Graded rings of Hermitian modular forms with singularities

We study graded rings of meromorphic Hermitian modular forms of degree two whose poles are supported on an arrangement of Heegner divisors. For the group $$\mathrm {SU}_{2,2}({\mathcal {O}}_K)$$ SU 2 , 2 ( O K ) where K is the imaginary-quadratic number field of discriminant $$-d$$ - d , $$d \in \{4, 7,8,11,15,19,20,24\}$$ d ∈ { 4 , 7 , 8 , 11 , 15 , 19 , 20 , 24 } we obtain a polynomial algebra without relations. In particular the Looijenga compactifications of the arrangement complements are weighted projective spaces.

Existence of parabolic minimizers to the total variation flow on metric measure spaces

We give an existence proof for variational solutions u associated to the total variation flow. Here, the functions being considered are defined on a metric measure space $$({\mathcal {X}}, d, \mu )$$ ( X , d , μ ) satisfying a doubling condition and supporting a Poincaré inequality. For such parabolic minimizers that coincide with a time-independent Cauchy–Dirichlet datum $$u_0$$ u 0 on the parabolic boundary of a space-time-cylinder $$\Omega \times (0, T)$$ Ω × ( 0 , T ) with $$\Omega \subset {\mathcal {X}}$$ Ω ⊂ X an open set and $$T > 0$$ T > 0 , we prove existence in the weak parabolic function space $$L^1_w(0, T; \mathrm {BV}(\Omega ))$$ L w 1 ( 0 , T ; BV ( Ω ) ) . In this paper, we generalize results from a previous work by Bögelein, Duzaar and Marcellini by introducing a more abstract notion for $$\mathrm {BV}$$ BV -valued parabolic function spaces. We argue completely on a variational level.

Generalized flatness constants, spanning lattice polytopes, and the Gromov width

In this paper we motivate some new directions of research regarding the lattice width of convex bodies. We show that convex bodies of sufficiently large width contain a unimodular copy of a standard simplex. Following an argument of Eisenbrand and Shmonin, we prove that every lattice polytope contains a minimal generating set of the affine lattice spanned by its lattice points such that the number of generators (and the lattice width of their convex hull) is bounded by a constant which only depends on the dimension. We also discuss relations to recent results on spanning lattice polytopes and how our results could be viewed as the beginning of the study of generalized flatness constants. Regarding symplectic geometry, we point out how the lattice width of a Delzant polytope is related to upper and lower bounds on the Gromov width of its associated symplectic toric manifold. Throughout, we include several open questions.

The degeneration formula for stable log maps

We give a direct proof for the degeneration formula of Gromov–Witten invariants including its cycle version for degenerations with smooth singular locus in the setting of stable log maps of Abramovich-Chen, Chen, Gross–Siebert.

The perfect cone compactification of quotients of type IV domains

The perfect cone compactification is a toroidal compactification which can be defined for locally symmetric varieties. Let $$\overline{D_{L}/\widetilde{O}^{+}(L)}^{p}$$ D L / O ~ + ( L ) ¯ p be the perfect cone compactification of the quotient of the type IV domain $$D_{L}$$ D L associated to an even lattice L. In our main theorem we show that the pair $${ (\overline{D_{L}/\widetilde{O}^{+}(L)}^{p}, \Delta /2) }$$ ( D L / O ~ + ( L ) ¯ p , Δ / 2 ) has klt singularities, where $$\Delta $$ Δ is the closure of the branch divisor of $${ D_{L}/\widetilde{O}^{+}(L) }$$ D L / O ~ + ( L ) . In particular this applies to the perfect cone compactification of the moduli space of 2d-polarised K3 surfaces with ADE singularities when d is square-free.