Yu. G. Prokhorov, V. V. Shokurov
We prove the boundedness of complements modulo two conjectures: Borisov-Alexeev conjecture and effective adjunction for fibre spaces. We discuss the last conjecture and prove it in two particular cases.
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Combinatorial Nonresonance Theorems for Hyperplane Arrangement Complements
本文研究超平面排列补集上复秩一局部系统的非共振现象，提出组合充分条件，并改进Bailet、Dimca和Yoshinaga的定理，同时发展限制与提升技术证明线排列的非共振定理。
The aim of this paper is to clarify and generalize techniques of works alg-geom/9711024 (see also math.AG/9810097 and math.AG/9901004). Roughly speaking, we prove that for local Fano contractions the existence of complements can be reduced to the existence of complements for lower dimensional projective Fano varieties.
On existence of bounded relative-global complements for Fano fibrations
本文证明了固定维数且具 $ε$-lc 奇点的 Fano 纤维化存在有界相对全局补，并在基为一维时进一步证明了有界相对全局 klt 补的存在性。
In this note we extend some of the results of a previous paper \url{arXiv:math/0511593} to algebraically closed fields of finite characteristic. In particular, we show that there is an explicit expression in $n$ and $d$ which is divisible by the prime to $p$ part of the order of the the automorphism group of a smooth degree $d$ hypersurface $\subset \mathbb{P}^n_k$ for $k$ an algebraically closed field of characteristic $p$.
An elementary proof of Hilbert's theorem on ternary quartics: Some complements
本文在2012年提出的初等证明基础上，进一步简化了Hilbert关于实系数非负三元四次型可表示为三个二次型平方和的定理证明。
We formulate and prove a weighted version of Zariski's hyperplane section theorem on the topological fundamental groups of the complements of hypersurfaces in a projective space. As an application, we calculate fundamental groups of the complements of certain singular projective plane curves.
We give an accessible introduction to global singularity theory including the basic structure theorems by Damon, and Pragacz and Weber.
We prove finiteness results on integral points on complements of large divisors in projective varieties over finitely generated fields of characteristic zero. To do so, we prove a function field analogue of arithmetic finiteness results of Corvaja-Zannier and Levin using Wang's function field Subspace Theorem. We then use a method of Evertse-Győry for concluding finiteness of integral points over finitely generated fields from known finiteness results over number fields.
Local and global invariant cycle theorems for Hodge modules
该研究论证了Hodge模的局部与整体不变循环定理可通过一般理论直接推导，并补充了相关研究的评述内容。
We establish a global Torelli theorem for the complete family of Calabi-Yau threefolds arising from cyclic triple covers of $\mathbb P^3$ branched along stable hyperplane arrangements.
More strong version of the main inductive theorem about the complements on surfaces is proved and the models of exceptional log del Pezzo surfaces with $δ=0$ are constructed
Survey written for the Proceedings of the AMS Meeting on Algebraic Geometry, Seattle, 2005. Based on the talk delivered at this occasion, but a few comments on recent developments are added.
We propose a generalization of Verbitsky's global Torelli theorem in the framework of compact Kähler irreducible holomorphically symplectic orbifolds by adapting Huybrechts' proof (arXiv:1106.5573). As intermediate step needed, we also provide a generalization of the twistor space and the projectivity criterion based on works of Campana (arXiv:math/0402243) and Huybrechts (arXiv:math/0106014) respectively.
The Prym variety for a branched double covering of a nonsingular projective curve is defined as a polarized abelian variety. We prove that any double covering of an elliptic curve which has more than $4$ branch points is recovered from its Prym variety.
A Strong Global Torelli theorem for OG6 type varieties
本文证明 OG6 型超凯勒流形由其次积分上同调及其 Hodge 结构和 Beauville--Bogomolov--Fujiki 形式唯一决定，并由此推出双亚纯等价即双全纯等价。
Algebraic surfaces of general type with $q=0$, $p_g=2$ and $K^2=1$ were described by Enriques and then studied in more detail by Horikawa. In this paper we consider a $16$-dimensional family of special Horikawa surfaces which are certain bidouble covers of $\mathbb{P}^2$. The construction is motivated by that of special Kunev surfaces which are counterexamples for infinitesimal Torelli and generic global Torelli problem. The main result of the paper is a generic global Torelli theorem for special Horikawa surfaces. To prove the theorem, we relate the periods of special Horikawa surfaces to the…
Hyperbolicity of Fermat-type curves and their complements
本文利用广义Borel定理证明了Fermat型曲线及其补集的双曲性，改进了现有度数界结果。
The main purpose of this paper is to show that the mixed Hodge polynomial of the ``space of equations'' for smooth complete intersections of given multidegree in $\mathbb{C} P^n$ is divisible by the mixed Hodge polynomial of the group $\mathrm{GL}_{n+1}(\mathbb{C})$, the quotient being the mixed Hodge polynomial of the corresponding quotient space. As a by-product of the method used in the proof, we obtain expressions divisible by the order the automorphism group of any smooth projective hypersurface of given dimension and degree
Towards the Global Torelli Theorem
本文证明所有维数下K-平凡簇的有限性结果，利用极化K3曲面的整体托雷利定理建立纤维同构的霍奇理论充分条件，并应用于双有理等价纤维稠密的射影族，得出所有纤维同构的结论。
We describe the proof that the period map from the Torelli space of Calabi-Yau manifolds to the classifying space of polarized Hodge structures is an embedding. The proof is based on the constructions of holomorphic affine structure on the Teichmüller space and Hodge metric completion of the Torelli space. A canonical global holomorphic section of the holomorphic $(n, 0)$ class on the Teichmüller space is constructed.
We prove a "purity implies formality" statement in the context of the rational homotopy theory of smooth complex algebraic varieties, and apply it to complements of hypersurface arrangements. In particular, we prove that the complement of a toric arrangement is formal. This is analogous to the classical formality theorem for complements of hyperplane arrangements, due to Brieskorn, and generalizes a theorem of De Concini and Procesi.
Complements, index theorem, and minimal log discrepancies of foliated surface singularities
本文将配对与簇的若干结果推广到叶状曲面对，证明局部补集有界性、局部指标定理及mlds一致有界性，否定Cascini-Spice的两个问题，并恢复叶状曲面mlds升链条件猜想的证明。
We prove Kollár's injectivity theorem for globally $F$-regular varieties.
We prove a global effective membership result for polynomials on a non-reduced algebraic subvariety of $\C^N$. It can be seen as a global version of a recent local result of Sznajdman, generalizing the Briançon-Skoda-Huneke theorem for the local ring of holomorphic functions at a point on a reduced analytic space.
Let C_1 and C_2 be algebraic plane curves in the complex plane such that the curves intersect in d_1\cdot d_2 points where d_1,d_2 are the degrees of the curves respectively. Oka and Sakamoto proved that the fundamental group of the complement of C_1 \cup C_2 is isomorphic to the direct of product of the fundamental group of the complement of C_1 and the fundamental group of the complement of C_2. In this paper we prove the converse of Oka and Sakamoto's result for line arrangements. Let A_1 and A_2 be non-empty arrangements of lines in complex plane such that the fundamental group of the comp…
We prove global effective versions of the Brian\ccon-Skoda-Huneke theorem. Our results extend, to singular varieties, a result of Hickel on the membership problem in polynomial ideals in $\mathbf C^n$, and a related theorem of Ein and Lazarsfeld for smooth projective varieties. The proofs rely on known geometric estimates and new results on multivariable residue calculus.
Weak Akizuki-Nakano vanishing theorem for globally $F$-split 3-folds
本文证明了全局 $F$-分裂 3-折上的弱 Akizuki-Nakano 消没定理，并据此研究了全局 $F$-分裂 Fano 3-折的形变及局部完全交全局 $F$-正则 3-折增厚的 Kodaira 消没定理。
We prove that the integral points are potentially Zariski dense in the complement of a reduced effective singular anticanonical divisor in a smooth del Pezzo surface, with the exception of $\mathbb{P}^2$ minus three concurrent lines (for which potential density does not hold). This answers positively a question raised by Hassett and Tschinkel and, combined with previous results, completes the proof of the potential density of integral points for complements of anticanonical divisors in smooth del Pezzo surfaces. We then classify the complements which are simply connected and for these we prove…
Compact hyperkaehler manifolds are higher-dimensional generalizations of K3 surfaces. The classical Global Torelli theorem for K3 surfaces, however, does not hold in higher dimensions. More precisely, a compact hyperkaehler manifold is in general not determined by its natural weight-two Hodge structure. The text gives an account of a recent theorem of M. Verbitsky, which can be regarded as a weaker version of the Global Torelli theorem phrased in terms of the injectivity of the period map on the connected components of the moduli space of marked manifolds.
Error terms for the motives of discriminant complements and a Cayley-Bacharach theorem
本文在简化假设下证明了关于Cayley-Bacharach集的相关猜想，推导其位于低次曲线的结论，并以此估计光滑超曲面空间归一化motive的误差项，同时证明高维下motive的收敛性等判别式补空间相关结果。
The main result of the paper is a boundedness for $n$-complements on algebraic surfaces. In addition, applications of this theorem to a classification of log Del Pezzo surfaces and of birational contractions for 3-folds are formulated.
As a consequence of our recently established generalized Schmidt's subspace theorem for closed subschemes in general position, we prove a degeneracy theorem for integral points on the complement of a union of nef effective divisors. A novel aspect of our result is the attainment of a strong degeneracy conclusion (arithmetic quasi-hyperbolicity) under weak positivity assumptions on the divisors. The proof hinges on applying our recent theorem with a well-situated ample divisor realizing a certain lexicographical minimax. We also explore the connections with earlier work by other authors and mak…
The result of this paper is proved in arXiv:1112.1163
We systematically study the moduli theory of symplectic varieties (in the sense of Beauville) which admit a resolution by an irreducible symplectic manifold. In particular, we prove an analog of Verbitsky's global Torelli theorem for the locally trivial deformations of such varieties. Verbitsky's work on ergodic complex structures replaces twistor lines as the essential global input. In so doing we extend many of the local deformation-theoretic results known in the smooth case to such (not-necessarily-projective) symplectic varieties. We deduce a number of applications to the birational geomet…
A complex hypersurface D in complex affine n-space C^n is a linear free divisor (LFD) if its module of logarithmic vector fields has a global basis of linear vector fields. We classify all LFDs for n at most 4. Analogous to Grothendieck's comparison theorem, we say that the global logarithmic comparison theorem (GLCT) holds for D if the complex of global logarithmic differential forms computes the complex cohomology of the complement of D in C^n. We develop a general criterion for the GLCT for LFDs and prove that it is fulfilled whenever the Lie algebra of linear logarithmic vector fields is r…
A mapping class group of an oriented manifold is a quotient of its diffeomorphism group by the isotopies. In the published version of "Mapping class group and a global Torelli theorem for hyperkahler manifolds" I made an error based on a wrong quotation of Dennis Sullivan's famous paper "Infinitesimal computations in topology". I claimed that the natural homomorphism from the mapping class group to the group of automorphims of cohomology of a simply connected Kahler manifold has finite kernel. In a recent preprint arXiv:1907.05693, Matthias Kreck and Yang Su produced counterexamples to this st…
In 1933, van Kampen described the fundamental groups of the complements of plane complex projective algebraic curves. Recently, Chéniot-Libgober proved an analogue of this result for higher homotopy groups of the complements of complex projective hypersurfaces with isolated singularities. Their description is in terms of some "homotopical variation operators". We generalize here the notion of "homotopical variation" to (singular) quasi-projective varieties. This is a first step for further generalizations of van Kampen's theorem. A conjecture, with a first approach, is stated in the special ca…
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