We obtain several results about stability of the Bergman kernel on a tower of coverings on complex manifolds. An effective estimate for stability of the Bergman kernel is given for a tower of coverings on a compact Riemann surface of genus $\geq 2$. Stability of the Bergman kernel is established for towers of coverings on all hyperbolic Riemann surfaces and on complete Kähler manifolds that satisfy certain potential conditions. As a consequence, stability of the Bergman kernel is established for any tower of coverings of Riemann surfaces.

We show that for an immersed two-sided minimal surface in $\mathbb{R}^3$, there is a lower bound on the index depending on the genus and number of ends. Using this, we show the nonexistence of an embedded minimal surface in $\mathbb{R}^3$ of index $2$, as conjectured by Choe. Moreover, we show that the index of an immersed two-sided minimal surface with embedded ends is bounded from above and below by a linear function of the total curvature of the surface.

In an earlier work, we investigated some consequences of the existence of a Kähler metric of negative holomorphic sectional curvature on a projective manifold. In the present work, we extend our results to the case of semi-negative (i.e., non-positive) holomorphic sectional curvature. In doing so, we define a new invariant that records the largest codimension of maximal subspaces in the tangent spaces on which the holomorphic sectional curvature vanishes. Using this invariant, we establish lower bounds for the nef dimension and, under certain additional assumptions, for the Kodaira dimension of the manifold. In dimension two, a precise structure theorem is obtained.

Let $\mathscr{Y}$ be the double cover of the quintic symmetric determinantal hypersurface in $\mathbb{P}^{14}$. We consider Calabi–Yau threefolds $Y$ defined as smooth linear sections of $\mathscr{Y}$. In our previous works, we have shown that these Calabi–Yau threefolds $Y$ are naturally paired with Reye congruence Calabi–Yau threefolds $X$ by the projective duality of $\mathscr{Y}$, and observed that these Calabi–Yau threefolds have several interesting properties from the viewpoint of mirror symmetry and also projective geometry. In this paper, we prove the derived equivalence between the linear sections $Y$ of $\mathscr{Y}$ and the corresponding Reye congruences $X$.

We prove that Ruan’s Cohomological Crepant Resolution Conjecture holds for the Hilbert–Chow morphisms. There are two main ideas in the proof. The first one is to use the representation theoretic approach proposed in [QW] which involves vertex operator techniques. The second is to prove certain universality structures about the $3$-pointed genus-$0$ extremal Gromov–Witten invariants of the Hilbert schemes by using the indexing techniques from [LiJ], the product formula from [Beh2] and the co-section localization from [KL1, KL2, LL]. We then reduce Ruan’s Conjecture from the case of an arbitrary surface to the case of smooth projective toric surfaces which has already been proved in [Che].