automorphisms of moduli spaces of principal bundles over a smooth curve
本文研究特征零代数闭域上殆单群G在亏格≥4光滑射影曲线上半稳定G-丛模空间的自同构群，通过分析Hitchin纤维化奇异纤维得到结果，并附带刻画了Hitchin基中两类奇异除子的不可约分支。
Xiao's Degree-Four Fibration and the Polizzi Model
本文从Polizzi分支模型和Xiao的奇异纤维分类出发，推导局部球面辫及其Picard-Lefschetz提升，证明基本群为Z^2，计算Albanese核，并给出奇异CP^2#7CP^2-bar的扭曲双构造。
We study the geometry of elliptic fibrations satisfying the conditions of Step 2 of Tate's algorithm with a discriminant of valuation 4. We call such geometries USp(4)-models, as the dual graph of their special fiber is the twisted affine Dynkin diagram of type C$_2$. These geometries are used in string theory to model gauge theories with the non-simply-laced Lie group USp(4) on a smooth divisor S of the base. Starting with a singular Weierstrass model of a USp(4)-model, we present a crepant resolution of its singularities. We study the fiber structure of this smooth elliptic fibration and ide…
Lagrangian fibrations of holomorphic-symplectic varieties of K3^[n]-type
本文研究 K3^[n] 型紧 Kähler 全纯辛流形上满足本原、零最高交条件的 nef 线丛，证明其线性系给出拉格朗日纤维化，并将一般情形双有理刻画为 K3 曲面上一维稳定挠层模空间的 Tate-Shafarevich 扭曲。
We briefly review the formal picture in which a Calabi-Yau $n$-fold is the complex analogue of an oriented real $n$-manifold, and a Fano with a fixed smooth anticanonical divisor is the analogue of a manifold with boundary, motivating a holomorphic Casson invariant counting bundles on a Calabi-Yau 3-fold. We develop the deformation theory necessary to obtain the virtual moduli cycles of \cite{LT}, \cite{BF} in moduli spaces of stable sheaves whose higher obstruction groups vanish. This gives, for instance, virtual moduli cycles in Hilbert schemes of curves in $\Pee^3$, and Donaldson-- and Grom…
Given a formal flat meromorphic connection over an excellent scheme over a field of characteristic zero, in a previous paper we established existence of good formal structures and a good Deligne-Malgrange lattice after suitably blowing up. In this paper, we reinterpret and refine these results by introducing some related structures. We consider the turning locus, which is the set of points at which one cannot achieve a good formal structure without blowing up. We show that when the polar divisor has normal crossings, the turning locus is of pure codimension 1 within the polar divisor, and henc…
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