arXiv subjects

Michael T. Schultz

Publications and source records attributed to Michael T. Schultz.

6 recordsLinked to original sources

arXiv2026

Line congruences associated to Appell's hypergeometric functions of rank-4

Line congruences are the genesis of important examples of transformations of projective surfaces, such as the Laplace transform. We survey and review results related to this historical subject, then derive original formulae for the Laplace transform of the entire rank-4 linear system associated to such an immersed projective surface. We apply our results to study the geometry of surfaces defined by Appell's hypergeometric functions of rank-4: namely, $F_2$ and $F_4$. We show that the sequence of Laplace invariants for each is determined respectively by the Euler-Poisson-Darboux equation for $F_2$, and Darboux's Harmonic equation for $F_4$. Further, we show the natural line congruences generated by the Laplace transforms of each constitute a $W$-congruence, an important example of line congruence in which a surface and its Laplace transform are simultaneously locally conformally equivalent.

math.DG↗

arXiv2025

Geometric aspects of rank-3 vector bundles over surfaces and 2-plane distributions on 5-manifolds

We study geometric aspects of horizontal 2-plane distributions on the complement of the zero section in the 5-dimensional total space of a rank-3 vector bundle equipped with connection over a surface. We show that any surface in 3-dimensional projective space can be associated to such a geometric structure in 5-dimensions, and establish a dictionary between the projective differential geometry of the surface and the growth vector of the 2-plane distribution.

math.DG↗

arXiv2024

Twists, Eisenstein series, and Instantons in Local Mirror Symmetry

We propose a mechanism for computing the genus zero invariants of local Calabi-Yau fourfolds arising as the total space of the canonical bundle of a rank-1 Fano threefold. Our method relies heavily on modular parameterizations of the associated Landau-Ginzburg model, as well as extension regulator classes and higher normal functions studied by Doran and Kerr in this setting. The generating function of the local invariants is proposed as a functional inverse of a weight-4 modular form of Eisenstein type that is naturally computed from the mirror data, in analogy with some known results for local threefolds that computes genus zero Gopakumar-Vafa invariants. Using the twist construction of Doran and the first named author, these two constructions are connected via Doran's generalized functional invariant map on the periods.

math.AG↗

arXiv2024

On holomorphic conformal structures associated with lattice polarized K3 surfaces

We discuss the connection between Picard-Fuchs equations for certain families of lattice polarized K3 surfaces and the construction of integrable holomorphic conformal structures on their period domains. We then compute an explicit example of a locally conformally flat holomorphic metric associated with generic Jacobian Kummer surfaces, which allows for a novel description of the local variation of complex structure.

math.AG↗

arXiv2022

On the mixed-twist construction and monodromy of associated Picard-Fuchs systems

We use the mixed-twist construction of Doran and Malmendier to obtain a multi-parameter family of K3 surfaces of Picard rank $ρ\ge 16$. Upon identifying a particular Jacobian elliptic fibration on its general member, we determine the lattice polarization and the Picard-Fuchs system for the family. We construct a sequence of restrictions that lead to extensions of the polarization by two-elementary lattices. We show that the Picard-Fuchs operators for the restricted families coincide with known resonant hypergeometric systems. Second, for the one-parameter mirror families of deformed Fermat hypersurfaces we show that the mixed-twist construction produces a non-resonant GKZ system for which a basis of solutions in the form of absolutely convergent Mellin-Barnes integrals exists whose monodromy we compute explicitly.

math.AG↗

arXiv2020

From the signature theorem to anomaly cancellation

We survey the Hirzebruch signature theorem as a special case of the Atiyah-Singer index theorem. The family version of the Atiyah-Singer index theorem in the form of the Riemann-Roch-Grothendieck-Quillen (RRGQ) formula is then applied to the complexified signature operators varying along the universal family of elliptic curves. The RRGQ formula allows us to determine a generalized cohomology class on the base of the elliptic fibration that is known in physics as (a measure of) the local and global anomaly. Combining several anomalous operators allows us to cancel the local anomaly on a Jacobian elliptic surface, a construction that is based on the construction of the Poincaré line bundle over an elliptic surface.

math.DG↗