Spin polynomial invariants of smooth structures on algebraic surfaces

A. N. Tyurin

Izvestiya. Mathematics , Tome 42 (1994) no. 2, pp. 333-369.

Résumé

A construction of spin polynomial invariants of smooth structures on algebraic surfaces is given, and a precise formula for their computation is obtained.

Bibliographie

[1] Bertram A., “Moduli of rank 2 vector bundles, theta divisors, and the geometry of curves in projective space”, J. Diff. Geom., 35 (1992), 429–469 | MR | Zbl

[2] Bredlow S., Daskalopoulos G., “Moduli of stable pairs for holomorphic bundles over Riemann surfaces”, Int. J. Math., 2 (1991), 477–513 | DOI | MR

[3] Donaldson S., “Polynomial invariants for smooth 4-manifolds”, Topology, 29 (1990), 257–315 | DOI | MR | Zbl

[4] Donaldson S., “Instantons and Yang–Mills Theory”, Interface Math. and Physics, Oxford, 1990

[5] Donaldson S., Kronheimer P., The Geometry of Four–Manifolds, Clarendon Press, Oxford, 1990 | MR | Zbl

[6] Fintushel R., Stern R. J., Surgery in cusp neighborhoods and the Geography of irreducible 4-manifolds, Preprint, 1992 | Zbl

[7] Freed D., Uhlenbeck K., Instantons and four-manifolds, Mathematical Sciences Research Institute Publications, 1, Springer, N.Y., 1984 ; Frid D., Ulenbek K., Instantony i chetyrekhmernye mnogoobraziya, Mir, M., 1988 | MR | Zbl | MR

[8] Fulton W., Intersection theory, Springer-Verlag, N.Y., 1987 ; Fulton U., Teoriya peresechenii, Mir, M., 1989 | MR | MR

[9] Gieseker D., “On the moduli of vector bundles on an algebraic surface”, Ann. Math., 106 (1977), 45–60 | DOI | MR | Zbl

[10] O'Grady K. G., “Algebro-geometric analogues of Donaldson's polinomials”, Inv. Math., 107 (1992), 351–395 | DOI | MR

[11] Kotschick D., “$SO(3)$-invariants for 4-manifolds with $b_2^+=1$”, Proc. London Math. Soc., 3:63 (1991), 426–448 | DOI | MR

[12] Kotschick D., Morgan J., “$SO(3)$-invariants for 4-manifolds with $b_2^+=1$” (to appear)

[13] Li J., Algebraic geometric interpretation of Donaldson's polynomial invariants of algebraic surfaces, Preprint, UCLA, 1992

[14] Morgan J., Comparison of the Donaldson polynomial invariants with their algebro-geometric analogues, Preprint, Princeton, 992 | MR

[15] Pidstrigach V., Cutting along 3-spheres for Spin-invariants, Preprint, Göttingen, 1992

[16] Pidstrigach V., Tyurin A. N., “Invarianty gladkikh struktur algebraicheskoi poverkhnosti, zadavaemye operatorom Diraka”, Izv. AN SSSR. Ser. matem., 56:2 (1992), 279–371 | MR | Zbl

[17] Thaddeus M., Stable pairs, linear systems and the Verlinde formula, Preprint, Oxford, 1992 | MR

[18] Tyurin A. N., “Algebro-geometricheskie aspekty gladkosti”, UMN, 44:3 (1989), 93–143 | MR | Zbl

[19] Tyurin A., “The geometry of the special components of moduli space of vector bundles over algebraic surfaces of general type”, Complex Algebraic Varieties, Proceedings of the Conference (Bayreuth, 1990), Lecture Notes in Math., 1507, Springer, Berlin, 1992, 166–175 | MR

[20] Tyurin A., “A slight generalization of the theorem of Mehta–Ramanathan”, Algebraic geometry, Lecture Notes in Math., 1479, Springer, Berlin, 1989, 258–272 | MR

[21] Tyurin A., “The simple method of distinguishing the underlying differentiable structures of algebraic surfaces”, Mathematica Göttingenis, Sonderforschungsbereichs Geometry and Analysis, Heft 25, 1992, 1–24

[22] Tyurin A., “Cycles, curves and vector bundles on an algebraic surface”, Duke Math. J., 52:4 (1987), 813–851 | MR

[23] Shafarevich I. R. i dr., Algebraicheskie poverkhnosti, Tr. MIAN, 65, 1975 | MR | Zbl

[24] Uhlenbeck K., “Connections with $L^p$ bounds on curvature”, Comm. Math. Phys., 83 (1982), 31–42 | DOI | MR | Zbl