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We introduce the notion of positivity for a real basic $(1,1)$ class in basic Bott-Chern cohomology group on foliated manifolds, and study the relationship between this positivity and the negativity of transverse holomorphic sectional curvature and give some geometric applications., Comment: 29 pages

Teissier problem aims to characterize the equality case of Khovanskii-Teissier type inequality for $(1,1)$-classes on a compact K\"ahler manifold. When each of the involved $(1,1)$-classes is assumed to be nef and big, this problem has been solved by the previous works of Boucksom-Favre-Jonsson, Fu-Xiao and Li. In this note, we shall settle the case that the involved $(1,1)$-classes are just assumed to be nef. We also extend the results to some settings where some of the $(1,1)$-classes are not necessarily nef. By constructing examples, it is shown that our results are optimal., Comment: V5: title modified; presentation improved

A recent celebrated theorem of Diverio-Trapani and Wu-Yau states that a compact K\"ahler manifold admitting a K\"ahler metric of quasi-negative holomorphic sectional curvature has an ample canonical line bundle, confirming a conjecture of Yau. In this paper we shall consider a natural notion of almost quasi-negative holomorphic sectional curvature and extend this theorem to compact K\"ahler manifolds of almost quasi-negative holomorphic sectional curvature. We also obtain a gap-type theorem for the inequality $\int_Xc_1(K_X)^n>0$ in terms of the holomorphic sectional curvature. In the discussions, we introduce a capacity notion for the negative part of holomorphic sectional curvature, which plays a key role in studying the relation between the almost quasi-negative holomorphic sectional curvature and ampleness of the canonical line bundle., Comment: v4: main results improved

We shall discuss a higher-rank Khovanskii-Teissier inequality, generalizing a theorem of Li. In the course of the proof, we develop new Hodge-Riemann bilinear relations in certain mixed settings, which in themselves slightly extend the existing results and imply new Khovanskii-Teissier type inequalities and log-concavity results., Comment: V4: presentation reorganized, some details added, typos fixed

We derive some integral inequalities for holomorphic maps between complex manifolds. As applications, some rigidity and degeneracy theorems for holomorphic maps without assuming any pointwise curvature signs for both the domain and target manifolds are proved, in which key roles are played by total integration of the function of the first eigenvalue of second Ricci curvature and an almost nonpositivity notion for holomorphic sectional curvature introduced in our previous work. We also apply these integral inequalities to discuss the infinite-time singularity type of the K\"ahler-Ricci flow. The equality case is characterized for some special settings., Comment: V4: Introduction section refined; accepted by Transactions of the American Mathematical Society

We study the local curvature estimates of long-time solutions to the normalized K\"ahler-Ricci flow on compact K\"ahler manifolds with semi-ample canonical line bundles. Using these estimates, we prove that on such a manifold, the set of singular fibers of the semi-ample fibration on which the Riemann curvature blows up at time-infinity is independent of the choice of the initial K\"ahler metric. Moreover, when the regular fiber of the semi-ample fibration is not a finite quotient of a torus, we determine the exact curvature blow-up rate of the K\"ahler-Ricci flow near the regular fiber., Comment: Small changes; accepted by Advances in Mathematics

On a compact K\"ahler manifold with semi-ample canonical line bundle and Kodaira dimension one, we observe a relation between the infinite-time singularity type of the K\"ahler-Ricci flow and the characteristic indexes of singular fibers of the semi-ample fibration., Comment: V2: some details added; accepted by Mathematical Research Letters

On a compact K\"ahler manifold, we introduce a notion of almost nonpositivity for the holomorphic sectional curvature, which by definition is weaker than the existence of a K\"ahler metric with semi-negative holomorphic sectional curvature. We prove that a compact K\"ahler manifold of almost nonpositive holomorphic sectional curvature has a nef canonical line bundle, contains no rational curves and satisfies some Miyaoka-Yau type inequalities. In the course of the discussions, we attach a real value to any fixed K\"ahler class which, up to a constant factor depending only on the dimension of manifold, turns out to be an upper bound for the nef threshold., Comment: Small changes; accepted by Mathematische Zeitschrift

In this paper, we discuss diameter bound and Gromov-Hausdorff convergence of a twisted conical K\"ahler-Ricci flow on the total spaces of some holomorphic submersions. We also observe that, starting from a model conical K\"ahler metric with possibly unbounded scalar curvature, the conical K\"ahler-Ricci flow will instantly have bounded scalar curvature for $t>0$, and the bound is of the form $\frac{C}{t}$. Several key results will be obtained by direct arguments on the conical equation without passing to a smooth approximation. In the last section, we present several remarks on a twisted K\"ahler-Ricci flow and its convergence., Comment: V3, small changes; accepted by Annals of Global Analysis and Geometry

For the K\"ahler-Ricci flow on a compact K\"ahler manifold with semi-ample canonical line bundle, we prove the singularity type at infinity does not depend on the choice of the initial metric. We also provide new simple proofs for some existing classification results on infinite-time singularity type of the K\"ahler-Ricci flow., Comment: V2, modified Abstract, refined Introduction section, updated references; accepted by The Journal of Geometric Analysis

We consider the K\"ahler-Ricci flow on certain Calabi-Yau fibration, which is a Calabi-Yau fibration with one dimensional base or a product of two Calabi-Yau fibrations with one dimensional bases. Assume the K\"ahler-Ricci flow on total space admits a uniform lower bound for Ricci curvature, then the flow converges in Gromov-Hausdorff topology to the metric completion of the regular part of generalized K\"ahler-Einstein current on the base, which is a compact length metric space homeomorphic to the base. The analogue results for the continuity method on such Calabi-Yau fibrations are also obtained. Moreover, we show the continuity method starting from a suitable K\"ahler metric on the total space of a Fano fibration with one dimensional base converges in Gromov-Hausdorff topology to a compact metric on the base. During the proof, we show the metric completion of the regular part of a generalized K\"ahler-Einstein current on a Riemann surface is compact., Comment: V3, some incorrect sentences removed, typos fixed, references updated; accepted by Mathematische Annalen

We study finite-time collapsing limits of the continuity method. When the continuity method starting from a rational initial K\"ahler metric on a projective manifold encounters a finite-time volume collapsing, this projective manifold admits a Fano fibration over a lower dimensional base. In this case, we prove the continuity method converges to a singular K\"ahler metric on the base in the weak sense; moreover, if the base is smooth and the fibration has no singular fibers, then the convergence takes place in Gromov-Hausdorff topology., Comment: V3, small changes; accepted by International Mathematics Research Notices

We prove that, under a semi-ampleness type assumption on the twisted canonical line bundle, the conical K\"ahler-Ricci flow on a minimal elliptic K\"ahler surface converges in the sense of currents to a generalized conical K\"ahler-Einstein on its canonical model. Moreover, the convergence takes place smoothly outside the singular fibers and the chosen divisor., Comment: V2; accepted by Acta Mathematica Scientia, English Series

We prove that, on a minimal elliptic K\"ahler surface of Kodaira dimension one, the continuity method introduced by La Nave and Tian in \cite{LT} starting from any initial K\"ahler metric converges in Gromov-Hausdorff topology to the metric completion of the generalized K\"ahler-Einstein metric on its canonical model constructed by Song and Tian in \cite{ST06}., Comment: V3, small changes, accepted by International Mathematics Research Notices

In this paper, we develop an elementary proof of the change of variables in multiple integrals. Our proof is based on an induction argument. Assuming the formula for (m-1)-integrals, we define the integral over hypersurface in Rm, establish the divergent theorem and then use the divergent theorem to prove the formula for m-integrals. In addition to its simplicity, an advantage of our approach is that it yields the Brouwer Fixed Point Theorem as a corollary., Comment: 6 pages