Related papers: Fourier-Mukai transforms and the decomposition the…

In this expository article, we outline the theory of harmonic differential forms and its consequences. We provide self-contained proofs of the following important results in differential geometry: (1) Hodge theorem, which states that for a…

We realize explicit symmetries of Bridgeland stability conditions on any abelian threefold given by Fourier-Mukai transforms. In particular, we extend the previous joint work with Maciocia to study the slope and tilt stabilities of sheaves…

We use homological methods to establish a formal criterion for Generic Vanishing, in the sense originated by Green and Lazarsfeld and pursued further by Hacon and the first author, but in the context of an arbitrary Fourier-Mukai…

Let $S$ be an irreducible smooth projective surface defined over an algebraically closed field $k$. For a positive integer $d$, let ${\rm Hilb}^d(S)$ be the Hilbert scheme parametrizing the zero-dimensional subschemes of $S$ of length $d$.…

We focus on a class of Weierstrass elliptic threefolds that allows the base of the fibration to be a Fano surface or a numerically $K$-trivial surface. We define the notion of limit tilt stability, and show that the Fourier-Mukai transform…

To every reduced (projective) curve X with planar singularities one can associate many fine compactified Jacobians, depending on the choice of a polarization on X, which are birational (possibly non-isomorphic) Calabi-Yau projective…

We study some arithmetic properties of the mirror maps and the quantum Yukawa coupling for some 1-parameter deformations of Calabi-Yau manifolds. First we use the Schwarzian differential equation, which we derived previously, to…

We interpret symplectic geometry as certain sheaf theory by constructing a sheaf of curved A_\infty algebras which in some sense plays the role of a "structure sheaf" for symplectic manifolds. An interesting feature of this "structure…

We extend to singular schemes with Gorenstein singularities or fibered in schemes of that kind Bondal and Orlov's criterion for an integral functor to be fully faithful. We also contemplate a criterion for equivalence. We offer a proof that…

Given a Fourier-Mukai transform $\Phi$ between the bounded derived categories of two smooth projective curves, we verifiy that the induced map between the Jacobian varieties preserves the principal polarization if and only if $\Phi$ is an…

We formulate a version of the integral Hodge conjecture for categories, prove the conjecture for two-dimensional Calabi-Yau categories which are suitably deformation equivalent to the derived category of a K3 or abelian surface, and use…

The purpose of this paper is to establish injectivity theorems for higher direct image sheaves of canonical bundles twisted by pseudo-effective line bundles and multiplier ideal sheaves. As applications, we generalize Koll'ar's torsion…

In general, if M is a moduli space of stable sheaves on X, there is a unique alpha in the Brauer group of M such that a pi_M^* alpha^{-1}-twisted universal sheaf exists on X times M. In this paper we study the situation when X and M are K3…

In this note we prove that the Fourier-Mukai transform $\Phi_{\mathcal{U}}$ induced by the universal family of the moduli space $\mathcal{M}_{\mathbb{P}^2}(4,1,3)$ is not fully faithful.

The Baum-Connes map for finitely generated free abelian groups is a K-theoretic analogue of the Fourier-Mukai transform from algebraic geometry. We describe this K-theoretic transform in the language of topological correspondences, and…

Let $G$ be a compact, connected, simply-connected Lie group. We use the Fourier-Mukai transform in twisted $K$-theory to give a new proof of the ring structure of the $K$-theory of $G$.

We shall introduce a stability condition for a coherent sheaf associated to an elliptic surface. Then we study the behavior under relative Fourier-Mukai transforms.

We analyse infinitesimal deformations of pairs $(X,\mathcal{F})$ with $\mathcal{F}$ a coherent sheaf on a smooth projective manifold $X$ over an algebraic closed field of characteristic $0$. We describe a differential graded Lie algebra…

We show that every exact fully faithful functor from the category of perfect complexes on the spectrum of dual numbers to the bounded derived category of a noetherian separated scheme is of Fourier-Mukai type. The kernel turns out to be an…

Let $K$ be a discrete valuation field with perfect residue field, we study the functor from weakly admissible filtered $(\varphi,N,G_K)$-modules over $K$ to the isogeny category of Breuil-Kisin-Fargues $G_K$-modules. This functor is the…