Supersymmetric Proof of the Hirzebruch– Riemann–Roch Theorem for Non-Kähler Manifolds

Lecture no

Last time: Real 4-manifolds M with almost (many) complex structures but with no integrable almost complex structure, no complex structure. In understanding this situation , we observed the importance of the Riemann-Roch –Hirzebruch formula, and of the fact that it actually holds in the non-Kähler case(from the Atiyah-Singer Theorem). Namely for a compact complex manifold of any dimension and wi...

Riemann-roch Theorems for Differentiable Manifolds

A path integral derivation of χy-genus

The formula for the Hirzebruch χy-genus of complex manifolds is a consequence of the Hirzebruch–Riemann–Roch formula. The classical index formulae for Todd genus, Euler number and signature correspond to the case when the complex variable y = 0,−1 and 1 respectively. Here we give a direct derivation of this nice formula based on supersymmetric quantum mechanics. PACS numbers: 12.60.Jv, 02.40.Ma...

Riemann–roch Theorem for Operations in Cohomology of Algebraic Varieties

The Riemann–Roch theorem for multiplicative operations in oriented cohomology theories for algebraic varieties is proved and an explicit formula for the corresponding Todd classes is given. The formula obtained can also be applied in the topological situation, and the theorem can be regarded as a change-of-variables formula for the integration of cohomology classes. The classical Riemann–Roch t...

THE STRUCTURE OF THE THREE-STRANDED HELiX. POLY (A + 2U)* BY H. TODD MILES

8Ibid., Definition 8.1. 'Ibid., section 11. 10Ibid., Theorem 10.2. '1 Kodaira, 'K., L. Nirenberg, and D. C. Spencer, "On the existence of deformations of complex analytic structures," Ann. of Math., 68, 450-459 (1958). 12 Atiyah, It. F., and F. Hirzebruch, "Riemann-Roch theorems for differentiable manifolds," Bull. Amer. Math. Soc., 65, 276-281 (1959). 13 Enriques, F., Le Superficie Algebriche ...