Maximal time existence of unnormalized conical Kähler–Ricci flow

We generalize the maximal time existence of Kähler–Ricci flow in [G. Tian and Z. Zhang, On the Kähler–Ricci flow on projective manifolds of general type, Chin. Ann. Math. Ser. B 27 (2006), no. 2, 179–192] and [J. Song and G. Tian, The Kähler–Ricci flow through singularities, Invent. Math. 207 (2017), no. 2, 519–595] to the conical case. Furthermore, if the log canonical bundle {K_{M}+(1-\beta)[D]} is big or big and nef, we can examine the limit behaviors of such conical Kähler–Ricci flow. Moreover, these results still hold when D is a simple normal crossing divisor.

SNC Log Symplectic Structures on Fano Products

This paper classifies Poisson structures with the reduced simple normal crossing divisor on a product of Fano varieties of Picard number 1. The characterization of even-dimensional projective spaces from the viewpoint of Poisson structures is given by Lima and Pereira. In this paper, we generalize the characterization of projective spaces to any dimension.

Asymptotically conical Calabi-Yau metrics with cone singularities along a compact divisor

We construct Asymptotically Locally Euclidean (ALE) and, more generally, asymptotically conical Calabi–Yau metrics with cone singularities along a compact simple normal crossing divisor. In particular, this includes the case of the minimal resolution of 2D quotient singularities for any finite subgroup $\Gamma \subset U(2)$ acting freely on the three-sphere, hence generalizing Kronheimer’s construction of smooth ALE gravitational instantons.

LOGARITHMIC DE RHAM COMPARISON FOR OPEN RIGID SPACES

In this note, we prove the logarithmic $p$ -adic comparison theorem for open rigid analytic varieties. We prove that a smooth rigid analytic variety with a strict simple normal crossing divisor is locally $K(\unicode[STIX]{x1D70B},1)$ (in a certain sense) with respect to $\mathbb{F}_{p}$ -local systems and ramified coverings along the divisor. We follow Scholze’s method to produce a pro-version of the Faltings site and use this site to prove a primitive comparison theorem in our setting. After introducing period sheaves in our setting, we prove aforesaid comparison theorem.

Symmetric differentials and variations of Hodge structures

Let D be a simple normal crossing divisor in a smooth complex projective variety X. We show that the existence on X-D of a non-trivial polarized complex variation of Hodge structures with integral monodromy implies that the pair (X,D) has a non-zero logarithmic symmetric differential (a section of a symmetric power of the logarithmic cotangent bundle). When the corresponding period map is generically immersive, we show more precisely that the logarithmic cotangent bundle is big.

Positivity properties of the bundle of logarithmic tensors on compact Kähler manifolds

Let $X$ be a compact Kähler manifold, endowed with an effective reduced divisor $B=\sum Y_{k}$ having simple normal crossing support. We consider a closed form of $(1,1)$-type $\unicode[STIX]{x1D6FC}$ on $X$ whose corresponding class $\{\unicode[STIX]{x1D6FC}\}$ is nef, such that the class $c_{1}(K_{X}+B)+\{\unicode[STIX]{x1D6FC}\}\in H^{1,1}(X,\mathbb{R})$ is pseudo-effective. A particular case of the first result we establish in this short note states the following. Let $m$ be a positive integer, and let $L$ be a line bundle on $X$, such that there exists a generically injective morphism $L\rightarrow \bigotimes ^{m}T_{X}^{\star }\langle B\rangle$, where we denote by $T_{X}^{\star }\langle B\rangle$ the logarithmic cotangent bundle associated to the pair $(X,B)$. Then for any Kähler class $\{\unicode[STIX]{x1D714}\}$ on $X$, we have the inequality $$\begin{eqnarray}\displaystyle \int _{X}c_{1}(L)\wedge \{\unicode[STIX]{x1D714}\}^{n-1}\leqslant m\int _{X}(c_{1}(K_{X}+B)+\{\unicode[STIX]{x1D6FC}\})\wedge \{\unicode[STIX]{x1D714}\}^{n-1}.\end{eqnarray}$$ If $X$ is projective, then this result gives a generalization of a criterion due to Y. Miyaoka, concerning the generic semi-positivity: under the hypothesis above, let $Q$ be the quotient of $\bigotimes ^{m}T_{X}^{\star }\langle B\rangle$ by $L$. Then its degree on a generic complete intersection curve $C\subset X$ is bounded from below by $$\begin{eqnarray}\displaystyle \biggl(\frac{n^{m}-1}{n-1}-m\biggr)\int _{C}(c_{1}(K_{X}+B)+\{\unicode[STIX]{x1D6FC}\})-\frac{n^{m}-1}{n-1}\int _{C}\unicode[STIX]{x1D6FC}.\end{eqnarray}$$ As a consequence, we obtain a new proof of one of the main results of our previous work [F. Campana and M. Păun, Orbifold generic semi-positivity: an application to families of canonically polarized manifolds, Ann. Inst. Fourier (Grenoble) 65 (2015), 835–861].

The second main theorem for entire curves into Hilbert modular surfaces

The main goal of this article is to prove a second main theorem for entire curves into Hilbert modular surfaces. As an application of our main theorem, we obtain a condition ensuring that entire curves in a Hilbert modular surface of general type are contained in the exceptional divisors. We also show a second main theorem for a simple normal crossing divisor which is tangent to a holomorphic distribution of codimension one on a smooth projective algebraic manifold.

Simple normal crossing Fano varieties and log Fano manifolds

A projective log variety (X, D) is called alog Fano manifoldifXis smooth and ifDis a reduced simple normal crossing divisor onΧwith − (KΧ+D) ample. Then-dimensional log Fano manifolds (X, D) with nonzeroDare classified in this article when the log Fano indexrof (X, D) satisfies eitherr≥n/2withρ(X) ≥ 2 orr≥n− 2. This result is a partial generalization of the classification of logarithmic Fano 3-folds by Maeda.