We prove that stable degeneration, the canonical degeneration associated to the normalized volume minimizer of a Kawamata log terminal (klt) singularity, preserves non-degenerate reflexive differential forms. In particular, the stable degeneration of a symplectic singularity is again symplectic. Combining this with a deformation-theoretic rigidity result for symplectic degenerations, we confirm Kaledin's conjecture that the formal completion of any symplectic singularity is conical. As applications, we show that the natural base of any normalized nilpotent orbit closure is a K-semistable Fano variety, and that the normalized volume minimizer of a hypertoric singularity is induced by the standard dilation.
We give a short proof of Kaledin's theorem on the degeneration of the noncommutative Hodge-to-de Rham spectral sequence. Our approach is based on topological Hochschild homology and the theory of cyclotomic spectra. As a consequence, we also obtain relative versions of the degeneration theorem, both in characteristic zero and for regular bases in characteristic $p$.
Linear degenerate flag varieties are degenerations of flag varieties as quiver Grassmannians. For type A flag varieties, we obtain characterizations of flatness, irreducibility and normality of these degenerations via rank tuples. Some of them are shown to be isomorphic to Schubert varieties and can be realized as highest weight orbits of partially degenerate Lie algebras, generalizing the corresponding results on degenerate flag varieties. To study normality, cell decompositions of quiver Grassmannians are constructed in a wider context of equioriented quivers of type A.
We study degenerations of non-simple principally polarized abelian surfaces to the boundary in the toroidal compactification of $\mathcal{A}_2$, and describe the degenerate abelian surfaces as well as the degenerate elliptic curves that live inside them.
Two interesting questions in algebraic geometry are: (i) how can a smooth projective varieties degenerate? and (ii) given two such degenerations, when can we say that one is "more singular/degenerate" than the other? Schmid's Nilpotent Orbit Theorem yields Hodge-theoretic analogs of these questions, and the Hodge-theoretic answers in turn provide insight into the motivating algebro-geometric questions, sometimes with applications to the study of moduli. Recently the Hodge-theoretic questions have been completely answered. This is an expository survey of that work.
The degeneration of the hyperelliptic sigma function is studied. We use the Sato Grassmannian for this purpose. A simple decomposition of a rational function gives a decomposition of Plücker coordinates of a frame of the Sato Grassmannian. It then gives a decomposition of the tau function corresponding to the degeneration of a hyperelliptic curve of genus $g$ in terms of the tau functions corresponding to a hyperelliptic curve of genus $g-1$. Since the tau functions are described by sigma functions, we get the corresponding formula for the degenerate hyperelliptic sigma function.
In this article we prove that the log Hodge de Rham spectral sequences of certain proper log smooth schemes of Cartier type in characteristic $p>0$ degenerate at $E_1$. We also prove that the log Kodaira vanishings for them hold when they are projective. We formulate the log weak Lefschetz conjecture for log crystalline cohomologies and prove that it is true in certain cases.
Starting from an anti-symplectic involution on a K3 surface, one can consider a natural Lagrangian subvariety inside the moduli space of sheaves over the K3. One can also construct a Prymian integrable system following a construction of Markushevich--Tikhomirov, extended by Arbarello--Saccà--Ferretti, Matteini and Sawon--Chen. In this article we address a question of Sawon, showing that these integrable systems and their associated natural Lagrangians degenerate, respectively, into fix loci of involutions considered by Heller--Schaposnik, Garcia-Prada--Wilkins and Basu--Garcia-Prada. Along the…
We characterize quotient of a non-degenerate abelian fibration by a finite étale equivalence relation. We show that non-uniruled degenerations of each such quotient tend to be almost non-degenerate.
We use local Hamiltonian torus actions to degenerate a symplectic manifold to a normal crossings symplectic variety in a smooth one-parameter family. This construction, motivated in part by the Gross-Siebert and B. Parker's programs, contains a multifold version of the usual (two-fold) symplectic cut construction and in particular splits a symplectic manifold into several symplectic manifolds containing normal crossings symplectic divisors with shared irreducible components in one step.
Degenerations of multisingularities and Artin algebras
Institute of Mathematics, University of Warsaw（华沙大学数学研究所）；Department of Mathematics, UNC Chapel Hill（北卡罗来纳大学教堂山分校数学系）；HUN-REN Rényi Institute of Mathematics（匈牙利研究与创新网络雷尼数学研究所）
There is a large number of different ways of constructing Calabi-Yau manifolds, as well as related non-geometric formulations, relevant in string compactifications. Showcasing this diversity, we discuss explicit deformation families of discretely distinct Hirzebruch hypersurfaces in $\mathbb{P}^n \times \mathbb{P}^1$ and identify their toric counterparts in detail. This precise isomorphism is then used to investigate some of their special divisors of interest, and in particular the secondary deformation family of their Calabi-Yau subspaces. In particular, most of the above so called Hirzebruch…
