On surjective morphisms to abelian varieties and a generalization of the Iitaka conjecture

We explore the relationship between fibrations arising naturally from a surjective morphism to an abelian variety. These fibrations encode geometric information about the morphism. Our study focuses on the interplay of these fibrations and presents several applications. Then we propose a generalization of the Iitaka conjecture which predicts an equality of Kodaira dimension of fibrations, and prove it when the base is a smooth projective variety of maximal Albanese dimension.Comment: 28 pages, comments welcome. arXiv admin note: text overlap with arXiv:2207.0835

A generalized optimal power flow program for distribution system analysis and operation with distributed energy resources and solid state transformers

The present distribution system is gradually trending towards a smart grid paradigm with massive development of distributed energy resources (DER), advanced power electronics interfaces, and a digitalized communication platform. Such profound changes bring challenges as well as opportunities for an entity like the distribution network operator (DNO) to optimally operate DERs and other controllable elements to achieve higher levels of energy efficiency, economic benefits, supply reliability and power quality. The major contribution of this dissertation is in the development of a generalized three-phase optimal power flow (OPF) program in a novel control scheme for future distribution system optimization and economic operation. It is developed based on primal-dual interior point method (PDIPM). The program is general enough to model comprehensive system components and topologies. The program can also be customized by user-defined cost functions, system constraints, and new device, such as solid state transformers (SST). An energy storage optimal control using dynamic programming is also proposed to coordinate with the OPF based on a pricing signal called the distribution locational marginal price (DLMP). The proposed OPF program can be used by the DNO in an open access competitive control scheme to optimally aggregate the energy mix by combining the profitability of each resource while satisfying system security constraints --Abstract, page iv

On Nonvanishing for uniruled log canonical pairs

We prove the Nonvanishing conjecture for uniruled log canonical pairs of dimension $n$, assuming the Nonvanishing conjecture for smooth projective varieties in dimension $n-1$. We also show that the existence of good models for non-uniruled klt pairs in dimension $n$ implies the existence of good models for log canonical pairs in dimension $n$.Comment: This paper supersedes arXiv:1906.1145

MMP for locally stable families and wall crossing for moduli of stable pairs

We construct reduction and wall-crossing morphisms between the moduli spaces of stable pairs as the coefficients vary, generalizing the earlier work of Ascher, Bejleri, Inchiostro and Patakfalvi which deals with the klt case. Along the proof, we show that one can run the MMP with scaling on normal locally stable families over a normal base, and that the existence of good minimal models is preserved when reducing coefficients away from zero.Comment: 35 pages, comments welcom

On global ACC for foliated threefolds

In this paper, we prove the rational coefficient case of the global ACC for foliated threefolds. Specifically, we consider any lc foliated log Calabi-Yau triple $(X,\mathcal{F},B)$ of dimension $3$ whose coefficients belong to a set $\Gamma$ of rational numbers satisfying the descending chain condition, and prove that the coefficients of $B$ belong to a finite set depending only on $\Gamma$. To prove our main result, we introduce the concept of generalized foliated quadruples, which is a mixture of foliated triples and Birkar-Zhang's generalized pairs. With this concept, we establish a canonical bundle formula for foliations in any dimension. As for applications, we extend Shokurov's global index conjecture in the classical MMP to foliated triples and prove this conjecture for threefolds with nonzero boundaries and for surfaces. Additionally, we introduce the theory of rational polytopes for functional divisors on foliations and prove some miscellaneous results.Comment: 22 pages. Add a paragraph on pages 3-4. Proposition 6.4 and Lemma 7.2 strengthened. Small modification of the proof of 8.1. Reference update

Infinitesimal structure of log canonical thresholds

We show that log canonical thresholds of fixed dimension are standardized. More precisely, we show that any sequence of log canonical thresholds in fixed dimension $d$ accumulates in a way which is i) either similar to how standard and hyperstandard sets accumulate, or ii) to log canonical thresholds in dimension $\leq d-2$. This provides an accurate description on the infinitesimal structure of the set of log canonical thresholds. We also discuss similar behaviors of minimal log discrepancies, canonical thresholds, and K-semistable thresholds.Comment: 20 page