Related papers: Semisimple Quantum Cohomology and Blow-ups

In our joint paper with W. Fulton (math.AG/9804041) we prove a formula for the cohomology class of a quiver variety. This formula involves a new class of generalized Littlewood-Richardson coefficients, all of which surprisingly seem to be…

We show how blowing up varieties in base loci of linear systems gives a procedure for creating new homological projective duals from old. Starting with a HP dual pair $X,Y$ and smooth orthogonal linear sections $X_L,Y_L$, we prove that the…

We prove an analogue of the de Rham theorem for polar homology; that the polar homology $HP_q(X)$ of a smooth projective variety $X$ is isomorphic to its $H^{n,n-q}$ Dolbeault cohomology group. This analogue can be regarded as a geometric…

It has been known for a long time that Ext's between IC-sheaves may often be expressed in terms of Hom's between cohomology groups. We prove a more general result under weaker assumptions. The result is used to describe the action of the…

We initiate the study of Nash blowups in prime characteristic. First, we show that a normal variety is non-singular if and only if its Nash blowup is an isomorphism, extending a theorem by A. Nobile. We also study higher Nash blowups, as…

We prove that the group of area-preserving diffeomorphisms of the 2-sphere admits a non-trivial homogeneous quasimorphism to the real numbers with the following property. Its value on any diffeomorphism supported in a sufficiently small…

Let U be the tautological subbundle on the Grassmannian $\mathrm{Gr}(k, n)$. There is a natural morphism $\mathrm{Tot}(U) \to \mathbb{A}^n$. Using it, we give a semiorthogonal decomposition for the bounded derived category…

We give a short proof of the following result: Let $X$ be a complex surface of general type. If the canonical divisor of the minimal model of $X$ has selfintersection $= 1$, then $X$ is not diffeomorphic to a rational surface. Our proof is…

An important question in mathematical relativity theory is that of the nature of spacetime singularities. The equations of general relativity, the Einstein equations, are essentially hyperbolic in nature and the study of spacetime…

For a proper, smooth scheme $X$ over a $p$-adic field $K$, we show that any proper, flat, semistable $\mathcal{O}_K$-model $\mathcal{X}$ of $X$ whose logarithmic de Rham cohomology is torsion free determines the same $\mathcal{O}_K$-lattice…

We prove that if X is a Grassmannian of type A, then the Schubert basis of the (small) quantum cohomology ring QH(X) is the only homogeneous deformation of the Schubert basis of the ordinary cohomology ring of X that multiplies with…

We give elementary proofs of the main theorems about (small) quantum cohomology of Grassmannians, including the quantum Giambelli and quantum Pieri formulas, the rim-hook algorithm, Siebert and Tian's presentation, and a recent theorem of…

We determine the two-point invariants of the equivariant quantum cohomology of the Hilbert scheme of points of surface resolutions associated to type A_n singularities. The operators encoding these invariants are expressed in terms of the…

The dth symmetric product of a curve of genus g is a smooth projective variety. This paper is concerned with the little quantum cohomology ring of this variety, that is, the ring having its 3-point Gromov-Witten invariants as structure…

The goal of this article is twofold. On one hand, we study the subvarieties of projective varieties which possess partially ample normal bundle; we prove that they are G2 in the ambient space. This generalizes results of Hartshorne and…

This article is concerned with a semilinear time-fractional diffusion equation with a superlinear convex semilinear term in a bounded domain $\Omega$ with the homogeneous Dirichlet, Neumann, Robin boundary conditions and non-negative and…

We prove that if a Hamiltonian diffeomorphism of a closed monotone symplectic manifold with semisimple quantum homology has more contractible fixed points, counted homologically, than the total dimension of the homology of the manifold,…

We show that for a given holomorphic noncharacteristic surface S in two-dimensional complex space, and a given holomorphic function on S, there exists a unique meromorphic solution of Burgers' equation which blows up on S. This proves the…

For a subvariety of a smooth projective variety, consider the family of smooth hypersurfaces of sufficiently large degree containing it, and take the quotient of the middle cohomology of the hypersurfaces by the cohomology of the ambient…

A refined form of the `Folk Theorem' that a smooth action by a compact Lie group can be (canonically) resolved, by iterated blow up, to have unique isotropy type was established by the authors in the context of manifolds with corners; the…