Related papers: Stringy Hodge numbers of Pfaffian double mirrors a…
We study the connection between stringy Betti numbers of Gorenstein toric varieties and the generating functions of the Ehrhart polynomials of certain polyhedral regions. We use this point of view to give counterexamples to Hibi's…
In a previous paper we showed that any variety with log-terminal singularities admits a crepant resolution by a smooth Artin stack. In this paper we prove the converse, thereby proving that a variety admits a crepant resolution by a smooth…
Mirror symmetry relates type IIB string theory on a Calabi-Yau 3-fold to type IIA on the mirror CY manifold, whose complex structure and Kaehler moduli spaces are exchanged. We show that the mirror map is a particular case of a more general…
We compute the stringy $E$-function of the affine cone over a Grassmannian. If the Grassmannian is not a projective space then its cone does not admit a crepant resolution. Nonetheless the stringy $E$-function is sometimes a polynomial and…
Let $Y$ be a projective submanifold of the total space of the inverse of a very ample line bundle $\pi:L^{-1}\rightarrow B$ over a projective manifold $B$. Any section of $L^{-1}\rightarrow B$ is isomorphic to $B$ and the Hodge numbers of…
We study the action of mirror symmetry on two-dimensional N=(2,2) effective theories obtained by compactifying Type IIA string theory on Calabi-Yau fourfolds. Our focus is on fourfold geometries with non-trivial three-form cohomology. The…
Kuznetsov has conjectured that Pfaffian varieties should admit non-commutative crepant resolutions which satisfy his Homological Projective Duality. We prove half the cases of this conjecture, by interpreting and proving a duality of…
We introduce a notion of Homological Projective Duality for smooth algebraic varieties in dual projective spaces, a homological extension of the classical projective duality. If algebraic varieties $X$ and $Y$ in dual projective spaces are…
One of the dualities in string theory, the F-theory/heterotic string duality in eight dimensions, predicts an interesting correspondence between two seemingly disparate geometrical objects. On one side of the duality there are elliptically…
We give a spectral sequence to compute the logarithmic Hodge groups on a hypersurface type toric log Calabi-Yau space, compute its E_1 term explicitly in terms of tropical degeneration data and Jacobian rings and prove its degeneration at…
Using Saito's theory of mixed Hodge modules, we study a generalization of Hellus-Schenzel's "cohomologically complete intersection" property. This property is equivalent to perversity of the shifted constant sheaf. We relate the generalized…
We consider the conjectures from the paper by Katzarkov, Kontsevich, and Pantev about Landau-Ginzburg Hodge numbers associated to tamely compactifiable Landau-Ginzburg models. We test these conjectures in case of dimension two, verifying…
We leverage the results of the prequel in combination with a theorem of D. Orlov to yield some results in Hodge theory of derived categories of factorizations and derived categories of coherent sheaves on varieties. In particular, we…
Mirror symmetry of the type II string has a beautiful generalization to the heterotic string. This generalization, known as (0,2) mirror symmetry, is a field still largely in its infancy. We describe recent developments including the ideas…
We reformulate the construction of Kontsevich's completion and use Lawson homology to define many new motivic invariants. We show that the dimensions of subspaces generated by algebraic cycles of the cohomology groups of two $K$-equivalent…
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…
This thesis is about the study of Lie groupoids endowed with a compatible (multiplicative) differential 1-form. The motivation and scope of the present work is to study the geometry of PDEs using the formalism of Lie groupoids and…
String theory has already motivated, suggested, and sometimes well-nigh proved a number of interesting and sometimes unexpected mathematical results, such as mirror symmetry. A careful examination of the behavior of string propagation on…
We analyze the map between heterotic and type II N=2 supersymmetric string theories for certain two and three moduli examples found by Kachru and Vafa. The appearance of elliptic j-functions can be traced back to specializations of the…
We compute certain one-loop corrections to F^4 couplings of the heterotic string compactified on T^2, and show that they can be characterized by holomorphic prepotentials. We then discuss how some of these couplings can be obtained in…