Let $X^{[n]}$ be the Hilbert scheme of $n$ points on the smooth quasi-projective surface $X$, and let $L^{[n]}$ be the tautological bundle on $X^{[n]}$ naturally associated to the line bundle $L$ on $X$. As a corollary of Haiman's results, we express the image $\Phi (L^{[n]})$ of the tautological bundle $L^{[n]}$ for the Bridgeland-King-Reid equivalence $\Phi :D^b(X^{[n]})\to D_{{\mathfrak{S}}_n}^b(X^n)$ in terms of a complex $C_L^{•}$ of ${\mathfrak{S}}_n$-equivariant sheaves in $D_{{\mathfrak{S}}_n}^b(X^n)$ and we characterize the image $\Phi (L^{[n]}\otimes \cdot \cdot \cdot \otimes L^{[n]})$ in terms of the hyperderived spectral sequence $E_1^{p,q}$ associated to the derived $k$-fold tensor power of the complex $C_L^{•}$. The study of the ${\mathfrak{S}}_n$-invariants of this spectral sequence allows us to get the derived direct images of the double tensor power and of the general $k$-fold exterior power of the tautological bundle for the Hilbert-Chow morphism, providing Danila-Brion-type formulas in these two cases. This easily yields the computation of the cohomology of $X^{[n]}$ with values in $L^{[n]}\otimes L^{[n]}$ and ${\Lambda }^k{L}^{[n]}$

We extend to manifolds of arbitrary dimension the Castelnuovo–de Franchis inequality for surfaces. The proof is based on the theory of generic vanishing and on the Evans-Griffith syzygy theorem in commutative algebra. Along the way we give a positive answer, in the setting of Kähler manifolds, to a question of Green and Lazarsfeld on the vanishing of higher direct images of Poincaré bundles. We indicate generalizations to arbitrary integral transforms

We investigate existence and asymptotic completeness of the wave operators for the nonlinear Klein-Gordon equation with a defocusing exponential nonlinearity in two space dimensions. A certain threshold is defined based on the value of the conserved Hamiltonian, below which the exponential potential energy is dominated by the kinetic energy via a Trudinger-Moser-type inequality. We prove that if the energy is below or equal to the critical value, then the solution approaches a free Klein-Gordon solution at the time infinity. An interesting feature in the critical case is that the Strichartz estimate together with Sobolev-type inequalities cannot control the nonlinear term uniformly on each time interval: it crucially depends on how much the energy is concentrated. Thus we have to trace concentration of the energy along time, in order to set up favorable nonlinear estimates, and only after that we can apply Bourgain's induction argument (or any other similar one)

Let $S$ be a closed, orientable surface of genus at least $2$. The space $T_H\times ML$, where $T_H$ is the “hyperbolic” Teichmüller space of $S$ and $\mathrm{ML}$ is the space of measured geodesic laminations on $S$, is naturally a real symplectic manifold. The space $\mathrm{CP}$ of complex projective structures on $S$ is a complex symplectic manifold. A relation between these spaces is provided by Thurston's grafting map $\mathrm{Gr}$. We prove that this map, although not smooth, is symplectic. The proof uses a variant of the renormalized volume defined for hyperbolic ends

We investigate the complement of the discriminant in the projective space $P{\mathrm{Sym}}^d{C}^{n+1}$ of polynomials defining hypersurfaces of degree $d$ in $P^n$. Following the ideas of Zariski, we are able to give a presentation for the fundamental group of the discriminant complement which generalises the well-known presentation in case $n=1$ (i.e., of the spherical braid group on $d$ strands).

In particular, our argument proceeds by a geometric analysis of the discriminant polynomial as proposed in [Be] and draws on results and methods from [L1] addressing a comparable problem for any versal unfolding of Brieskorn-Pham singularities