Luke Cherveny

Bentley University

Department of Mathematical Sciences

Research Interests

My expertise is differential geometry, especially complex differential geometry. I am interested in questions concerning the existence of special metrics on Kähler manifolds, which often also draws on techniques in complex algebraic geometry and geometric analysis. In the past I worked in the neighboring areas of Gromov-Witten theory and mathematical mirror symmetry.

More generally, I'm interested in topics across geometry and topology, as well as statistics and data science.

Papers

• The Futaki Invariant of Kähler Blowups with Isolated Zeros via Localization (submitted)

  The Calabi-Futaki invariant assigns a complex number to each holomorphic vector field on a compact Kähler manifold with chosen Kähler class Ω. Its vanishing is necessary for Ω to contain a Kähler metric of constant scalar curvature. In this paper, I used localization to establish an asymptotic expansion for the Calabi-Futaki invariant of a Kähler blow-up, at least when only isolated zeros appear. Degenerate localization calculations naturally arise in this situation, requiring the results of my earlier paper.

• On localizing Futaki-Morita integrals at isolated degenerate zeros. Differential Geometry and Applications 56 (2018), p. 1--12. (arXiv)

  Localization may be used on a complex manifold to reduce the computation of Futaki-Morita integrals, which includes Chern numbers and the Futaki invariant obstructing cscK metrics as special cases, to the zero locus of a holomorphic vector field. In this paper I established a formula giving the localization contribution to such an integral due to isolated and degenerate zeros of a holomorphic vector field. As an application, localizing at a maximally degenerate zero on CP^n leads to a general formula for Futaki-Morita integrals that does not involve a summation (unlike the standard non-degenerate calculation).

• An Explicit Genus-Zero Mirror Principle With Two Marked Points (preprint)

  A prediction of mirror symmetry is that a certain generating function for Gromov-Witten invariants on a Calabi-Yau manifold should be expressible in terms of hypergeometric-type data. In this paper based on my dissertation work, I extended the Lian-Liu-Yau mirror principle program to allow for the explicit construction of such hypergeometric data for GW invariants involving up to two markings. These invariants encode interesting enumerative data and are related to famous integrality conjectures, which I verify in several calculations.

• (with R.C. Smith and M. Salapaka) A Preisach Model for Quantifying Hysteresis in an Atomic Force Microscope. Proc. SPIE 4693, Smart Structures and Materials 2002: Modeling, Signal Processing, and Control, (10 July 2002)