In this paper, we present two kinds of total Chern forms c(E,G) and c(E,G) as well as a total Segre form c(E,G) of a holomorphic Finsler vector bundle c(E,G) expressed by the Finsler metric c(E,G), which answers a question of Faran [The equivalence problem for complex Finsler Hamiltonians, in <i>Finsler Geometry</i>, Contemporary Mathematics, Vol. 196 (American Mathematical Society, Providence, RI, 1996), pp. 133144] to some extent. As some applications, we show that the signed Segre forms c(E,G) are positive c(E,G)-forms on c(E,G) when c(E,G) is of positive Kobayashi curvature; we prove, under an extra assumption, that a FinslerEinstein vector bundle in the sense of Kobayashi is semi-stable; we introduce a new definition of a flat Finsler metric, which is weaker than Aikous one [Finsler geometry on complex vector bundles, in <i>A Sampler of RiemannFinsler Geometry</i>, MSRI Publications, Vol. 50 (Cambridge

Smooth valuations on manifolds are studied by establishing a link with the Rumin-de Rham complex of the co-sphere bundle. Several operations on differential forms induce operations on smooth valuations: signature operator, Rumin-Laplace operator, Euler-Verdier involution and derivation operator. As an application, Alesker’s Hard Lefschetz Theorem for even translation invariant valuations on a finite-dimensional Euclidean space is generalized to all translation invariant valuations. The proof uses K¨ahler identities, the Rumin-de Rham complex and spectral geometry.

We develop the foundation of the complex symplectic geometry of Lagrangian subvarieties in a hyperk¨ahler manifold. We establish a characterization, a Chern number inequality, topological and geometrical properties of Lagrangian submanifolds. We discuss a category of Lagrangian subvarieties and its relationship with the theory of Lagrangian intersection. We also introduce and study extensively a normalized Legendre transformation of Lagrangian subvarieties under a birational transformation of projective hyperk¨ahler manifolds. We give a Pl¨ucker type formula for Lagrangian intersections under this transformation.

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Let$M$be a smooth hypersurface of constant signature in$CP$^{$n$},$n$≥3. We prove the regularity foron$M$in bidegree (0,1). As a consequence, we show that there exists no smooth hypersurface in$CP$^{$n$},$n$≥3, whose Levi form has at least two zero-eigenvalues.