Laurent Manivel
This is a survey paper about a selection of results in complex algebraic geometry that appeared in the recent and less recent litterature, and in which rational homogeneous spaces play a prominent r{ô}le. This selection is largely arbitrary and mainly reflects the interests of the author.
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Lecture notes of an algebraic geometry graduate course. The topics covered are as follows. Cohomology: ext sheaves and groups, cohomology with support, local cohomology, local duality. Duality: relative duality, Cohen-Macaulay schemes. Morphisms of schemes: flatness and base change and related topics. Hilbert and Quotient schemes.
Given a connected algebraic group G over an algebraically closed field and a G-homogeneous space X, we describe the Chow ring of G and the rational Chow ring of X, with special attention to the Picard group. Also, we investigate the Albanese and the "anti-affine" fibrations of G and X.
This is a survey paper about a selection of recent results on the geometry of a special class of Fano varieties, which are called of K3 type. The focus is mostly Hodge-theoretical, with an eye towards the multiple connections between Fano varieties of K3 type and hyperkähler geometry.
Let $B$ be a curve defined over an algebraically closed field $k$ and let $X\to B$ be an elliptic surface with base curve $B$. We investigate the geometry of everywhere locally trivial principal homogeneous spaces for $X$, i.e. elements of the Tate-Shafarevich group. If $Y$ is such a principal homogeneous space of order $n$, we find strong restrictions on the $\mathbb{P}^{n-1}$ bundle over $B$ into which $Y$ embeds. Examples for small values of $n$ show that, in at least some cases, these restrictions are sharp. Finally, we determine these bundles in case $k$ has characteristic zero, $B = \mat…
We compute Hoelder Complexes,i.e. the complete bi-Lipschitz invariants, for germs of real weighed homogeneous algebraic or semialgebraic surfaces.
In our work we focus on the geometry of elliptic normal curves of degree 6 embedded in $\mathbb{P}^5$. We determine the space of quadric hypersurfaces through an elliptic normal curve of degree 6 and find the explicit equations of generators of $I(\text{Sec}(C_6))$. We study the images $C_p$ and $C_{pq}$ of a sextic $C_6$ under the projection from a general point $P \in \mathbb{P}^5$ and a general line $\overline{PQ} \subset \mathbb{P}^5$. In particular, we show that $C_p$ is $k$-normal for all $k \geq 2$ and $I(C_p)$ is generated by three homogeneous polynomials of degree 2 and two homogeneou…
This is a report on some of the main developments in birational geometry in the last few years focusing on the minimal model program, Fano varieties, singularities and related topics, in characteristic zero.
Adventures in algebraic geometry
基于2024年春季罗格斯大学代数几何入门课程，以简化技术细节为代价介绍相对高级的主题。
Convexity and Concavity from Complex Variables to Algebraic Geometry
本文针对复分析中核心的凸性、凹性及包络主题，探索其在代数几何中的对应内容，并呈现了该方向的初步研究结果。
We characterize the canonical algebras such that for all dimension vectors of homogeneous modules the corresponding module varieties are complete intersections (respectively, normal). We also investigate the sets of common zeros of semi-invariants of non-zero degree in important cases. In particular, we show that for sufficiently big vectors they are complete intersections and calculate the number of their irreducible components.
In the first half of twentieth century the theory of complex analytic functions and of their zerosets was fully developed. The definition of holomorphic function has a local nature. Germs of holomorphic functions form a distinguished subring of the ring of germs of continuous functions. This way came out the notion of analytic space. The definition of a complex analytic set is by local models as in the case of complex manifolds. But while local models for manifolds are open sets of ${\mathbb C}^n$, a local model of an analytic space is the zeroset of finitely many analytic functions on an open…
The article is dedicated to the memory of O.N. Vvedenskii. Vvedenskii's results are presented as well as selected new results of arithmetic algebraic geometry. Elements of ontology of Vvedenskii's research also given.
We show that a weighted homogeneous complex surface singularity is metrically conical (i.e., bi-Lipschitz equivalent to a metric cone) only if its two lowest weights are equal. We also give an example of a pair of weighted homogeneous complex surface singularities that are topologically equivalent but not bi-Lipschitz equivalent.
Collections of parabolic orbits in homogeneous spaces, homogeneous dynamics and hyperkahler geometry
该研究针对满足$b_2(M)\u003e=5$的超凯勒流形，运用齐性空间遍历理论改进Morrison-Kawamata锥猜想相关结果，证明本原MBM类的Beauville-Bogomolov平方绝对值有仅依赖流形形变类的上界。
This paper is a survey on arc spaces, a recent topic in algebraic geometry and singularity theory. The geometry of the arc space of an algebraic variety yields several new geometric invariants and brings new light to some classical invariants.
Topics in Adic Spaces
本文为海德堡春季学校迷你课程的扩展讲义，系统介绍 adic 空间中的凝聚层、态射性质及其与形式几何和 Berkovich 空间的联系。
Growth of the dimension of the homogeneous components of Color Lie Superalgebras
本文综述代数齐次分量维数增长的理论意义，重点讨论其在多项式环、代数簇及射影簇线丛截面空间中的应用、已有结果与未来方向。
Equivariant geometry of low-dimensional quadrics
该研究聚焦低维二次曲面的等变几何问题，提出了有限群在齐性空间及低维二次曲面上正则作用的新型稳定线性化构造方案。
Formulated problems concern the following topics: (1) Birationally nonequivalent linear actions; (2) Cayley degrees of simple algebraic groups; (3) Singularities of two-dimensional quotients.
We present a new proof of the classification of complex simple Lie algebras via the projective geometry of homogeneous varieties. Our proof proceeds by constructing homogeneous varieties using the ideals of the secant and tangential varieties of homogeneous varieties already constructed. Our algorithms make no reference to root systems. Our proofs use properties of root systems, but not their classification.
On the geometry of some quiver zero loci Fano fourfolds (with an appendix by E. Kalashnikov and F. Tufo)
本文对 Kalashnikov 提出的 170 族箭图旗零轨迹 Fano 四维流形进行研究，将其视为齐性向量丛截面的零轨迹，并给出全部 170 族的双有理与双正则刻画。
We study the projective geometry of homogeneous varieties $X= G/P\subset P(V)$, where $G$ is a complex simple Lie group, $P$ is a maximal parabolic subgroup and $V$ is the minimal $G$-module associated to $P$. Our study began with the observation that Freudenthal's magic chart could be derived from Zak's theorem on Severi varieties and standard geometric constructions. Our attempt to understand this observation led us to discover further connections between projective geometry and representation theory. Among other things, we calculate the variety of tangent directions to lines on $X$ through …
Let $M$ be complete flat pseudo-Riemannian homogeneous manifold and $Γ\subset\Iso(\RR^n_s)$ its fundamental group. We show that $M$ is a trivial fiber bundle $G/Γ\to M\to\RR^{n-k}$, where $G$ is the Zariski closure of $Γ$ in $\Iso(\RR^n_s)$. Moreover, we show that the $G$-orbits in $\RR^n_s$ are affinely diffeomorphic to $G$ endowed with the (0)-connection. If the induced metric on the $G$-orbits is non-degenerate, then $G$ (and hence $Γ$) has linear abelian holonomy. If additionally $G$ is not abelian, then $G$ contains a certain subgroup of dimension 6. In particular, for non-abelian $G$ orb…
We established the associativity of the quantum cohomologies of homogeneous varieties by using degeneration method in algebraic geometry.
We prove the existence of canonical tubular neighbourhoods around complex submanifolds of Kähler manifolds that are adapted to both the holomorphic and symplectic structure. This is done by solving the complex Homogeneous Monge-Ampère equation on the deformation to the normal cone of the submanifold. We use this to establish local regularity for global weak solutions, giving local smoothness to the (weak) geodesic rays in the space of (weak) Kähler potentials associated to a given complex submanifold. We also use it to get an optimal regularity result for naturally defined plurisubharmonic env…
This is an expanded and updated version of a lecture series I gave at Seoul National University in September 1997. It is in some sense an update of the 1979 Griffiths and Harris paper with a similar title. I discuss: Homogeneous varieties, Topology and consequences Projective differential invariants, Varieties with degenerate Gauss images, When can a uniruled variety be smooth?, Dual varieties, Linear systems of bounded and constant rank, Secant and tangential varieties, Systems of quadrics with tangential defects, Recognizing uniruled varieties, Recognizing intersections of quadrics, Recogniz…
What is... Random Algebraic Geometry?
本文综述随机代数几何，该领域将概率视角引入实代数几何，为希尔伯特第十六问题等经典难题提供现代解决途径。
The Veronese Geometry of Dziobek Configurations and Generic Finiteness for Homogeneous Potentials
本文通过Veronese几何证明了Dziobek中心配置的通用有限性，并给出了其数量的统一上界，该界仅依赖于配置空间的维度。
This is a survey article prepared for the submission to "Handbook of moduli". The following topics are discussed: (i) Basic facts and examples of resolutions for nilpotent orbit (ii) Q-factorial terminalizations of nilpotent orbit closures and related birational geometry (iii) Poisson deformations of nilpotent orbit closures.
We give an elementary introduction to our papers relating the geometry of rational homogeneous varieties to representation theory. We also describe related work and recent progress.
Notes on Siegfried Bosch's Algebraic Geometry and Commutative Algebra (dedicated to Grothendieck)
本文系统整理交换代数与代数几何的核心内容，涵盖环模层概形及平展上同调等进阶主题，作为对格罗滕迪克的致敬。
We introduce W-spin structures on a Riemann surface and give a precise definition to the corresponding W-spin equations for any quasi-homogeneous polynomial W. Then, we construct examples of nonzero solutions of spin equations in the presence of Ramond marked points. The main result of the paper is a compactness theorem for the moduli space of the solutions of W-spin equations when W is a non-degenerate, quasi-homogeneous polynomial whose variables all have weight (or fractional degree) wt(x_i) < 1/2. In particular, the compactness theorem holds for the A,D, and E superpotentials.
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