LIU Xiaolong

LIU Xiaolong

I’m a second-year graduate student in the area of the algebraic geometry.

Some Interesting Papers, Notes and Websites

Published:

Here we will introduce some interesting papers and notes here. I may or may not read them.

1. Notes and Books

1.1. Algebraic geometry

Introduction. A very detailed note about moduli space, providing a detailed introduction to the stack theory and its applications. This note including the basic theory of Deligne-Mumford stacks, algebraic stacks and the construction of moduli space of curves. Furthermore, this book introduce the theory of good moduli space and GIT theory.

Introduction. Mostly aimed at an audience with backgrounds in geometry and homological algebra, these notes offer an introduction to derived geometry based on a lecture course given by the second author. The focus is on derived algebraic geometry, mainly in characteristic $0$, but we also see the tweaks which extend most of the content to analytic and differential settings.

Introduction. A nice introduction text book of birational geometry from basic MMP theory to recent BCHM theory.

Introduction. The main goal of this book is to provide a comprehensive overview of the algebraic theory of K-stability for Fano varieties. In the past decade, it has become clear that the machinery of higher dimensional geometry, centered around the minimal model program, provides a powerful tool for studying K-stability and K-moduli theory of Fano varieties purely algebraically.

Introduction. These notes have been written for distribution to the participants to the summer school in Algebraic Geometry organized by the Scuola Matematica Universitaria in Cortona in the period 13-26 August 1995. The aim was to describe the classification of rational homogeneous varieties and to provide the theorems of Borel-Weil and Bott.

Introduction. A fundamental problem in algebraic geometry is to determine which varieties are rational, that is, birational to the projective space. Several important developments in the field have been motivated by this question. The main goal of the note is to describe two recent directions of study in this area. One approach goes back to Iskovskikh and Manin, who proved that smooth, $3$-dimensional quartic hypersurfaces are not rational. This relies on ideas and methods from higher-dimensional birational geometry. The second approach, based on recent work of Claire Voisin and many other people, relies on a systematic use of decomposition of the diagonal and invariants such as Chow groups, Brauer groups, etc. to prove irrationality. Both directions will give us motivation to introduce and discuss some important concepts and results in algebraic geometry.

Introduction. The goal of the first half of this note is to introduce and study the basic structure theory of abelian varieties, as covered in (say) Mumford’s book. In the second half of the course, we shall discuss derived categories and the Fourier–Mukai transform, and give some geometric applications.

Introduction. We will aim to hit the highlights of the theory: complex-analytic case, general algebraic theory over an arbitrary field, duality and endomorphism algebras, and special results over interesting fields (finite fields, local fields, global fields). We will discuss heights and prove the Mordell–Weil theorem for an abelian variety over a global field.

Introduction. We will begin with an overview of the nature of the structure theory for various classical groups, with an emphasis on some examples and the remarkable properties of tori. We will then turn to a systematic development of basic generalities in the theory of smooth affine groups over an arbitrary field: group-theoretic constructions (matrix realizations, closed orbit lemma, commutator subgroups, normalizers, centralizers, Lie algebras, quotients), Jordan decomposition, structure of solvable groups (unipotent groups, diagonalizable groups, Lie-Kolchin theorem), and the solvable and unipotent radicals. The theory really takes off with the Borel fixed-point theorem concerning the action of a smooth connected (split) solvable group on a smooth quasi-projective variety. That will lead into the important conjugacy theorems (for maximal tori and Borel subgroups), which then takes us on the long road toward the structure theory of connected reductive groups (in terms of root systems), one of the real gems of pure mathematics. We will work through lots of examples, and begin to understand the special role of SL2 in the general theory of split reductive groups (to be developed in a sequel course).

Introduction. The aim of the note is to cover the structure theory of connected reductive groups over an arbitrary field $k$, including (relative) root systems, rational conjugacy theorems, and the Galois-action on Dynkin diagrams, illustrated with a variety of examples (both split and nonsplit). Some results need to be established over algebraically closed fields early in the course as a prelude to refinements over general fields. For $k = \mathbb{R}$ and $k = \mathbb{C}$ this illuminates the theory of semisimple Lie groups, for finite $k$ it clarifies the structure of finite groups of Lie type, and for $k$ a local or global field it leads to vast generalizations of classical results on the arithmetic of quadratic forms and central simple algebras over such fields.

Introduction. We develop the relative theory of reductive group schemes, using dynamic techniques and algebraic spaces to streamline the original development in SGA3.

  • Brian Conrad. Alterations, notes typed by Tony Feng and Aaron Landesman.

Introduction. In 1994, Johan de Jong astonished the algebraic geometry world by proving a weakened form of resolution of singularities (using so-called “alterations” in the role of blow-ups) that works in arbitrary characteristic and is sufficient for most applications (even in characteristic $0$). Moreover, his method is so conceptual that it also applies over discrete valuation rings, giving a “semi-stable reduction theorem” for higher-dimensional varieties. These results immediately had many wonderful applications (such as Serre’s positivity conjecture in commutative algebra, and the potentially semistable property for Galois representations arising from the $p$-adic étale cohomology of varieties over $p$-adic fields). In addition to the pretty geometric ideas introduced by de Jong, his proof uses many important general concepts in algebraic geometry, such as the theory of stable reduction for curves, the étale topology, Artin approximation, algebraic stacks/spaces, Raynaud–Gruson flattening, resolution of singularities for abstract surfaces, and so on. So this topic is an ideal vehicle for seeing many important methods and results being used. In this note we will work through the proofs of de Jong’s main theorems over fields and along the way explain as much of the background.

Introduction. This is the most general version of Keel-mori theorem about the existence of coarse moduli spaces. In this note, he use stacks instead of groupoids to give a streamlined proof of the Keel-Mori existence theorem for coarse moduli spaces (useless bonus: noetherian hypotheses eliminated).

Introduction. We provide a proof of Nagata’s compactification theorem: any separated map of finite type between quasi-compact and quasi-separated schemes (e.g., noetherian schemes) factors as an open immersion followed by a proper morphism. This is a detailed exposition of private notes of Deligne that translate Nagata’s method into modern terms, and includes some applications of general interest in the theory of rational maps, such as refined versions of Chow’s Lemma and the elimination of indeterminacies in a rational map, as well as a blow-up characterization of when a proper morphism (to a rather general base scheme) is birational.

Introduction. A nice introduction of intersection theory developed by Fulton etc. This including chapter 1-6 and chapter 15 in the Fulton’s book, including the basic construction and the theorem of Grothendieck-Riemann-Roch.

Introduction. A standard and nice introduction of fundamental theory of algebraic geometry, including scheme theory and its cohomology.

Introduction. A breif and fast introduction of Lie Algebras and Algebraic Groups.

Introduction. In comparison with his book, the emphasis is on heuristic arguments rather than formal proofs and on varieties rather than schemes. The notes also discuss the proof of the Weil conjectures (Grothendieck and Deligne).

Introduction. This note follows from a course which was taught in Bonn, Germany over the Wintersemester 2016/17, by Prof. Dr. Peter Scholze. This note including the basics of algebraic geometry, so about sheaves, schemes, $\mathscr{O}_X$-modules, affine/separated/proper morphisms, and eventually to show that proper normal curves over $k$ can be naturally associated to a type of field extension of $k$, and separated curves are quasi-projective.

Introduction. This note follows from a course which was taught in Bonn, Germany over the Sommersemester 2017, by Prof. Dr. Peter Scholze. This note is started by looking at properties of flat maps between schemes, smooth, unramified, and étale morphisms, as well as the sheaf of Kähler differentials. Also sheaf cohomology in general will be introduced. After proving some technical statements about the cohomology of coherent sheaves, and various base change properites, we had all the fire-power we needed to state and prove the Riemann-Roch Theorem and Serre Duality. We wrapped up the course with the theory of formal functions, Zariski’s main theorem and Stein factorisation.

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1.2. Other Areas

Introduction. A graduate-level textbook on algebraic topology from a fairly classical point of view which including theory of fundamental groups, covering spaces, homology groups, cohomology groups and basic homotopy theory.

Introduction. This is a note for a course in MIT (18.905 and 18.906). Topics include: Singular homology, CW complexes, Homological algebra, Cohomology, Poincare duality, basic homotopy theory, obstruction theory, classifying spaces, spectral sequences, characteristic classes and Steenrod operations.

Introduction. 介绍基本的代数学基础的很好的书，详尽的介绍了基础的范畴理论还有代数学基础。

Introduction. 全面的介绍了同调代数的内容，比其他的书都要多出很多有趣的东西。

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2. Papers

Introduction. Let $X$ be an irreducible projective variety and $f$ a morphism $X\to\mathbb{P}^n$. They give a new proof of the fact that the preimage of any linear variety of dimension $k\geq n + 1 − \dim f(X)$ is connected, by using the Generalized Hodge Index Theorem and de Jong’s altration theorem that hold in any characteristic. They also prove the connectedness Theorem of Fulton and Hansen as application of our main theorem.

Introduction. We formulate a version of the integral Hodge conjecture for categories, prove the conjecture for two-dimensional Calabi-Yau categories which are suitably deformation equivalent to the derived category of a K3 or abelian surface, and use this to deduce cases of the usual integral Hodge conjecture for varieties. Along the way, we prove a version of the variational integral Hodge conjecture for families of two-dimensional Calabi-Yau categories, as well as a general smoothness result for relative moduli spaces of objects in such families. Our machinery also has applications to the structure of intermediate Jacobians, such as a criterion in terms of derived categories for when they split as a sum of Jacobians of curves.

Introduction. We show that rationality does not specialize in flat projective families of complex fourfolds with terminal singularities. This answers a question of Totaro, who established the analogous result in all dimensions greater than $4$.

Introduction. We study the intersection of two copies of $\mathsf{Gr(2, 5)}$ embedded in $\mathbb{P}^9$, and the intersection of the two projectively dual Grassmannians in the dual projective space. These intersections are deformation equivalent, derived equivalent Calabi–Yau threefolds. We prove that generically they are not birational. As a consequence, we obtain a counterexample to the birational Torelli problem for Calabi–Yau threefolds. We also show that these threefolds give a new pair of varieties whose classes in the Grothendieck ring of varieties are not equal, but whose difference is annihilated by a power of the class of the affine line. Our proof of non-birationality involves a detailed study of the moduli stack of Calabi–Yau threefolds of the above type, which may be of independent interest.

Introduction. We perform a systematic study of Gushel-Mukai varieties—quadratic sections of linear sections of cones over the Grassmannian $\mathsf{Gr(2,5)}$. This class of varieties includes Clifford general curves of genus $6$, Brill-Noether general polarized K3 surfaces of genus 6, prime Fano threefolds of genus $6$, and their higher-dimensional analogues.

We establish an intrinsic characterization of normal Gushel-Mukai varieties in terms of their excess conormal sheaves, which leads to a new proof of the classification theorem of Gushel and Mukai. We give a description of isomorphism classes of Gushel-Mukai varieties and their automorphism groups in terms of linear algebraic data naturally associated to these varieties.

We carefully develop the relation between Gushel-Mukai varieties and Eisenbud-Popescu-Walter sextics introduced earlier by Iliev-Manivel and O’Grady. We describe explicitly all Gushel-Mukai varieties whose associated EPW sextics are isomorphic or dual (we call them period partners or dual varieties respectively). Finally, we show that in dimension 3 and higher, period partners/dual varieties are always birationally isomorphic.

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3. Surveys

Introduction. In these lecture notes we give an introduction to Bridgeland stability conditions on smooth complex projective varieties with a particular focus on the case of surfaces. This includes basic definitions of stability conditions on derived categories, basics on moduli spaces of stable objects and variation of stability. These notes originated from lecture series by the first author at the summer school “Recent advances in algebraic and arithmetic geometry”, Siena, Italy, August 24-28, 2015 and at the school “Moduli of Curves”, CIMAT, Guanajuato, Mexico, February 22 - March 4, 2016.

Introduction. In 1991 Campana and Peternell proposed, as a natural algebro-geometric extension of Mori’s characterization of the projective space, the problem of classifying the complex projective Fano manifolds whose tangent bundle is nef, conjecturing that the only varieties satisfying these properties are rational homogeneous. In this paper we review some background material related to this problem, with special attention to the partial results recently obtained by the authors, such as the case of big nef and $1$-ample tangent bundle, and the criterion of complete flags $G/B$ by FT-manifolds.

Introduction. This is a survey written in an expositional style on the topic of symplectic singularities and symplectic resolutions, which could also serve as an introduction to this subject.

Introduction. We give a survey of the recent progress on the study of K-stability of Fano varieties by an algebro-geometric approach. 也可见许晨阳教授的K-稳定性的另一个中文简介.

Introduction. We review some of the methods used in the classification of Fano varieties and the description of their birational geometry. Mori theory brought important simplifications to this classical theory which we will illustrate with a few examples.

Introduction. The aim of these notes is to acquaint the reader with important objects in complex algebraic geometry: K3 surfaces and their higher-dimensional analogs, hyperkähler manifolds. These manifolds are interesting from several points of view: dynamical (some have interesting automorphism groups), arithmetical (although we will not say anything on this aspect of the theory), and geometric. It is also one of those rare cases where the Torelli theorem allows for a powerful link between the geometry of these manifolds and lattice theory.

Introduction. Gushel-Mukai varieties are smooth (complex) dimensionally transverse intersections of a cone over the Grassmannian $\mathsf{Gr(2,5)}$ with a linear space and a quadratic hypersurface. They occur in each dimension $1$ through $6$ and they are Fano varieties (their anticanonical bundle is ample) in dimensions $3, 4, 5$, and $6$. The aim of this survey is to discuss the geometry, moduli, Hodge structures, and categorical aspects of these varieties. It is based on joint work with Alexander Kuznetsov and earlier work of Logachev, Iliev, Manivel, O’Grady, and Perry.

Introduction. 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 rˆole. This selection is largely arbitrary and mainly reflects the interests of the author.

Introduction. This lecture is an introduction to my joint project with N. Mok where we develop a geometric theory of Fano manifolds of Picard number 1 by studying the collection of tangent directions of minimal rational curves through a generic point. After a sketch of some historical background, the fundamental object of this project, the variety of minimal rational tangents, is defined and various examples are examined. Then some results on the variety of minimal rational tangents are discussed including an extension theorem for holomorphic maps preserving the geometric structure. Some applications of this theory to the stability of the tangent bundles and the rigidity of generically finite morphisms are given.

Introduction. This survey paper discusses some of the recent progress in the study of rational curves on algebraic varieties.To start, we discuss existence results for rational curves on foliated manifolds. We construct the tangent morphism, define the variety of minimal rational tangents (VMRT) and name a number of results that show that many of the geometry properies of a uniruled variety are encoded in the projective geometry of the VMRT. The results are then applied in two different settings. For one, we discuss the geometry of chains of rational curves on uniruled varieties, show how the length of a variety can be determined from the VMRT and discuss the concrete example of the moduli space of vector bundles on a curve. On the other hand, the existence results for rational curves can be also be used to study manifolds which are known NOT to contain rational curves. We employ this method to study deformations of morphisms. Finally, we look at families of varieties of canonically polarized manifolds over complex surfaces, and use non-existence results for rational curves on the base to relate the variation of the family with the logarithmic Kodaira dimension of the base.

Introduction. Grothendieck introduced the notion of a “motif” in a letter to Serre in 1964. Later he wrote that, among the objects he had been privileged to discover, they were the most charged with mystery and formed perhaps the most powerful instrument of discovery. In this article, I shall explain what motives are, and why Grothendieck valued them so highly. （亦有中文版，刊于数学译林，2009年第3期，徐克舰译，付保华校对）

Introduction. This text is a survey of derived algebraic geometry. It covers a variety of general notions and results from the subject with a view on the recent developments at the interface with deformation quantization.

Introduction. A survey about derived algebraic geometry.

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4. Websites

Introduction. The Stacks project is an ever growing open source textbook and reference work on algebraic stacks and the algebraic geometry needed to define them.

Introduction. Kerodon is an online textbook on categorical homotopy theory, infinity categories and related mathematics.

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updated at 2024-03-26.