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... Then for any 1 One reason for this absence may be the sense that establishing the Artin theorems in supergeometry would require the tedious repetition of various difficult arguments in commutative algebra, deformation theory, and algebraic geometry, e.g. for the original approach [Sch68,Rim72,Nér64,Art69a,Art69b,Art74] (with some of these substituteable by later approaches, see e.g. [Fle81,Pop86,CdJ02,Hal17,HR19]). In any case one can find in the literature various assertions that certain results hold in supergeometry "by the same argument as" something in these works. 2 Usually with some details added and of course with reference and credit given to the original source. ...

... Formal versality is an open condition if ξ R formally versal at t ∈ Spec R implies that there exists a Zariski open subset t ∈ U ⊂ Spec R such that ξ R is formally versal at every closed point u ∈ U . Following Flenner [Fle81] we deduce criteria for openness of formal versality by considering a certain sheaf Ex(ξ R , M) on the superscheme Spec R whose vanishing locus is the set of points at which ξ R is formally versal. Observe that if Ex(ξ R , M) were a coherent module, then openness of formal versality would follow automatically from the fact that coherent modules have closed support; however, in general, Ex(ξ R , M) is just a sheaf of pointed sets. ...

... Openness of versality (Theorem 1.7 above) is proven as Theorem 7.7 below. We follow [Fle81], except that the proof of the key lemma about half-exact functors (Lemma B.6, the counterpart of [Fle81, Theorem 6.1]) is done by reduction to the bosonic case. 3 We show in Theorem 8.2 that (under appropriate hypotheses), formal versality in fact implies versality; here we begin by following Artin's original proof [Art74, Theorem 3.3], but then avoid a convergence question by reducing to the bosonic case. ...

We generalize Artin's three main algebraicity theorems to the setting of supergeometry: Artin approximation, algebraization of formal moduli, and algebraization of stacks.

... Relation with other work. The idea of using Exal functors to simplify M. Artin's results should be attributed to H. Flenner [Fle81]. Our results and techniques are quite different, however. ...

... To be precise, for any scheme T , together with an object ξ ∈ X(T ), homogeneity produces an additive functor Exal X (ξ, −) : QCoh(T ) → Ab sharply controlling the theory of ξ. The author learnt these ideas from J. Wise (in person) and his paper [Wis11], though they are likely well-known, and go back at least as far as the work of H. Flenner [Fle81]. ...

... The results of this section are well-known to experts, and similar to those obtained by H. Flenner [Fle81] and J. ...

We give a proof of openness of versality using coherent functors. As an
application, we streamline Artin's criterion for algebraicity of a stack. We
also introduce multi-step obstruction theories, employing them to produce
obstruction theories for the stack of coherent sheaves, the Quot functor, and
spaces of maps in the presence of non-flatness.

... It is not difficult to prove that formal Kuranishi maps exist in a very general setting , namely for groupoïds or functors with a so-called obstruction theory which means that the possibility of extending an object to an infinitesimal thickening of its artinian base is controlled by the vanishing of a certain obstruction element in some vector space (which has to be functorial; compare [1], [5]). Under rather mild assumptions (which are grouppoïd versions of Schlessinger's conditions), there is even a canonical obstruction theory (see [5] p.458). ...

... It is not difficult to prove that formal Kuranishi maps exist in a very general setting , namely for groupoïds or functors with a so-called obstruction theory which means that the possibility of extending an object to an infinitesimal thickening of its artinian base is controlled by the vanishing of a certain obstruction element in some vector space (which has to be functorial; compare [1], [5]). Under rather mild assumptions (which are grouppoïd versions of Schlessinger's conditions), there is even a canonical obstruction theory (see [5] p.458). The formal semi-universal deformation (in the sense of Schlessinger) can be obtained as the zero set of a formal map between two vector spaces ...

... where T denotes the tangent space of the semi-universal deformation and Ob an obstruction space (which has to be the same for all infinitesimal extensions of length one). There does not seem to exist a direct reference for this fact, but κ can be easily constructed inductively, only by using (standard) definitions in formal deformation groupoïds (as for instance contained in [5]). So the existence of a formal Kuranishi map is valid under rather mild assumptions. ...

We explain, in a non technical way, several general methods for constructing semi-universal deformations, especially by Kuranshi
maps. Moreover, we give (standard) criteria of universality and smoothness of the semi-universal deformation, discuss the
existence of operations on the base germ and describe intrinsically the Massey products.

... Before formulating the main results we review some basic notation and facts about deformation theories. In contrast to [Sch] we will not use the language of deformation functors but instead employ deformation groupoids as in [Rim,Fle2,BFl]. Most deformations will take place over An S , Art S , or An^S, the categories of germs of complex spaces, Artinian complex spaces, or formal complex spaces respectively, over a fixed germ S ¼ ðS; 0Þ. ...

... The condition of homogeneity can be weakened to so-called semihomogeneity, see [Rim]. We remark that the main applications of this section remain true under this weaker condition in view of the results of [Fle2]. Note, however, that in all reasonable geometric situations condition (H) above is satisfied. ...

... We recall the following facts, see, e.g., [Fle1,Fle2]. Note that a functor G: CohðS Þ ! ...

This paper contains the details and complete proofs of our earlier announcement in math.AG/9907004 . We construct a general semiregularity map for algebraic cycles as asked for by S. Bloch in 1972. The existence of such a semiregularity map has well known consequences for the structure of the Hilbert scheme and for the variational Hodge conjecture. Aside from generalizing and extending considerably previously known results in this direction, we give new applications to deformations of modules that encompass, for example, results of Artamkin and Mukai. The formation of the semiregularity map here involves powers of the cotangent complex, Atiyah classes, and trace maps, and is defined not only for subspaces of manifolds but for perfect complexes on arbitrary complex spaces. It generalizes in particular Illusie's treatment of the Chern character to the analytic context and specializes to Bloch's earlier description of the semiregularity map for locally complete intersections as well as to the infinitesimal Abel-Jacobi map for submanifolds.

... For any morphism f : X → S of complex analytic germs or spaces, let L X/S ∈ D − (X) denote an (analytic) cotangent complex of X over S, or rather of f . Up to isomorphism, such cotangent complex is a well defined object in the indicated derived category, see [20] or [13,12]. ...

... 1.9. A morphism f : X → S is versal at s ∈ S, if it is flat and induces a formally versal deformation of the fibre X(s) = f −1 (s) at s in S. It is versal if it is so at every point in S. Versality is an open property on S by [20]; see also [12]. ...

... (1) The versality criterion, see [20] or [12], states that for any coherent O Smodule N the support of the cokernel of the Kodaira-Spencer map δ N X/S is contained in the locus of S where f is not versal. In particular, if f is versal, then δ N X/S is surjective for any coherent O S -module N . ...

We present versal complex analytic families, over a smooth base and of fibre dimension zero, one, or two, where the discriminant constitutes a free divisor. These families include finite flat maps, versal deformations of reduced curve singularities, and versal deformations of Gorenstein surface singularities in C^5. It is shown that such free divisors often admit a "fast normalization", obtained by a single application of the Grauert-Remmert normalization algorithm. For a particular Gorenstein surface singularity in C^5, namely the simple elliptic singularity of type \tilde A_4, we exhibit an explicit discriminant matrix and show that the slice of the discriminant for a fixed j-invariant is the cone over the dual variety of an elliptic curve. Comment: 29 pages, misprints and references corrected

... For the proof of the following theorem, we refer to [Fl78,Fl81]. Hence, if φ : (X , x) → (S, s) is a versal deformation of φ −1 (s), x then, for a sufficiently small representative φ : X → S, the multigerm φ : x ∈φ −1 (t) (X , x ) → (S, t), t ∈ S, is a versal deformation of its fibre, the multigerm x ∈φ −1 (t) φ −1 (t), x . ...

... For the proof see[Fl81, Satz 5.2]. It is based on the following useful result (c.f. ...

We give a survey on some aspects of deformations of isolated singularities. In addition to the presentation of the general theory, we report on the question of the smoothability of a singularity and on relations between different invariants, such as the Milnor number, the Tjurina number, and the dimension of a smoothing component.

... X i /S , i = 1, 2, and the kernel and cokernel of (4) are locally free, we have a morphism of vector bundles (5) ρ ϕ : ...

... A is a small extension with kernel ε such that Y ∈ Def es,µ (see [5] Satz 5.2). The second requirement follows from Theorem 6.4. ...

We construct equisingular semiuniversal deformations of Legendrian curves.

... To show that the given deformation is versal (semiuniversal) along the trivial sections, it suffices to show that it is formally versal (semiuniversal), according to [Fl,(5.2) Satz]. Thus, it is sufficient to consider a small extension χ : C ։ C in A K with kernel εK, and equisingular deformations η, η over C, C respectively. ...

... [Sch] for the existence of a formal versal deformation (the first by Theorem 3.1, for the second this is well-known), the same holds for Def es R . Now, it follows from [Fl,(5.2) Satz] that Def es R has a semiuniversal deformation with smooth base, and that Def es R←P → Def es R is smooth. ...

In this paper we develop the theory of equisingular deformations of plane curve singularities in arbitrary characteristic. We study equisingular deformations of the parametrization and of the equation and show that the base space of its semiuniveral deformation is smooth in both cases. Our approach through deformations of the parametrization is elementary and we show that equisingular deformations of the parametrization form a linear subfunctor of all deformations of the parametrization. This gives additional information, even in characteristic zero, the case which was treated by J. Wahl. The methods and proofs extend easily to good characteristic, that is, when the characteristic does not divide the multiplicity of any branch of the singularity. In bad characteristic, however, new phenomena occur and we are naturally led to consider weakly trivial respectively weakly equisingular deformations, that is, those which become trivial respectively equisingular after a finite and dominant base change. The semiuniversal base space for weakly equisingular deformations is, in general, not smooth but becomes smooth after a finite and purely inseparable base extension. For the proof of this fact we introduce some constructions which may have further applications in the theory of singularities in positive characteristic.

... Definition 2.27([4, p. 594]) Let us consider two types of cuirasses, namely I =(I • , (K i ), (K i ), (K i )) andÎ = (I • , (K i ), (K i ), (K i ))(8), which have the same underlying simplicial set. We write ...

We show that every compact complex analytic space endowed with a fine logarithmic structure and every morphism between such spaces admit a semi-universal deformation. These results generalize the analogous results in complex analytic geometry first independently proved by A. Douady and H. Grauert in the ’70. We follow Douady’s two steps process approach consisting of an infinite-dimensional construction of the deformation space followed by a finite-dimensional reduction.

... In literature, a versal and effective deformation is called semi-universal or miniversal. By a general result of H. Flenner ([Fle81], Satz 5.2), every versal deformation gives a semi-universal deformation. ...

We show that every compact complex analytic space endowed with a fine logarithmic structure and every morphism between such spaces admit a semi-universal deformation. These results generalize the analogous results in complex analytic geometry first independently proved by A. Douady and H. Grauert in the '70. We follow Douady's two steps process approach consisting of an infinite-dimensional construction of the semi-universal deformation space followed by a finite-dimensional reduction.

... Artin gives criteria under which a stack is locally cut out by polynomial equations inside its tangent space, thereby ensuring the stack is algebraic. Since Artin's original formulation, there have been a number of improvements [Fle81,Lur12,Pri12,Hal12,HR13]. The statement we give here is close to the form given by Hall [Hal12], but with some hypotheses strengthened for the sake of transparency: ...

This article is based in part on lecture notes prepared for the summer school "The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles" at the Institute for Mathematical Sciences at the National University of Singapore in July of 2014. The aim is to provide a brief introduction to algebraic stacks, and then to give several constructions of the moduli stack of Higgs bundles on algebraic curves. The first construction is via a "bootstrap" method from the algebraic stack of vector bundles on an algebraic curve. This construction is motivated in part by Nitsure's GIT construction of a projective moduli space of semi-stable Higgs bundles, and we describe the relationship between Nitsure's moduli space and the algebraic stacks constructed here. The third approach is via deformation theory, where we directly construct the stack of Higgs bundles using Artin's criterion.

... Theorem 3.4 ([4], Theorem 5.2). Let F → C be a fibered groupoid. ...

We construct versal and equimultiple versal deformations of the parametrization of a Legendrian curve.

... C.5.1, Cor. C.5.2] and related references as [LiS,Flen,Fle1,Illu1,Illu2,Buc]). The aim of this section is to describe the vector space ...

In this paper we give a description of the first order deformation space of a regular embedding X → Y of reduced algebraic schemes. We compare our result with results of Ran (in particular [Ran, Prop. 1.3]).

... I.4.11, Satz III.8.1], also see [51]. ...

In this thesis we study singular curves on K3 surfaces. Let $\mathcal{B}_g$
denote the stack of polarised K3 surfaces of genus $g$ and set
$p(g,k)=k^2(g-1)+1$. There is a stack $ \mathcal{T}^n_{g,k} \to \mathcal{B}_g$
with fibre over the polarised surface $(X,L)$ parametrising all unramified
morphisms $f: C \to X$, birational onto their image, with $C$ an integral
smooth curve of genus $ p(g,k)-n$ and $f_*C \sim kL$. One can think of $
\mathcal{T}^n_{g,k}$ as parametrising all singular curves on K3 surfaces such
that the normalisation map is unramified (or equivalently such that the curve
has "immersed" singularities).
The stack $ \mathcal{T}^n_{g,k}$ comes with a natural moduli map $$\eta \; :
\;\mathcal{T}^n_{g,k} \to \mathcal{M}_{p(g,k)-n}$$ to the Deligne-Mumford stack
of curves, defined by forgetting the map to the K3 surface. We first show that
$\eta$ is generically finite (to its image) on at least one component of
$\mathcal{T}^n_{g,k} $, in all but finitely many values of $p(g,k)-n$. We also
consider related questions about the Brill-Noether theory of singular curves on
K3 surfaces as well as the surjectivity of twisted Gaussian maps on
normalisations of singular curves. Lastly, we apply the deformation theory of
$\mathcal{T}^n_{g,k}$ to a seemingly unrelated problem, namely the
Bloch-Beilinson conjectures on the Chow group of points of K3 surfaces with a
symplectic involution.

... By Lemma 3.4, p * ((K X/Y ) m ⊗ L ⊗ I m (ϕ L )) is coherent. Using [Fle81] (cf. also [BDIP02, Thm 10.7, page 47]), there exists a subvariety Z of Y of codimension at least 1 such that p is smooth on Y \ Z and for every point t ∈ Y \ Z, we have ...

Our main goal in this article is to prove a new extension theorem for
sections of the canonical bundle of a weakly pseudoconvex K\"ahler manifold
with values in a line bundle endowed with a possibly singular metric. We also
give some applications of our result.

... 2.8. On the other hand, if ϕ is a flat morphism, then the Kodaira-Spencer map is surjective, if, and only if, ϕ represents a versal deformation of the fibre X 0 = ϕ −1 (0) ⊆ X of ϕ over the origin ( [Fle81]). ...

Let $\varphi :X\to S$ be a morphism between smooth complex analytic spaces and let $f=0$ define a free divisor on $S$. We prove that if the deformation space $T^1_{X/S}$ of $\varphi $ is a Cohen–Macaulay $\mathcal {O}_X$-module of codimension 2, and all of the logarithmic vector fields for $f=0$ lift via $\varphi $, then $f\circ \varphi =0$ defines a free divisor on $X$; this is generalized in several directions.
Among applications we recover a result of Mond–van Straten, generalize a construction of Buchweitz–Conca, and show that a
map $\varphi :\mathbb {C}^{n+1}\to \mathbb {C}^n$ with critical set of codimension 2 has a $T^1_{X/S}$ with the desired properties. Finally, if $X$ is a representation of a reductive complex algebraic group $G$ and $\varphi $ is the algebraic quotient $X\to S=X\!{/\!/} G$ with $X\!{/\!/} G$ smooth, then we describe sufficient conditions for $T^1_{X/S}$ to be Cohen–Macaulay of codimension 2. In one such case, a free divisor on $\mathbb {C}^{n+1}$ lifts under the operation of ‘castling’ to a free divisor on $\mathbb {C}^{n(n+1)}$, partially generalizing work of Granger–Mond–Schulze on linear free divisors. We give several other examples of such representations.

... After completing this paper we also located in the literature two very nice papers of H. Flenner addressing similar results for analytic spaces. In particular, if S is excellent and of finite Krull dimension, then Theorem A follows from the results of [Fle81,§7] and Theorems B and D are the main results of [Fle82] (this is in the analytic category, however, thus M is assumed to be coherent in Theorem D). Without further assumptions on f and M q (e.g. ...

We prove that a wide class of Ext-functors on algebraic stacks are coherent
(in the sense of M. Auslander). As applications, we show that many Hom-functors
are represented by affine schemes, and that Cohomology and Base Change also
holds for algebraic stacks.

... Assigning to a pair (E, ϕ) the sheaf E gives a functor P → Q. It is well known that there are semiuniversal deformations in Q (see [ST] or [BK]) and that versality is open is Q (see e.g., [Fl1]). The fibre of P → Q over a given object E ∈ Q(T ) is the groupoid P E → An T as explained in [Bi,Sect. ...

Let X be a complex space and F a coherent O
x
-module, A F-(co)framed sheaf on X is a pair (ε, ϕ) with a coherent O
x
-module ε and a morphism of coherent sheaves ϕ: F
F ε (resp. ϕ: ε → F). Two such pairs (ε, ϕ) and (ε′,ϕ′) are said to be isomorphic if there exists an isomorphism of sheaves α: ε →ε′ with α° ϕ = ϕ′ (resp. ϕ′° α = ϕ). A pair (α, ϕ) is called simple if its only automorphism is the identity on ε. In this note we prove a representability theorem in a relative framework, which implies in particular that there is a moduli space of simple F- (co) framed sheaves on a given compact complex space X.

The supermoduli space $\frak{M}_{g,0,2r}$ is not projected for all $g \ge 5r +1 \ge 6$.

We study the cohomology of Jacobians and Hilbert schemes of points on reduced and locally planar curves, which are however allowed to be singular and reducible. We show that the cohomologies of all Hilbert schemes of all subcurves are encoded in the cohomologies of the fine compactified Jacobians of connected subcurves, via the perverse Leray filtration. We also prove, along the way, a result of independent interest, giving sufficient conditions for smoothness of the total space of the relative compactified Jacobian of a family of locally planar curves.

This volume consists of ten articles which provide an in-depth and reader-friendly survey of some of the foundational aspects of singularity theory. Authored by world experts, the various contributions deal with both classical material and modern developments, covering a wide range of topics which are linked to each other in fundamental ways.
Singularities are ubiquitous in mathematics and science in general. Singularity theory interacts energetically with the rest of mathematics, acting as a crucible where different types of mathematical problems interact, surprising connections are born and simple questions lead to ideas which resonate in other parts of the subject.
This is the first volume in a series which aims to provide an accessible account of the state-of-the-art of the subject, its frontiers, and its interactions with other areas of research.
The book is addressed to graduate students and newcomers to the theory, as well as to specialists who can use it as a guidebook.

In [22] we established axiomatic parametrised Cohen-Macaulay approximation which in particular was applied to pairs consisting of a finite type flat family of Cohen-Macaulay rings and modules. In this sequel we study the induced maps of deformation functors and deduce properties like smoothness and injectivity under general, mainly cohomological conditions on the module.

In a previous article (J. Algebra 367 (2012), 142-165) we established axiomatic parametrised Cohen-Macaulay approximation which in particular was applied to pairs consisting of a finite type flat family of Cohen-Macaulay rings and modules. In this sequel we study the induced maps of deformation functors and deduce properties like smoothness and injectivity under general, mainly cohomological conditions on the module.

For a pair (algebra, module) with isolated singularity we establish the existence of a versal henselian deformation. Obstruction theory in terms of an Andr\'e-Quillen cohomology for pairs is a central ingredient in the Artin theory used. Cohen-Macaulay approximation induces maps between versal base spaces for pairs and cohomology conditions ensure properties like smoothness and isomorphism.

This book arises from the INdAM Meeting "Complex and Symplectic Geometry", which was held in Cortona in June 2016. Several leading specialists, including young researchers, in the field of complex and symplectic geometry, present the state of the art of their research on topics such as the cohomology of complex manifolds; analytic techniques in Kähler and non-Kähler geometry; almost-complex and symplectic structures; special structures on complex manifolds; and deformations of complex objects. The work is intended for researchers in these areas.

Our main goal in this article is to prove an extension theorem for sections of the canonical bundle of a weakly pseudoconvex Kähler manifold with values in a line bundle endowed with a possibly singular metric. We also give some applications of our result.

We construct examples of primitive contractions of Calabi–Yau threefolds with exceptional locus being P 1 ×P 1, P 2, and smooth del Pezzo surfaces of degrees ≤ 5. We describe the images of these primitive contractions and find their smoothing families. In particular, we give a method to compute the Hodge numbers of a generic fiber of the smoothing family of each Calabi–Yau threefold with one isolated singularity obtained after a primitive contraction of type II. As an application, we get examples of natural conifold transitions between some families of Calabi–Yau threefolds. 1.

Singularity theory is a field of intensive study in modern mathematics with fascinating relations to algebraic geometry, complex analysis, commutative algebra, representation theory, theory of Lie groups, topology, dynamical systems, and many more, and with numerous applications in the natural and technical sciences.
This book presents the basic singularity theory of analytic spaces, including local deformation theory, and the theory of plane curve singularities. Plane curve singularities are a classical object of study, rich of ideas and applications, which still is in the center of current research and as such provides an ideal introduction to the general theory. Deformation theory is an important technique in many branches of contemporary algebraic geometry and complex analysis. This introductory text provides the general framework of the theory while still remaining concrete.
In the first part of the book the authors develop the relevant techniques, including the Weierstras preparation theorem, the finite coherence theorem etc., and then treat isolated hypersurface singularities, notably the finite determinacy, classification of simple singularities and topological and analytic invariants. In local deformation theory, emphasis is laid on the issues of versality, obstructions, and equisingular deformations. The book moreover contains a new treatment of equisingular deformations of plane curve singularities including a proof for the smoothness of the mu-constant stratum which is based on deformations of the parameterization. Computational aspects of the theory are discussed as well. Three appendices, including basic facts from sheaf theory, commutative algebra, and formal deformation theory, make the reading self-contained.
The material, which can be found partly in other books and partly in research articles, is presented from a unified point of view for the first time. It is given with complete proofs, new in many cases. The book thus can serve as source for special courses in singularity theory and local algebraic and analytic geometry.

We prove that cohomology and base change holds for algebraic stacks,
generalizing work of Brochard in the tame case. We also show that
Hom-spaces on algebraic stacks are represented by abelian cones,
generalizing results of Grothendieck, Brochard, Olsson, Lieblich, and
Roth--Starr. To accomplish all of this, we prove that a wide class of
Ext-functors in algebraic geometry are coherent (in the sense of M.
Auslander).

This article deals with two topics: the first, which has a general character,
is a variation formula for the the determinant line bundle in non-K\"ahlerian
geometry. This formula, which is a consequence of the non-K\"ahlerian version
of the Grothendieck-Riemann Roch theorem proved recently by Bismut, gives the
variation of the determinant line bundle corresponding to a perturbation of a
Fourier-Mukai kernel ${\cal E}$ on a product $B\times X$ by a unitary flat line
bundle on the fiber $X$. When this fiber is a complex surface and ${\cal E}$ is
a holomorphic 2-bundle, the result can be interpreted as a Donaldson invariant.
A finer variation formula has been proved by J. Grivaux.
The second topic concerns a geometric application of our variation formula,
namely we will study compact complex subspaces of the moduli spaces of stable
bundles considered in our program for proving existence of curves on minimal
class VII surfaces. Such a moduli space comes with a distinguished point
$a=[{\cal A}]$ corresponding to the canonical extension ${\cal A}$ of $X$. The
compact subspaces $Y\subset {\cal M}^\st$ containing this distinguished point
play an important role in our program. We will prove a non-existence result:
there exists no compact complex subspace of positive dimension $Y\subset {\cal
M}^\st$ containing $a$ with an open neighborhood $a\in Y_a\subset Y$ such that
$Y_a\setminus\{a\}$ consists only of non-filtrable bundles. In other words,
within any compact complex subspace of positive dimension $Y\subset {\cal
M}^\st$ containing $a$, the point $a$ can be approached by filtrable bundles.
Specializing to the case $b_2=2$ we obtain a new way to complete the proof of a
theorem in a previous article: any minimal class VII surface with $b_2=2$ has a
cycle of curves. Applications to class VII surfaces with higher $b_2$ will be
be discussed in a forthcoming article.

Let $\pi:M\to B$ be a proper holomorphic submersion between complex manifolds
and ${\cal E}$ a holomorphic bundle on $M$. We study and describe explicitly
the torsion subsheaf $\Tors(R^1\pi_*({\cal E}))$ of the first direct image
$R^1\pi_*({\cal E})$ under the assumption $R^0\pi_*({\cal E})=0$. We give two
applications of our results. The first concerns the locus of points in the base
of a generically versal family of complex surfaces where the family is
non-versal. The second application is a vanishing result for
$H^0(\Tors(R^1\pi_*({\cal E})))$ in a concrete situation related to our program
to prove existence of curves on class VII surfaces.

Using notions of homogeneity we give new proofs of M. Artin's algebraicity
criteria for functors and groupoids. Our methods give a more general result,
unifying Artin's two theorems and clarifying their differences.

This article deals with families of automorphisms of rational surfaces and
related rigidity properties. We introduce the notions of rigidity and
infinitesimal rigidity, and give a bound on the number of parameters of a
family of rational surfaces endowed with an automorphism. This yields in
particular a cohomological criterion to prove the rigidity of an automorphism.
We study automorphisms on rational surfaces X whose anticanonical divisor is
effective but also automorphisms of rational Kummer surfaces. Finally we
construct a theoretical framework to compute effectively in any concrete
example the action of a rational surface automorphism on the space of
infinitesimal deformations of the surface.

Ein interessantes Problem ist es, die Geometrie und Struktur des Douadyraums H eines komplexen Raumes X zu verstehen. M. Lehn hat gezeigt, dass sich die Garbe der Kählerschen 1-Formen ΩH durch eine relative Ext-Garbe beschreiben lässt, wenn X kompakt und glatt ist. Ziel der vorliegenden Dissertation ist es, eine solche Beschreibung für singuläre Räume X zu zeigen und diese für Deformationen von kompakten komplexen Räumen und Modulräumen von Vektorbündeln zu verallgemeinern. Dazu betrachten wir allgemeine Deformationstheorien. Für Deformationen über einem Basisraum S führen wir S- Auflösungen ein und zeigen, dass für semiuniverselle Deformationen ohne infinitesimale Automorphismen ΩS isomorph zur (1)-ten Kohomologie einer solchen S-Auflösung ist. Etwas allgemeiner erhalten wir eine ähnliche Beschreibung von ΩS auch für verselle Deformationen mit infinitesimalen Automorphismen.

In recent years I have computed versal deformations of various singularities, partly by hand, but mostly with the program Macaulay. I explain here how to do these computations. As an application I discuss the smoothability of a certain curve singularity, a case I had not been to settle with general methods. As a result I find an example of a reduced curve singularity with several smoothing components.

In this work, modular subspaces corresponding to the ordinary deformation functor as well as the functor of deformations with section of a singularity. After developing the necessary facts from deformation theory and the techniques of cotangent cohomology, several criteria for modularity are given, in terms of relative cotangent cohomology modules. Based on these theoretical results, an algorithm is explained to obtain the modular stratum of an isolated singularity up to a given order. As applications, the modular strata of isolated hypersurface singularities of small modality are determined and several examples - mainly of (semi-)quasihomogeneous singularities - are presented which exhibit interesting and surprising phenomena.

In dieser Arbeit werden Deformationen Riemannscher Flächen auf elementare Weise mittels Integration von Vektorfeldern konstruiert. Diese Deformationen werden zum feinen Modulraum der Riemannschen Flächen mit Teichmüller-Markierung zusammengeklebt. Die Vorgehensweise unterscheidet sich wesentlich von den analytischen Beweisen Teichmüllers und den algebraisch geometrischen Beweisen Grothendiecks.

We consider flat families of reduced curves on a smooth surface 5 such that each member C has the same number of singularities and each singularity has a fixed singularity type (up to analytic resp. topological equivalence). We show that these families are represented by a scheme H and give sufficient conditions for the smoothness of H (at C). Our results improve previously known criteria for families with fixed analytic singularity type and seem to be quite sharp for curves in ℙ2 of small degree. Moreover, for families with fixed topological type this paper seems to be the first in which arbitrary singularities are treated.

We construct several new examples of Calabi–Yau threefolds with Picard group of rank 1. Each of these examples is obtained
by smoothing the image of a primitive contraction with exceptional divisor being a del Pezzo surface of degree 4, 5, 6, or
7, or ℙ1×ℙ1.

This is a survey of the main directions in the theory of deformations of complex spaces. It touches on related questions: deformations of maps, cohomology of algebras, topology of singularities. The construction of the tangent complex and the tangent cohomology of a complex space is set out. The generalized Kodaira-Spencer class and the obstruction are defined as elements of the tangent cohomology; the latter is calculated in terms of the cohomology of the structure sheaf of the space. The constancy of the Euler characteristic of the tangent cohomology is established for deformations that realize versal and universal deformations of compact spaces and the connection of the geometry of a base of a versal deformation with the Massey operation in tangent cohomology is explained.

Es handelt sich um einen Beweis der folgenden Stze, die zuerst von Grauert angegeben wurden (Publ. Math. I.H.E.S. No. 5, 1960; vgl. dies Zbl.100, 80 (1963)):Es seif:X Y eine eigentliche holomorphe Abbildung komplexer Rume, sei einef-platte kohrente analytische Garbe berX; es bezeichneX
y
die Faser vonf ber einem Punkty Y und die analytische Einschrnkung von aufX
y
. Dann gilt: (I) Die Funktionend
q
(y)=dimH
q
(X
y
, sind halbstetig nach oben. (II) Ist fr einq die Funktiond
q
(y) konstant undY reduziert, so ist dieq-te direkte Bildgarbe von unterf lokal frei berY. (III) Die Euler-Poincar-Charakteristikx(y)=(–1)
q
dimH
q
(X
y
,) ist lokal konstant berY. — Der Beweis benutzt systematisch den Begriff des Steinschen Kompaktums (= kompakte semianalytische Menge mit Steinscher Umgebungsbasis). Mit Hilfe der von Frisch bewiesenen Tatsache, da die Algebra der Schnitte in der Strukturgarbe eines komplexen Raumes ber einem Steinschen Kompaktum noethersch ist (Invent. Math.4, 118–138 (1967); vgl. dies Zbl.167, 68 (1969)), gelingt es, die Grothendieckschen Methoden im algebraischen Fall (EGA III) auf die analytische Situation zu bertragen.

We prove here a theorem, which generalizes Grauert's comparison theorem ([4], Hauptsatz IIa; cf. also Knorr [7], Vergleichssatz) and which is an analogue of a Grothndieck's result in Algebraic Geometry ([6], Chap. III., 4.1.5). The proof makes essential use of a coherence theorem for sheaves of polynomials: Let X,Y be complex spaces, : XY a proper holomorphic map and T=(T1,...,TN) a system of indeterminates. Then, for everyO
X[T] graded sheafm, all direct image sheaves Rq*
m areY[T]-coherent. The proof is as in [2].

TABLA DE CONTENIDO
TABLA DE CONTENIDO
Cohomologie de Diabgrammes. Deformations Equivariantes de G-Schemas et Deformations de Schemas en Groupes. Categories Formelles Complexes de Rham et Cohomologie Cristalline.

Tabla de contenido Tabla de contenido
Chapitre 1: Algebre Homotopique relative.
Chapitre 2: Complexe cotangent: Definition et premieres proprietes .
Chapitre 3: complexe cotangent et extension infinitesimales.
Chapitre 4: Deformations de modules.
Bibliographies.
Index.

D6formation de singularit6s isol6es

- G Pourcin

Pourcin, G.: D6formation de singularit6s isol6es. Asterisque 16, 161-173 (1974)

Der Kotangentenkomplex in der analytischen Geometrie

- H Flenner

Über Deformationen holomorpher Abbildungen

- H Flenner