Orbifold Riemann--Roch for threefolds with an application to Calabi--Yau geometry

Abstract

We prove an orbifold Riemann-Roch formula for a polarized 3-fold (X, D). As an application, we construct new families of projective Calabi-Yau threefolds.

Ice cream and orbifold Riemann - Roch

“…Thus using ice cream P orb (Q, k X ) in place of the more obvious r periodicity contribution ∆ 1−t r effectively cuts the dissident points out of the curve strata C, modifying their degrees and those of the isotypical components of their normal bundle to be integral. We believe that for 1-dimensional orbifold locus, the formal statement and proof of Conjecture 4.4 should be within reach of the strategies of Buckley's thesis [1]. Her proof in the Calabi-Yau case involves two ingredients: taking cyclic covers in the style of [3] introduces the Dedekind sums at the dissident points.…”

“…We now have abundant experience of working with these, and definitive results in special cases, such as the Calabi-Yau 3-fold orbifolds of Buckley's thesis [1]. Our conjectures in the case of 1-dimensional orbifold locus are fairly specific, and in principle within reach of our methods, although we do not yet venture formal proofs.…”

“…Experiments and the results of [1] and [2] suggest that for these, the fractional strictly periodic contribution ∆ 1−t r given by generalized Dedekind sums (Definition 2.5) can be replaced by an integral term, at the expense of modifying the contributions corresponding to adjacent strata. (a 1 , a 2 , a 3 ) with a 1 + a 2 + a 3 ≡ 0 mod r; if hcf(r, a i ) = s i > 1, then Q is a dissident point on a 1 si curve C. To handle the periodicity of the Hilbert series of X, we have a choice between two alternative strategies.…”

“…Section 4 discusses briefly what we hope to do in this direction, and the difficulties associated with positive dimensional orbifold loci, especially their dissident strata (where the inertia group jumps); we exemplify this with Buckley's results on orbifold RR for polarized Calabi-Yau 3-folds [1]. The paper is backed up by a website, http://warwick.ac.uk/staff/Miles.Reid/Ice , containing additional material that does not fit in the paper, including implementations of our main algorithms in computer algebra and links to other papers.…”

“…The orbifold RR formula of [3] has found numerous subsequent extensions and applications; see for example A. R. Iano-Fletcher [4], G. Brown, S. Altınok and M. Reid [5], A. Buckley and B. Szendrői [1], J. J. Chen, J. A. Chen and M. Chen [6] and M. Kawakita [7], and we expect these ideas to be equally applicable in the study of higher dimensional varieties.…”