Gueorgui Todorov
In this paper we will prove that for a threefold of general type and large volume the second plurigenera is positive and the fifth canonical map is birational.
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This note mainly studies the generic finiteness of ϕ_m of a complex projective 3-fold of general type. A new result on the classification to bicanonical pencil for Gorenstein 3-folds is attached in the last section.
This paper aims to investigate effectivity problems of pluricanonical systems on varieties of general type in positive characteristic. In practice, we will consider a sub-linear system $|S^0_{-}(X, K_X + nK_X)| \subseteq |H^0(X, K_X +nK_X)|$ generated by certain Frobenius stable sections, and prove that for a minimal terminal threefold $X$ of general type with either $q(X)>0$ or Gorenstein singularities, if $n\geq 28$ then $|S^0_{-}(X, K_X + nK_X)| \neq \emptyset$; if $n\geq 42$ then the linear system $|S^0_{-}(X, K_X + nK_X)|$ defines a birational map.
We investigate birationality of the anti-pluricanonical map $ϕ_{-m}$, the rational map defined by the anti-pluricanonical system $|-mK|$, on $\mathbb{Q}$-Fano threefolds.
We explicitly find lower bounds on the volume of threefolds and fourfolds of general type in order to have nonvanishing of pluricanonical systems and birationality of pluricanonical maps. In the case of threefolds of large volume, we also give necessary and sufficient conditions for the fourth canonical map to be birational.
On generic finiteness of pluricanonical maps of threefolds of general type
本文证明了对满足P2(X)≥2的一般型极小三维流形X，|6K_X|定义通用有限映射，且该结果最优；同时证明对所有一般型极小三维流形X，当n≥38时|nK_X|定义通用有限映射。关键技术是曲面的新通用有限性准则和特殊解消下的有效比较不等式。
We prove that the r-th pluricanonical maps of threefolds of general type have birationally bounded fibers if $r\geqslant 2$. Similarly, we prove that the r-th pluricanonical maps of fourfolds of general type have birationally bounded fibers if $r\geqslant 4$. We extend these results to higher dimensions in terms of constants arising naturally from the birational geometry of varieties of general type.
We prove that the anti-pluricanonical map $Φ_{|-mK_{X}|}$ is birational when $m\geq 16$ for 5-fold $X$ whose anticanonical divisor is nef and big.
We prove that there is a fixed constant r=r_n, such that if X is a variety of general type, then the rth pluricanonical map is birational.
We prove that for any smooth complex projective threefold of Kodaira dimension one, the $m$-th pluricanonical map is birational to the Iitaka fibration for every $m\geq5868$ and divisible by $12$.
We give a method for the computation of the plurigenera of a product-quotient manifold. We give two different types of applications to it: to the construction of Calabi-Yau threefolds and to the determination of the minimal model of a product-quotient surface of general type.
In this paper we will prove a uniformity result for the Iitaka fibration $f:X \rightarrow Y$, provided that the generic fiber has a good minimal model and the variation of $f$ is zero or that $κ(X)=\rm{dim}(X)-1$.
In this paper we prove that for any smooth projective variety of Albanese fiber dimension two and of general type, the 6-canonical map is birational. And we also show that the 5-canonical map is birational for any such variety with some geometric restrictions.
Let $X$ be a smooth complex projective algebraic variety of maximal Albanese dimension. We give a characterization of $κ(X)$ in terms of the set $V^0(X,ω_{X})$ $:=\{P\in {\text{\rm Pic}}^0(X)|h^0(X, ω_X \otimes P) \ne 0\}$. An immediate consequence of this is that the Kodaira dimension $κ(X)$ is invariant under smooth deformations. We then study the pluricanonical maps $ϕ_m:X -> \Bbb{P} (H^0(X,mK_X))$. We prove that if $X$ is of general type, $ϕ_m$ is generically finite for $m\geq 5$ and birational for $m\geq 5 \text{\rm dim} (X) +1$. More generally, we show that for $m\geq 6$ the image of $ϕ_…
We describe examples showing the sharpness of Fujita's conjecture on adjoint bundles also in the general type case, and use these examples to formulate related bold conjectures on pluricanonical maps of varieties of general type.
This paper aims to improve a theorem of Janos Kollar. For a given Complex projective threefold X of general type, suppose the plurigenus p_k(X)\ge 2, Kollar proved that the (11k+5)-canonical map is birational. Here we show that either the (7k+3)-canonical map or the (7k+5)-canonical map is birational and that the m-canonical map is stably birational for m\ge 13k+6. If P_k(X)\ge 3, then the m-canonical map is stably birational for m\ge 10k+8. In particular, the 12-canonical map is birational when p_g(X)\ge 2 and the 11-canonical map is birational when p_g(X)\ge 3.
In this article we prove the following boundedness result: Fix a DCC set $I\subset [0, 1]$. Let $\mathfrak{D}$ be the set of all log pairs $(X, Δ)$ satisfying the following properties: (i) $X$ is a projective surface defined over an algebraically closed field, (ii) $(X, Δ)$ is log canonical and the coefficients of $Δ$ are in $I$, and (iii) $K_X+Δ$ is big. Then there is a positive integer $N=N(I)$ depending only on the set $I$ such that the linear system $|\lceil m(K_X+Δ)\rceil|$ defines a birational map onto its image for all $m\geq N$ and $(X, Δ)\in\mathfrak{D}$
Pluricanonical Geometry of Varieties Isogenous to a Product and Abelian Covers
研究同源于积的三次曲面的典范映射，证明当$p_g \ge 5$时4-典范映射为理性映射，并构造最大典范度的例子。
On bicanonical maps of threefolds of general type with large volumes
研究一般型三维流形大体积的双CANONICAL映射的维度以及大canonical体积的pluricanonical映射的纤维结构
Stable surfaces and their log analogues are the type of varieties naturally occuring as boundary points in moduli spaces. We extend classical results of Kodaira and Bombieri to this more general setting: if $(X,Δ)$ is a stable log surface with reduced boundary (possibly empty) and $I$ is its global index, then $4I(K_X+Δ)$ is base-point-free and $8I(K_X+Δ)$ is very ample. These bounds can be improved under further assumptions on the singularities or invariants, for example, $5(K_X+Δ)$ is very ample if $(X,Δ)$ has semi-canonical singularities.
In this thesis we looked into three different problems which share, as a common factor, the exstensive use of the Fourier--Mukai transform as research tool. In the first Part we investigated the syzygies of Kummer varieties (i.e. quotients of abelian varieties by the $\mathbb{Z}/2\Z$-action induced by the group operation), extending to higher syzygies results on projective normality and degree of equations of Sasaki, Kempf, and Khaled. The second Part of this Thesis (partially written in collaboration with Z.Jiang and M. Lahoz) is dedicated to the study of pluricanonical linear systems on vari…
In this paper we study pluricanonical maps of minimal projective 3-folds of general type with geometric genus $1$, $2$ and $3$. We go in the direction pioneered by Enriques and Bombieri, and other authors, pinning down, for low projective genus, a finite list of exceptions to the birationality of some pluricanonical map. In particular, apart from a finite list of weighted baskets, we prove the birationality of $φ_{16}$, $φ_{6}$ and $φ_{5}$ respectively.
On pluricanonical boundedness of varieties of general type
本文给出 Chen–Jiang 定理的新证明，修正原证明漏洞，并利用该方法改进 Lacini 的有界性定理，证明一般型 n 维簇的 r-典范映射具有双有理有界纤维，从而肯定回答 Chen 和 Jiang 2014 年提出的问题。
The minimal volume of stable surfaces of rank one
本文确定了秩为一的稳定表面的最小体积，并证明达到此最小值的表面在同构意义下是唯一的。该结果解决了Alexeev和第二作者的猜想。
We pose some questions about spaces parametrizing rational curves on rationally connected varieties. We give a partial answer for cubic threefolds. Many of our results were previously proved by Iliev, Markushevich and Tikhimirov by analyzing vector bundles on cubic threefolds. Our proofs use residuation rather than vector bundles.
The dominant rational maps of finite degree from a fixed variety to varieties of general type, up to birational isomorphisms, form a finite set. This has been known as the Iitaka-Severi conjecture, and is nowdays an established result, in virtue of some recent advances in the theory of pluricanonical maps. We study the question of finding some effective estimate for the finite number of maps, and to this aim we provide some update and refinement of the classical treatment of the subject.
Let $S$ be a nonsingular minimal complex projective surface of general type and the canonical map of $S$ is generically finite. Beauville showed that the geometric genus of the image of the canonical map is vanishing or equals the geometric genus of $S$ and discussed the canonical degrees for these two cases. We generalize his results to nonsingular minimal complex projective threefolds.
It has long been known that to a complex cubic surface or threefold one can canonically associate a principally polarized abelian variety. We give a construction which works for cubics over an arithmetic base. This answers, away from the prime 2, an old question of Deligne and a recent question of Kudla and Rapoport. We further classify the Mumford-Tate groups of the abelian varieties which arise, and give additional arithmetic applications.
Geography and Deformations of $\mathbb{Z}_2^s$-Covers of General Type Over Weighted Projective Threefolds
研究一般类型三维流形作为$\mathbb{Z}_2^s$-覆盖的加权投影空间的构造，探讨其不变量、变形理论及$m$-canonical映射行为，提出新的模空间构造准则并分类$\mathbb{Z}_2^s$-覆盖的平滑性条件。
If X is a nonsingular curve in a Calabi--Yau threefold Y whose normal bundle N_{X/Y} is a generic semistable bundle, are the local Gromov-Witten invariants of X well defined? For X of genus two or higher, the issues are subtle. We will formulate a precise line of inquiry and present some results, some positive and some negative.
We study the complexity of birational self-maps of a projective threefold $X$ by looking at the birational type of surfaces contracted. These surfaces are birational to the product of the projective line with a smooth projective curve. We prove that the genus of the curves occuring is unbounded if and only if $X$ is birational to a conic bundle or a fibration into cubic surfaces. Similarly, we prove that the gonality of the curves is unbounded if and only if $X$ is birational to a conic bundle.
This paper has two aims. The former is to give an introduction to our earlier work on the Hodge theory of algebraic maps and more generally to some of the main themes of the theory of perverse sheaves and to some of its geometric applications. Particular emphasis is put on the topological properties of algebraic maps. The latter is to prove a motivic version of the decomposition theorem for the resolution of a threefold Y. This result allows to define a pure motive whose Betti realization is the intersection cohomology of Y. We provide a few explicit computations of perverse truncations and in…
We give a class of examples of reducible (d-semistable) threefolds of CY type with two irreducible components for which (it is reasonably easy to prove that) no family of admissible genus zero stable maps sweeps out a surface, yet such stable maps occur in infinitely many degrees.
We characterise smooth curves in a smooth cubic threefold whose blow-ups produce a weak-Fano threefold. These are curves $C$ of genus $g$ and degree $d$, such that (i) $2(d-5) \le g$ and $d\le 6$; (ii) $C$ does not admit a 3-secant line in the cubic threefold. Among the list of ten possible such types $(g,d)$, two were previously left as open numerical possibilities, namely $(g,d) = (0,5)$ and $(2,6)$. Using the Sarkisov link associated with a curve of type $(2,6)$, we are able to produce the first example of a pseudo-automorphism with dynamical degree greater than $1$ on a smooth threefold wi…
We prove that the image of the pluricanonical representation of a group of bimeromorphic automorphisms of a complex manifold has bounded finite subgroups. As a consequence, we show that the group of bimeromorphic automorphisms of an $n$-dimensional complex manifold whose Kodaira dimension is at least $n-2$, satisfies the Jordan property.
We classify minimal projective 3-folds of general type with $p_g = 2$ by studying the birationality of their 6-canonical maps.
Prym varieties and cubic threefolds over $\mathbb{Z}$
本文发展了特征2域上的Prym簇与三次三维簇理论，证明了光滑三次三维簇在任意域上的非有理性及Torelli定理，并解决了Deligne关于算术Torelli映射的猜想。
We introduce new obstructions to rationality for geometrically rational threefolds arising from the geometry of curves and their cycle maps.
Orbifold Aspects of Certain Occult Period Maps
本文刻画了三次三维簇和四次簇的霍奇结构自同构群，并确定了小维数射影流形自同构群在循环覆盖霍奇结构上的作用，证实了Kudla和Rapoport的猜想。
In this article we exhibit certain projective degenerations of smooth $K3$ surfaces of degree $2g-2$ in $\Bbb P^g$ (whose Picard group is generated by the hyperplane class), to a union of two rational normal scrolls, and also to a union of planes. As a consequence we prove that the general hyperplane section of such $K3$ surfaces has a corank one Gaussian map, if $g=11$ or $g\geq 13$. We also prove that the general such hyperplane section lies on a unique $K3$ surface, up to projectivities. Finally we present a new approach to the classification of prime Fano threefolds of index one, which doe…
The second fundamental form of the moduli space of cubic threefolds in $\mathcal A_5$
研究 $\mathcal A_5$ 中 Siegel 度量限制在三次三维流形的中间雅可比轨迹上的第二基本形式，证明其像包含于某个乘法映射的核中。
Extendability of general $K3$ surfaces without Gaussian maps and classification of non-prime Fano threefolds
本文利用退化到 $K3$ 地毯的方法，无需 Gauss 映射即证明一般非素 $K3$ 曲面可延展性、非素 Fano 三维簇分类及 Hilbert 概形不可约性，并给出素 $K3$ 曲面无穷多不可延展亏格。
Infinitesimal Torelli problems for special Gushel-Mukai and related Fano threefolds: Hodge theoretical and categorical perspectives
本文研究特殊Gushel-Mukai及相关Fano三维簇的无穷小Torelli问题，证明特殊Gushel-Mukai三维簇的无穷小周期映射不变部分单射，并给出周期映射微分核的几何描述。
Quartic curves in the quintic del Pezzo threefold
本文证明五次 del Pezzo 三维簇 X5 上有理四次曲线的 Hilbert 概形 H4(X5) 同构于 X5 上直线的 Hilbert 概形上的 Grassmannian 丛，因此光滑且不可约。
A Conditional Reduction of the Rational Hodge Conjecture for Threefolds and Deformation-Theoretic Verifications in Several Families
本文提出几何逼近假设 Hypothesis~BB，证明其蕴含三流形的有理 Hodge 猜想，并利用形变理论判据在 Calabi--Yau、Fano 及一般五次三流形族中验证了该假设，将其约化为可计算条件。
GIT stable cubic threefolds and certain fourfolds of $K3^{[2]}$-type
本文研究光滑与奇异节点三次三维簇与 $K3^{[2]}$ 型超凯勒四维簇模空间之间的同构在节点超平面上的退化行为，推广至 $A_i$ 型奇点情形，并引入 $K$ 型凯勒锥截面概念以处理 $A_2$ 情形。
Let $C \subset P^{g-1}$ be a smooth canonical curve of genus $g \geq 3$. The purpose of this article is to further develop a method to classify varieties having $C$ as their curve section, using Gaussian map computations. In a previous article a careful analysis of the degeneration to the cone over the hyperplane section was made for _prime_ Fano threefolds, that is Fano threefolds whose Picard group is generated by the hyperplane bundle. In this article we extend this method and classify Fano threefolds of higher index (which still have Picard number one). We are also able to classify Mukai v…
Weakly-special threefolds and non-density of rational points
本文验证了Campana猜想的一部分，通过建立轨道映射模空间性质并利用Kobayashi-Ochiai定理的推广，证明了Bogomolov-Tschinkel构造的弱特殊三维簇上有理点不稠密。
We present the first explicit examples of a rational threefold and a Calabi-Yau threefold, admitting biregular automorphisms of positive entropy not preserving any dominant rational maps to lower positive dimensional varieties. The most crucial part is the rationality of the quotient threefold of a 3-dimensional torus of product type.
This is the second of a series of papers studying real algebraic threefolds using the minimal model program. The main result is the following. Let $X$ be a smooth projective real algebraic 3-fold. Assume that the set of real points is an orientable 3-manifold (this assumption can be weakened considerably). Then there is a fairly simple description on how the topology of real points changes under the minimal model program. The first application is to study the topology of real projective varieties which are birational to projective 3-space (Nash conjecture). The second application is a factoriz…
First we find effective bounds for the number of dominant rational maps $f:X \rightarrow Y$ between two fixed smooth projective varieties with ample canonical bundles. The bounds are of the type $\{A \cdot K_X^n\}^{\{B \cdot K_X^n\}^2}$, where $n=dimX$, $K_X$ is the canonical bundle of $X$ and $A,B $ are some constants, depending only on $n$. Then we show that for any variety $X$ there exist numbers $c(X)$ and $C(X)$ with the following properties: For any threefold $Y$ of general type the number of dominant rational maps $f:X \r Y$ is bounded above by $c(X)$. The number of threefolds $Y$, modu…
Decomposition of the diagonal, intermediate Jacobians, and universal codimension-2 cycles in positive characteristic
本文将复三维簇稳定有理性的两类对角上同调分解障碍推广到正特征有理链连通三维簇，证明相关代数代表元的典范自对偶性，并指出特征>2时带7结点的非常一般四次双立体消解不存在泛余维2闭链类。
Algebraic hyperbolicity of very general hypersurfaces in projective spaces
本文证明了射影空间中非常一般超曲面代数双曲性的最后未解情形，即四次三维簇，完成了分类，方法基于Coskun--Riedl构造及稳定映射模空间和对数几何。
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