This article presents a new approach to the unobstructedness result for deformations of Calabi-Yau varieties by introducing the tangent lifting property for a functor on artinian local algebras. The verification that the deformation functor of a Calabi-Yau variety in characteristic 0 fulfills the tangent lifting property uses, just as the verification of the $T^1$-lifting property, the degeneration of the Hodge to de Rham spectral sequence. Our main use of our methods is however to the mixed characteristic case. In that case to be able to verify the conditions needed one needs an extension of the criterion introducing divided powers.
We study an operation, that we call lifting, creating non-isomorphic monomial curves from a single monomial curve. Our main result says that all but finitely many liftings of a monomial curve have Cohen-Macaulay tangent cones even if the tangent cone of the original curve is not Cohen-Macaulay. This implies that the Betti sequence of the tangent cone is eventually constant under this operation. Moreover, all liftings have Cohen-Macaulay tangent cones when the original monomial curve has a Cohen-Macaulay tangent cone. In this case, all the Betti sequences are nothing but the Betti sequence of t…
We provide the main results of a deformation theory of smooth formal schemes. First we deal with the case of global lifting of smooth morphisms. We prove that the obstruction to the existence of a global lifting lies in a Ext^1 group. Then we study uniqueness and existence of lifting of smooth formal schemes. The set of isomorphism classes of smooth liftings is classified by a Ext^1 group and there exists an obstruction in a Ext^2 group whose vanishing characterizes the existence of smooth liftings.
We show that on an Abelian variety over an algebraically closed field of positive characteristic, the obstruction to lifting an automorphism to an Abelian variety over a field of characteristic zero as a morphism vanishes if and only if it vanishes for lifting it as an autoequivalence. We also compare the deformation space of these two types of deformations.
We prove existence and nonexistence results for certain differential forms in positive characteristic, called {\em good deformation data}. Some of these results are obtained by reduction modulo $p$ of Belyi maps. As an application, we solve the local lifting problem for groups with Sylow $p$-subgroup of order $p$.
We construct a quasi-canonical lifting of a $K3$ surface of finite height over a finite field of characteristic $p\geq3$. Such results are previously obtained by Nygaard-Ogus when $p\geq5$. For this purpose, we use the display-theoretic deformation theory developed by Langer, Zink, and Lau. We study the display structure of the crystalline cohomology of deformations of a $K3$ surface of finite height in terms of the Dieudonné display of the enlarged formal Brauer group.
We provide a method to compute the dimension of the tangent space to the global infinitesimal deformation functor of a curve together with a subgroup of the group of automorphisms. The computational techniques we developed are applied to several examples including Fermat curves, $p$-cyclic covers of the affine line and to Lehr-Matignon curves.
We give an alternative approach to the computation of the dimension of the tangent space of the deformation space of curves with automorphisms. A homological version of the local-global principle similar to the one of J.Bertin, A. Mézard is proved, and a computation in the case of ordinary curves is obtained, by application of the results of S. Nakajima for the Galois module structure of the space of 2-holomorphic differentials on them.
In this paper, we study deformations of pairs (C,G) where G is a height 2 BT(or BT_n) group over a complete curve on algebraically closed field k of characteristic p. We prove that, if the curve C is a versal deformation of G, then there exists a unique lifting of the pair to the Witt ring W. We apply this result in the case of Shimura curves to obtain a lifting criterion.
Building on Schlessinger's work, we define a framework for studying geometric deformation problems which allows us to systematize the relationship between the local and global tangent and obstruction spaces of a deformation problem. Starting from Schlessinger's functors of Artin rings, we proceed in two steps: we replace functors to sets by categories fibered in groupoids, allowing us to keep track of automorphisms, and we work with deformation problems naturally associated to a scheme X, and which naturally localize on X, so that we can formalize the local behavior. The first step is already …
Let $f$ be a holomorphic mapping between compact complex manifolds. We give a criterion for $f$ to have {\it unobstructed deformations}, i.e. for the local moduli space of $f$ to be smooth: this says, roughly speaking, that the group of infinitesimal deformations of $f$, when viewed as a functor, itself satisfies a natural lifting property with respect to infinitesimal deformations. This lifting property is satisfied e.g. whenever the group in question admits a `topological' or Hodge-theoretic interpretation, and we give a number of examples, mainly involving Calabi-Yau manifolds, where that i…
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 …
We investigate when the tangent bundle of a projective manifold has a non-trivial first order (or positive-dimensional) deformation. This leads to a new conjectural characterization of the complex projective space.
We study complex compact Kaehler manifolds $X$ carrying a contact structure. If $X$ is almost homogeneous and $b_2(X) \geq 2$, then $X$ is a projectivised tangent bundle (this was known in the projective case even without assumption on the existence of vector fields). We further show that a global projective deformation of the projectivised tangent bundle over a projective space is again of this type unless it is the projectivisation of a special unstable bundle over a projective space. Examples for these bundles are given in any dimension.
Classical Serre-Tate theory describes deformations of ordinary abelian varieties. It implies that every such variety has a canonical lift to characteristic zero and equips its local moduli space with a Frobenius lifting and canonical multiplicative coordinates. A variant of this theory has been obtained for ordinary K3 surfaces by Nygaard and Ogus. In this paper, we construct canonical liftings modulo $p^2$ of varieties with trivial canonical class which are ordinary in the weak sense that the Frobenius acts bijectively on the top cohomology of the structure sheaf. Consequently, we obtain a Fr…
In this paper, we will outline computations of quantum sheaf cohomology for deformations of tangent bundles of toric varieties, for those deformations describable as deformations of toric Euler sequences. Quantum sheaf cohomology is a heterotic analogue of quantum cohomology, a quantum deformation of the classical product on sheaf cohomology groups, that computes nonperturbative corrections to analogues of (27*)^3 couplings in heterotic string computations. Previous computations have relied on either physics-based GLSM techniques or computation-intensive brute-force Cech cohomology techniques.…
We consider matrices with entries in a local ring, Mat(m,n,R). Fix a group action, G on Mat(m,n,R), and a subset of allowed deformations, Σ\subseteq Mat(m,n,R). The standard question of Singularity Theory is the finite-(Σ,G)-determinacy of matrices. Finite determinacy implies algebraizability and is equivalent to a stronger notion: stable algebraizability. In our previous work this determinacy question was reduced to the study of the tangent spaces to Σand to the orbit, T_{(Σ,A)}, T_{(GA,A)} , and their quotient, the tangent module to the miniversal deformation. In particular, the order of det…
We consider matrices with entries in a local ring, Mat(m,n;R). Fix an action of group G on Mat(m,n;R), and a subset of allowed deformations, Σin Mat(m,n;R). The standard question (along the lines of Singularity Theory) is the finite-(Σ,G)-determinacy of matrices. In our previous work this determinacy question was reduced to the study of the tangent spaces to Σand to the orbit, T_{(Σ,A)}, T_{(GA,A)}, and their quotient: the tangent module to the miniversal deformation. In particular, the order of determinacy is controlled by the annihilator of this tangent module. Then we have studied this tang…
The moduli space of generalized deformations of a Calabi-Yau hypersurface is computed in terms of the Jacobian ring of the defining polynomial. The fibers of the tangent bundle to this moduli space carry algebra structures, which are identified using subalgebras of a deformed Jacobian ring.
While the tangent space to an equisingular family of curves can be discribed by the sections of a twisted ideal sheaf, this is no longer true if we only prescribe the multiplicity which a singular point should have. However, it is still possible to compute the dimension of the tangent space with the aid of the equimulitplicity ideal. In this note we consider families L_m={(C,p) | mult_p(C)=m} with C in some linear system |L| on a smooth projective surface S and for a fixed positive integer m, and we compute the dimension of the tangent space to L_m at a point (C,p) depending on whether p is a …
For a singular variety X, an essential step to determine its smoothability and study its deformations is the understanding of the tangent sheaf and of the sheaf T^1_X:=ext^1(Omega_X,O_X). A variety is semi-smooth if its singularities are étale locally the product of a double crossing point (uv=0) or a pinch point (u^2-v^2w=0) with affine space; equivalently, if it can be obtained by gluing a smooth variety along a smooth divisor via an involution with smooth quotient. Our main result is the explicit computation of the tangent sheaf and the sheaf T^1_X for a semi-smooth variety X in terms of th…
Let $X$ be a compact complex manifold, consider a small deformation $ϕ: \mathcal{X} \to B$ of $X$, the dimensions of the cohomology groups of tangent sheaf $H^q(X_t,\mathcal{T}_{X_t})$ may vary under this deformation. This paper will study such phenomenons by studying the obstructions to deform a class in $H^q(X,\mathcal{T}_X)$ with the parameter $t$ and get the formula for the obstructions.
We reduce the study of the Krull dimension d of the deformation ring of the functor of deformations of curves with automorphisms to the study of the tangent space of the deformation functor of a class of matrix representations of the p-part of the decomposition groups at wild ramified points, and we give a method in order to compute d.
Let $X$ and $Y$ be proper birational varieties, say with only rational double points over a perfect field $k$ of positive characteristic. If $X$ lifts to $W_n(k)$, is it true that $Y$ has the same lifting property? This is true for smooth surfaces, but we show by example that this is false for smooth varieties in higher dimension, and for surfaces with canonical singularities. We also answer a stacky analogue of this question: given a canonical surface $X$ with minimal resolution $Y$ and stacky resolution $\mathcal{X}$, we characterize when liftability of $Y$ is equivalent to that of $\mathcal…
We give an elementary construction of the tangent-obstruction theory of the deformations of the pair $(X,L)$ with $X$ a reduced local complete intersection scheme and $L$ a line bundle on $X$. This generalizes the classical deformation theory of pairs in case $X$ is smooth. A criteria for sections of $L$ to extend is also given.
We consider dominant, generically algebraic, and tamely ramified (if the characteristic is positive) morphisms $π: X/S \to Y/S$, where Y,S are Noetherian and integral and X is a Krull scheme (e.g. normal Noetherian), and study the sheaf of tangent vector fields on Y that lift to tangent vector fields on X. We give an easily computable description of these vector fields using valuations along the critical locus. We generalise to positive characteristic Seidenberg's theorem on lifting tangent vector fields to the normalisation. If $π$ is a blow-up of a coherent ideal I we show that tangent vecto…
In this note we study standard and good determinantal schemes. We show that there exist arithmetically Cohen-Macaulay schemes that are not standard determinantal, and whose general hyperplane section is good determinantal. We prove that if a general hyperplane section of a scheme is standard (resp. good) determinantal, then the scheme is standard (resp. good) determinantal up to flat deformation. We also study the transfer of the property of being standard or good determinantal under basic double links.
We consider the polar curves $\PSO$ arising from generic projections of a germ $(S,0)$ of complex surface singularity onto $\C^2$. Taking $(S,0)$ to be a minimal singularity of normal surface (i.e. a rational singularity with reduced tangent cone), we give the $δ$-invariant of these polar curves, as well as the equisingularity-type of their generic plane projections, which are also the discriminants of generic projections of $(S,0)$. These two (equisingularity)-data for $\PSO$ are described in term, on the one side of the geometry of the tangent cone of $(S,0)$ and on the other side of the lim…
Motivated by a remark and a question of Nicholas Katz, we characterize the tangent space of the space of Fuchsian equations with given generic exponents inside the corresponding moduli space of logarithmic connections: we construct a weight 1 Hodge structure on the tangent space of the moduli of logarithmic connections such that deformations of Fuchsian equations correspond to the $(1,0)$-part.
A construction of the tangent dg Lie algebra of a sheaf of operad algebras on a site is presented. The requirements on the site are very mild; the requirements on the algebra are more substantial. A few applications including the description of deformatins of a scheme and equivariant deformations are considered. The construction is based upon a model structure on the category of presheaves which should be of an independent interest.
We consider matrices with entries in a local ring, Mat(R). Fix a group action, G on Mat(R), and a subset of allowed deformations, Σ. The traditional objects of study in Singularity Theory and Algebraic Geometry are the tangent spaces T_{(Σ,A)}, T_{(GA,A)}, and their quotient, the tangent module to the miniversal deformation, T^1_{(Σ,G,A)}. This module plays the key role in various deformation problems, e.g., deformations of maps, of modules, of (skew-)symmetric forms. In particular, the first question is to determine the support/annihilator of this tangent module. In [Belitski-Kerner.1] we hav…
We apply the known results on the Galois module structure of the sheaf of polydifferentials in order to study the dimension of the tangent space of the deformation functor of curves with automorphisms. We are able to find the dimension for the case of weakly ramified covers and for the case of the action of a cyclic group of order $p^v$.
Let $k$ be a field of characteristic zero, $\CO$ be a dg operad over $k$ and let $A$ be an $\CO$-algebra. In this note we define formal deformations of $A$, construct the deformation functor $$\Def_A:\dgar(k)\to\simpl$$ from the category of artinian local dg algebras to the category of simplicial sets. In the case $\CO$ and $A$ are non-positively graded, we prove that $\Def_A$ is governed by the tangent Lie algebra $T_A$ (defined as derivations of a cofibrant resolution of $A$). A very easy example shows that the result does not hold without this condition.
We introduce a precise notion, in terms of few Schlessinger's type conditions, of extended deformation functors which is compatible with most of recent ideas in the Derived Deformation Theory (DDT) program and with geometric examples. With this notion we develop the (extended) analogue of Schlessinger and obstruction theories. The inverse mapping theorem holds for natural transformations of extended deformation functors and all such functors with finite dimensional tangent space are prorepresentable in the homotopy category.
University of Illinois at Chicago（伊利诺伊大学芝加哥分校）；University of California San Diego（加州大学圣地亚哥分校）；Duke University（杜克大学）；Stony Brook University（石溪大学）；Princeton University（普林斯顿大学）；Università degli Studi di Milano（米兰大学）；Tsinghua University（清华大学）；University of Maryland（马里兰大学）
