The first subject of the paper concerns skew invariant theory. Let \(V\) be a finite-dimensional complex vector space with a skew symmetric non-degenerate form and let \(\text{Sp}(V)\) be the corresponding symplectic group. For a complex vector space \(W\) the author considers the action of \(\text{Sp}(V)\) on the exterior algebra \(\bigwedge(V\otimes W)\) (the action being trivial on \(W\)). He gives generators and relations for the algebra of \(\text{Sp}(V)\)-invariants for this action. Then, the author shows that there is a natural action of \(\text{Sp}(g)\), where \(g= \dim_{\mathbb C} W\), on \(\bigwedge(V\otimes W)\), by considering the spin representation of the orthogonal algebra associated to \(V\otimes (W\otimes W^*)\). The action of \(\text{Sp}(V)\times\text{Sp}(g)\) on \(\bigwedge(V\otimes W)\) is multiplicity free, and the author determines the highest weights of the representations which occur.
The second subject of the paper concerns Dolbeault cohomology groups with values in specific vector bundles. Let \(X\) be a compact closed complex manifold. Let \(\Omega^p_X\) be the sheaf of holomorphic \(p\)-forms on \(X\). The cohomology groups \(H^q(X, \Omega^{p_1}_X\otimes\cdots\otimes \Omega^{p_g}_X)\) are called pluri-Hodge groups in the paper. The author gives a precise formula for the Chern numbers of \(X\) in terms of the dimensions of the pluri-Hodge groups. In particular, a pluri-\(\chi_Y\) genus is also defined.
In Section 5 the author specializes to holomorphic symplectic manifolds. He shows that the graded trace of an \(\text{Sp}(g)\) element is essentially, up to normalization, the pluri-\(\chi_Y\) genus. This generalizes the situation for the usual \(\chi_Y\) genus. Then, he determines the dimensions of the pluri-Hodge groups in the case of \(K3\) surfaces, showing that they do not depend on the specific \(K3\) surface. Finally, the author applies the obtained results to the study of certain 3-manifold invariants.