Related papers: Rouquier dimension of some blow-ups

We construct a new class of asymptotically self-similar finite-time blowups that have two collapsing spatial scales for the 1D Constantin-Lax-Majda model. The larger spatial scale measures the decreasing distance between the bulk of the…

Let $X$ be a projective irreducible holomorphic symplectic manifold. We associate with any big $\mathbf{R}$-divisor $D$ on $X$ a convex polygon $\Delta_E^{\mathrm{num}}(D)$ of dimension 2, whose Euclidean volume is…

This paper is concerned with a cubic nonlinear Schr\"odinger system modeling the interaction between an optical beam and its third harmonic in a material with Kerr-type nonlinear response. We are mainly interested in the so-called…

We consider the space of holomorphic maps from a compact Riemann surface to a projective space blown up at finitely many points. We show that the homology of this mapping space equals that of the space of continuous maps that intersect the…

We consider blowups at a general point of weighted projective planes and, more generally, of toric surfaces with Picard number one. We give a unifying construction of negative curves on these blowups such that all previously known families…

We study behaviors of scalar quantities near the possible blow-up time, which is made of smooth solutions of the Euler equations, Navier-Stokes equations and the surface quasi-geostrophic equations. Integrating the dynamical equations of…

The first consistent phenomenological theory for two and three dimensional Rayleigh--Taylor (RT) turbulence has recently been presented by Chertkov [Phys. Rev. Lett. {\bf 91} 115001 (2003)]. By means of direct numerical simulations we…

For a smooth projective surface X the finite dimensionality of the Chow motive h(X), as conjectured by S.I Kimura, has several geometric consequences. For a complex surface of general type with p_g = 0 it is equivalent to Bloch's…

An odd-dimensional version of the Goldberg conjecture was formulated and proved by Boyer and Galicki, using an orbifold analogue of Sekigawa's formulas, and an approximation argument of K-contact structures with quasi-regular ones. We…

Let $X$ be a smooth $n$-dimensional Fano hypersurface in $\mathbb P^{n+1}$ where $n \geq 3$. Let $\Gamma$ be a smooth positive-dimensional complete intersection of $X$, a hypersurface and one of more hyperplanes in $\mathbb P^{n+1}$. Let $Y…

We show the existence of nonautonomous invariant manifolds for planar, asymptotically autonomous differential equations, that have equilibrium solutions with zero Lyapunov spectrum. These invariant manifolds correspond to the stable and…

An orbifold version of Bogomolov decomposition theorem is established for compact K\"ahler spaces with quotient singularities and first Chern class zero.The proof is a direct adaptation of the classical smooth case, using Ricci-flat…

We prove a blow-up criterion for the solutions to the $\nu$-dimensional Patlak-Keller-Segel equation in the whole space. The condition is new in dimension three and higher. In dimension two it is exactly Dolbeault's and Perthame's blow-up…

Consider a mechanical system with a real analytic potential. We prove that in dimension three, there is an open and dense subset of the set of non strict local minimums of the potential such that every one of its points is a Lyapunov…

We give a condition for certain divisors on the blow up of P^3 at points in general position to be ample. The result extends a theorem of G. Xu on the blow up of the projective plane.

The size function for a number field is an analogue of the dimension of the Riemann-Roch spaces of divisors on an algebraic curve. It was conjectured to attain its maximum at the trivial class of Arakelov divisors. This conjecture was…

The following conjecture arose out of discussions between B. Harbourne, J. Ro\'e, C. Cilberto and R. Miranda: for a smooth projective surface $X$ there exists a positive constant $c_X$ such that $h^1(\mathcal O_X(C))\le c_X h^0(\mathcal…

Let $X\subset \mathbb P^r$ be a projective factorial variety of dimension $3$, degree $n$, with at worst isolated singularities. Assume that the Picard group of $X$ is generated by the hyperplane section class. Let $C\subset X$ be a…

We view Dolbeault-Morse-Novikov cohomology H^{p,q}_\eta(X) as the cohomology of the sheaf \Omega_{X,\eta}^p of \eta-holomorphic p-forms and give several bimeromorphic invariants. Analogue to Dolbeault cohomology, we establish the…

Let $X$ be a fixed projective scheme which is flat over a base scheme $S$. The association taking a quasi-projective $S$-scheme $Y$ to the scheme parametrizing $S$-morphisms from $X$ to $Y$ is functorial. We prove that this functor…