Vladimir Baranovsky
We consider the algebraization problem for principal bundles with reductive structure group, defined on the complement of a closed subset Z in a proper formal scheme. We show that, when Z is of codimension at least 3, an algebraization always exists. For codimension 2 we show that an algebraization exists precisely when a certain additional condition is satisfied.
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We present a conjecture in Diophantine geometry concerning the construction of line bundles over smooth projective varieties over $\bar{\mathbb Q}}$. This conjecture, closely related to the Grothendieck Period Conjecture for cycles of codimension 1, is also motivated by classical algebraization results in analytic and formal geometry and in transcendence theory. Its formulation involves the consideration of $D$-group schemes attached to abelian schemes over algebraic curves over $\bar{\mathbb Q}}$. We also derive the Grothendieck Period Conjecture for cycles of codimension 1 in abelian varieti…
Algebraization of rigid analytic varieties and formal schemes via perfect complexes
本文将Toën和Vaquié的定理扩展到非阿基米德和形式环境，证明了光滑且良好的刚性解析簇代数化当且仅当其完美复形的范畴也是光滑且良好的。
Generalizing homogeneous spectra for rings graded by natural numbers, we introduce multihomogeneous spectra for rings graded by abelian groups. Such homogeneous spectra have the same completeness properties as their classical counterparts, but are possibly nonseparated. We relate them to ample families of invertible sheaves and simplicial toric varieties. As an application, we generalize Grothendieck's Algebraization Theorem and show that formal schemes with certain ample families are algebraizable.
We study the properness of the functor of $F$-trivial bundles by relating it to the base change question for the fundamental group scheme of Nori.
Let X be a proper scheme over a field k which satisfies Serre's condition S2 and G a reductive group over k. We prove that the functor of principal G-bundles defined away from a non-fixed closed subset in X of codimension at least 3, is an algebraic stack in the sense of Artin.
Let X be a smooth projective variety defined over an algebraically closed field k. Nori constructed a category of vector bundles on X, called essentially finite vector bundles, which is reminiscent of the category of representations of the fundamental group (in characteristic zero). In fact, this category is equivalent to the category of representations of a pro--finite group scheme which controls all finite torsors. We show that essentially finite vector bundles coincide with those which become trivial after being pulled back by some proper and surjective morphism to X.
Let $X$ be a normal, connected and projective variety over an algebraically closed field $k$. It is known that a vector bundle $V$ on $X$ is essentially finite if and only if it is trivialized by a proper surjective morphism $f:Y\to X$. In this paper we introduce a different approach to this problem which allows to extend the results to normal, connected and strongly pseudo-proper algebraic stack of finite type over an arbitrary field $k$.
Using combinatorial methods, we determine that a projective coordinate ring of the moduli of parabolic principal $\mathrm{SL}_2$-bundles on a marked projective curve is not Gorenstein when the genus and number of marked points are greater than $1$.
Vector bundles on proper toric 3-folds and certain other schemes
本研究证明满足特定修改条件的n维正常代数概形（含非射影情形）存在秩为n且顶陈数任意大的非平凡向量丛，应用于环面簇得出三维正常环面簇均有秩3的这类向量丛，并刻画了适用该结果的高维环面簇类。
Let $X$ be a projective and smooth variety over an algebraically closed field $k$. Let $f:Y\rightarrow X$ be a proper and surjective morphism of $k$-varieties. Assuming that $f$ is separable, we prove that the Tannakian category associated to the vector bundles $E$ on $X$ such that $f^*E$ is trivial is equivalent to the category of representations of a finite and etale group scheme. We give a counterexample to this conclusion in the absence of separability.
Ample vector bundles on non-proper schemes
该研究解决Hartshorne提出的两个公开问题，证明任意特征下丰富向量丛张量积的丰富性，还构造出光滑拟射影曲面中丰富线丛的非丰富自扩张。
We consider a canonical bundle formula for generically finite proper surjective morphisms and obtain subadjunction formulae for minimal log canonical centers of log canonical pairs. We also treat related topics and applications.
Pullback of principal bundles along proper morphisms
本文研究主丛沿紧合满态射拉回的行为，涵盖特征0代数闭域上的概形与复解析空间两类情形。
Given a compact complex manifold $M$, we investigate the holomorphic vector bundles $E$ on $M$ such that $φ^* E$ is trivial for some surjective holomorphic map $φ$, to $M$, from some compact complex manifold. We prove that these are exactly those holomorphic vector bundles that admit a flat holomorphic connection with finite monodromy homomorphism. A similar result is proved for holomorphic principal $G$-bundles, where $G$ is a connected reductive complex affine algebraic group.
Momentum maps and the Kähler property for base spaces of reductive principal bundles
本文研究紧底空间上约化主丛全空间的复几何，建立底空间Kähler性质、极大紧子群作用的动量映射与特殊等变紧化Kähler性质间的紧密联系，并提供示例说明结果的最优性。
Let $X$ be a Fano manifold which is the zero scheme of a general global section $s$ in an irreducible homogenous vector bundle over a Grassmannian. We prove that the restriction of the Plücker embedding embeds $X$ projectively normal, and that every small deformation of $X$ comes from a deformation of the section $s$. These results are strengthened in the case of Fano 4-folds.
Tensor-generating line bundles on toric varieties
该研究在环簇上刻画了张量生成线丛的组合判据，证明了Kleiman型的生成性判据，给出了相关例子并研究了典范丛生成的阻碍子簇及其对导出范畴自等价的影响。
We show that principal bundles for a semisimple group on an arbitrary affine curve over an algebraically closed field are trivial, provided the order of $π_1$ of the group is invertible in the ground field, or if the curve has semi-normal singularities. Several consequences and extensions of this result (and method) are given. As an application, we realize conformal blocks bundles on moduli stacks of stable curves as push forwards of line bundles on (relative) moduli stacks of principal bundles on the universal curve.
In this paper, we study a variation of a conjecture of Debarre on positivity of cotangent bundles of complete intersections. We establish the ampleness of Schur powers of cotangent bundles of generic complete intersections in projective manifolds, with high enough explicit codimension and multi-degrees. Our approach is naturally formulated in terms of flag bundles and allows one to reach the optimal codimension. On complex manifolds, this ampleness property implies intermediate hyperbolic properties. We give a natural application of our main result in this context.
We prove that the CM line bundle is ample on the proper moduli space which parametrizes KSBA stable varieties.
Special K-stability and positivity of CM line bundles
该研究证明了参数化极大变异特殊K-稳定簇的真族上CM线丛的丰沛性，并应用该结论推导出曲线上一致绝热K-稳定klt-平凡纤维化粗模空间的任意真子空间具有射影性。
For $G$ a split semi-simple group scheme and $P$ a principal $G$-bundle on a relative curve $X\to S$, we study a natural obstruction for the triviality of $P$ on the complement of a relatively ample Cartier divisor $D \subset X$. We show, by constructing explicit examples, that the obstruction is nontrivial if $G$ is not simply connected but it can be made to vanish, if $S$ is the spectrum of a dvr (and some other hypotheses), by a faithfully flat base change. The vanishing of this obstruction is shown to be a sufficient condition for etale local triviality if $S$ is a smooth curve, and the si…
Let X be a compact connected Riemann surface equipped with an anti-holomorphic involution σ. Let G be a connected complex reductive affine algebraic group, and let σ_G be a real form of G. We consider holomorphic principal G-bundles on X satisfying compatibility conditions with respect to σand σ_G. We prove that the points defined over $\mathbb R$ of the smooth locus of a moduli space of principal G-bundles on X are precisely these objects, under the assumption that {\rm genus}(X) > 2. Stable, semistable and polystable bundles are defined in this context. Relationship between any of these prop…
We study Miyaoka-type semistability criteria for principal Higgs G-bundles E on complex projective manifolds of any dimension. We prove that E has the property of being semistable after pullback to any projective curve if and only if certain line bundles, obtained from some characters of the parabolic subgroups of G, are numerically effective. One also proves that these conditions are met for semistable principal Higgs bundles whose adjoint bundle has vanishing second Chern class. In a second part of the paper, we introduce notions of numerical effectiveness and numerical flatness for principa…
In this paper, the first of a series of three, we classify holomorphic principal G-bundles over an elliptic curve, where G is a reductive group. We also study the local and global properties of the moduli space of semistable G-bundles. We identify canonical representatives for each S-equivalence class of semistable G-bundles, and study their automorphism groups.
Berndtsson-Lempert-Szőke Fields Associated to Proper Holomorphic Families of Vector Bundles
本文为未必构成全纯向量丛的 Hilbert 空间族引入 Berndtsson-Lempert-Szőke 复解析结构及相容曲率，并据此重新证明 Berndtsson 正像曲率定理及其 Liu–Yang 高秩推广。
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