The weighted log canonical threshold

In this note, we show how to apply the original $L^2$-extension theorem of Ohsawa and Takegoshi to the standard basis of a multiplier ideal sheaf associated with a plurisubharmonic function. In this way, we are able to reprove the strong openness conjecture and to obtain an effective version of the semicontinuity theorem for weighted log canonical thresholds.Comment: Final versio

Commutative algebraic groups and pp-adic linear forms

Let $G$ be a commutative algebraic group defined over a number field $K$ that is disjoint over $K$ to $\mathbb G_a$ and satisfies the condition of semistability. Consider a linear form $l$ on the Lie algebra of $G$ with algebraic coefficients and an algebraic point $u$ in a $p$-adic neighbourhood of the origin with the condition that $l$ does not vanish at $u$. We give a lower bound for the $p$-adic absolute value of $l(u)$ which depends up to an effectively computable constant only on the height of the linear form, the height of the point $u$ and $p$.Comment: This is a preprint of the Materials accepted for publication in "Acta Arithmetica

Singularity invariants of plurisubharmonic functions

In this paper, we combine tools in pluripotential theory and commutative algebra to study singularity invariants of plurisubharmonic functions. We obtain some relationships between singularity invariants of plurisubharmonic functions and holomorphic functions. These results lead a sharp lower bound for the log canonical threshold of a plurisubharmonic function. It simultaneously improves the main result in Demailly and Pham (Acta. Math 212: 1-9, 2014), the classical result due to Skoda (Bull. Soc. Math. France 100: 353-408, 1972), as well as the lower estimate in Fernex, Ein and Mustata (Math. Res. Lett: 10 219-236, 2003) which has received crucial applications to birational geometry in recent years.Comment: 17 page

Coaching early-career social sciences researchers to publish their first indexed publications: the Research Coach in Social Sciences program as a model

In this short essay, I describe how our Vietnam-based continuing education program, Research Coach in Social Sciences (RCISS), supports early-career researchers to (co)publish their first international indexed publications (i.e., publications in Clarivate Web of Science [WoS] or Sco- pus-indexed journals). In developed countries, junior researchers often seek help from univer- sity professors to publish their first publication. However, given the chronic shortage of senior social sciences scholars with international publishing experience in Vietnam, along with out- dated and ill-designed PhD programs, early-career researchers in the social sciences in Viet- nam often face challenges in international publishing

Some applications of the pp-adic analytic subgroup theorem

We use a $p$-adic analogue of the analytic subgroup theorem of W\"ustholz to deduce the transcendence and linear independence of some new classes of $p$-adic numbers. In particular we give $p$-adic analogues of results of W\"ustholz contained in [G. W\"ustholz, Some remarks on a conjecture of Waldschmidt, Diophantine approximations and transcendental numbers, Progress in Mathematics 31, Birkh\"auser Boston, Boston, MA, (1983), 329-336] and generalizations of results obtained by Bertrand in [D. Bertrand, Sous-groupes \`a un param\`etre $p$-adique de vari\'et\'es de groupe, Invent. Math. 40 (1977), no. 2, 171-193] and [D. Bertrand, Probl\`emes locaux, Soci\'et\'e Math\'ematique de France, Ast\'erisque 60-70 (1979), 163-189].Comment: This is a preprint of the Materials accepted for publication in "Glasnik Matematicki