Research

Research statement

My long-term project is to study autoequivalence groups of derived categories of K3 surfaces by any method in mathematics.

It is rather difficult to compute the autoequivalecne group of K3 surface. One of the reasons is that, in contrast to the case of automorphism groups, autoequivalence groups cannot be embedded into the Hodge isometry group of the cohomology group. Hence it is not enough to study only the actions of the autoequivalence groups on the cohomology group.

Homological mirror symmetry, originating the string theory in physics, predicts that there is a relation between autoequivalence groups of derived categories of coherent sheaves on Calabi-Yau varieties, and the (symplectic) mapping class groups of symplectic manifolds. For example, as an analogue of Dehn twists along Lagrangian spheres, Seidel-Thomas introduced spherical objects and an autoequivalence called the spherical twist along a spherical object. Furthermore, via homological mirror symmetry, the space of Bridgeland stability conditions can be seen as an analogue of the Teichmüller spaces.

Then it is effective to study the autoequivalence groups by comparing it with the mapping class groups of real surfaces. A rich research history of the mapping class groups yields rich ideas to study the autoequivalence groups. Actually, I have been studying autoequivalence groups as an analogue of mapping class groups. Certainly there's still a lot to work on this project.

My works (see "Papers" in this webpage) can be divided into three aspects of autoequivalence groups: group theoretic aspects, metric geometric aspects and dynamical aspects.

(1) Group theoretic aspects: [7]

In [7], as an analogue of Dehn twists for closed oriented real surfaces, we study spherical twists for dg-enhanced triangulated categories. We introduce the intersection number and relate it to group-theoretic properties of spherical twists. We show an inequality analogous to a fundamental inequality in the theory of mapping class groups about the behavior of the intersection number via iterations of Dehn twists. We also classify the subgroups generated by two spherical twists using the intersection number. These results are similar to one in the theory of mapping class groups. In passing, we prove a structure theorem for finite dimensional dg-modules over the graded dual numbers and use this to describe the autoequivalence group. As an application, we compute the center of autoequivalence groups of derived categories of K3 surfaces.

(2) Metric geometric aspects: [5,8]

In [5], motivated by the curvature properties of metrics on Teichmüller spaces, we study a metric on the space Stab(D) of stability conditions and the isometric action of autoequivalence groups. To introduce a topology on Stab(D), Bridgeland defined the metric d_B on Stab(D). We show that this metric d_B is neither CAT(0) nor Gromov hyperbolic and that the categorical entropy is lower bounded by the translation length with respect to a quotient metric of d_B by the natural C-action. We also prove the hyperbolicity of pseudo-Anosov autoequivalences with respect to the quotient metric, and completely classify pseudo-Anosov autoequivalences in the case of curves.

In [8], we construct a compactification of the space of Bridgeland stability conditions on a smooth projective curve, as an analogue of Thurston compactifications in Teichmüller theory. In the case of elliptic curves, we compare our results with the classical one of the torus via homological mirror symmetry and give the Nielsen-Thurston classification of autoequivalences using the compactification. We also observe an interesting phenomenon in the case of the projective line.

(3) Dynamical aspects: [1,2,3,4,6]

The categorical entropy, introduced by Dimitrov-Haiden-Katzarkov-Kontsevich, is a categorification of the topological entropy and measures the dynamical complexity of the iteration of an autoequivalence.

In [1,2,3], we study Gromov-Yomdin type equality, i.e. one between the categorical entropy and the spectral radius of the action on the numerical Grothendieck group. We show the Gromov-Yomdin type equality for curves and varieties with the ample (anti)canonical bundle. We also show the compatibility of two entropies associated with an automorphism: the topological entropy and the categorical entropy of pullback autoequivalences. Furthermore, the inequality on the lower bound of the categorical entropy by the spectral radius is proved in general settings.

In [4], we study Serre dimension of triangulated category. We prove that Serre dimension is upper bounded by the infimum of global dimension of stability conditions due to Ikeda-Qiu in general settings, and classify triangulated categories of Serre dimension lower than one.

In [6], we introduce the notion of Hochschild entropy. Using the study of autoequivaolence groups and the homological mirror symmetry of K3 surfaces established by Sheridan-Smith, we construct symplectice Torelli mapping class of positive categorical entropy. This is an categorical analogue of a Torelli mapping class of positive topological entropy constructed by Thurston. It is technically difficult to prove the invariance of the categorical entropy between a K3 surface over Novikov field and its complex model.