Bounds for the number of moduli for irregular varieties of general type

A variational characterization for the shift of eigenvalues caused by a general type of perturbation is derived for second order self-adjoint elliptic differential operators. This result allows the direct extension of asymptotic formulae from simple eigenvalues to repeated ones. Some examples of particular interest are presented theoretically and numerically for the Laplacian operator for the following domain perturbations: excision of a small hole, local change of conductivity, small boundary deformation.

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Automorphisms of threefolds of general type acting trivially in cohomology

Geometriae Dedicata ◽

10.1007/s10711-021-00619-w ◽

2021 ◽

Author(s):

Hang Zhao

Keyword(s):

General Type

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Variation of singular Kähler–Einstein metrics: Positive Kodaira dimension

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2021-0028 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Junyan Cao ◽

Henri Guenancia ◽

Mihai Păun

Keyword(s):

General Type ◽

Einstein Metrics ◽

Kodaira Dimension ◽

Einstein Metric ◽

Canonical Bundle ◽

Fiber Space ◽

Generic Fiber ◽

Lelong Numbers ◽

Kähler Einstein Metrics

Abstract Given a Kähler fiber space p : X → Y {p:X\to Y} whose generic fiber is of general type, we prove that the fiberwise singular Kähler–Einstein metric induces a semipositively curved metric on the relative canonical bundle K X / Y {K_{X/Y}} of p. We also propose a conjectural generalization of this result for relative twisted Kähler–Einstein metrics. Then we show that our conjecture holds true if the Lelong numbers of the twisting current are zero. Finally, we explain the relevance of our conjecture for the study of fiberwise Song–Tian metrics (which represent the analogue of KE metrics for fiber spaces whose generic fiber has positive but not necessarily maximal Kodaira dimension).

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LINEAR SYSTEMS ON IRREGULAR VARIETIES

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748019000069 ◽

2019 ◽

Vol 19 (6) ◽

pp. 2087-2125 ◽

Cited By ~ 3

Author(s):

Miguel Ángel Barja ◽

Rita Pardini ◽

Lidia Stoppino

Keyword(s):

Continuous Function ◽

Line Bundle ◽

Linear Systems ◽

Projective Variety ◽

Geometrical Properties ◽

Complex Projective Variety ◽

Wide Range ◽

Image Position ◽

Irregular Varieties ◽

Complex Projective

Let $X$ be a normal complex projective variety, $T\subseteq X$ a subvariety of dimension $m$ (possibly $T=X$) and $a:X\rightarrow A$ a morphism to an abelian variety such that $\text{Pic}^{0}(A)$ injects into $\text{Pic}^{0}(T)$; let $L$ be a line bundle on $X$ and $\unicode[STIX]{x1D6FC}\in \text{Pic}^{0}(A)$ a general element.We introduce two new ingredients for the study of linear systems on $X$. First of all, we show the existence of a factorization of the map $a$, called the eventual map of $L$ on $T$, which controls the behavior of the linear systems $|L\otimes \unicode[STIX]{x1D6FC}|_{|T}$, asymptotically with respect to the pullbacks to the connected étale covers $X^{(d)}\rightarrow X$ induced by the $d$-th multiplication map of $A$.Second, we define the so-called continuous rank function$x\mapsto h_{a}^{0}(X_{|T},L+xM)$, where $M$ is the pullback of an ample divisor of $A$. This function extends to a continuous function of $x\in \mathbb{R}$, which is differentiable except possibly at countably many points; when $X=T$ we compute the left derivative explicitly.As an application, we give quick short proofs of a wide range of new Clifford–Severi inequalities, i.e., geographical bounds of the form $$\begin{eqnarray}\displaystyle \text{vol}_{X|T}(L)\geqslant C(m)h_{a}^{0}(X_{|T},L), & & \displaystyle \nonumber\end{eqnarray}$$ where $C(m)={\mathcal{O}}(m!)$ depends on several geometrical properties of $X$, $L$ or $a$.

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