Hi, I am a postdoc at Imperial College London, mainly interested in algebraic geometry and complex geometry. My mentor is Paolo Cascini. Before that, I was a graduate student at the University of Michigan. My advisor was Mircea Mustaţă.
(8) Hodge theory and K-stability of some very symmetric hypersurfaces, (arXiv)
(See Abstract)
We study some interesting hypersurfaces that naturally arise when studying the period map on the moduli space of hypersurfaces, in the context of Sung Gi Park's recent work on studying the GIT moduli space of hypersurfaces via the minimal exponent. We compute the Hodge structure on the singular cohomology and the intersection cohomology of these hypersurfaces, and also show the K-polystability of certain mildly singular degenerate hypersurfaces among them. In particular, the following hypersurface is K-polystable for l≥2:
\{ x_{11}\cdots x_{1d} + \ldots + x_{l1} \cdots x_{ld} = 0\} \subset \PP^{ld-1}.
(7) Trace for the Du Bois complex, Journal of Singularities (arXiv)
(See Abstract)
We construct some version of the trace morphism between the Du Bois complexes, with applications towards the behavior of the local cohomological dimension and some Hodge theoretic aspects of singularities under finite morphisms.
(6) Lefschetz morphisms on singular cohomology and local cohomological dimension of toric varieties, (with Sridhar Venkatesh), International Mathematics Research Notices (arXiv)
(See Abstract)
Given a proper toric variety and a line bundle on it, we describe the morphism on singular cohomology given by the cup product with the Chern class of that line bundle in terms of the data of the associated fan. Using that, we relate the local cohomological dimension of an affine toric variety with the Lefschetz morphism on the singular cohomology of a projective toric variety of one dimension lower. As a corollary, we show that the local cohomological defect is not a combinatorial invariant. We also produce numerous examples of toric varieties in every dimension with any possible local cohomological defect, by showing that the local cohomological defect remains unchanged under taking a pyramid.
(5) Local cohomology and singular cohomology of toric varieties via mixed Hodge modules, (with Sridhar Venkatesh), to appear in Journal of Algebraic Geometry. (arXiv)
(See Abstract)
Given an affine toric variety X embedded in a smooth variety, we prove a general result about the mixed Hodge module structure on the local cohomology sheaves of X. As a consequence, we prove that the singular cohomology of a proper toric variety is mixed of Hodge-Tate type. Additionally, using these Hodge module techniques, we derive a purely combinatorial result on rational polyhedral cones that has consequences regarding the depth of reflexive differentials on a toric variety. We then study in detail two important subclasses of toric varieties: those corresponding to cones over simplicial polytopes and those corresponding to cones over simple polytopes. Here, we give a comprehensive description of the local cohomology in terms of the combinatorics of the associated cones, and calculate the Betti numbers (or more precisely, the Hodge-Du Bois diamond) of a projective toric variety associated to a simple polytope.
(4) Canonical bundle formula and a conjecture on certain algebraic fiber spaces by Schnell, to appear in Pure Appl. Math. Q. (arXiv)
(See Abstract)
We observe what the canonical bundle formula gives towards a conjecture of Schnell on algebraic fiber spaces, a question concerning the equivalence between the non-vanishing conjecture and the Campana--Peternell conjecture. As a result, we give a partial result on Schnell's conjecture under two independent assumptions. One weakens Schnell's assumption of the pseudo-effectivity of the canonical bundle of the base by adding some effective divisor supported on the ramification locus. The other is analogous to results on algebraic fiber spaces where the existence of good minimal models of a general fiber is assumed, but we use a priori a weaker assumption. More precisely, we prove Schnell's conjecture when the canonical class of the general fiber is represented by a rigid current.
(3) A remark on Fujino's work on the canonical bundle formula via period maps, Nagoya Mathematical Journal (arXiv)
(See Abstract)
Fujino gave a proof for the semi-ampleness of the moduli part in the canonical bundle formula in the case when the general fibers are K3 surfaces or Abelain varieties. We show a similar statement when the general fibers are primitive symplectic varieties. This answers a question of Fujino raised in the same article. Moreover, using the structure theory of varieties with trivial first Chern class, we reduce the question of semi-ampleness in the case of families of K-trivial varieties to a question when the general fibers satisfy a slightly weaker Calabi-Yau condition.
(2) The intersection cohomology Hodge module of toric varieties (with Sridhar Venkatesh), to appear in Épijournal de Géométrie Algébrique (arXiv)
(See Abstract)
We study the Hodge filtration of the intersection cohomology Hodge module for toric varieties. More precisely, we study the cohomology sheaves of the graded de Rham complex of the intersection cohomology Hodge module and give a precise formula relating it with the stalks of the intersection cohomology as a constructible complex. The main idea is to use the Ishida complex in order to compute the higher direct images of the sheaf of reflexive differentials.
(1) L2 approach to the Saito vanishing theorem, Ann. Sc. Norm. Super. Pisa Cl. Sci. (arXiv)
(See Abstract)
We give an analytic approach to the Saito vanishing theorem going back to the original idea for the proof of the Kodaira-Nakano vanishing theorem. The key ingredient is the curvature formula for Hodge bundles and the Higgs field estimates for degenerations of Hodge structures.