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From the work of Lian, Liu, and Yau on" Mirror Principle", in the explicit computation of the Euler data Q=\{Q_0, Q_1,...\} for an equivariant concavex bundle Q=\{Q_0, Q_1,...\} over a toric manifold, there are two places the structure of the bundle comes into play:(1) the multiplicative characteric class Q=\{Q_0, Q_1,...\} of Q=\{Q_0, Q_1,...\} one starts with, and (2) the splitting type of Q=\{Q_0, Q_1,...\} . Equivariant bundles over a toric manifold has been classified by Kaneyama, using data related to the linearization of the toric action on the base toric manifold. In this article, we relate the splitting type of Q=\{Q_0, Q_1,...\} to the classifying data of Kaneyama. From these relations, we compute the splitting type of a couple of nonsplittable equivariant vector bundles over toric manifolds that may be of interest to string theory and mirror symmetry. A code in Mathematica that carries out the computation of some of these examples is attached.

We construct a smooth Artin stack parameterizing the stable weighted curves of genus one with twisted fields and prove that it is isomorphic to the blowup stack of the moduli of genus one weighted curves studied by Hu and Li. This leads to a blowup-free construction of Vakil-Zinger's desingularization of the moduli of genus one stable maps to projective spaces. This construction provides the cornerstone of the theory of stacks with twisted fields, which is thoroughly studied in arXiv:2005.03384 and leads to a blowup-free resolution of the stable map moduli of genus two.

We give an example of a dominant rational self-map of the projective plane whose dynamical degree is a transcendental number.

Let (X, g) be a compact Riemannian manifold with quasi-positive Riemannian scalar curvature. If there exists a complex structure J compatible with g, then the Kodaira dimension of (X, J) is equal to −∞ and the canonical bundle KX is not pseudo-effective. We also introduce the complex Yamabe number λc(X) for compact complex manifold X, and show that if λ_c(X) is greater than 0, then κ(X) is equal to −∞; moreover, if X is also spin, then the Hirzebruch A-hat genus \hat{A}(X) is zero.