Donatella Iacono
In this paper, we use (bi)semicosimplicial language to study the classical problem of infinitesimal deformations of a closed subscheme in a fixed smooth variety, defined over an algebraically closed field of characteristic 0. In particular, we give an explicit description of the differential graded Lie algebra controlling this problem.
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We generalize Voisin's theorem on deformations of pairs of a symplectic manifold and a Lagrangian submanifold to the case of Lagrangian normal crossing subvarieties. Partial results are obtained for arbitrary Lagrangian subvarieties. We apply our results to the study of singular fibers of Lagrangian fibrations.
We study the tropicalizations of analytic subvarieties of normal toric varieties over complete non-archimedean valuation fields. We show that a Zariski closed analytic subvariety of a normal toric variety is algebraic if its tropicalization is a finite union of polyhedra. Previously, the converse direction was known by the theorem of Bieri and Groves. Over the field of complex numbers, Madani, L. Nisse, and M. Nisse proved similar results for analytic subvarieties of tori.
On the subvarieties with nonsingular real loci of a real algebraic variety
针对光滑射影实代数簇，研究了用子簇实轨迹逼近实流形及由非奇异或空实轨迹子簇生成的Chow群子群问题，给出了新的正反结果。
We investigate deformations of lagrangian manifolds with singularities. We introduce a complex similar to the de Rham-complex whose cohomology calculates deformation spaces. Examples of singular lagrangian varieties are presented and deformations are calculated explicitly.
We prove a criterion stating when a line bundle on a smooth coisotropic subvariety Y of a smooth variety X with an algebraic Poisson structure, admits a first order deformation quantization.
We consider a smooth Lagrangian subvariety Y in a smooth algebraic variety X with an algebraic symplectic from. For a vector bundle E on Y and a choice Oh of deformation quantization of the structure sheaf of X, we establish when E admits a deformation quantization to a module over Oh. If the necessary conditions hold, we describe the set of equivalence classes of such quantizations.
We construct examples of algebraic surfaces with interesting fundamental groups.
Algebraic hyperbolicity of subvarieties of homogeneous varieties
本文将超曲面的代数双曲性结果推广到高余维完全交，证明在射影空间中非常一般的完全交在次数和满足特定阈值条件时具有或不具有代数双曲性。
Using the Atiyah class we give a criterion for a vector bundle on a coisotropic subvariety, $Y$, of an algebraic Poisson variety $X$ to admit a first and second order noncommutative deformation. We also show noncommutative deformations of a vector bundle are governed by a curved dg Lie algebra which reduces to the classical relative Hochschild complex when the Poisson structure on $X$ is trivial.
Let $M$ be a compact complex manifold equipped with a hyperkähler metric, and $X$ be a closed complex analytic subvariety of $M$. In alg-geom/9403006, we proved that $X$ is trianalytic, i. e., complex analytic with respect to all complex structures induced by the hyperkähler structure, provided that $M$ is generic in its deformation class. Here we study the complex analytic deformations of trianalytic subvarieties. We prove that all deformations of $X$ are trianalytic and naturally isomorphic to $X$ as complex analytic varieties. We show that this isomorphism is compatible with the metric indu…
In this note we study two features of submanifolds (subvarieties) with ample normal bundles in a compact Kähler manifold X. First, we study how algebraic X can be, i.e. we investigate the algebraic dimension. Second, we study curves with ample normal bundles in case X is projective. We prove in various cases that the class of the curve is in the interior of the Mori cone.
Using deformation theory of rational curves, we prove a conjecture of Sommese on the extendability of morphisms from ample subvarieties when the morphism is a smooth (or mildly singular) fibration with rationally connected fibers. We apply this result in the context of Fano fibrations and prove a classification theorem for projective bundle and quadric fibration structures on ample subvarieties.
We study compactifications of subvarieties of algebraic tori defined by imposing a sufficiently fine polyhedral structure on their non-archimedean amoebas. These compactifications have many nice properties, for example any k boundary divisors intersect in codimension k. We consider some examples including $M_{0,n}\subset\bar M_{0,n}$ (and more generally log canonical models of complements of hyperplane arrangements) and compact quotients of Grassmannians by a maximal torus.
We study horizontal subvarieties $Z$ of a Griffiths period domain $\mathbb D$. If $Z$ is defined by algebraic equations, and if $Z$ is also invariant under a large discrete subgroup in an appropriate sense, we prove that $Z$ is a Hermitian symmetric domain $\mathcal D$, embedded via a totally geodesic embedding in $\mathbb D$. Next we discuss the case when $Z$ is in addition of Calabi-Yau type. We classify the possible VHS of Calabi-Yau type parametrized by Hermitian symmetric domains $\mathcal D$ and show that they are essentially those found by Gross and Sheng-Zuo, up to taking factors of sy…
We give a new practical method for computing subvarieties of projective hypersurfaces. By computing the periods of a given hypersurface X, we find algebraic cohomology cycles on X. On well picked algebraic cycles, we can then recover the equations of subvarieties of X that realize these cycles. In practice, a bulk of the computations involve transcendental numbers and have to be carried out with floating point numbers. However, if X is defined over algebraic numbers then the coefficients of the equations of subvarieties can be reconstructed as algebraic numbers. A symbolic computation then ver…
We investigate algebraic and analytic subvarieties of C^n with automorphisms which can not be extended to the ambient space.
Smooth subvarieties of Jacobians
本文研究了在光滑拟射影复变种上存在非光滑子簇线性组合的代数整数上同调类的新例子，其中一些例子的维数为6，是最小可能的。所考虑的类是非常一般曲线Jacobian上的最小上同调类。主要工具是复 bordism。
We show that for a variety which admits a quasi-finite period map, finiteness (resp.~non-Zariski-density) of $S$-integral points implies finiteness (resp.~non-Zariski-density) of points over all $\mathbb{Z}$-finitely generated integral domains of characteristic zero. Our proofs rely on foundational results in Hodge theory due to Deligne, Griffiths, and Schmid, and Bakker-Brunebarbe-Tsimerman. We give straightforward applications to Shimura varieties, locally symmetric varieties, the moduli space of smooth hypersurfaces in projective space, and the moduli of smooth divisors in an abelian variet…
Let Y be a subvariety of a smooth projective variety X, and V a vector bundle on X. Given that the restriction of V to Y splits into a direct sum of line bundles, we ask whether V splits on X. I answer this question in affirmative if holds: Y is a q-ample subvariety of X (for appropriate q), it admits sufficiently many embedded deformations, and is very general within its own deformation space. The result goes beyond the previously known splitting criteria for vector bundles corresponding to restrictions. It allows to treat in a unified way examples arising in totally different situations. I d…
Let $X$ be a quasi-affine algebraic variety isomorphic to the complement of a closed subvariety of dimension at most $n-3$ in $\C^n$. We find some conditions under which an isomorphism of two closed subvarieties of $X$ can be extended to an automorphism of $X$.
Sets of Special Subvarieties of Bounded Degree
本文针对代数簇族的 Hodge 轨迹，提出计算给定次数内全部弱特殊子簇的算法，并据此证明特殊与弱特殊子簇有界次数集合的有限性猜想。
On Subvarieties of Degenerations of Fano Varieties
本文研究Fano簇及可分有理连通簇退化上的几何不可约子概型，证明其维数存在由一般纤维Fano指数给出的下界，拓展了相关子概型存在性结论的几何性质研究。
By a $B$-regular variety, we mean a smooth projective variety over $C$ admitting an algebraic action of the upper triangular Borel subgroup $B \subset SL_2(C)$ such that the unipotent radical in $B$ has a unique fixed point. A result of M. Brion and the first author describes the equivariant cohomology algebra (over $C$) of a $B$-regular variety $X$ as the coordinate ring of a remarkable affine curve in $X \times P^1$. The main result of this paper uses this fact to classify the $B$-invariant subvarieties $Y$ of a $B$-regular variety $X$ for which the restriction map $i_Y:H^*(X) \to H^*(Y)$ is…
Smoothing low-dimensional algebraic cycles [after Kollár and Voisin]
本文综述了 Kollár 与 Voisin 对 Borel–Haefliger 问题的肯定解答：在维数小于整体一半的条件下，任何代数子簇都同调等价于光滑代数子簇的整系数线性组合。
In earlier papers it was shown that the generic tropical variety of an ideal can contain information on algebraic invariants as for example the depth in a direct way. The existence of generic tropical varieties has so far been proved in the constant coefficient case for the usual notion of genericity. In this paper we generalize this existence result to include the case of non-constant coefficients in certain settings. Moreover, we extend the notion of genericity to arbitrary closed subvarieties of the general linear group. In addition to including the concept of genericity on algebraic groups…
Consider a flag variety $Fl$ over an algebraically closed field, and a subvariety $V$ whose cycle class is a multiplicity-free sum of Schubert cycles. We show that $V$ is arithmetically normal and Cohen-Macaulay, in the projective embedding given by any ample invertible sheaf on $Fl$.
Let $Z$ be an algebraic subvariety of a Shimura variety. We extend results of the first author to prove an effective upper bound for the degree of a non-facteur maximal special subvariety of $Z$.
Real Legendrian subvarieties are classical objects of differential geometry and classical mechanics and they have been studied since antiquity. However, complex Legendrian subvarieties are much more rigid and have more exceptional properties. The most remarkable case is the Legendrian subvarieties of projective space and prior to the author's research only few smooth examples of these were known. The first series of results of this thesis is related to the automorphism group of any Legendrian subvariety in any projective contact manifold. The connected component of this group (under suitable m…
A nilmanifold is a (left) quotient of a nilpotent Lie group by a cocompact lattice. A hypercomplex structure on a manifold is a triple of complex structure operators satisfying the quaternionic relations. A hypercomplex nilmanifold is a compact quotient of a nilpotent Lie group equipped with a left-invariant hypercomplex structure. Such a manifold admits a whole 2-dimensional sphere $S^2$ of complex structures induced by quaternions. We prove that for any hypercomplex nilmanifold $M$ and a generic complex structure $L\in S^2$, the complex manifold $(M,L)$ has algebraic dimension 0. A stronger …
Tropicalization is a procedure that assigns polyhedral complexes to algebraic subvarieties of a torus. If one fixes a weighted polyhedral complex, one may study the set of all subvarieties of a toric variety that have that complex as their tropicalization. This gives a "tropical realization" moduli functor. We use rigid analytic geometry and the combinatorics of Chow complexes as studied by Alex Fink to prove that when the ambient toric variety is quasiprojective, the moduli functor is represented by a rigid space. As an application, we show that if a polyhedral complex is the tropicalization …
We extend tropicalization and tropical compactification of subvarieties of algebraic tori to subvarieties of spherical homogeneous spaces. Given a tropical compactification of a subvariety, we show that the support of the colored fan of the ambient spherical variety agrees with the tropicalization of the subvariety. The proof is based on our equivariant version of the attening by blow-up theorem. We provide many examples.
Topological Gysin Coherence for Algebraic Characteristic Classes of Singular Spaces
本文证明了Gysin凝聚示性类框架适用于Hodge理论型示性类$IT_{1,\ast}$，从而证明了关于Grassmannian中Schubert簇的Brasselet等人猜想，并研究了Chern类和Todd类。
Let $f:X\to X $ be a dominant self-morphism of an algebraic variety over an algebraically closed field of characteristic zero. We consider the set $Σ_{f^{\infty}}$ of $f$-periodic (irreducible closed) subvarieties of small dynamical degree, the subset $S_{f^{\infty}}$ of maximal elements in $Σ_{f^{\infty}}$, and the subset $S_f$ of $f$-invariant elements in $S_{f^{\infty}}$. When $X$ is projective, we prove the finiteness of the set $P_f$ of $f$-invariant prime divisors with small dynamical degree, and give an optimal upper bound (of cardinality) $$\sharp P_{f^n}\le d_1(f)^n(1+o(1))$$ as $n\to…
A flag domain D = G/V for G a simple real non-compact group G with compact Cartan subgroup is non-classical if it does not fiber holomorphically or anti-holomorphically over a Hermitian symmetric space. We prove that any two points in a non-classical domain D can be joined by a finite chain of compact subvarieties of D. Then we prove that for F an infinite, finitely generated discrete subgroup of G, the analytic space F\D does not have an algebraic structure.
We introduce an algebraicity criteria. It has the following form: under certain conditions, an analytic subvariety of some algebriac variety over a global field $K$, if it contains many $K$-points, then it is algebraic over $K.$ This gives a way to show the transcendence of points via the transcendence of analytic subvarieties. Such a situation often appears when we have a dynamical system, because we can often produce infinitely many points from one point via iterates. Combing this criteria and the study of invariant subvarieties, we get some results on the transcendence in arithmetic dynamic…
First, we study constructible subsets of $\A^n_k$ which contain a line in any direction. We classify the smallest such subsets in $\A^3$ of the type $R\cup\{g\neq 0\},$ where $g\in k[x_1,...,x_n]$ is irreducible of degree $d$, and $R\subset V(g)$ is closed. Next, we study subvarieties $X\subset\A^N$ for which the set of directions of lines contined in $X$ has the maximal possible dimension. These are variants of the Kakeya problem in an algebraic geometry context.
Bounded geometry for PCF-special subvarieties
研究PCF映射在模空间中的几何配置，证明有限多个PCF特殊子集的存在，并给出子集闭包的度数界。
Over a smooth complex projective curve, we study an algebraic versal deformation space with fixed determinant of a coherent sheaf. The algebraic versal deformation space decomposes into a disjoint union of Shatz strata, namely locally closed subschemes which parametrize coherent sheaves with common Harder-Narasimhan types. We study the geometry and local topology of large unstable strata and their behavior along boundaries.
We study a formal deformation problem for rational algebraic cycle classes motivated by Grothendieck's variational Hodge conjecture. We argue that there is a close connection between the existence of a Chow-Künneth decomposition and the existence of expected deformations of cycles. This observation applies in particular to abelian schemes.
We review the theory of non-commutative deformations of sheaves and describe a versal deformation by using an A-infinity algebra and the change of differentials of an injective resolution. We give some explicit non-trivial examples.
New dimensional estimates for subvarieties of linear algebraic groups
本文为足够大域上连通几乎单线性代数群的一般子簇证明了改进的维数界，将参数对维数的依赖从指数塔型降至双重指数与简单指数，并由此得到非扭经典群直径界的直接推论。
On the distribution of the Hodge locus
本文证明 level 至少为 3 的 Hodge 轨迹（正周期维数）是有限并集因而是代数的，而 level 1 或 2 时若含典型特殊子簇则解析稠密，并完整阐明了特殊子簇的典型/非典型分布。
A differential analogue of the wild automorphism conjecture
建立了代数动力学中Reichstein-Rogalski-Zhang猜想的微分类比：若射影簇X上存在整体代数向量场v使得(X,v)无真不变子簇，则X为阿贝尔簇；同时研究了阿贝尔簇上具有该性质的向量场。
We study the p-adic deformation properties of algebraic cycle classes modulo rational equivalence. We show that the crystalline Chern character of a vector bundle lies in a certain part of the Hodge filtration if and only if, rationally, the class of the vector bundle lifts to a formal pro-class in K-theory on the p-adic scheme.
Given a pair of derived-equivalent Calabi--Yau manifolds of dimension more than two, we prove that the derived equivalence can be extended to general fibers of versal deformations. As an application, we give a new proof of the Pfaffian--Grassmannian derived equivalence.
Let $X$ be a projective Frobenius split variety over an algebraically closed field with splitting $θ: F_* Ø_X \to Ø_X$. In this paper we give a sharp bound on the number of subvarieties of $X$ compatibly split by $θ$. In particular, suppose $\sL$ is a sufficiently ample line bundle on $X$ (for example, if $\sL$ induces a projectively normal embedding) with $n = \dim H^0(X, \sL)$. We show that the number of $d$-dimensional irreducible subvarieties of $X$ that are compatibly split by $θ$ is less than or equal to ${n \choose d+1}$. This generalizes a well known result on the number of closed poin…
Matroids and the space of torus-invariant subvarieties of the Grassmannian with given homology class
本文研究了复Grassmannian中具有给定同调类的环面不变代数子簇的刻画问题，利用拟阵理论技术给出了部分解答，并将其应用于计算特定Grassmannian的Euler-Chow级数。
An algebraic zip datum is a tuple $\CZ := (G,P,Q,ϕ)$ consisting of a reductive group $G$ together with parabolic subgroups $P$ and $Q$ and an isogeny $ϕ\colon P/R_uP\to Q/R_uQ$. We study the action of the group $E := \{(p,q)\in P{\times}Q | ϕ(π_{P}(p)) =π_Q(q)\}$ on $G$ given by $((p,q),g)\mapsto pgq^{-1}$. We define certain smooth $E$-invariant subvarieties of $G$, show that they define a stratification of $G$. We determine their dimensions and their closures and give a description of the stabilizers of the $E$-action on $G$. We also generalize all results to non-connected groups. We show tha…
What makes an algebraic curve special?
本文综述了代数曲线及模空间 $\mathcal{M}_g$ 中特殊子簇的多种“特殊”含义，并结合具体例子重点介绍了 Hodge 理论与 Teichmüller 理论中非典型交点的双代数视角。
We consider (small) algebraic deformations of germs of real-algebraic CR submanifolds in complex space and study the biholomorphic equivalence problem for such deformations. We show that two algebraic deformations of minimal holomorphically nondegenerate real-algebraic CR submanifolds are holomorphically equivalent if and only if they are algebraically equivalent.
In this thesis, we study deformations of compact holomorphic Poisson manifolds and algebraic Poisson schemes in the framework of Kodaira-Spencer's analytic deformation theory and Grothendieck's algebraic deformation theory.
Let X be a smooth algebraic variety over a field K containing the real numbers. We introduce the notion of twisted associative (resp. Poisson) deformation of the structure sheaf of X. These are stack-like versions of usual deformations. We prove that there is a twisted quantization operation from twisted Poisson deformations to twisted associative deformations, which is canonical and bijective on gauge equivalence classes. This result extends work of Kontsevich, and our own earlier work, on deformation quantization of algebraic varieties.
In a recent paper, J.-B. Bost establishes a criterion for certain ``formal subvarieties'' of algebraic varieties to be algebraic. His theorem unifies and generalizes results of Chudnovsky's and Y. André, motivated by an arithmetic conjecture of Grothendieck that predicts that the solutions of certain differential equations are algebraic functions. The proof makes use of the diophantine approximation techniques introduced by these authors but with a systematic geometric point of view, notably via Arakelov geometry and the formalism of ``slopes''.
This note contains a correction of the proofs of the main results of the paper [A. Yekutieli, Deformation quantization in algebraic geometry, Adv. Math. 198 (2005), 383-432]. The results are correct as originally stated.
Let X be a smooth algebraic variety over a field of characteristic 0. We introduce the notion of twisted associative (resp. Poisson) deformation of the structure sheaf O_X. These are stack-like versions of usual deformations. We prove that there is a twisted quantization map from twisted Poisson deformations to twisted associative deformations, which is canonical and bijective on equivalence classes.
The paper is devoted to peculiarities of the deformation quantization in the algebro-geometric context. A direct application of the formality theorem to an algebraic Poisson manifold gives a canonical sheaf of categories deforming coherent sheaves. The global category is very degenerate in general. Thus, we introduce a new notion of a semi-formal deformation, a replacement in algebraic geometry of an actual deformation (versus a formal one). Deformed algebras obtained by semi-formal deformations are Noetherian and have polynomial growth. We propose constructions of semi-formal quantizations of…
We give the definitions of affine algebraic supervariety and affine algebraic group through the functor of points and we relate them to the other definitions present in the literature. We study in detail the algebraic supergroup $SL(m|n)$ and give explicitly the Hopf algebra structure of the algebra representing its functor of points. At the end we give also the quantization of $SL(m|n)$ together with its coactions on suitable quantum spaces according to Manin's philosophy.
By analytic deformations of complex structures, we mean perturbations of the Dolbeault operator. By algebraic deformations of complex structures, we mean deformations of holomorphic glueing data. For complex manifolds there is, infinitesimally, a correspondence between these two types deformations. In this article we argue that an analogous correspondence holds between the analytic and algebraic deformations of a super Riemann surface.
We give a proof of Yekutieli's global algebraic deformation quantization result which does not rely on the choice of local sections of the bundle of affine coordinate systems. Instead we use an argument inspired by algebraic De Rham cohomology.
This is a report on recent progress concerning the interactions between derived algebraic geometry and deformation quantization. We present the notion of derived algebraic stacks, of shifted symplectic and Poisson structures, as well as the construction of deformation quantization of shifted Poisson structures. As an application we propose a general construction of the quantization of the moduli space of $G$-bundles on an oriented space of arbitrary dimension.
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