In this article we give a sufficient condition for a morphism $φ$ from a smooth variety $X$ to projective space, finite onto a smooth image, to be deformed to an embedding. This result puts some theorems on deformation of morphisms of curves and surfaces such as $K3$ and general type, obtained by ad hoc methods, in a new, more conceptual light. One of the main interests of our result is to apply it to the construction of smooth varieties in projective space with given invariants. We illustrate this by using our result to construct canonically embedded surfaces with $c_1^2=3p_g-7$ and derive some interesting properties of their moduli spaces. Another interesting application of our result is the smoothing of ropes. We obtain a sufficient condition for a rope embedded in projective space to be smoothable. As a consequence, we prove that canonically embedded carpets satisfying certain conditions can be smoothed. We also give simple, unified proofs of known theorems on the smoothing of $1$--dimensional ropes and $K3$ carpets. Our condition for deforming $φ$ to an embedding can be stated very transparently in terms of the cohomology class of a suitable first order infinitesimal deformation of $φ$. It holds in a very general setting (any $X$ of arbitrary dimension and any $φ$ unobstructed with an algebraic formally semiuniversal deformation). The simplicity of the result can be seen for instance when we specialize it to the case of curves.
We make a systematic study of the infinitesimal lifting conditions of a pseudo finite type map of noetherian formal schemes. We recover the usual general properties in this context, and, more importantly, we uncover some new phenomena. We define a completion map of formal schemes as the one that arises canonically by performing the completion of a noetherian formal scheme along a subscheme, following the well-known pattern of ordinary schemes. These maps are etale in the sense of this work (but not adic). They allow us to give a local description of smooth morphisms. This morphisms can be fact…
This a first step to develop a theory of smooth, etale and unramified morphisms between noetherian formal schemes. Our main tool is the complete module of differentials, that is a coherent sheaf whenever the map of formal schemes is of pseudo finite type. Among our results we show that these infinitesimal properties of a map of usual schemes carry over into the completion with respect to suitable closed subsets. We characterize unramifiedness by the vanishing of the module of differentials. Also we see that a smooth morphism of noetherian formal schemes is flat and its module of differentials …
We construct infinitesimal deformations on an open domain of a smooth projective surface given by a complement of plumbings of disjoint linear chains of smooth rational curves. We show that the infinitesimal deformations are not small deformations, that is, they change the complex structure away from the boundary of the domain.
In this note, we study the infinitesimal forms of Deligne cycle class maps. As an application, we prove that the infinitesimal form of a conjecture by Beilinson is true.
We continue our study on infinitesimal lifting properties of maps between locally noetherian formal schemes started in math.AG/0604241. In this paper, we focus on some properties which arise specifically in the formal context. In this vein, we make a detailed study of the relationship between the infinitesimal lifting properties of a morphism of formal schemes and those of the corresponding maps of usual schemes associated to the directed systems that define the corresponding formal schemes. Among our main results, we obtain the characterization of completion morphisms as pseudo-closed immersi…
We show that the Hilbert scheme which parameterises bitangent lines to a general quartic surface is a counterexample to the infinitesimal Torelli claim and is a smooth regular surface with no rational curves and very ample canonical divisor.
We identify dglas that control infinitesimal deformations of the pairs (manifold, Higgs bundle) and of Hitchin pairs. As a consequence, we recover known descriptions of first order deformations and we refine known results on obstructions. Secondly we prove that the Hitchin map is induced by a natural L-infinity morphism and, by standard facts about L-infinity algebras, we obtain new conditions on obstructions to deform Hitchin pairs.
The purpose of this paper is to prove dimension formulas for $T^1$ and $T^2$ for rational surface singularities. These modules play an important role in the deformation theory of isolated singularities in analytic and algebraic geometry. The first may be identified as the Zariski tangent space of the versal deformation of the singularity; i.e. it is the space of infinitesimal deformations. The second contains the obstruction space -- in all known cases it is the whole obstruction space for rational surface singularities.
This paper gives a canonical construction, in terms of additive cohomological functors, of the universal formal deformation of a compact complex manifold without vector fields (more generally of a faithful $g$-module, where $g$ is a sheaf of Lie algebras without sections). The construction is based on a certain (multivariate) Jacobi complex $J(g)$ associatd to $g$: indeed ${\mathbb C}\oplus {\mathbb H}^0(J(g))^*$ is precisely the base ring of the universal deformation.
The notions of $\mathbb Q$-Gorenstein scheme and of $\mathbb Q$-Gorenstein morphism are introduced for locally Noetherian schemes by dualizing complexes and (relative) canonical sheaves. These cover all the previously known notions of $\mathbb Q$-Gorenstein algebraic variety and of $\mathbb Q$-Gorenstein deformation satisfying Kollár condition, over a field. By studies on relative $\mathbf S_{2}$-condition and base change properties, valuable results are proved for $\mathbb Q$-Gorenstein morphisms, which include infinitesimal criterion, valuative criterion, $\mathbb Q$-Gorenstein refinement, a…
This paper is concerned with a refinement of the Stein factorization, and with applications to the study of deformations of surjective morphisms. We show that every surjective morphism f:X->Y between normal projective varieties factors canonically via a finite cover of Y that is etale in codimension one. This "maximally etale factorization" is characterized in terms of positivity of the push-forward of the structure sheaf and satisfies a functorial property. It turns out that the maximally etale factorization is stable under deformations, and naturally decomposes an etale cover of the Hom-sche…
The goal of this paper is to give a method to compute the space of infinitesimal deformations of a double cover of a smooth algebraic variety. The space of all infinitesimal deformations has a representation as a direct sum of two subspaces. One is isomorphic to the space of simultaneous deformations of the branch locus and the base of the double covering. The second summand is the subspace of deformations of the double covering which induce trivial deformations of the branch divisor. The main result of the paper is a description of the effect of imposing singularities in the branch locus. As …
In this article we study the deformation of finite maps and show how to use this deformation theory to construct varieties with given invariants in a projective space. Among other things, we prove a criterion that determines when a finite map can be deformed to a one--to--one map. We use this criterion to construct new simple canonical surfaces with different $c_1^2$ and $χ$. Our general results enable us to describe some new components of the moduli of surfaces of general type. We also find infinitely many moduli spaces $\mathcal M_{(x',0,y)}$ having one component whose general point correspo…
The strong homotopy Lie algebra, controlling simultaneous deformations of a morphism of associative algebras and its domain and codomain is constructed. Isomorphism of the cohomology of this strong homotopy Lie algebra with the classical construction is shown.
We describe infinitesimal deformations of complex naturally graded filiform Leibniz algebras. It is known that any $n$-dimensional filiform Lie algebra can be obtained by a linear integrable deformation of the naturally graded algebra $F_n^3(0)$. We establish that in the same way any $n$-dimensional filiform Leibniz algebra can be obtained by an infinitesimal deformation of the filiform Leibniz algebras $F_{n}^1,$ $F_{n}^2$ and $F_{n}^3(α)$. Moreover, we describe the linear integrable deformations of above-mentioned algebras with a fixed basis of $HL^2$ in the set of all $n$-dimensional Leibni…
If $X$ is a rational surface without nonzero holomorphic vector field and $f$ is an automorphism of $X$, we study in several examples the Zariski tangent space of the local deformation space of the pair $(X, f)$.
Questions related to deformations of germs of finite morphisms of smooth surfaces are discussed. A classification of the four-sheeted germs of finite covers $F: (U,o')\to (V,o)$ is given up to smooth deformations, where $(U,o')$ and $(V,o)$ are two connected germs of smooth complex-analytic surfaces. The singularity types of their branch curves and the local monodromy groups are investigated also.
Let X be a smooth projective curve over a field of characteristic p>0. We show that the Hitchin morphism, which associates to a Higgs bundle its characteristic polynomial, has a non-trivial deformation over the affine line. This deformation is constructed by considering the moduli stack of t-connections on vector bundles on X and an analogue of the p-curvature, and by observing that the associated characteristic polynomial is, in a suitable sense, a p-th power.
In this article, we study the Hilbert scheme of generically non-reduced curves in $\mathbb{P}^3$. We prove the existence of generically non-reduced curves in $\mathbb{P}^3$ for which there exist infinitesimal deformations of the curve that do not induce deformations of the associated reduced scheme. We show that such infinitesimal deformations contribute to the non-reducedness of the corresponding Hilbert scheme. We introduce the notion of extension of curves and prove that such infinitesimal deformations (such that the associated reduced scheme does not deform) are inherited by the extended c…
We compute the restricted infinitesimal deformations of the restricted simple Lie algebras over an algebraically closed field of characteristic different from 2 and 3.
$G_\infty$-structure is shown to exist on the deformation complex of a morphism of associative algebras. The main step of the construction is extension of a $B_\infty$-algebra by an associative algebra. Actions of $B_\infty$-algebras on associative and $B_\infty$-algebras are analyzed, extensions of $B_\infty$-algebras by associative and $B_\infty$-algebras, that they act upon, are constructed. The resulting $G_\infty$-algebra on the deformation complex of a morphism is shown to be quasi-isomorphic to the $G_\infty$-algebra on deformation complex of the corresponding diagram algebra.
The parity of theta characteristics is preserved by infinitesimal deformations
Fundação para a Ciência e Tecnologia (FCT), Portugal, through CAMGSD, IST-ID（葡萄牙科学与技术基金会（FCT），通过CAMGSD，IST-ID）；European Union - Next Generation EU, Mission 4 Component 2（欧盟-下一代欧盟，使命4组成部分2）；Italian MUR（意大利教育部大学与研究部）；Centro de Análise Matemática, Geometria e Sistemas Dinâmicos（数学分析、几何与动力系统中心）；INDAM（意大利国家高等研究学院数学研究所）
We show that the classification of simple finite group schemes over an algebraically closed field reduces to the classification of abstract simple finite groups and of simple restricted Lie algebras in positive characteristic. Both these two simple objects have been classified. We review this classification. Finally, we address the problem of determining the infinitesimal deformations of simple finite group schemes.
In this article we study the deformations of hyperelliptic polarized varieties $(X,L)$ of dimension $m$ and sectional genus $g$ such that the image $Y$ of the morphism $φ$ induced by $|L|$ is smooth. If $L^m < 2g-2$, it is known that, by adjunction and the Clifford's theorem, any deformation of $(X,L)$ is hyperelliptic. Thus, we focus on when $L^m=2g-2$ or $L^m=2g$. We prove that, if $(X,L)$ is Fano-K3, then, except when $Y$ is a hyperquadric, all deformations of $(X,L)$ are again hyperelliptic (if $Y$ is a hyperquadric, the general deformation of $φ$ is an embedding). This contrasts with the …
The purpose of this article is to study the deformations of smooth surfaces $X$ of general type whose canonical map is a finite, degree 2 morphism onto a minimal rational surface or onto $\mathbf F_1$, embedded in projective space by a very ample complete linear series. Among other things, we prove that any deformation of the canonical morphism of such surfaces $X$ is again a morphism of degree 2. A priori, this is not at all obvious, for the invariants $(p_g(X),c_1^2(X))$ of most of these surfaces lie on or above the Castelnuovo line; thus, in principle, a deformation of such $X$ could have a…
In this paper we describe the infinitesimal deformations of null-filiform Leibniz superalgebras over a field of zero characteristic. It is known that up to isomorphism in each dimension there exist two such superalgebras $NF^{n,m}$. One of them is a Leibniz algebra (that is $m=0$) and the second one is a pure Leibniz superalgebra (that is $m\neq 0$) of maximum nilindex. We show that the closure of union of orbits of single-generated Leibniz algebras forms an irreducible component of the variety of Leibniz algebras. We prove that any single-generated Leibniz algebra is a linear integrable defor…
We use the Thom-Whitney construction to show that infinitesimal deformations of a coherent sheaf F are controlled by the differential graded Lie algebra of global sections of an acyclic resolution of the sheaf End(E), where E is any locally free resolution of F. In particular, one recovers the well known fact that the tangent space to deformations of F is Ext^1(F,F), and obstructions are contained in Ext^2(F,F). The main tool is the identification of the deformation functor associated with the Thom-Whitney DGLA of a semicosimplicial DGLA whose cohomology is concentrated in nonnegative degrees …
We realize the infinitesimal Abel-Jacobi map as a morphism of formal deformation theories, realized as a morphism in the homotopy category of differential graded Lie algebras. The whole construction is carried out in a general setting, of which the classical Abel-Jacobi map is a special example.
In 2010, Hrushovski--Loeser showed that the Berkovich analytification of a quasi-projective variety over a non-Archimedean valued field admits a deformation retraction onto a finite simplicial complex. In this article, we adapt the tools and methods developed by Hrushovski--Loeser to study if such deformation retractions can be obtained to be compatible with respect to a given morphism. Amongst other results, we show that compatible deformation retractions exist over a constructible partition of the base and prove the general statement in the case of a morphism of relative dimension 1 where th…
We propose an alternative definition for families of stable pairs $(X,D)$ over a possibly non-reduced base when $D$ is reduced, by replacing $(X,D)$ with an appropriate orbifold pair $(\mathcal X,\mathcal D)$. This definition of a stable family ends up being equivalent to previous ones, but has the advantage of being more amenable to the tools of deformation theory. Moreover, adjunction for $(\mathcal X,\mathcal D)$ holds on the nose; there is no correction term coming from the different. This leads to the existence of functorial gluing morphisms for families of stable surfaces and functorial …
In this article, we investigate deformations of a Calabi-Yau manifold $Z$ in a toric variety $F$, possibly not smooth. In particular, we prove that the forgetful morphism from the Hilbert functor $H^F_Z$ of infinitesimal deformations of $Z$ in $F$ to the functor of infinitesimal deformations of $Z$ is smooth. This implies the smoothness of $H^F_Z $ at the corresponding point in the Hilbert scheme. Moreover, we give some examples and include some computations on the Hodge numbers of Calabi-Yau manifolds in Fano toric varieties.
We show that deformations of a surjective morphism onto a Fano manifold of Picard number 1 are unobstructed and rigid modulo the automorphisms of the target, if the variety of minimal rational tangents of the Fano manifold is non-linear or finite. The condition on the variety of minimal rational tangents holds for practically all known examples of Fano manifolds of Picard number 1, except the projective space. When the variety of minimal rational tangents is non-linear, the proof is based on an earlier result of N. Mok and the author on the birationality of the tangent map. When the varieties …
Let $X$ be a compact connected Riemann surface of genus $g$, with $g\geq 2$, and ${\cal M}_ξ$ a smooth moduli space of fixed determinant semistable vector bundles of rank $n$, with $n\geq 2$, over $X$. Take a smooth anticanonical divisor $D$ on ${\cal M}_ξ$. So $D$ is a Calabi-Yau variety. We compute the number of moduli of $D$, namely $\dim H^1(D, T_D)$, to be $3g-4 + \dim H^0({\cal M}_ξ, K^{-1}_{{\cal M}_ξ})$. Denote by $\cal N$ the moduli space of all such pairs $(X',D')$, namely $D'$ is a smooth anticanonical divisor on a smooth moduli space of semistable vector bundles over the Riemann su…
We construct a general semiregularity map for cycles on a complex analytic or algebraic manifold and show that such semiregularity map can be obtained from the classical tool of the Atiyah-Chern character. The first part of the paper is fairly detailed, deducing the existence and explicit form of a generalized semiregularity map from known results and constructions. We obtain as well a description of the infinitesimal Abel-Jacobi map for smooth cycles as the leading term of this generalized semiregularity map, indicate why for locally complete intersections the appropriate component of our sem…
Let Y be a compact reduced subspace of a complex manifold X, and let F be a subsheaf of the tangent bundle T_X which is closed under the Lie bracket. This paper discusses criteria to guarantee that infinitesimal deformations of the inclusion morphism Y -> X give rise to positive-dimensional deformation families, deforming the inclusion map "along the sheaf F". In case where X is complex-symplectic and F is the sheaf of Hamiltonian vector fields, this partially reproduces known results on unobstructedness of deformations of Lagrangian submanifolds. Written for the IMPANGA Lecture Notes series, …
According to a well-known theorem of Serre and Tate, the infinitesimal deformation theory of an abelian variety in positive characteristic is equivalent to the infinitesimal deformation theory of its Barsotti-Tate group. We extend this result to $1$-motives.
We consider deformations of singular Lagrangian varieties in symplectic spaces. We show the coherence of the direct image sheaves of relative infinitesimal Lagrangian deformations. Using this result, we prove that, under some assumptions, a Lagrangian deformation is analytically rigid provided that this is the case infinitesimally. This gives generalisations of recent results by Colin de Verdière concerning Lagrangian curves.
In this article we present a unified way to smooth certain multiple structures called ropes on smooth varieties. We prove that most ropes of arbitrary multiplicity, supported on smooth curves can be smoothed. By a rope being smoothable we mean that the rope is the flat limit of a family of smooth, irreducible curves. To construct a smoothing, we connect, on the one hand, deformations of a finite morphism to projective space and, on the other hand, morphisms from a rope to projective space. We also prove a general result of independent interest, namely, finite covers onto smooth irreducible cur…
The study of infinitesimal deformations of a variety embedded in projective space requires that of deformations of a collection of points, as specified by a zero-dimensional scheme. Further, basic problems in infinitesimal interpolation correspond directly to the analysis of such a scheme We interpret conjectures of Froberg and Iarrobino on the Hilbert function of a general collection of infinitesimal neighbourhoods of a collection of points in projective space. A main result gives a method for verifying the Weak Conjecture prescribed. Further, we present results on establishing the Strong Con…
Fialowski and Schlichenmaier constructed examples of global deformations of Lie algebras of vector fields from deforming the underlying variety. We formulate their approach in a conceptual way. Namely, we construct a stack of deformations and a morphism form the moduli stack of stable marked curves. The morphism associates to a family of marked curves the family of Lie algebras obtained by taking the Lie algebra of vertical vector fields on the family where one has extracted the marked points. We show that this morphism is almost a monomorphism by Pursell-Shanks theory.
An analytic approach and description are presented for the moduli cotangent sheaf for suitable stable curve families including noded fibers. For sections of the square of the relative dualizing sheaf, the residue map at a node gives rise to an exact sequence. The residue kernel defines the vanishing residue subsheaf. For suitable stable curve families, the direct image sheaf on the base is locally free and the sequence of direct images is exact. Recent work of Hubbard-Koch and a formal argument provide that the direct image sheaf is naturally identified with the moduli cotangent sheaf. The res…
