Kirill Zainoulline
In the present notes we provide a new uniform way to compute a canonical p-dimension of a split algebraic group G for a torsion prime p using degrees of basic polynomial invariants described by V.Kac. As an application, we compute the canonical p-dimensions for all split exceptional algebraic groups.
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We define and study a new numerical invariant of an algebraic group action which we call the canonical dimension. We then apply the resulting theory to the problem of computing the minimal number of parameters required to define a generic hypersurface of degree d in P^{n-1}.
We present the elementary properties of log canonical centers of log varieties.
In this paper we study certain homogeneous spaces, which we call generalised affine Stiefel algebraic varieties. The main aim is to characterise the canonical divisor of generalised affine Stiefel algebraic varieties in terms of group representations. Affine Stiefel algebraic varieties and in particular $S^{n}$ are two special cases of the generalised affine Stiefel algebraic varieties.
In this note we construct an unlimited family of irregular algebraic surfaces of general type with canonical map of degree $ 8 $, irregularity $ 1 $ and arbitrarily large geometric genus such that the image of the canonical map is not a surface of minimal degree.
We give a necessary and sufficient condition for the canonical divisor to vanish on a quasi-homogeneous affine algebraic variety.
We prove the base point free theorem for big line bundles on a three-dimensional log canonical projective pair defined over the algebraic closure of a finite field.
In this note, we construct a minimal surface of general type with geometric genus p g = 4, self-intersection of the canonical divisor K^2 = 32 and irregularity q = 1 such that its canonical map is an abelian cover of degree 16 of P^1 x P^1.
We survey some recent developments in the study of canonical Kähler metrics on algebraic varieties and their relation with stability in algebraic geometry.
In this note, we construct two minimal surfaces of general type with geometric genus p_g= 3, irregularity q = 0, self-intersection of the canonical divisor K^22 =20,24 such that their canonical map is of degree 20. In one of these surfaces, the canonical linear system has a non-trivial fixed part. These surfaces, to our knowledge, are the first examples of minimal surfaces of general type with canonical map of degree 20.
We study actions of connected algebraic groups on normal algebraic varieties, and show how to reduce them to actions of affine subgroups.
In this note, we construct some minimal smooth surfaces of general type with canonical map of degree $ 13, 15, 17, 18, 21, 22 $. These surfaces are constructed as $ \mathbb{Z}_{3}^2$-covers of a blow-up of $ \mathbb{P}^1 \times \mathbb{P}^1 $.
On base point free theorem and Mori dream spaces for log canonical threefolds over the algebraic closure of a finite field
本文在特征大于5时去除了大性条件，将有限域代数闭包上三维对数典范射影对的基点自由定理推广至一般线丛，并讨论了此类域上的Mori dream空间。
Smooth surfaces have finitely generated canonical rings and projective canonical models. For normal surfaces, however, the graded ring of multicanonical sections is possibly nonnoetherian, such that the corresponding homogeneous spectrum is noncompact. I construct a canonical compactification by adding finitely many non-Q-Gorenstein points at infinity, provided that each Weil divisor is numerically equivalent to a Q-Cartier divisor. Similar results hold for arbitrary Weil divisors instead of the canonical class.
In this paper, we collect the fundamental basic properties of jet modules in algebraic geometry and related properties of differential operators. We claim no originality but we want to provide a reference work for own research and the research of other people.
For nonsingular projective 3-folds $X$ (of general type) whose geometric genus $p_g=h^0(X, K_X)$ is $\geq 5$, the birationality of the fourth canonical map $φ_{4,X}=Φ_{|4K_X|}$ was characterized by D.-Q. Zhang and the first author in 2008. This paper aims at characterizing the birationality of $φ_{4,X}$ for those $X$ with $p_g=4$.
Given a reflexive sheaf on a mildly singular projective variety, we prove a flatness criterion under certain stability conditions. This implies the algebraicity of leaves for sufficiently stable foliations with numerically trivial canonical bundle such that the second Chern class does not vanish. Combined with the recent work of Druel and Greb-Guenancia-Kebekus this establishes the Beauville-Bogomolov decomposition for minimal models with trivial canonical class.
This note aims to improve known numerical bounds proved earlier by Chen \cite{PAMS} and Chen-Hacon \cite{Chen-Hacon} and to present some new examples of smooth minimal 3-folds canonically fibred by surfaces (resp. curves) of geometric genus as large as 19 (resp. 13). As an interesting by-product, we present a new class of general type surfaces which are canonically fibred by curves of genus 13.
Canonical bundle formula and a conjecture on certain algebraic fiber spaces by Schnell
本文利用典范丛公式，在两种独立假设下证明了Schnell关于代数纤维空间的猜想，其中一种假设涉及刚性流表示的典范类。
The Hamiltonian theory of isomonodromy equations for meromorphic connections with irregular singularities on algebraic curves is constructed. An explicit formula for the symplectic structure on the space of monodromy and Stokes matrices is obtained. The Whitham equations for the isomonodromy equations are derived. It is shown that they provide a flat connection on the space of the spectral curves of the Hitchin systems.
In this article we present a 3-dimensional analogue of a well-known theorem of E. Bombieri (in 1973) which characterizes the bi-canonical birationality of surfaces of general type. Let $X$ be a projective minimal 3-fold of general type with $\mathbb{Q}$-factorial terminal singularities and the geometric genus $p_g(X)\ge 5$. We show that the 4-canonical map $ϕ_4$ is {\it not} birational onto its image if and only if $X$ is birationally fibred by a family $\mathscr{C}$ of irreducible curves of geometric genus 2 with $K_X\cdot C_0=1$ where $C_0$ is a general irreducible member in $\mathscr{C}$.
Remarks on Relative Canonical Bundles and Algebraicity Criteria for Foliations in Kähler context
受W. Ou近期预印本启发，本研究在Kähler背景下推进相对典范丛正性研究，证明Ou的叶状结构代数性准则，并小幅推广其单直纹性准则。
The aim of this paper is to give another proof to a result on the image of a homogeneous quadratic map which is positive with respect to a homogeneous cone, given by Graczyk and Ishi in 2014. The new proof depends on a purely algebraic method, whereas the original depends on analytic arguments. This enables us to give explicit formulas of the basic relative invariants of homogeneous cones, obtained by the previous paper [J. Lie Theory 24 (2014), 1013--1032] without analytic arguments.
Quantization of canonical cones of algebraic curves
本文利用形式伪微分算子对代数曲线典范锥上的函数分次代数进行量子化，并在 Poincaré 单值化情形下借助 Rankin-Cohen 括号给出等价构造，还描述了有理曲线时的量子代数表示。
We show that a minimal surface of general type has a canonical symplectic structure (unique up to symplectomorphism) which is invariant for smooth deformation. We show that the symplectomorphism type is also invariant for deformations which allow certain normal singularities, provided one remains in the same smoothing component. We use this technique to show that the Manetti surfaces yield examples of surfaces of general type which are not deformation equivalent but are canonically symplectomorphic.
Singularities of Foliations and Good Moduli Spaces of Algebraic Stacks
本文基于最小模型程序理论，研究叶状结构的奇点与代数堆的良模空间，证明log-canonical奇点的稳定子群为有限扩张的代数torus，从而局部存在良模空间，且canonical奇点的稳定点locus非空。
Let $(X,D)$ be a log smooth pair of dimension $n$, where $D$ is a reduced effective divisor such that the log canonical divisor $K_X + D$ is pseudo-effective. Let $G$ be a connected algebraic subgroup of $\mathrm{Aut}(X,D)$. We show that $G$ is a semi-abelian variety of dimension $\le \min\{n-\barκ(V), n\}$ with $V := X\setminus D$. In the dimension two, Shigeru Iitaka claimed in his 1979 Osaka J. Math. paper that $\dim G\le \bar{q}(V)$ for a log smooth surface pair with $\barκ(V) = 0$ and $\bar{p}_g(V) = 1$. We (re)prove and generalize this classical result for all surfaces with $\barκ=0$ wit…
We describe the basic cohomology ring of the canonical holomorphic foliation on a moment-angle manifold, LVMB-manifold or any complex manifold with a maximal holomorphic torus action. Namely, we show that the basic cohomology has a description similar to the cohomology ring of a complete simplicial toric variety due to Danilov and Jurkiewicz. This settles a question of Battaglia and Zaffran, who previously computed the basic Betti numbers for the canonical holomorphic foliation in the case of a shellable fan. Our proof uses an Eilenberg-Moore spectral sequence argument; the key ingredient is t…
We introduce the notion of generalized MR log canonical surfaces and establish the minimal model theory for generalized MR log canonical surfaces in full generality.
The paper is a colloquial-style discussion of invariants of algebraic surfaces analogous to the Donaldson polynomials, arising from moduli spaces of ``jumping'' Yang--Mills instantons, or moduli spaces of jumping vector bundles. The invariants have the following applications: (1) to the Van de Ven conjecture that the Kodaira dimension is a diffeomorphism invariant; (2) to proving that algebraic surfaces with $p_g > 0$ have a proper sublattice of $H^2(X,\Z)$ invariant under diffeomorphism; (3) to proving the same result as (2) for surfaces with $p_g = 0$, in particular the Barlow surface.
In this paper, we study the relative anti-canonical divisor $-K_{X/Y}$ of an algebraic fiber space $ϕ: X \to Y$, and we reveal relations among positivity conditions of $-K_{X/Y}$, certain flatness of direct image sheaves, and variants of the base loci including the stable (augmented, restricted) base loci and upper level sets of Lelong numbers. This paper contains three main results: The first result says that all the above base loci are located in the horizontal direction unless they are empty. The second result is an algebraic proof for Campana--Cao--Matsumura's equality on Hacon--$\rm{M^c}$…
We show that the Kahler-Ricci flow on an algebraic manifold of positive Kodaira dimension and semi-ample canonical line bundle converges to a unique canonical metric on its canonical model. It is also shown that there exists a canonical measure of analytic Zariski decomposition on an algebraic manifold of positive Kodaira dimension. Such a canonical measure is unique and invariant under birational transformations under the assumption of the finite generation of canonical rings.
We prove that the canonical bundle of any holomorphic family of compact complex algebraic manifolds carries a singular Hermitian metric having non-negative curvature current and such that every holomorphic section of the canonical bundle of the central fiber is L^2 with respect to this metric. This result was proved by Siu when the members of the family are general type, and used to establish the deformation invariance of plurigenera in that case. In fact, we prove our result in a more general setting of a smooth hypersurface with not-necessarily normal trivial bundle in a manifold that is Ste…
Let X be a projective 3-fold with at most Q-factorial terminal singularities on which K_X is nef and big. Suppose the canonical index r(X)>1. For any positive integer m, it is interesting to consider the base point freeness and birationality of the divisor mK_X. For example, we know the following results: (1) the system |5rK_X| is base point free (Ein-Lazarsfeld-Lee); (2) |mK_X| gives a birational map for all m>4r+2 (M. Hanamura). This article aims to present a better result in direction (2). As far as our method can tell here, |mK_X| gives a birational map for all m>2r+5. (Q-divisor method + …
Non-projective complete log canonical surfaces
本文在任意特征代数闭域（除有限域的代数闭包外）上构造了典范除子半丰沛的非射影完全对数典范代数曲面，并统一处理了所有非负小平维数（0,1,2），同时证明了小平维数为负无穷的完全对数典范代数曲面必为射影的。
Reconstructing real algebraic maps locally like moment-maps with prescribed images and compositions with the canonical projections to the $1$-dimensional real affine space
本研究提出一类局部类似矩映射、原像为单点集或球面乘积的非正余维数显式实代数映射，补充了Nash-Tognoli理论下显式实代数映射实例不足的问题，并给出其与典范投射复合的显式函数族。
Let C be a curve (possibly non reduced or reducible) lying on a smooth algebraic surface. We show that the canonical ring R(C, ω_C) is generated in degree 1 if C is numerically 4-connected, not hyperelliptic and even (i.e. with K_C of even degree on every component). As a corollary we show that on a smooth algebraic surface of general type with p_g(S)>0 and q(S)=0 the canonical ring R(S, K_S) is generated in degree \leq 3 if there exists a curve C in |K_S| numerically 3-connected and not hyperelliptic.
Splitting of algebraic fiber spaces with nef relative anti-canonical divisor and decomposition of $F$-split varieties
本文证明在完美域上具有nef相对反典范除子的代数纤维空间经有限覆盖基变换后分裂为乘积，并推广了正特征Beauville-Bogomolov分解，应用于丰沛猜想与有理点问题。
Given a compact complex manifold X of dimension n, we define a bimeromorphic invariant $κ_+(X)$ as the maximum p for which there is a saturated line subsheaf L of the sheaf of holomorphic p forms whose Kodaira dimension $κ(L)$ equals p. We call X special if $κ_+(X)=0$. We give some evidence that this condition characterizes X to have identically vanishing Kobayashi pseudometric. We use the well-known construction of F. Campana to give, for each projective X a canonical fibration $f: X\to Y$ holomorphic outside a proper subvariety of Y and whose general fibers are special. We show that the inhe…
We impose a rather unknown algebraic structure called a `hyperstructure' to the underlying space of an affine algebraic group scheme. This algebraic structure generalizes the classical group structure and is canonically defined by the structure of a Hopf algebra of global sections. This paper partially generalizes the result of A.Connes and C.Consani
We study the spaces of stable real and quaternionic vector bundles on a real algebraic curve. The basic relationship is established with unitary representations of an extension Z/2 by the fundamental group. By comparison with the space of real or quaternionic connections, some of the basic topological invariants of these spaces are calculated.
We construct a complex algebraic surface with geometric genus $p_g=3$, irregularity $q=0$, self-intersection of the canonical divisor $K^2=24$ and canonical map of degree $24$ onto $\mathbb P^2$.
Some algebraic surfaces with canonical map of degree 10, 12, 14
本文通过$\mathbb{Z}_2^3$-覆盖有理曲面，构造了典范映射次数为10、12、14且具有特定不规则性的代数曲面，结合已有结果证明了d=10和12时所有不规则性均可实现，d=14时不规则性为2的情形仍待解决。
Some remark on real algebraic maps which are topologically special generic maps and generalize the canonical projections of the unit spheres
本文针对推广单位球面典型投影的拓扑特殊一般映射，构造了像为光滑嵌入流形的实代数映射，作为先前构造的补充注记。
The main motivation for the study of cluster algebras initiated in math.RT/0104151, math.RA/0208229 and math.RT/0305434 was to design an algebraic framework for understanding total positivity and canonical bases in semisimple algebraic groups. In this paper, we introduce and explicitly construct the canonical basis for a special family of cluster algebras of rank 2.
Let $Y$ be a generic link of a subvariety $X$ of a nonsingular variety $A$. We give a description of the Grauert-Riemenschneider canonical sheaf of $Y$ in terms of the multiplier ideal sheaves associated to $X$ and use it to study the singularities of $Y$. As the first application, we give a criterion when $Y$ has rational singularities and show that log canonical threshold increases and log canonical pairs are preserved in generic linkage. As another application we give a quick and simple liaison method to generalize the results of de Fernex-Ein and Chardin-Ulrich on the Castelnuovo-Mumford r…
In the present work we establish the equality of the canonical metric of two commuting maps on an algebraic variety X. As a consequence the canonical height and measure associated to both maps are identical.
Let G be a semisimple algebraic group over an algebraically closed field of positive characteristic. In this note, we show that an irreducible closed subvariety of the flag variety of G is compatibly split by the unique canonical Frobenius splitting if and only if it is a Richardson variety, i.e. an intersection of a Schubert and an opposite Schubert variety.
We discuss the classical problem of counting planes tangent to general canonical sextic curves at three points. We determine the number of real tritangents when such a curve is real. We then revisit a curve constructed by Emch with the greatest known number of real tritangents, and conversely construct a curve with very few real tritangents. Using recent results on the relation between algebraic and tropical theta characteristics, we show that the tropicalization of a canonical sextic curve has 15 tritangent planes.
Let $(S,D)$ be a minimal log pair of general type with $S$ a smooth projective surface and $D$ a simple normal corssing reduced divisor on $S$. We assume that its log canonial linear system $|K_S+D|$ is composed of a penciel, let $f\colon S\to B$ be the fiberation induced by the linear system $|K_S+D|$ and $F$ be a general fiber of $f$. Let $b$ (resp. $g$) be the genus of the base curve $B$ (resp. general fiber $F$) and $k=D\cdot F$ the intersection number. We show that 1. If $k>0$ and $b\geq 2$ then $2\leq g+k \leq 3$, when $g+k=3$ we have $b=2$ and $h^1(S,K_S+D)=0$. 2. Suppose $p_a(D)\leq 2(…
The purpose of this note is to review an algebraic proof of the finite generation theorem due to Birkar-Cascini-Hacon-McKernan whose method is based on the Minimal Model Program. A survey article for Current Development in Mathematics 2007.
We give the basic definitions of group actions on (algebraic) stacks, and prove the existence of fixed points and quotients as (algebraic) stacks.
In this note we present examples of complex algebraic surfaces of general type with canonical maps of degree $10$, $11$ and $14$. They are constructed as quotients of a product of two Fermat septics using certain free actions of the group $\mathbb Z_7^2$.
We prove that the log canonical ring of a klt pair of dimension $3$ with $\mathbb{Q}$-boundary over an algebraically closed field of characteristic $p>5$ is finitely generated. In the process we prove log abundance for such pairs in the case $κ=2$.
The Deligne-Lusztig varieties associated to the Coxeter classes of the algebraic groups 2A2, 2B2 and 2G2 are affine algebraic curves. We produce explicit projective models of the closures of these curves. Furthermore for $d$ the Coxeter number of these groups, we find polynomials for each of these models that cut out the $\F_q$-points, the $\F_{q^d}$-points and the $\F_{q^{d+1}}$-points, and demonstrate a relation satisfied by these polynomials.
Descent for Algebraic Schemes
本文以初等方式系统阐述了域上代数概形的基本下降定理，涵盖 Grothendieck、Weil 等经典结果。
Let X be a projective variety which is algebraic Lang hyperbolic. We show that Lang's conjecture holds (one direction only): X and all its subvarieties are of general type and the canonical divisor K_X is ample at smooth points and Kawamata log terminal points of X, provided that K_X is Q-Cartier, no Calabi-Yau variety is algebraic Lang hyperbolic and a weak abundance conjecture holds.
These are lecture notes from the IMPANGA 2010 Summer school. They give an introduction to log canonical thresholds, covering some basic properties, examples, and some recent results and open questions.
This set of notes provides some additional explanatory material on the analytic proof of the finite generation of the canonical ring for a compact complex algebraic manifold of general type.
For the coordinate algebras of connected affine algebraic groups, we explore the problem of finding a presentation by generators and relations canonically determined by the group structure.
We survey known results on the canonical bundle formula and its applications in algebraic geometry.
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