Let m ≥ 1 be an integer. For any Z in the Siegel upper half-space we consider the multivariate theta series

We study certain moduli spaces of stable vector bundles of rank two on cubic and quartic threefolds. In many cases under consideration, it turns out that the moduli space is complete and irreducible and a general member has vanishing intermediate cohomology. In one case, all except one component of the moduli space has such vector bundles.

We make a number of conjectures about the geometry of continuous moduli parameterizing the string landscape. In particular we conjecture that such moduli are always given by expectation value of scalar fields and that moduli spaces with finite non-zero diameter belong to the swampland. We also conjecture that points at infinity in a moduli space correspond to points where an infinite tower of m...

In this paper, we study a fundamental domain for the Siegel-Jacobi space Sp(g,Z)⋉H (g,h) Z \Hg × C .

We show that the half-line m functions associated with vector-valued discrete Schrödinger operators are elements in Siegel upper half space. introduce a metric on space of to these operators. Then, we action transfer matrices is distance decreasing.

We prove a localization formula for the moduli space of stable relative maps. As an application, we prove that all codimension i tautological classes on the moduli space of stable pointed curves vanish away from strata corresponding to curves with at least i− g + 1 genus 0 components. As consequences, we prove and generalize various conjectures and theorems about various moduli spaces of curves...

We study generalisations to the structure groups U(n) of the familiar (abelian) Seiberg-Witten monopole equations on a four-manifold X and their moduli spaces. For n = 1 one obtains the classical monopole equations. For n > 1 our results indicate that there should not be any non-trivial gauge-theoretical invariants which are obtained by the scheme ‘evaluation of cohomology classes on the fundam...

We prove that Donaldson-Thomas type invariants are equal to weighted Euler characteristics of their moduli spaces. In particular, such invariants depend only on the scheme structure of the moduli space, not the symmetric obstruction theory used to define them. We also introduce new invariants generalizing Donaldson-Thomas type invariants to moduli problems with open moduli space. These are usef...

We introduce a notion of stable spherical variety which includes the spherical varieties under a reductive group G and their flat equivariant degenerations. Given any projective space P where G acts linearly, we construct a moduli space for stable spherical varieties over P, that is, pairs (X, f), where X is a stable spherical variety and f : X → P is a finite equivariant morphism. This space i...

The study of moduli spaces has been a source of much research activity in physics, topology and algebraic topology. See for example [10, 12]. The classical example is the moduli space of smooth n-pointed stable curves of genus g, denoted Mg,n. This moduli space was introduced by Deligne, Mumford and Knudsen [4, 9, 16] and gives a natural compactification of Mumford’s [15] moduli space of nonsin...