相关论文: Arithmetic of degenerating principal variations of…
We study the predictions of mirror symmetry for the 1-parameter family of Calabi-Yau 3-folds $\tilde{X}$ with hodge numbers $h^{11}=31,h^{21}=1$ constructed in \cite{BN}. We calculate the Picard-Fuchs differential equation associated to…
We study the Hodge theory of twisted derived categories and its relation to the period-index problem. Our main contribution is the development of a theory of twisted Mukai structures for topologically trivial Brauer classes on arbitrary…
We prove an analog of Schmid's $\text{\rm SL}_2$-orbit theorem for a class of variations of mixed Hodge structure which includes logarithmic deformations, degenerations of 1-motives and archimedean heights. In particular, as consequence…
Motivated by classical Alexander invariants of affine hypersurface complements, we endow certain finite dimensional quotients of the homology of abelian covers of complex algebraic varieties with a canonical and functorial mixed Hodge…
This thesis details the results of four interrelated projects. The first of these presents a new proof of the theorem of Cooper, Danciger and Wienhard classifying the limits under conjugacy of the orthogonal groups in GL(n; R). The second…
We introduce a family of algebras which are multiplicative analogues of preprojective algebras, and their deformations, as introduced by M. P. Holland and the first author. We show that these algebras provide a natural setting for the…
We introduce a variety $\hat{G}_2$ parameterizing isotropic five-spaces of a general degenerate four-form in a seven dimensional vector space. It is in a natural way a degeneration of the variety $G_2$, the adjoint variety of the simple Lie…
We explore 6-dimensional compactifications of F-theory exhibiting (2,0) superconformal theories coupled to gravity that include discretely charged superconformal matter. Beginning with F-theory geometries with Abelian gauge fields and…
We compute the Hodge numbers of the variation of Hodge structure of the middle cohomology (with compact support) of the Landau-Ginzburg model dual to a weighted projective space. We state a conjectural formula for the Hodge numbers of…
We introduce a canonical isomorphism from the space of pure-type complex differential forms on a compact complex manifold to the one on its infinitesimal deformations. By use of this map, we generalize an extension formula in a recent work…
We study zeta-functions for a one parameter family of quintic threefolds defined over finite fields and for their mirror manifolds and comment on their structure. The zeta-function for the quintic family involves factors that correspond to…
We study global sections of Hodge bundles arising from two complementary constructions: a deformation-theoretic construction, which yields global geometric consequences for period maps, and a construction from the matrix representation of…
Compact K\"{a}hler manifolds satisfy several nice Hodge-theoretic properties such as the Hodge symmetry, the Hard Lefschetz property and the Hodge-Riemann bilinear relations, etc. In this note, we investigate when such nice properties hold…
The recent result of Strominger, Yau and Zaslow relating mirror symmetry to the quantum field theory notion of T-duality is reinterpreted as providing a way of geometrically characterizing which Calabi-Yau manifolds have mirror partners.…
We prove that the mirror map is trivial for the canonical formal families of Calabi-Yau varieties constructed by Gross and the second author. In other words, the natural coordinate in a canonical Calabi-Yau family is a canonical coordinate…
For a smooth, projective complex variety, we introduce several mixed Hodge structures associated to higher algebraic cycles. Most notably, we introduce a mixed Hodge structure for a pair of higher cycles which are in the refined normalized…
We introduce an "extended locus of Hodge classes" that also takes into account integral classes that become Hodge classes "in the limit". More precisely, given a polarized variation of integral Hodge structure of weight zero on a…
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 say that an abelian variety is degenerate if its Hodge ring is not generated by divisor classes. Degeneracy leads to some interesting challenges when computing Sato-Tate groups, and there are currently few examples and techniques…
We apply the Renormalisation Group Evolution (RGE) to analyze the phenomenological implications of an extended supersymmetric model, for the value of the unification scale and the strong coupling at the electroweak scale. The model we…