Vector bundles on complex projective spaces and systems of partial differential equations. I

Stiller, Peter F.

doi:10.1090/S0002-9947-1986-0860379-0

Vector bundles on complex projective spaces and systems of partial differential equations. I
HTML articles powered by AMS MathViewer

by Peter F. Stiller

Trans. Amer. Math. Soc. 298 (1986), 537-548

DOI: https://doi.org/10.1090/S0002-9947-1986-0860379-0

PDF | Request permission

Abstract:

This paper establishes and investigates a relationship between the space of solutions of a system of constant coefficient partial differential equations and the cohomology (${H^1}$ in particular) of an associated vector bundle/reflexive sheaf on complex projective space. Using results of Grothendieck and Shatz on vector bundles over projective one-space, the case of partial differential equations in two variables is completely analyzed. The final section applies results about vector bundles on higher-dimensional projective spaces to the case of three or more variables.

References

Mei-Chu Chang, Stable rank $2$ bundles on $\textbf {P}^{3}$ with $c_{1}=0,$ $c_{2}=4,$ and $\alpha =1$, Math. Z. 184 (1983), no. 3, 407–415. MR 716286, DOI 10.1007/BF01163513
Charles K. Chui and Ren Hong Wang, Multivariate spline spaces, J. Math. Anal. Appl. 94 (1983), no. 1, 197–221. MR 701458, DOI 10.1016/0022-247X(83)90014-8
A. Grothendieck, Sur la classification des fibrés holomorphes sur la sphère de Riemann, Amer. J. Math. 79 (1957), 121–138 (French). MR 87176, DOI 10.2307/2372388
Robin Hartshorne, On the classification of algebraic space curves, Vector bundles and differential equations (Proc. Conf., Nice, 1979), Progr. Math., vol. 7, Birkhäuser, Boston, Mass., 1980, pp. 83–112. MR 589222
Christian Okonek, Michael Schneider, and Heinz Spindler, Vector bundles on complex projective spaces, Progress in Mathematics, vol. 3, Birkhäuser, Boston, Mass., 1980. MR 561910
Stephen S. Shatz, On subbundles of vector bundles over $\textbf {P}^{1}$, J. Pure Appl. Algebra 10 (1977/78), no. 3, 315–322. MR 469920, DOI 10.1016/0022-4049(77)90010-X
Peter F. Stiller, Certain reflexive sheaves on $\textbf {P}^{n}_{\textbf {C}}$ and a problem in approximation theory, Trans. Amer. Math. Soc. 279 (1983), no. 1, 125–142. MR 704606, DOI 10.1090/S0002-9947-1983-0704606-6

Algebra

Wolfgang Dahmen and Charles A. Micchelli, On the local linear independence of translates of a box spline, Studia Math. 82 (1985), no. 3, 243–263. MR 825481, DOI 10.4064/sm-82-3-243-263
Wolfgang Dahmen and Charles A. Micchelli, On the optimal approximation rates for criss-cross finite element spaces, J. Comput. Appl. Math. 10 (1984), no. 3, 255–273. MR 755803, DOI 10.1016/0377-0427(84)90038-4

On the solutions of certain systems of partial difference equations and linear dependence of translates of box splines

Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157