Algebraic Curves and Projective Geometry. Proc. conf Trento, 1988

Non-special scrolls with general moduli

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In 1 , the case where G acts freely is trivial in the holomorphic context, and taken care of by proposition 2, p. We only provide an argument for the last statement in 1 , for which we do not know of a precise reference. The question is local, hence we may assume that P is a smooth point of X , and that G is equal to the stabilizer of P , hence it is a finite group. This is clear on the open set which is the complement of the branch set p R. At the points of R which are smooth points, then the G action can be linearized locally, and it is a pseudoreflection; so in this case the action of G involves only the last variable, hence the result follows from the dimension 1 case, where every torsion free module is locally free.

Step II. Thus, the question of the existence of a linearization is reduced to the algebraic question of the splitting of the central extension given by the Theta group. This question is addressed by group cohomology theory, as follows see [ ]. The divisor class of 2 O is never represented by a G -invariant divisor, since all the G -orbits consist of 4 points, and the degree of 2 O is not divisible by 4.

The nice part of the story is the following useful result, which was used by Atiyah in the case of elliptic curves, to study vector bundles over these [ 12 ]. In the next subsection we shall give an explicit example where the Heisenberg group is used for a geometrical construction. Now let us return to the general discussion of cohomology of G -linearized sheaves.

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The names change a little bit: instead of a linearized vector bundle one talks here of G -equivariant vector bundles, and there is a theory of equivariant characteristic classes, i. This construction is used in [ 37 ] to prove the following. We argue similarly to [ 35 ], Step 4 of theorem 0. Cases 1-I and 1-II can be excluded as case 0 , since then D would be reducible. By the description of Bagnera—De Franchis varieties given in Sect. Therefore, there are exactly 4 invariant divisors in the linear system L. In order to establish that the general fibre of the Albanese map is non hyperelliptic, it suffices to prove the following lemma.

There are two cases. Hence D has two double points and geometric genus equal to 3. Assume that C is hyperelliptic, and denote by h the hyperelliptic involution, which lies in the centre of Aut C. QED for the lemma. We may now apply the previous theorem in order to obtain the classification. The first variation of the energy function vanishes precisely when f is a harmonic map, i. The energy functional enters also in the study of geodesics and Morse theory see [ ]. The next theorem is one of the most important results, first obtained in [ ]. Not only the condition about the curvature is necessary for the existence of a harmonic representative in each homotopy class, but moreover it constitutes the main source of connections with the concept of classifying spaces, in view of the classical see [ 76 , ] theorem of Cartan—Hadamard establishing a deep link between curvature and topology.

Thus in complex dimension 1 one cannot hope for a stronger result, to have a holomorphic map rather than just a harmonic one. Unfortunately, this is not the case, as one sees, already in the case of elliptic curves; but then one may restrict the hope to proving that f is either holomorphic or antiholomorphic. Let X be a cKM, i. Then X is a complex torus if and only if it has the same integral cohomology algebra of a complex torus, i. Then X is a complex torus if and only if it has the same rational cohomology algebra of a complex torus, i.

B is an ample divisor, and A and X are projective. The first question is to show that the canonical divisor, which is just the ramification divisor R , is an ample divisor. Added in proof : Will Sawin observed that case i does not occur for varieties of general type, since for these, by a result of Popa and Schnell, holomorphic 1- forms always have zeros.

Instead, Debarre, Jiang and Lahoz showed that case ii leads to the occurrence of certain iterated torus bundles, so that only a proper modification of our conjecture can be true. Y is a complex deformation of X if and only if X and Y have the same Hodge type.

Without loss of generality, we may assume that G contains no translations. By our hypothesis f induces an isomorphism of cohomology groups and has degree equal to 1. Therefore Y is also a Generalized Hyperelliptic manifold of complex dimension n.

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The following is one more characterization of complex tori and Abelian varieties. Let Z be a projective variety. Hence T is a group of translations, and we obtain that Z is a complex torus. Assume further that the real rank of the derivative Df is at least 4 in some point of M. Then f is either holomorphic or antiholomorphic. One needs to show that at some point the real rank of the derivative DF is at least 4.

Let Y be a subanalytic compact set, or just a compact set of Hausdorff dimension at most h.

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Second proof of the lemma: Y is a compact of Hausdorff dimension at most h. From part I of the theorem one knows that f is holomorphic or antiholomorphic in short, one says that f is dianalytic. There remains only to prove that f is biholomorphic. Step 1: f is a finite map; otherwise, since f is proper, there would be a complex curve C such that f C is a point. On the other hand, since f is a homotopy equivalence, and f C is a point, this class must be zero. Step 3: f is holomorphic, finite and of degree 1. Then the inverse of f is defined on V , the complement of a complex analytic set, and by the Riemann extension theorem normality of smooth varieties the inverse extends to N , showing that f is biholomorphic.

Let us try however to describe precisely the main hypothesis of strong negativity of the curvature, which is a stronger condition than the strict negativity of the sectional curvature.

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Indeed, the bulk of the calculations is to see that there is an upper bound for the nullity of the Hermitian sectional curvature, i. The first result in this direction was the theorem of [ ], also obtained by Jost and Yau see [ ] and also [ ] for other results. In this case the homomorphism leads to a harmonic map to a curve, and one has to show that the Stein factorization yields a map to some Riemann surface which is holomorphic for some complex structure on the target.

Holomorphic forms are closed, i.

More precisely, one gets the following theorem [ 87 ]:. A simpler example was then found by Kotschick [ 2 ], ex. We do not mention in detail generalisations of the Castelnuovo—de Franchis theory to higher dimensional targets see [ 87 , , , ] , since these shall not be used in the sequel. We want however to mention another result [ ], see also [ ] for a weaker result which again, like the isotropic subspace theorem, determines explicitly the genus of the target curve a result which is clearly useful for classification and moduli problems.

Which are the projective groups, i. Serre himself proved see [ ] that the answer to the first question is positive for every finite group.

In this chapter we shall only limit ourselves to mention some examples and results to give a general idea, especially about the use of harmonic maps, referring the reader to the book [ 2 ], entirely dedicated to this subject. An obvious observation is that the crucial hypothesis is projectivity. Arapura, Bressler and Ramachandran answer in particular one question raised by Johnson and Rees, namely they show [ 7 ]:. We say instead that Y is deformation equivalent to X if Y is equivalent to X for the equivalence relation generated by the symmetric relation of direct deformation.

The counterexamples by Claire Voisin were based on topological ideas, namely on the integrality of some multilinear algebra structures on the cohomology of projective varieties. Hodge Symmetry holds true, i. We need here perhaps to recall once more that deformation equivalent complex manifolds are diffeomorphic. We refer to the original paper for details of the construction and proof. The article [ ] contains several proofs, according to the taste of the several authors. One can see the relation of the above lemma with the vanishing of Massey products, which we now define in the simplest case.

Instead things are different for torsion coefficients, as shown by Torsten Ekedahl [ ]. Hence the well known fact that the set of fundamental groups of smooth projective varieties is just the set of fundamental groups of smooth algebraic surfaces. Serre to show that any finite group G occurs as the fundamental group of a smooth projective surface S.

One of the many reasons of the beauty of the theory of curves is given by the uniformization theorem, stating that any complex manifold C of dimension 1 which is not of special type i. In higher dimensions there are simply connected projective varieties that have a positive dimensional moduli space: already in the case of surfaces, e.

So, if there is some analogy, it must be a weaker one, and the first possible direction is to relate somehow Kodaira dimension with curvature. Moreover, it is known, by a theorem of Siegel [ ], cf. We recall these concepts see for instance [ 6 , ]. In his book [ ] Shafarevich answered the following. But, as already mentioned, a classical result of J. Hano see [ ] Theorem IV, p.

On the Enumeration of Algebraic Curves — from Circles to Instantons

By taking finite ramified coverings of such locally symmetric varieties, and blow ups of points of the latter, we easily obtain many examples which are holomorphically convex but not Stein. The existence of a curve satisfying these two properties would then give a negative answer to the Shafarevich question. So, we can define a subvariety to be Shafarevich bad if. But still, could they exist, giving a negative answer to the Shafarevich question? Indeed, for each projective surface, after blowing up a finite number of points, we can always obtain such a fibration. Assume instead that this image is infinite and look at components of the singular fibres: we are interested to see whether we find some Shafarevich bad cycles.

In this way they propose to construct counterexamples to the Shafarevich question with non residually finite fundamental group , provide certain group theoretic questions have an affirmative answer. We want to point out a topological consequence of the Shafarevich property, for simplicity we consider only the case of a projective surface X with infinite fundamental group.

In the first case we have a fibration with compact fibres of complex dimension 1, in the second case we have a discrete set such that the fibre has complex dimension 1, and the conclusion is that:.

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