An analytic proof of Riemann-Roch-Hirzebruch theorem for Kaehler manifolds.

*(English)*Zbl 0219.53054Let \(X\) 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.

\[ (1)\qquad 0 \rightarrow C^\infty(\xi_0) \overset{d_0} \to C^\infty(\xi_1) \overset{d_1}\to \cdots \overset{d_{k-1}} \to C^\infty(\xi_k)\rightarrow 0 \] 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 |