Related papers: Birational smooth minimal models have equal Hodge …

We show that birational smooth complex projective varieties with numerically effective canonical bundles along the exceptional loci have the same Betti numbers. In particular, birational smooth minimal models share the same Betti numbers.…

Let X and Y be two smooth projective n-dimensional algebraic varieties X and Y over C with trivial canonical line bundles. We use methods of p-adic analysis on algebraic varieties over local number fields to prove that if X and Y are…

Mirrors $X^{\vee}$ of quasi-smooth Calabi-Yau hypersurfaces $X$ in weighted projective spaces ${\Bbb P}(w_0, \ldots, w_d)$ can be obtained as Calabi-Yau compactifications of non-degenerate affine toric hypersurfaces defined by Laurent…

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…

This is the first part in a series of papers on counting surfaces on Calabi-Yau 4-folds. Besides the Hilbert scheme of 2-dimensional subschemes, we introduce \emph{two} types of moduli spaces of stable pairs. We show that all three moduli…

We perform the mirror transformations of Calabi-Yau manifolds with one moduli whose Hodge numbers $(h^{11}, h^{21})$ are minimally small. Since the difference of Hodge numbers is the generation of matter fields in superstring theories made…

Let $\mathcal{A}$ be a smooth proper C-linear triangulated category Calabi-Yau of dimension 3 endowed with a (non-trivial) rank function. Using the homological unit of $\mathcal{A}$ with respect to the given rank function, we define Hodge…

This proceedings note introduces aspects of the authors' work relating mirror symmetry and integral variations of Hodge structure. The emphasis is on their classification of the integral variations of Hodge structure which can underly…

We consider homological mirror symmetry in the context of hypertoric varieties, showing that appropriate categories of B-branes (that is, coherent sheaves) on an additive hypertoric variety match a category of A-branes on a Dolbeault…

We investigate Hodge-theoretic properties of Calabi-Yau complete intersections $V$ of $r$ semi-ample divisors in $d$-dimensional toric Fano varieties having at most Gorenstein singularities. Our main purpose is to show that the…

We prove a representation-theoretic version of Borisov-Batyrev mirror symmetry, and use it to construct infinitely many new pairs of orbifolds with mirror Hodge diamonds, with respect to the usual Hodge structure on singular complex…

We prove the Topological Mirror Symmetry Conjecture by Hausel-Thaddeus for smooth moduli spaces of Higgs bundles of type $\operatorname{SL}_n$ and $\operatorname{PGL}_n$. More precisely, we establish an equality of stringy Hodge numbers for…

We provide results on the smoothness of normalisers in connected reductive algebraic groups $G$ over fields $k$ of positive characteristic $p$. Specifically we we give bounds on $p$ which guarantee that normalisers of subalgebras of…

The aim of this paper is to give an application of p-adic Hodge theory to stringy Hodge numbers introduced by V. Batyrev for a mathematical formulation of mirror symmetry. Since the stringy Hodge numbers of an algebraic variety are defined…

Fix a smooth projective family of curves $C \to S$ and a split reductive group scheme $G$ over a Noetherian base scheme $S$. For any (possibly nonreduced) fixed relative Cartier divisor $D$, we provide a treatment of the moduli of…

A normal variety $X$ is called Calabi-Yau if $K_X \sim_{\mathbb Q} 0$. The index of $X$ is the smallest positive integer $m$ so that $m K_X \sim 0$. We construct smooth, projective Calabi-Yau varieties in every dimension with doubly…

This paper continues the authors' program of studying mirror symmetry via log geometry and toric degenerations, relating affine manifolds with singularities, log Calabi-Yau spaces, and toric degenerations of Calabi-Yaus. The main focus of…

We show that smooth varieties of general type which are well formed weighted complete intersections of Cartier divisors have maximal Hodge level, that is, their the rightmost middle Hodge numbers do not vanish. We show that this does not…

The predictions of the Mirror Symmetry are extended in dimensions n>3 and are proven for projective complete intersections Calabi-Yau varieties. Precisely, we prove that the total collection of rational Gromov-Witten invariants of such…

We prove that, for every totally real number field E_0, there exists a weight three variation of Hodge structure of Calabi-Yau type defined over the rational numbers with associated endomorphism algebra E_0 such that the unique irreducible…