The holomorphic torsion forms of the de Rham complex
Let \(\pi: M \to S\) be a holomorphic submersion of complex varieties with compact fiber \(X\), and let \(E\) be a holomorphic fiber bundle on \(M\). Let \(h^E\) be a Hermitian metric on \(E\), and let \(\omega^M\) be a (1,1) closed form on \(M\) that induces a Hermitian metric \(h^{TX}\) on the holomorphic tangent fiber bundle \(TX=TM/S\). Let \(G\) be a compact Lie group acting holomorphically on \(M\) along the fibers \(X\). For \(g\in G\), consider the forms of equivariant holomorphic torsion \(T_g(\omega^M, h^E)\), which are the sums of forms of type \((p, p)\) on \(S\) such that \[ \frac{\overline{\partial}\partial}{2i\pi}T_g\left(\omega^M, h^E\right)= \text{ch}_g\left(R^\cdot\pi_\ast E,h^{R^\cdot\pi_\ast E}\right)-\int_{X_g} \text{Td}_g\left(TX,h^{TX}\right)\text{ch}_g\left(E,h^E\right). \tag{1} \] The characteristic classes on the right of (1) are the classes appearing in the Lefschetz-Atiyah-Bott formula. Apply the previous construction to a \(\mathbb Z\)-graduated fiber bundle \(\Lambda^\cdot(T^\ast X)\). Let \(T_g(\omega^M)\) be the forms of the corresponding holomorphic torsion. It is easily verified that in this case (1) becomes \[ \frac{\overline{\partial}\partial} {2i\pi} T_g(\omega^M)=0.\tag{2} \] If \(S\) is compact and Kählerian, it is deduced from (2) that \(T_g(\omega^M)\) defines a class of cohomology on \(S\). The objective of this note is to announce that in general, the forms \(T_g(\omega^M)\) are not only closed but exact.
vanishing of the holomorphic torsion forms
forms of analytic torsion of de Rham complex
scientific article; zbMATH DE number 1854346