Related papers: On $m$--fold Holomorphic Differentials and Modular…

We introduce deformations of the space of (multi-diagonal) harmonic polynomials for any finite complex reflection group of the form W=G(m,p,n), and give supporting evidence that this space seems to always be isomorphic, as a graded…

We give explicit structure of the graded ring of modular forms with respect to Gamma(N) (N=1,2,3,4,5,6,7,8,9,10,12,16,18) and for some other congruence groups. We also study the modular forms of half-integer weight for certain groups.

We construct the deformation functor associated to a couple of morphisms of differential graded Lie algebras, and use it to study the infinitesimal deformations of a holomorphic map of compact complex manifolds. In particular, in the case…

We study second-order modular differential equations whose solutions transform equivariantly under the modular group. In the reducible case, we construct all such solutions using an explicit ansatz involving Eisenstein series and the…

We construct the deformation functor associated with a pair of morphisms of differential graded Lie algebras, and use it to study infinitesimal deformations of holomorphic maps of compact complex manifolds. In particular, using L-infinity…

We study birational maps among 1) the moduli space of semistable torsion sheaves of Hilbert polynomial $4m+2$ on a smooth quadric surface, 2) the moduli space of semistable torsion sheaves of Hilbert polynomial $m^{2}+3m+2$ on…

For transcendental values of q the quantum tangent spaces of all left-covariant first order differential calculi of dimension less than four on the quantum group $\SLq 2$ are given. All such differential calculi $\Gamma $ are determined and…

An eta-quotient of level $N$ is a modular form of the shape $f(z) = \prod_{\delta | N} \eta(\delta z)^{r_{\delta}}$. We study the problem of determining levels $N$ for which the graded ring of holomorphic modular forms for $\Gamma_{0}(N)$…

Let $\Gamma$ be an $N^2$-dimensional bicovariant first order differential calculus on a Hopf algebra $SL_q(N)$. There are three possibilities to construct a differential Z-graded Hopf algebra $\Gamma^\wedge$ which contains $\Gamma$ as its…

In this work we revisit the "holomorphic modular bootstrap", i.e. the classification of rational conformal field theories via an analysis of the modular differential equations satisfied by their characters. By making use of the…

For any compact almost complex manifold $(M,J)$, the last two authors defined two subgroups $H_J^+(M)$, $H_J^-(M)$ of the degree 2 real de Rham cohomology group $H^2(M, \mathbb{R})$ in arXiv:0708.2520. These are the sets of cohomology…

Dynamical degrees and spectra can serve to distinguish birational automorphism groups of varieties in quantitative, as opposed to only qualitative, ways. We introduce and discuss some properties of those degrees and the Cremona degrees,…

We give a summary of results for dimensions of spaces of cuspidal Siegel modular forms of degree 2. These results together with a list of dimensions of the irreducible representations of the finite groups GSp(4,Fp) are then used to produce…

We give all possible holomorphic Eisenstein series on $\Gamma_0(p)$, of rational weights greater than $2$, and with multiplier systems the same as certain rational-weight eta-quotients at all cusps. We prove they are modular forms and give…

Manifolds with exceptional holonomy play an important role in string theory, supergravity and M-theory. It is explained how one can find the holonomy algebra of an arbitrary Riemannian or Lorentzian manifold. Using the de~Rham and Wu…

We derive a holomorphic anomaly equation for the Vafa-Witten partition function for twisted four-dimensional $\mathcal{N} =4$ super Yang-Mills theory on $\mathbb{CP}^{2}$ for the gauge group $SO(3)$ from the path integral of the effective…

We explain the basic ideas, describe with proofs the main results, and demonstrate the effectiveness, of an evolving theory of vector-valued modular forms (vvmf). To keep the exposition concrete, we restrict here to the special case of the…

In this paper we answer a question posed by Horikawa in 1978, who showed that the above moduli space is composed of 11 locally closed strata building up 4 irreducible components and having at most 3 connected components. We prove that the…

We propose a conjectural semiorthogonal decomposition for the derived category of the moduli space of stable rank 2 bundles with fixed determinant of odd degree, independently formulated by Narasimhan. We discuss some evidence for, and…

Let F be the imaginary quadratic field of discriminant -3 and OF its ring of integers. Let Gamma be the arithmetic group GL_3 (OF), and for any ideal n subset OF let Gamma_0 (n) be the congruence subgroup of level n consisting of matrices…