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Fabrizio Catanese

Publications and source records attributed to Fabrizio Catanese.

123 records · Page 7Linked to original sources
arXiv2004

Moduli spaces of surfaces and real structures

We give infinite series of groups Gamma and of compact complex surfaces of general type S with fundamental group Gamma such that 1) Any surface S' with the same Euler number as S, and fundamental group Gamma, is diffeomorphic to S. 2) The moduli space of S consists of exactly two connected components, exchanged by complex conjugation. Whence, i) On the one hand we give simple counterexamples to the DEF = DIFF question whether deformation type and diffeomorphism type coincide for algebraic surfaces. ii) On the other hand we get examples of moduli spaces without real points. iii) Another interesting corollary is the existence of complex surfaces S whose fundamental group Gamma cannot be the fundamental group of a real surface. Our surfaces are surfaces isogenous to a product; i.e., they are quotients (C_1 X C_2)/G of a product of curves by the free action of a finite group G. They resemble the classical hyperelliptic surfaces, in that G operates freely on C_1, while the second curve is a triangle curve, meaning that $C_2 / G \equiv \PP ^1 and the covering is branched in exactly three points.

math.AG↗
arXiv2003

Fibred K"ahler and quasi-projective Groups

We formulate a new theorem giving several necessary and sufficient conditions in order that a surjection of the fundamental group $π_1(X)$ of a compact Kähler manifold onto the fundamental group $Π_g$ of a compact Riemann surface of genus $g \geq 2$ be induced by a holomorphic map. For instance, it suffices that the kernel be finitely generated. We derive as a corollary a restriction for a group $G$, fitting into an exact sequence $ 1 \ra H \ra G \ra Π_g \ra 1$, where $H$ is finitely generated, to be the fundamental group of a compact Kähler manifold. Thanks to the extension by Bauer and Arapura of the Castelnuovo de Franchis theorem to the quasi-projective case (more generally, to Zariski open sets of compact Kähler manifolds) we first extend the previous result to the non compact case. We are finally able to give a topological characterization of quasi-projective surfaces which are fibred over a (quasi-projective) curve by a proper holomorphic map of maximal rank.

math.AG↗
arXiv2003

Deformations in the large of some complex manifolds, I

Main topic of the paper is the determination, for a compact complex manifold $M$, of the class of manifolds $X$ which are deformation equivalent to it. If $M$ is a complex torus, then also $X$ is so. After describing the structure of principal holomorphic torus bundles over curves, a similar result (stability by deformations in the large) is obtained also for the latter class of manifolds. A section of the paper is devoted to the structure of principal holomorphic torus bundles over tori, establishing the Riemann bilinear relations, and exhibiting a so called Appell Humbert family, which could conjecturally give all small deformations. Finally, a general definition is given of so called Blanchard -Calabi manifolds, which, using an old construction of Sommese, show that the family of complex structures on the differentiable manifold underlying the product of a curve of genus $g\geq 2$ with a 2-dimensional complex torus admits several distinct deformation types.

math.AG↗
arXiv2002

Canonical projections of irregular algebraic surfaces

A good canonical projection of a surface $S$ of general type is a morphism to the 3-dimensional projective space P^3 given by 4 sections of the canonical line bundle. To such a projection one associates the direct image sheaf F of the structure sheaf of S, which is shown to admit a certain length 1 symmetrical locally free resolution. The structure theorem gives necessary and sufficient conditions for such a length 1 locally free symmetrical resolution of a sheaf F on P^3 in order that $Spec (mathcal F)$ yield a canonically projected surface of general type. The result was found in 1983 by the first author in the regular case, and the main ingredient here for the irregular (= non Cohen-Macaulay) case is to replace the use of Hilbert resolutions with the use of Beilinson's monads. Most of the paper is devoted then to the analysis of irregular canonical surfaces with p_g=4 and low degree (the analysis in the regular case had been initiated by F. Enriques). For q=1 we classify completely the (irreducible) moduli space for the minimal value K^2=12. We also study other moduli spaces, q=2, K^2=18, and q=3, K^2=12. The last family is provided by the divisors in a (1,1,2) polarization on some Abelian 3-fold. A remarkable subfamily is given by the pull backs, under a 2-isogeny, of the theta divisor of a p.p.Abelian 3-fold: for those surfaces the canonical map is a double cover of a canonical surface (i.e., such that its canonical map is birational). The canonical image is a sextic surface with a plane cubic as double curve, and moreover with an even set of 32 nodes as remaining singular locus. Our method allows to write the equations of these transcendental objects, but not yet to construct new ones by computer algebra.

math.AG↗
arXiv2002

Symplectic structures of algebraic surfaces and deformation

Friedman and Morgan made the "speculation" that deformation equivalence and diffeomorphism should coincide for algebraic surfaces. Counterexamples, for the hitherto open case of surfaces of general type, have been given in the last years by Manetti, by Kharlamov-Kulikov and in my cited article. For the latter much simpler examples, it was shown that there are surfaces $S$ which are not deformation equivalent to their complex conjugate. However, by Seiberg-Witten theory, any (oriented) diffeomorphism of minimal surfaces carries the canonical class K to + K or to - K, and deformation equivalence implies the existence of a diffeomorphism carrying K to +K. In fact, as observed by a referee, the bulk of the proof was to show that our surfaces have no selfhomeomorphism carrying K to - K (the same for the K-K surfaces). In this note we show that Manetti's surfaces provide indeed a counterexample to the reinforced conjecture, since they are symplectomorphic. Our result is that a surface of general type has a canonical symplectic structure (up to symplectomorphism) which is invariant for deformation and for certain degenerations to normal surfaces. Since moreover no simply connected counterexamples to the conjecture are known, we provide explicit families of 1-connected surfaces, which are obtained by glueing together two fixed manifolds with boundary, are not deformation equivalent, but are homeomorphic under a homeomorphism carrying K to +K. We also give as application the existence of symplectically equivalent, but not deformation equivalent cuspidal plane curves.

math.AG↗
arXiv2001

Quadratic sheaves and self-linkage

Generalizing an old result proved by P. Rao [see MR 83i:14025] for arithmetically Cohen-Macaulay, self-linked subschemes of codimension 2 in the projective n-space P, we give a characterization of self-linked pure subschemes of codimension 2 in P satisfying a necessary parity condition, in characteristic different from 2. We make use of the theory of quadratic sheaves described in a previous paper of the authors [see MR 99h:14044].

math.AG↗
arXiv2001

Some remarks on the universal cover of an open K3 surface

We shall give, in an optimal form, a sufficient numerical condition for the finiteness of the fundamental group of the smooth locus of a normal K3 surface. We shall moreover prove that, if the normal K3 surface is elliptic and the above fundamental group is not finite, then there is a finite covering which is a complex torus.

math.AG↗
arXiv2000

Real hyperelliptic surfaces and the orbifold fundamental group

In this paper we finish the topological classification of real algebraic surfaces of Kodaira dimension zero and we make a step towards the Enriques classification of real algebraic surfaces, by describing in detail the structure of the moduli space of real hyperelliptic surfaces. Moreover, we point out the relevance in real geometry of the notion of the orbifold fundamental group of a real variety, and we discuss related questions on real varieties $(X, σ)$ whose underlying complex manifold $X$ is a $K (π, 1)$. Our first result is that if $(S, σ)$ is a real hyperelliptic surface, then the differentiable type of the pair $(S, σ)$ is completely determined by the orbifold fundamental group exact sequence. This result allows us to determine all the possible topological types of $(S, σ)$, and to prove that they are exactly 78. It follows also as a corollary that there are exactly eleven cases for the topological type of the real part of S. Finally, we show that once we fix the topological type of $(S, σ)$ corresponding to a real hyperelliptic surface, the corresponding moduli space is irreducible (and connected). We also give, through a series of tables, explicit analytic representations of the 78 components of the moduli space.

math.AG↗
arXiv2000

Focal Loci of Algebraic Varieties I

The focal locus $Σ_X$ of an affine variety $X$ is roughly speaking the (projective) closure of the set of points $O$ for which there is a smooth point $x \in X$ and a circle with centre $O$ passing through $x$ which osculates $X$ in $x$. Algebraic geometry interprets the focal locus as the branching locus of the endpoint map $ε$ between the Euclidean normal bundle $N_X$ and the projective ambient space ($ε$ sends the normal vector $O-x$ to its endpoint $O$), and in this paper we address two general problems : 1) Characterize the "degenerate" case where the focal locus is not a hypersurface 2) Calculate, in the case where $Σ_X$ is a hypersurface, its degree (with multiplicity)

math.AG↗
arXiv1997

On the Scalar Curvature of Einstein Manifolds

We show that there are high-dimensional smooth compact manifolds which admit pairs of Einstein metrics for which the scalar curvatures have opposite signs. These are counter-examples to a conjecture considered by Besse. The proof hinges on showing that the Barlow surface has small deformations with ample canonical line bundle.

dg-ga↗
arXiv1996

The orbifold fundamental group of Persson-Noether-Horikawa surfaces

The Noether-Horikawa surfaces are the minimal surfaces S with K^2=2p_g-4. For 8 | K^2 they belong to two families of respective type C and N (connected, resp. non connected branch locus for the canonical map). For 16 | K^2 the two types are homeomorphic. Ulf Persson constructed surfaces of type N with a maximally singular canonical model X, whose topology encodes information on the differentiable structure of S. A similar analysis was done by the first author for type C. In this paper we study the genus 2 fibration on X and, in particular, our main result is (X^# being the nonsingular locus of X) π_1(X^#)= Z_4 x Z_4 if 8 | K^2 but 16 does not | K^2 π_1(X^#)= Z_4 x Z_2 if 16 | K^2.

alg-geom↗
arXiv1996

Even sets of nodes are bundle symmetric

Let k be an algebraically closed field of characteristic p different from 2, and let F be a nodal surface of degree d in the projective 3-space P over k (i.e. the singularities of F are only ordinary quadratic, nodes for short). Let N be a subset of the set of nodes of F: then N is said to be n/2-even for n=0,1 if the following condition (*) holds. Namely, let F' be a minimal resolution of the singularities of F, let N' be the inverse image of N in F', let H be the inverse image of a plane section of F. Then N is n/2 if: (*) the class of N'+nH is 2-divisible in Pic(F'). In this paper we prove the following characterization of even sets, which has been conjectured by W. Barth and the second author in 1979. Let F be as above. Then the n/2-even set of nodes N on F are exactly the bundle symmetric sets, i.e. the degeneracy loci of symmetric maps s:E*(-d-n)->E of a suitable locally free sheaf E over P (i.e. F is the locus where rk(s)<rkE, N is the locus where rk(s)=rkE-2). We also give a classification of even sets in degree d=4,5.

alg-geom↗
arXiv1995

Rational surfaces in P^4 containing a plane curve

The families of smooth rational surfaces in $\PP^4$ have been classified in degree $\le 10$. All known rational surfaces in $\PP^4$ can be represented as blow-ups of the plane $\PP^2$. The fine classification of these surfaces consists of giving explicit open and closed conditions which determine the configurations of points corresponding to all surfaces in a given family. Using a restriction argument originally due independently to Alexander and Bauer we achieve the fine classification in two cases, namely non-special rational surfaces of degree 9 and special rational surfaces of degree 8. The first case completes the fine classification of all non-special rational surfaces. In the second case we obtain a description of the moduli space as the quotient of a rational variety by the symmetric group $S_5$. We also discuss in how far this method can be used to study other rational surfaces in $\PP^4$.

alg-geom↗