Related papers: Twisted Fourier-Mukai transforms for holomorphic s…
We investigate a construction providing pairs of Calabi-Yau varieties described as zero loci of pushforwards of a hyperplane section on a roof as described by Kanemitsu. We discuss the implications of such construction at the level of Hodge…
This paper is an introduction to Homological Mirror Symmetry, derived categories, and topological D-branes aimed mainly at a mathematical audience. In the paper we explain the physicists' viewpoint of the Mirror Phenomenon, its relation to…
We prove the Mirror Conjecture for Calabi-Yau manifolds equipped with a holomorphic symplectic form. Such manifolda are also known as complex manifolds of hyperkaehler type. We obtain that a complex manifold of hyperkaehler type is Mirror…
We study rank two locally-free Fourier-Mukai transforms on K3 surfaces and show that they come in two distinct types according to whether the determinant of a suitable twist of the kernel is positive or not. We show that a necessary and…
We construct an equivalence between the derived category of sheaves on an elliptic threefold without a section and a derived category of twisted sheaves (modules over an Azumaya algebra) on any small resolution of its relative Jacobian.
The SYZ approach to mirror symmetry for log Calabi-Yau manifolds starts from a Lagrangian torus fibration on the complement of an anticanonical divisor. A mirror space is constructed by gluing local charts (moduli spaces of local systems on…
We study homological mirror symmetry for toric varieties, exploring the relationship between various Fukaya-Seidel categories which have been employed for constructing the mirror to a toric variety. In particular, we realize tropical…
We study so-called non-syzygetic cubic fourfolds, i.e., smooth cubic fourfolds containing two cubic surface scrolls in distinct hyperplanes with intersection number between the two scrolls equal to $1$. We prove that a very general…
A typical large complex-structure limit for mirror symmetry consists of toric varieties glued to each other along their toric boundaries. Here we construct the mirror large volume limit space as a Weinstein symplectic manifold. We prove…
This paper gives a new way of constructing Landau-Ginzburg mirrors using deformation theory of Lagrangian immersions motivated by the works of Seidel, Strominger-Yau-Zaslow and Fukaya-Oh-Ohta-Ono. Moreover we construct a canonical functor…
We produce twisted derived equivalences between torsors under abelian varieties and their moduli spaces of simple semi-homogeneous sheaves. We also establish the natural converse to this result and show that a large class of twisted derived…
Recently, Hu and Xi have exhibited derived equivalent endomorphism rings arising from (relative) almost split sequences as well as AR-triangles in triangulated categories. We present a broader class of triangles (in algebraic triangulated…
This is the first in a series of papers that deals with duality statements such as Mukai-duality (T-duality, from algebraic geometry) and the Baum-Connes conjecture (from operator $K$-theory). These dualities are expressed in terms of…
Every Fourier--Mukai equivalence between the derived categories of two K3 surfaces induces a Hodge isometry of their cohomologies viewed as Hodge structures of weight two endowed with the Mukai pairing. We prove that this Hodge isometry…
Let $M$ be a holomorphically symplectic manifold, equipped with a Lagrangian fibration $\pi:\; M \to X$. A degenerate twistor deformation (sometimes also called ``a Tate-Shafarevich twist'') is a family of holomorphically symplectic…
We prove a conjectural correspondence of Cao-Maulik-Toda which relates Gopakumar-Vafa invariants of fiber classes on a smooth projective Calabi-Yau 4-fold fibered over a curve to the Gopakumar-Vafa invariants of a smooth fiber under an…
We study twisted derived equivalences of hyper-K\"ahler fourfolds. We describe when two hyper-K\"ahler fourfolds of $K3^{[2]}$-type of Picard rank $1$ with isomorphic transcendental lattices are derived equivalent. Then we present new…
We find that for many Calabi-Yau threefolds with elliptic or genus one fibrations mirror symmetry factorizes between the fiber and the base of the fibration. In the simplest examples, the generic CY elliptic fibration over any toric base…
This paper continues the study of two examples of extremal transitions between families of Calabi-Yau threefolds. In a previous paper we suggested that the "mirror transition" between mirror families predicted by Morrison could be achieved…
We give some simple examples of mirror Calabi-Yau fourfolds in Type II string theory. These are realised as toroidal orbifolds. Motivated by the Strominger, Yau, Zaslow argument we give explicitly the mirror transformation which maps Type…