Proper Morphism Research Articles

The geometric minimal models of analytic spaces

This paper shows that an analytic space X has a unique maximal model through which every proper surjective morphism from a non-singular analytic space to X factors. This is called the geometric minimal model of X and characterized by the contraction property of rational curves. Some other properties such as functoriality, the direct product property and the quotient property of the geometric minimal model are also studied here. The relation of the geometric minimal model with Mori's minimal model is discussed.

Degeneration of weight spectral sequences

A notion of morphism of semi-stable type is a higher dimensional analogue of semi-stable degeneration over the unit disc. For a proper surjective morphism of semi-stable type, the author constructed a cohomological mixed Hodge complex which gives a candidate of the limit of Hodge structures. In this article we define finite increasing filtrations on the cohomological mixed Hodge complex above and prove the E2-degeneracy of the spectral sequences obtained from these filtrations.

On Macaulayfication of Noetherian schemes

The Macaulayfication of a Noetherian scheme X is a birational proper morphism from a Cohen-Macaulay scheme to X. In 1978 Faltings gave a Macaulayfication of a quasi-projective scheme if its non-Cohen-Macaulay locus is of dimension 0 or 1. In the present article, we construct a Macaulayfication of Noetherian schemes without any assumption on the non-Cohen-Macaulay locus. Of course, a desingularization is a Macaulayfication and, in 1964, Hironaka already gave a desingularization of an algebraic variety over a field of characteristic 0. Our method, however, to construct a Macaulayfication is independent of the characteristic.

Holomorphic and algebraic vector bundles on 0-convex algebraic surfaces

LetX be a smooth complex algebraic surface such that there is a proper birational morphism/:X → Y withY an affine variety. Let Xhol be the 2-dimensional complex manifold associated toX. Here we give conditions onX which imply that every holomorphic vector bundle onX is algebraizable and it is an extension of line bundles. We also give an approximation theorem of holomorphic vector bundles on Xhol (X normal algebraic surface) by algebraic vector bundles.

On Macaulayfication of sheaves

Let X be a quasi-projective scheme and ℱ a coherent sheaf of modules over X such that its non-Cohen–Macaulay locus is at most one dimensional. We use and extend the techniques of Brodmann to construct proper birational morphisms of quasi-projective schemes f:Y→X and Cohen–Macaulay coherent sheaves of modules over Y that are isomorphic to the pull-back of ℱ away from the exceptional locus of f. Certain blow-ups of X at locally complete intersections subschemes which contain non-reduced scheme structures on the non-Cohen–Macaulay locus of ℱ are the main part of the construction.

SINGULAR LOCI IN HOLOMORPHIC FAMILIES

Let [Formula: see text] be a proper flat morphism between manifolds, and [Formula: see text] an analytic subset, such that the fibres Xt, for all [Formula: see text], determine a locally trivial deformation of a compact complex manifold. Non-generic fibres Xt, for t ∈ A, may be taken a priori to be singular spaces, or to have a smooth complex structure which is biholomorphically distinct from their generic neighbours. The main theorem of this article provides a sufficient condition for local triviality of the entire family [Formula: see text], in terms of the dimension of A and of the singular subvarieties of certain "degeneracy loci" in [Formula: see text]. Several specific applications of the main theorem are subsequently examined, some of which correspond sharply with examples of "structure-jumping" in complex deformations and "jumping loci" of vector bundles on complex projective space.

A new proof of Suzuki's formula

We give a different proof of a formula of Suzuki and its strengthening by Zaidenberg for the topological Euler characteristic of an affine surface fibered over a curve. We deduce this formula using the ideas for the proof of an analogous formula for a proper morphism.

The Grothendieck duality theorem via Bousfield’s techniques and Brown representability

Grothendieck proved that if f : X ⟶ Y f:X\longrightarrow Y is a proper morphism of nice schemes, then R f ∗ Rf_* has a right adjoint, which is given as tensor product with the relative canonical bundle. The original proof was by patching local data. Deligne proved the existence of the adjoint by a global argument, and Verdier showed that this global adjoint may be computed locally. In this article we show that the existence of the adjoint is an immediate consequence of Brown’s representability theorem. 1It follows almost as immediately, by “smashing” arguments, that the adjoint is given by tensor product with a dualising complex. Verdier’s base change theorem is an easy consequence.

An Extension of Baum-Fulton-MacPherson's Riemann-Roch Theorem for Singular Varieties

In [1] Baum, Fulton and MacPherson extend the Grothendieck-RiemannRoch theorem (abbr. GRR) for non-singular projective varieties to (possibly singular) quasi-projective varieties over any field and proper morphisms. More precisely, in [1, Chapter 1] they prove the Riemann-Roch (abbr. RR) in the case of (possibly singular) complex projective varieties with the ordinary singular homology theory with rational coefficients, by deformation to the normal bundle, and in [1, Chapter 2], in more general, they prove the RR in the case of (possibly singular) quasi-projective varieties over any field and proper morphisms with the Chow homology theory with rational coefficeints [3], by the Grassmannian graph construction. If the field is the complex numbers C, then the above Chow homology theory can be replaced by the singular homology theory (and Borel-Moore homology theory for non-compact varieties). Thus [1, Chap. 2] generalizes [1, Chap. 1].

Convexit� holomorphe intermediaire

The purpose of the present article is to give a generalisation of the notion of holomorphic convexity. We begin with an analogue of Remmert's reduction Theorem by showing that aq-holomorphically convex complex space is naturally endowed with a proper morphism on aq-complete space. We show after that theq-complete spaces arising in this situation are special: they enjoy in particular the property that each relatively compact open subset carries a Kahler form.

On Verdier's specialization formula for Chern classes

Let M be a complex manifold of complex dimension m, and let X C M be a proper analytic subvariety. Let D C ~ be the unit disk, and f : X ~ D a proper analytic morphism. Put X~: = f l(s), s e D. We know that, for r > 0 sufficiently small, the inclusion Xo C f I(D,) is a homotopy equivalence. Thus for small nonzero s e D we may define the specialization map on homology a. :H,(X~)~H.(Xo) as the composition ~, H,(X~) , H , ( f I(D,)) ~ n,(Xo).

Exceptional curves on rational numerically elliptic surfaces

A numerically elliptic surface is a complete smooth algebraic surface X over an algebraically closed field k with a proper morphism f: X-* C to a smooth curve C such that the general fiber off is an integral curve of arithmetic genus 1. If the generic fiber is smooth the surface is called elliptic; otherwise it is called quasi-elliptic. The latter only can occur if k has characteristic 2 or 3, in which case the general fiber is a rational curve with an ordinary cusp [BM]. If no fiber off contains an exceptional cwue (i.e., a smooth irreducible curve isomorphic to [Fp’ and having self-intersection l), then X is said to be minimal; all elliptic surfaces will hereafter be assumed to be minimal. One says that an elliptic surface X is Jacobian if the smooth points of the generic fiber X, comprise the Jacobian curve of X,,. It is equivalent for X, to have a rational point, or for f to have a section. For a rational Jacobian numerically elliptic surface, the exceptional curves are precisely the sections of the libration, which provides a tool by which an enumeration of the exceptional curves on a rational Jacobian numerically elliptic surface can be carried out (cf. [MP, HL, MOP]). Whether Jacobian or not, a rational

A formula for Mumford measure in superstring theory

~ superstring theory. In the first part we prove a conjecture of Manin (2, 5): ~/~ = ~/2 • As in the bosonic case (5), this proof will be used in the second part of the note to derive the fundamental formula. We will work with the definition of SUSY-curves (superconformal map- pings) due to Baranov and Shvarts (i). Polyakov supermeasure is defined in (i), but the super-version of the Belavin-Knizhnik theorem is as yet unknown and it is not clear how to do summation of Mumford superforms over spinor structures. i. Berezinians. Let v: X + S be a smooth proper morphism of complex supervarieties. Then, as in the even case, for any coherent sheaf .~ on X which is flat over S, one can de- fine on S a sheaf B6~) of rank ii0 or 011, with the following properties: • i) If all R~.,~ are locally free, then B (.~)= ® (BerR~.L~)(-I)~;

Semi-integral extensions and proper morphisms

Semi-integral extensions and proper morphisms

Real proper morphisms

Introduction. In this paper we study proper morphisms between real algebraic varieties in two different settings. In the first paragraph we use the concept of semi-integral extensions J BI[ to define semi-integral morphisms which turn out to be analogous (if we work in the family of semialgebraic subsets) of proper maps. In the second paragraph we use the language of real spectra to give an approach to proper morphisms as it is done classically in schemas over algebraically closed fields. Let us mention that proper morphisms have been also studied by Delfs and Knebusch [6] in their general development of semialgebraic spaces. Also I have just known of some related work by Ramanakoraisina [8] on the same topic.

On the Douady Space of a Compact Complex Space in the Category C(???), II

This is a continuation of our previous paper [4]. Let/ : X->S be a proper morphism of complex spaces and $ a coherent analytic sheaf on X. Then we denote by Dx/s($} the Douady space of flat quotients of g over S ; Dx/s($} represents the functor D x / $ ( ( ? } '• (An/5)°-~»Sets defined by Dx/s( <O (T) : =the set of coherent quotients & of g T (the pull-back of g to XxsT) such that ^ is flat over T. We say that / is a ^-morphism (/belongs to <g/S in the terminology of [4]) (resp. is Moishezon) if XTed is a meromorphic image over S of a complex space X which is proper and locally Kahler (resp. locally projective) over S (cf. [4, (2.1) (resp. (1.2))]). Then the purpose of the present paper is to prove the following theorem which generalizes Theorem in [4]:

Proper morphisms and excellent schemes

Let f: X → Y be a finite type morphism of locally noetherian schemes. It is well known ([3, IV, 7.8.6]) that the excellent property ascends from Y to X. On the other side there are counter-examples where X is excellent and Y is not. First of all it is easy to show that the condition on chains of prime ideals does not descend (see [3, IV, 7.8.4]), even by finite morphisms. Secondly in [2] it is produced an example where X is excellent while Y is not a G-scheme (i.e. it has not the good properties of formal fibers). However in [2] it is also proved that the property concerning the openness of regular loci (the so called “J-2”) descends by finite type surjective morphisms. Therefore we are led to the following question: When does the G-scheme property descend? I.e. what conditions do we need on f? A reasonable condition is conjectured (in [2]) as the following: f is proper surjective. The aim of the present paper is precisely to give an answer to such a question. What we really prove is the following. If X is a G-scheme and J-2 (quasi-excellent), then the same is true for Y, provided that f is proper surjective and moreover all the residue fields of Y have characteristic 0. We remark that the result is strongly based on Hironaka’s desingularization for quasi-excellent schemes defined over a field of characteristic 0.

Primitive holomorphic maps of curves

Let t: X→Y be a proper morphism of non-singular irreducible affine curves over ℌ. This paper shows that there is seldom a holomorphic function f such that (t,f): X → Y × ℌ is a holomorphic embedding.

Zariski-Offenheit von Eigentlichen, Flachen holomorphen Abbildungen

The main result of this paper is the theorem: Proper flat morphisms of complex spaces are Zariski-open. In general a flat epimorphism of complex spaces f:X → S need not be open relativ to the Zariski-topology on X and S (example 5.3). But it is shown in this paper that for flat morphisms f:X → S to every xOeX there is an open neighbourhood U of xo in X such that for any Zariski-open subset Z of X the set f (Z∩U) is Zariski-open in f (U). An important tool for the proof of this proposition is the notion of the “devissage relatif” introduced by M. RAYNAUD and L. GRUSON in [11]. In case that f is also proper this local result about flat morphisms yields the global one stated above.

The arithmetic plurigenera of surfaces

Let S0 be a complex projective surface with only isolated Gorenstein singularities (see Introduction to (12)). By Serre's criterion ((4), p. 185) this is equivalent to saying that S0 is normal and Gorenstein. By an algebraic smooth deformation of S0, we shall mean a flat, proper morphism of varieties, ρ: say, with fibre ρ−1(y0) = S0 for some y0 ∈ Y and with the general fibre ρ−1(y) = S being a smooth surface. In the paper (12), we studied such smooth deformations of S0 and in particular the behaviour of the plurigenera Pn of the surfaces in the family. The main result of (12) was the fact that Pn(S0) ≤ Pn(S) for all positive integers n, where the choice of the particular smooth surface was irrelevant by a result of Iitaka(5). To prove the above result we introduced what were called the arithmetic plurigenera of S0, which we define again below. In this paper we shall study more closely these arithmetic quantities, and in the process answer some of the questions posed in (11).