André Hirschowitz

Laboratoire J.-A. Dieudonné, Université de Nice - Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 02. Office tel: 04 92 07 62 04 - Dept fax: 04 93 51 79 74 - Home tel: 04 93 29 05 75; email: ah@math.unice.fr
  • G.Ellingsrud-A.Hirschowitz: Sur le fibré normal des courbes gauches. C.R.A.S. 299 (1984) 245-248. We apply degeneration techniques in order to prove that the normal bundle of most good (eg nonspecial) generic space curves is stable with natural cohomology.
  • Cohomology of a general instanton bundle (avec Hartshorne) Ann. Sc. ENS Paris,15, 365-390 (1982). The scope of the m\'ethode d'Horace is extended to the cohomology of vector bundles and a conjecture of Hartshorne is proved.
  • Hirschowitz, Andre On the convergence of formal equivalence between embeddings. (English) [J] Ann. Math., II. Ser. 113, 501-514 (1981). A new geometric method is introduced for proving convergence of formal equivalences, and the so-called formal principle is extended beyond earlier results of Griffiths and Hartshorne (among others).
  • Hirschowitz, A.: Sur la postulation generique des courbes rationnelles. Acta Math. 146, 209-230 (1981). In this paper is introduced the so-called "m\'ethode d'Horace" for postulation problems, which will be reused and enlarged many times. Here a conjecture of Hartshorne is settled.
  • Hirschowitz, A.; Piriou, A. Proprietes de transmission pour les distributions integrales de Fourier. Commun. Partial Differ. Equations 4, 113-217 (1979). A whole theory is developed for these symmetries of distributions; this theory incorporates original considerations concerning indices (or signatures) associated (à la Maslov) to configurations of Lagrangian subspaces of a symplectic vector space.
  • Hirschowitz, Andre Le probleme de Levi pour les espaces homogenes. Bull. Soc. math. France 103, 191-201 (1975). A truly geometric (and not constructive) proof of the existence of strictly plurisubharmonic functions on (most) locally pseudoconvex open subsets of homogeneous manifolds.
  • Hirschowitz, A.: Remarques sur les ouverts d'holomorphie d'un produit dénombrable de droites. Ann. Inst. Fourier 19, No.1, 219-229 (1969). Could be interesting from an ethnological point of view, or if you want to get a feeling about my mathematical roots. By the way, some results at the very end of the paper are litigious as was pointed out (in 1999 !) by Murielle Mauer (Liege).
    • We apply degeneration techniques in order to prove that the normal bundle of most good (eg nonspecial) generic space curves is stable with natural cohomology. \item {\bf Le groupe de Chow \'equivariant} [50]. %C.R.A.S. 298 (1984) 87-89. For a projective variety, I introduce the equivariant Chow group and prove it is equal to the ordinary one when the group is solvable. Using this, I compute the dimensions of Chow groups of the Hilbert scheme of the plane in the first nontrivial case $Hilb^3P^2$. \item {\bf La m\'ethode d'Horace pour l'interpolation \`a plusieurs variables} [52]. %Manus. Math. 50 (1985) 337-388. The scope of the Horace method is extended to fat points and it is proved in particular that generic unions of double points have maximal rank in the projective space of dimension three. ~ \item {\bf Smoothing algebraic space curves} [53] (with R. Hartshorne). %Algebraic geometry, Sitges 1983. L.N. in Math. 1124 (1985) 98-131. We develop the natural technique for smoothing nodal curves. (This is heavily used in [49].) ~ \item {\bf Sections planes et multis\'ecantes pour les courbes gauches g\'en\'eriques principales} [57]. %Space curves, Proceedings Rocca di Pappa 1985, Lect. %Notes in Math. 266 (1987) 124-155. Good (eg nonspecial) generic space curves are proved to possess all sorts of general position properties. \item {\bf About the proof of Calabi's conjectures on compact Kahler manifolds} [60] (with Ph. Delano\'e). %L' Enseignement Math\'ematique 34 (1988) 107-122. We try to smoothen Yau's proof of Calabi's conjecture, avoiding H\"older spaces, and driving computations of estimates in a coordinate-free way. ~ \item {\bf Nouvelles courbes de bon genre dans l' espace projectif} [61] (with R. Hartshorne). %Math. Ann. 280 (1988) 353-367. We show space curves of (conjectural) maximal genus (for a fixed degree) in the so-called B range. The existence of such curves had been conjectured by Harris. The construction relies heavily on [59]. \item {\bf Sym\'etries des surfaces rationnelles g\'en\'eriques} [62]. %Math. Ann. 281 (1988) 255-261. I prove that the group of $k$-automorphisms of a generic rational surface over $k$ is equal to the automorphism group of the Neron-Severi object (the Weyl group of the situation). This is a victory of the scheme language over disorder. \item {\bf Probl\`emes de Brill-Noether en rang sup\'erieur} [63]. % CRAS. 307 (1988) 153-156. I prove that the tensor product of two generic vector bundles over a curve is nonspecial; and that a generic vector bundle of degree $d'+d"$ over a smooth projective curve of genus $g$ is extension of a vector bundle of degree $d"$ and slope $\mu"$ by a vector bundle of degree $d'$ and slope $\mu'$ if and only if $g-1 \leq \mu" -\mu'$ holds. (The proof is in [90].) \item {\bf Coh\'erence et dualit\'e sur le gros site de Zariski} [65]. %Proceedings Trento 1988, Lect. Notes in Math. 1389 (1989) 91-102. Flat (higher) direct images of coherent sheaves on projective schemes are interpreted as "coherent" sheaves on the Zariski big site. The point is that duality: $ {\cal F} \mapsto {\cal H} om (\cal F, \cal O)$ is an involution for such "coherent" sheaves, yielding a pertinent formulation of relative Serre-duality without derived categories. This theory has been extended and used by Simpson for his theory of presentable $n$-stacks. A variant of this theory has been recently considered by Hartshorne. \item {\bf Fibr\'es g\'en\'eriques sur le plan projectif} [76] (with Y. Laszlo). %Math. Ann. 297 (1993) 85-102. We introduce prioritary sheaves (those satisfying $Hom (E, E(-2))=0$). The theory of stable bundles on the plane (Chern classes and irreducibility of moduli) is profitably embedded in the theory of prioritary sheaves. ~ \item {\bf Higher-Order Syntax and Induction in Coq} [80] (with J. Despeyroux). %in Proc. `Logic Programming and Automated Reasoning' 94, Springer-Verlag LNAI %822 (1994), 159--173. What is induction for higher-order abstract syntax and how to explain it to a high-level theorem-prover. ~ \item {\bf Polynomial interpolation in several variables} [82] (with J. Alexander). %JAG 4, 201-222, 1995. This is the last of a series of papers (starting with [52]) proving that (most) generic unions of double points in projective spaces have the maximal rank property. On the way, the Horace method has been equipped with a "differential" version. ~ \item {\bf La r\'esolution minimale d'un arrangement g\'en\'eral d'un grand nombre de points dans $P^n$} [84] (with C. Simpson). %Invent. Math. 319 (1996), 467-503. A vector-valued differential Horace method allows us to prove that generic configurations of sufficiently many points in any projective space have the expected resolution. Thanks to Eisenbud and Popescu, this statement cannot be extended to any number of points. \item {\bf New evidence for Green's conjecture on syzygies of canonical curves} [87] (with S. Ramanan). %Ann. ENS Paris 31, 145-152 (1998). We make some computations in the Picard group of the moduli stack showing, for odd genus, that if the generic curve has the expected canonical resolution then so does any curve outside the evident ($k$-gonal) divisor. {\bf Descente pour les n-champs} [92] (with C. Simpson). A new theory for higher categories and higher stacks, together with a descent theorem for complexes of sheaves of modules (to be glued via quasi-isomorphisms). \end{itemize} \end{document}