35 research outputs found
Some Siegel threefolds with a Calabi-Yau model II
In the paper [FSM] we described some Siegel modular threefolds which admit a
Calabi-Yau model. Using a different method we give in this paper an enlarged
list of such varieties that admits a Calabi-Yau model in the following weak
sense: there exists a desingularization in the category of complex spaces of
the Satake compactification which admits a holomorphic three-form without zeros
and whose first Betti number vanishes Basic for our method is the paper [GN] of
van Geemen and Nygaard.Comment: 23 pages, no figure
Classical theta constants vs. lattice theta series, and super string partition functions
Recently, various possible expressions for the vacuum-to-vacuum superstring
amplitudes has been proposed at genus . To compare the different
proposals, here we will present a careful analysis of the comparison between
the two main technical tools adopted to realize the proposals: the classical
theta constants and the lattice theta series. We compute the relevant Fourier
coefficients in order to relate the two spaces. We will prove the equivalence
up to genus 4. In genus five we will show that the solutions are equivalent
modulo the Schottky form and coincide if we impose the vanishing of the
cosmological constant.Comment: 21 page
Harmonic theta series and the kodaira dimension of a6
We construct a basis of the space S14(Sp12(ℤ)) of Siegel cusp forms of degree 6 and weight 14 consisting of harmonic theta series. One of these functions has vanishing order 2 at the boundary which implies that the Kodaira dimension of A6 is nonnegative
The vanishing of two-point functions for three-loop superstring scattering amplitudes
In this paper we show that the two-point function for the three-loop chiral
superstring measure ansatz proposed by Cacciatori, Dalla Piazza, and van Geemen
vanishes. Our proof uses the reformulation of ansatz in terms of even cosets,
theta functions, and specifically the theory of the linear system
on Jacobians introduced by van Geemen and van der Geer.
At the two-loop level, where the amplitudes were computed by D'Hoker and
Phong, we give a new proof of the vanishing of the two-point function (which
was proven by them). We also discuss the possible approaches to proving the
vanishing of the two-point function for the proposed ansatz in higher genera
Superstring scattering amplitudes in higher genus
In this paper we continue the program pioneered by D'Hoker and Phong, and
recently advanced by Cacciatori, Dalla Piazza, and van Geemen, of finding the
chiral superstring measure by constructing modular forms satisfying certain
factorization constraints. We give new expressions for their proposed ans\"atze
in genera 2 and 3, respectively, which admit a straightforward generalization.
We then propose an ansatz in genus 4 and verify that it satisfies the
factorization constraints and gives a vanishing cosmological constant. We
further conjecture a possible formula for the superstring amplitudes in any
genus, subject to the condition that certain modular forms admit holomorphic
roots.Comment: Minor changes; final version to appear in Comm. Math. Phy
A new geometric description for Igusa's modular form
The modular form notably appears in one of Igusa's classic
structure theorems as a generator of the ring of full modular forms in genus 2,
being exhibited by means of a complicated algebraic expression. In this work a
different description for this modular form is provided by resorting to a
peculiar geometrical approach.Comment: 10 page