# zbMATH — the first resource for mathematics

An analytic proof of Riemann-Roch-Hirzebruch theorem for Kaehler manifolds. (English) Zbl 0219.53054
Let $$X$$ be a $$d$$-dimensional closed oriented manifold of class $$C^\infty$$, $$\{\xi\}_{0\le i\le k}$$ a finite sequence of vector bundles over $$X$$ and $$d_i: C^\infty(\xi_i)\to C^\infty(\xi_{i+1})$$, $$0\le i\le k-1$$, be differential operators, each of order $$m$$. The complex
$(1)\qquad 0 \rightarrow C^\infty(\xi_0) \overset{d_0} \to C^\infty(\xi_1) \overset{d_1}\to \cdots \overset{d_{k-1}} \to C^\infty(\xi_k)\rightarrow 0$ is called elliptic if $$d_{i+1}\circ d_i=0$$ for $$0\le i\le k-2$$ and the symbol sequence is exact outside the zero section. Given an elliptic complex (1), the cohomology groups $$H_i$$ are defined as: $$H_i = \text{kernel}\, d_i/ \text{image}\,d_{i-1}$$. The cohomology groups are finite dimensional vector spaces and the index of the complex is by definition $$\sum_{i=0}^k (-1)^i$$ dimension $$H_i$$. Let us introduce hermitian inner products $$\xi_i$$ and a Riemannian structure in $$X$$ and let $$d_i^*: C^\infty(\xi_{i+1})\rightarrow C^\infty(\xi_i)$$ be the adjoints of the operators $$d_i$$ with respect to these inner products. The operators $$\Delta_ i= -(d_i^*d_i+d_{i-1}d_{i-1}^*)$$ are elliptic differential operators. Let $$e^i(t,x,y)$$ be the fundamental solution of the heat operator $$\partial/\partial t - \Delta_i$$. One can express the index as the integral of the alternating sum of the traces on the diagonal of the fundamental solutions
$e^i(t,x,y): \text{Index} = \sum_{i=0}^k (-1) \int_x^1 (\text{Trace}\, e^i(t,x,x))*1,$ $$*1$$ being the volume element. Moreover for any positive integer $$N$$, one has the following Minakshisundaram’s expansion:
$\text{Trace}\ e^i(t,x,x) = t^{-d/2n} \Bigl(\sum_{j=0}^N a_j^i t^{j/2n}\Bigr) + O(t^{N*1-d/2n}),\quad t\downarrow 0,$ where $$a_j^i$$’s are $$C^\infty$$-functions defined on $$X$$. Therefore to express the index of the elliptic complex in terms of some topological invariants, it is enough to do so for the element of $$H^1(X,\mathbb R)$$ defined by $$(\sum_{i=0}^k (-1)^ia_j^i)*1$$.
In a previous paper, “Curvature and the eigenforms of the Laplace operator” [J. Differ. Geom. 5, 233–249 (1971; Zbl 0211.53901)] the author proved that if (1) is the de Rham complex and the inner products in $$\xi_i= \wedge^i T^*(X)$$, $$T^*(X)$$ being the cotangent bundle, are the canonical inner products induced by the Riemannian structure in $$T^*(X)$$, then $$(\sum_{i=0}^d (-1)^ia_d^i)*1= C$$ where $$C$$ is Euler class.
In the present paper technique developed in the paper quoted above is used to prove the following: Let $$X$$ be a Kähler manifold, $$\xi$$ be a holomorphic vector bundle over $$\xi_i = \xi\otimes \wedge^{0,1}(T^*(X))$$. Let us fix a hermitian metric in $$\xi$$ and a Kähler metric in $$T^*(X)$$. Then there are canonical hermitian metrics in $$\xi_i$$. Let $$\text{ch}(t)$$ be the Chern character of $$\xi$$ and $$\mathcal T(X)$$ be the Todd class of $$X$$. By using hermitian connections in and $$T(X)$$, $$\text{ch}(\xi)$$ and $$\mathcal T(X)$$ can be expressed as explicit closed forms defined on $$X$$. The main result of the paper states:
$\Bigl(\sum_{i=0}^n (-1)^ia_d^i\Bigr)*1 = [\text{ch}(\xi) \wedge $$\mathcal T(X)]_{2n},$ where \([\text{ch}(\xi) \wedge \(\mathcal T(X)]_{2n}$$ denotes the $$2n$$-th component of the exterior product $$\text{ch}(\xi) \wedge \(\mathcal T(X)$$, $$n$$ being the complex dimension of $$X$$. As an immediate consequence, one gets the Riemann-Roch-Hirzebruch theorem for Kähler manifolds.
Reviewer: V. K. Patodi

##### MSC:
 53C55 Global differential geometry of Hermitian and Kählerian manifolds
Full Text: