Related papers: Toric mirrors and test configurations

We use Renormalization Group ideas to study stability of moving fronts in the Ginzburg-Landau equation in one spatial dimension. In particular, we prove stability of the real fronts under complex perturbations. This extends the results of…

Mabuchi solitons generalize K\"{a}hler-Einstein metrics on Fano manifolds, which constitute a Yau-Tian-Donaldson type correspondence with relative Ding stability. Comparing with K\"{a}hler-Ricci solitons, there is a distinct necessary…

We modify the standard perfect symmetric obstruction theory for moduli spaces of simple perfect complexes, to the situation of complexes on abelian threefolds with fixed determinant and Fourier-Mukai determinant. As outcome we attach…

In this paper, we study the moduli space of quasi-polarized complex K3 surfaces of degree 6 and 8 via geometric invariant theory. The general members in such moduli spaces are complete intersections in projective spaces and we have natural…

Let $(L, h)\to (X, \omega)$ denote a polarized toric K\"ahler manifold. Fix a toric submanifold $Y$ and denote by $\hat{\rho}_{tk}:X\to \mathbb{R}$ the partial density function corresponding to the partial Bergman kernel projecting smooth…

We propose a way of understanding homological mirror symmetry when a complex manifold is a smooth compact toric manifold. So far, in many example, the derived category $D^b(coh(X))$ of coherent sheaves on a toric manifold $X$ is compared…

Given a quiver algebra A with relations defined by a superpotential, this paper defines a set of invariants of A counting framed cyclic A-modules, analogous to rank-1 Donaldson-Thomas invariants of Calabi-Yau threefolds. For the special…

We prove that smooth Fano threefolds have toric Landau--Ginzburg models. More precise, we prove that their Landau--Ginzburg models, presented as Laurent polynomials, admit compactifications to families of K3 surfaces, and we describe their…

We consider a class of Lagrangian sections $L$ contained in certain Calabi-Yau Lagrangian fibrations (mirrors of toric weak Fano manifolds). We prove that a form of the Thomas-Yau conjecture holds in this case: $L$ is Hamiltonian isotopic…

In this note, we study Schwarz's conjecture on application of Q-algebras to strict quantization. We prove that in the case of a torus with a constant Poisson structure, Schwarz's formalism gives the same star product as Rieffel…

An introduction is provided to some current research trends in stability in geometric invariant theory and the problem of Kaehler metrics of constant scalar curvature. Besides classical notions such as Chow-Mumford stability, the emphasis…

In this note, we aim to prove the finite semi-algebraic chamber decomposition theorem for K-semi(poly)stability under the assumption of the log boundedness of K-semistable degenerations. This boundedness assumption is naturally arising from…

A system of functional equations relating the Euler characteristics of moduli spaces of stable representations of quivers and the Euler characteristics of (Hilbert scheme-type) framed versions of quiver moduli is derived. This is applied to…

We prove various results involving arcs - which generalise test configurations - within the theory of K-stability. Our main result characterises coercivity of the Mabuchi functional on spaces of Fubini-Study metrics in terms of uniform…

We generalize Nikulin's and Dolgachev's lattice-theoretical mirror symmetry for K3 surfaces to lattice polarized higher dimensional irreducible holomorphic symplectic manifolds. In the case of fourfolds of $K3^{\left[2\right]}-$type we then…

We define new invariants of 3d Calabi-Yau categories endowed with a stability structure. Intuitively, they count the number of semistable objects with fixed class in the K-theory of the category ("number of BPS states with given charge" in…

For moduli of polarized smooth K-trivial a.k.a., Calabi-Yau varieties in a general sense, we revisit a classical problem of constructing its "weak K-moduli" compactifications which parametrizes K-semistable (i.e., semi-log-canonical…

In this note, we consider a sequence of test configurations compatible with a Kaehler metric in $c_1(L)$ on a polarized algebraic manifold $(X,L)$. Then an explicit formula for the Donaldson-Futaki invariant $F_1$ for the sequence will be…

We study certain DT invariants arising from stable coherent sheaves in a nonsingular projective threefold supported on the members of a linear system of a fixed line bundle. When the canonical bundle of the threefold satisfies certain…

We generalise Dwork's theory of $p$-adic formal congruences from the univariate to a multi-variate setting. We apply our results to prove integrality assertions on the Taylor coefficients of (multi-variable) mirror maps. More precisely,…