Related papers: Motives and the Pfaffian-Grassmannian equivalence

Using intersections of two Grassmannians in ${\mathbb{P}}^9$, Ottem-Rennemo and Borisov-C\u{a}ld\u{a}raru-Perry have exhibited pairs of Calabi-Yau threefolds $X$ and $Y$ that are deformation equivalent, L-equivalent and derived equivalent,…

We give a new proof of the 'Pfaffian-Grassmannian' derived equivalence between certain pairs of non-birational Calabi-Yau threefolds. Our proof follows the physical constructions of Hori and Tong, and we factor the equivalence into three…

We argue that there exists a derived equivalence between Calabi-Yau threefolds obtained by taking dual hyperplane sections (of the appropriate codimension) of the Grassmannian G(2, 7) and the Pfaffian Pf(7). The existence of such an…

We study some conjectures about Chow groups of varieties of geometric genus one. Some examples are given of Calabi-Yau threefolds where these conjectures can be verified, using the theory of finite-dimensional motives.

We study the Gauss and Jacobi sums from a viewpoint of motives. We exhibit isomorphisms between Chow motives arising from the Artin-Schreier curve and the Fermat varieties over a finite field, that can be regarded as (and yield a new proof…

We consider Fano sevenfolds $Y$ obtained by intersecting the Grassmannian $\hbox{Gr}(3,6)$ with a codimension 2 linear subspace (with respect to the Pl\"ucker embedding). We prove that the motive of $Y$ is Kimura finite-dimensional. We also…

We conjecture an equivalence between the Gromov-Witten theory of 3-folds and the holomorphic Chern-Simons theory of Donaldson-Thomas. For Calabi-Yau 3-folds, the equivalence is defined by the change of variables, exp(iu)=-q, where u is the…

In this short note we show how results of Orlov and To\"en imply that any equivalence between the derived categories of coherent sheaves on two varieties lifts to an equivalence at the level of dg-categories. This establishes the link…

Motivic equivalence for algebraic groups was recently introduced in [9], where a characterization of motivic equivalent groups in terms of higher Tits indexes is given. As a consequence, if the quadrics associated to two quadratic forms…

The Pfaffian-Grassmannian correspondence relates certain pairs of derived equivalent non-birational Calabi-Yau 3-folds. Given such a pair, I construct a set of derived equivalences corresponding to mutations of an exceptional collection on…

We study the conjecture claiming that, over a flexible field, isotropic Chow groups coincide with numerical Chow groups (with ${\Bbb{F}}_p$-coefficients). This conjecture is essential for understanding the structure of the isotropic motivic…

In this note we consider the motivic aspect of the middle cohomology of more than 200 classes of quasi-smooth Calabi--Yau threefolds inside weighted projective 4-space which come with an action of a cyclic group of even order. The action…

Using geometrical correspondences induced by projections and two-steps flag varieties, and a generalization of Orlov's projective bundle theorem, we relate the Hodge structures and derived categories of subvarieties of different…

A conjecture of Orlov predicts that derived equivalent smooth projective varieties over a field have isomorphic Chow motives. The conjecture is known for curves, and was recently observed for surfaces by Huybrechts. In this paper we focus…

We develop a notion of exponential motives on general prestacks equipped with a $\mathbf{G}_a$-action, and compare them with Whittaker motives via Gaitsgory's Kirillov model. We then establish foundational results for exponential motives on…

In this paper we prove that the intersections of the levels of the dimension filtration on Voevodsky's motivic complexes over a field $k$ with the levels of the slice one are "as small as possible", i.e., that $Obj d_{\le m}DM^{eff}_{-,R}…

We solve Bershadsky-Cecotti-Ooguri-Vafa (BCOV) holomorphic anomaly equation to determine the higher genus Gromov-Witten invariants ($g \leq 5$) of the derived equivalent Calabi-Yau threefolds, which are of the appropriate codimensions in…

Let $X$ be a smooth cubic hypersurface, and let $F$ be the Fano variety of lines on $X$. We establish a relation between the Chow motives of $X$ and $F$. This relation implies in particular that if $X$ has finite-dimensional motive (in the…

The integral identity conjecture of Kontsevich and Soibelman plays an important role in proving the existence of motivic Donaldson-Thomas invariants for three-dimensional noncommutative Calabi-Yau manifolds. There are a number of different…

In this paper, we establish the convergence for Gromov-Witten invariant of elliptic orbifold $\mathbb{P}^1$ with type $(3,3,3), (4,4,2)$ and $(6,3,2)$. We also prove the mirror theorems of Gromov-Witten theory for those orbifolds and FJRW…