Changes of Variables and Integration of Forms

Integration is one of the most a fundamental concepts in mathematics. In calculus you began by learning how to integrate one-variable functions on $\mathbb {R}$. Then, you learned how to integrate two- and three-variable functions on $\mathbb {R}^2$ and $\mathbb {R}^3$. After this you learned how to integrate a function after a change-of-variables, and finally in vector calculus you learned how to integrate vector fields along curves and over surfaces. It turns out that differential forms are actually very nice things to integrate. Indeed, there is an intimate relationship between the integration of differential forms and the change-of-variables formulas you learned in calculus. In section one we define the integral of a two-form on $\mathbb {R}^2$ in terms of Riemann sums. Integrals of n-forms on $\mathbb {R}^n$ can be defined analogously. We then use the ideas from Chap. 6 along with the Riemann sum procedure to derive the change of coordinates formula from first principles. In section two we look carefully at a simple change of coordinates example. Section three continues by looking at changes from Cartesian coordinates to polar, cylindrical, and spherical coordinates. Finally in section four we consider a more general setting where we see how we can integrate arbitrary one- and two-forms on parameterized one- and two-dimensional surfaces

Bubbles on Manifolds with a U(1) Isometry

We investigate the construction of five-dimensional, three-charge supergravity solutions that only have a rotational U(1) isometry. We show that such solutions can be obtained as warped compactifications with a singular ambi-polar hyper-Kahler base space and singular warp factors. We show that the complete solution is regular around the critical surface of the ambi-polar base. We illustrate this by presenting the explicit form of the most general supersymmetric solutions that can be obtained from an Atiyah-Hitchin base space and its ambi-polar generalizations. We make a parallel analysis using an ambi-polar generalization of the Eguchi-Hanson base space metric. We also show how the bubbling procedure applied to the ambi-polar Eguchi-Hanson metric can convert it to a global AdS_2xS^3 compactification.Comment: 33 pages, 5 figures, LaTeX; references adde

Tangent cones to positive-(1,1) De Rham currents

We consider positive-(1,1) De Rham currents in arbitrary almost complex manifolds and prove the uniqueness of the tangent cone at any point where the density does not have a jump with respect to all of its values in a neighbourhood. Without this assumption, counterexamples to the uniqueness of tangent cones can be produced already in C^n, hence our result is optimal. The key idea is an implementation, for currents in an almost complex setting, of the classical blow up of curves in algebraic or symplectic geometry. Unlike the classical approach in C^n, we cannot rely on plurisubharmonic potentials.Comment: 37 pages, 2 figure

On compatibility between isogenies and polarisations of abelian varieties

We discuss the notion of polarised isogenies of abelian varieties, that is, isogenies which are compatible with given principal polarisations. This is motivated by problems of unlikely intersections in Shimura varieties. Our aim is to show that certain questions about polarised isogenies can be reduced to questions about unpolarised isogenies or vice versa. Our main theorem concerns abelian varieties B which are isogenous to a fixed abelian variety A. It establishes the existence of a polarised isogeny A to B whose degree is polynomially bounded in n, if there exist both an unpolarised isogeny A to B of degree n and a polarised isogeny A to B of unknown degree. As a further result, we prove that given any two principally polarised abelian varieties related by an unpolarised isogeny, there exists a polarised isogeny between their fourth powers. The proofs of both theorems involve calculations in the endomorphism algebras of the abelian varieties, using the Albert classification of these endomorphism algebras and the classification of Hermitian forms over division algebras

Syzygies of differentials of forms

Given a standard graded polynomial ring $R=k[x_1,...,x_n]$ over a field $k$ of characteristic zero and a graded $k$-subalgebra $A=k[f_1,...,f_m]\subset R$, one relates the module $\Omega_{A/k}$ of K\"ahler $k$-differentials of $A$ to the transposed Jacobian module $\mathcal{D}\subset \sum_{i=1}^n R dx_i$ of the forms $f_1,...,f_m$ by means of a {\em Leibniz map} \Omega_{A/k}\rar \mathcal{D} whose kernel is the torsion of $\Omega_{A/k}$. Letting \fp denote the $R$-submodule generated by the (image of the) syzygy module of $\Omega_{A/k}$ and \fz the syzygy module of $\mathcal{D}$, there is a natural inclusion \fp\subset \fz coming from the chain rule for composite derivatives. The main goal is to give means to test when this inclusion is an equality -- in which case one says that the forms $f_1,...,f_m$ are {\em polarizable}. One surveys some classes of subalgebras that are generated by polarizable forms. The problem has some curious connections with constructs of commutative algebra, such as the Jacobian ideal, the conormal module and its torsion, homological dimension in $R$ and syzygies, complete intersections and Koszul algebras. Some of these connections trigger questions which have interest in their own.Comment: 20 pages. Minor changes after referee's report and updated bibliograph

Elliptic Genera and 3d Gravity

We describe general constraints on the elliptic genus of a 2d supersymmetric conformal field theory which has a gravity dual with large radius in Planck units. We give examples of theories which do and do not satisfy the bounds we derive, by describing the elliptic genera of symmetric product orbifolds of $K3$, product manifolds, certain simple families of Calabi-Yau hypersurfaces, and symmetric products of the "Monster CFT." We discuss the distinction between theories with supergravity duals and those whose duals have strings at the scale set by the AdS curvature. Under natural assumptions we attempt to quantify the fraction of (2,2) supersymmetric conformal theories which admit a weakly curved gravity description, at large central charge.Comment: 50 pages, 9 figures, v2: minor corrections to section

Codimension Two Determinantal Varieties with Isolated Singularities

We study codimension two determinantal varieties with isolated singularities. These singularities admit a unique smoothing, thus we can define their Milnor number as the middle Betti number of their generic fiber. For surfaces in C^4, we obtain a L\^e-Greuel formula expressing the Milnor number of the surface in terms of the second polar multiplicity and the Milnor number of a generic section. We also relate the Milnor number with Ebeling and Gusein-Zade index of the 1- form given by the differential of a generic linear projection defined on the surface. To illustrate the results, in the last section we compute the Milnor number of some normal forms from A. Fr\"uhbis-Kr\"uger and A. Neumer [2] list of simple determinantal surface singularities