Let f: V --> U be a smooth non-isotrivial family of canonically polarized n-dimensional complex manifolds, where U is the complement of a normal crossing divisor S in a projective manifold Y. We show that some symmetric product of the sheaf of one-forms with logarithmic poles along S contains an invertible subsheaf of positive Kodaira dimension. As a corollary one finds that U can not be a complete intersection in the projective N space of codimension l < N/2, nor the complement of l general hyperplanes, for l < N. Moreover, as shown by S. Kovacs before, U can not be a projective manifold with a nef tangent bundle.
If the induced morphism to the moduli scheme is generically finite, one can also exclude U to be the product of more than n curves. Moreover, the existence of the family forces the automorphism group of U to be finite.
Most of those results carry over to smooth families f:V --> U, with an f-semi-ample dualizing sheaf, provided f is of maximal variation.
For a smooth family $V \to U$ of polarized manifolds with semi-ample canonical sheaves, we show the following result: any entire curve must be contained in the fibers of the classifying map from the base space $U$ to the moduli space. This settles the Relative Isotriviality Conjecture, \cite[Conjecture 1.5]{DLSZ}.
We show, using [14], that a smooth projective fibration f : X $\rightarrow$ Y between connected complex quasi-projective manifolds satisfies the equality $κ$(X) = $κ$(X y) + $κ$(Y) of Logarithmic Kodaira dimensions if its fibres X y admit a good minimal model. Without the last assumption, this was conjectured in [11]. Several cases are established in [13], which inspired the present text. Although the present results overlap with those of [13] in the projective case, the approach here is different, based on the r{ô}le played by birationally isotrivial fibrations, special manifolds and the core…
We construct infinitely many non-isotrivial families of abelian varieties over given four punctured projective lines. These families lead to algebraic solutions of Painleve VI equation. Finally, based on a recent paper by Lin-Sheng-Wang, we prove a complete characterization for the locus of motivic Higgs bundles in the moduli space as fixed points of an ``additive'' self-map. This is a note based on the lecture given by the second named author on 04 Nov. 2022 at Tsinghua University.
Rigidity Criterion for Certain Calabi-Yau Families
Xiamen University（厦门大学）；Fudan University and Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（复旦大学和上海数学与交叉学科研究院（SIMIS））；Wuhan University（武汉大学）；Institut für Mathematik, Universität Mainz（美因茨大学数学研究所）
Constructing families of abelian varieties of $\text{GL}_2$-type over $4$-punctured complex projective line via $p$-adic Hodge theory and Langlands correspondence and application to algebraic solutions of Painleve VI equation
Positive sheaves of differentials coming from coarse moduli spaces
考虑光滑拟射影复基U上的正则极化复流形的光滑射影族，并假设该族是非等平凡的。如果Y是U的光滑紧化，使得D：= Y U是一个简单的正规交叉因子，那么我们可以考虑沿D的具有对数极点的微分的微分层. Viehweg和Zuo证明了对于某个数m>0，这个层的m次对称幂允许有许多截面。更准确地说，m次对称幂包含一个可逆层，其Kodaira-Iitaka维数至少是该族的变分。我们改进了这个结果，并表明，这个“Viehweg-Zuo层”来自粗模空间与给定的家庭，至少一般。作为一个直接推论，如果U是一个曲面，我们看到非等平凡性假设意味着U在坎帕纳意义下不可能是特殊的。
We prove that the variation in a smooth projective family of varieties admitting a good minimal model forms a lower bound for the Kodaira dimension of the base, if the dimension of the base is at most five and its Kodaira dimension is non-negative. This gives an affirmative answer to the conjecture of Kebekus and Kovacs for base spaces of dimension at most five.
We show that, for any fixed weight, there is a natural system of Hodge sheaves, whose Higgs field has no poles, arising from a flat projective family of varieties parametrized by a regular complex base scheme, extending the analogous classical result for smooth projective families due to Griffiths. As an application, based on positivity of direct image sheaves, we establish a criterion for base spaces of rational Gorenstein families to be of general type. A key component of our arguments is centered around the construction of derived categorical objects generalizing relative logarithmic forms …
