Interpolation in non-positively curved K\"ahler manifolds

We extend to any simply connected K\"ahler manifold with non-positive sectional curvature some conditions for interpolation in $\mathbb{C}$ and in the unit disk given by Berndtsson, Ortega-Cerd\`a and Seip. The main tool is a comparison theorem for the Hessian in K\"ahler geometry due to Greene, Wu and Siu, Yau.Comment: 9 pages, Late

Section Extension from Hyperbolic Geometry of Punctured Disk and Holomorphic Family of Flat Bundles

The construction of sections of bundles with prescribed jet values plays a fundamental role in problems of algebraic and complex geometry. When the jet values are prescribed on a positive dimensional subvariety, it is handled by theorems of Ohsawa-Takegoshi type which give extension of line bundle valued square-integrable top-degree holomorphic forms from the fiber at the origin of a family of complex manifolds over the open unit 1-disk when the curvature of the metric of line bundle is semipositive. We prove here an extension result when the curvature of the line bundle is only semipositive on each fiber with negativity on the total space assumed bounded from below and the connection of the metric locally bounded, if a square-integrable extension is known to be possible over a double point at the origin. It is a Hensel-lemma-type result analogous to Artin's application of the generalized implicit function theorem to the theory of obstruction in deformation theory. The motivation is the need in the abundance conjecture to construct pluricanonical sections from flatly twisted pluricanonical sections. We also give here a new approach to the original theorem of Ohsawa-Takegoshi by using the hyperbolic geometry of the punctured open unit 1-disk to reduce the original theorem of Ohsawa-Takegoshi to a simple application of the standard method of constructing holomorphic functions by solving the d-bar equation with cut-off functions and additional blowup weight functions

Safety and clinical activity of durvalumab combined with tremelimumab in recurrent/metastatic head and neck squamous cell carcinoma: a multicenter phase I study.

BACKGROUND: Programmed cell death protein 1 (PD-1) inhibitors prolong survival versus chemotherapy in recurrent/metastatic head and neck squamous cell carcinoma (R/M HNSCC), which often expresses cytotoxic T-lymphocyte-associated protein 4 (CTLA-4) and programmed cell death-ligand 1 (PD-L1), providing a rationale for combined PD-(L)1 and CTLA-4 blockade. We report a phase I, open-label study of the PD-L1 inhibitor durvalumab plus the CTLA-4 inhibitor tremelimumab (NCT02262741). METHODS: In dose exploration, two cohorts of previously treated patients received durvalumab 10 mg/kg plus tremelimumab 3 mg/kg, or durvalumab 20 mg/kg plus tremelimumab 1 mg/kg, for up to 12 months. Dose expansion comprised two cohorts of previously untreated patients with R/M HNSCC having baseline PD-L1 tumor cell (TC) expression ≥25% and <25% and one cohort of immunotherapy-pretreated patients with any PD-L1 level. All received durvalumab 20 mg/kg plus tremelimumab 1 mg/kg, then durvalumab 10 mg/kg, for up to 12 months. The primary endpoint was safety. The secondary endpoints were objective response rate (ORR) by RECIST version 1.1, pharmacokinetics, pharmacodynamics, and immunogenicity. RESULTS: A total of 71 patients were treated. The median duration of exposure was 13.6 weeks for durvalumab and 13.1 weeks for tremelimumab. In dose exploration, no dose-limiting toxicities occurred. No maximum tolerated dose was identified. Treatment-related adverse events (TRAEs) occurred in 69.0% of patients; grade 3/4 and serious TRAEs occurred in 31.0% and 18.3%, respectively. TRAEs led to discontinuation in 9.9%. There were no treatment-related deaths. The ORR was 5.6% (95% confidence interval 1.6-13.8), including one complete response and three partial responses, all patients were in dose expansion with PD-L1 TC ≥25% and no prior immunotherapy exposure; three had ongoing responses ≥12 months. The median overall survival in the total population was 8.6 months. Soluble PD-L1 suppression was almost complete in all cohorts, suggesting target engagement. CD4+Ki67+ T cells were significantly elevated in all dose-expansion cohorts. CONCLUSIONS: Treatment was well tolerated. However, response rates were low despite target engagement, no drug-drug interactions, and no drug-neutralizing antibodies to durvalumab

Outcome Evaluation of a Short-Term Hospitalization and Community Support Program for People Who Abuse Ketamine

Ketamine is a popular recreational drug among young people in Hong Kong. Long-term abuse of ketamine can lead to acute urological and medical issues, which often require immediate care at emergency rooms. Many patients require short-term hospitalization for medical management. This opens a brief time window, within which mental health professionals could engage young people who abuses ketamine in psychosocial, functional, and lifestyle interventions. The Crisis Accommodation Program (CAP) is a short-term hospitalization and community support program that addresses the health care needs of young people who abuse ketamine. During short-term hospitalization, the patient participates in a range of cognitive and psychosocial assessments, motivational interviewing, emotions management, and lifestyle re-design interventions. Upon discharge, social work professionals of non-government agencies continue to work with the patients on their action plans in the community. This evaluation study uses a quasi-experimental non-equivalent group design, in which the outcomes of the treatment group (n = 84) are compared with a comparison group (n = 34) who have a history of ketamine abuse but who have not joined the treatment program. The results confirm that the treatment group showed significant increases in motivation for treatment, reduction in drug use, improvement in cognitive screening tests, healthy lifestyle scores, and self-efficacy in avoidance of drugs over 13 weeks. When compared with the comparison group, the treatment group had significant decreases in anxiety and treatment needs and had moved from pre-contemplation to the contemplation or preparation stage. However, there were no significant changes in outcome measures covering lifestyle or self-efficacy in drug avoidance. Overall, the CAP is effective in reducing drug use, anxiety, and helping patients to move from pre-contemplation to the contemplation or preparation stage of change. The study results suggest that health care professionals can successfully engage young people who abuse ketamine to participate in a package of psychosocial interventions, motivational interviewing, and lifestyle re-design during their hospital stay for management of urological problems. The CAP also highlights the importance of collaboration between hospitals and community social services in the management of addiction

Positivity of relative canonical bundles and applications

Given a family $f:\mathcal X \to S$ of canonically polarized manifolds, the unique K\"ahler-Einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle $\mathcal K_{\mathcal X/S}$. We use a global elliptic equation to show that this metric is strictly positive on $\mathcal X$, unless the family is infinitesimally trivial. For degenerating families we show that the curvature form on the total space can be extended as a (semi-)positive closed current. By fiber integration it follows that the generalized Weil-Petersson form on the base possesses an extension as a positive current. We prove an extension theorem for hermitian line bundles, whose curvature forms have this property. This theorem can be applied to a determinant line bundle associated to the relative canonical bundle on the total space. As an application the quasi-projectivity of the moduli space $\mathcal M_{\text{can}}$ of canonically polarized varieties follows. The direct images $R^{n-p}f_*\Omega^p_{\mathcal X/S}(\mathcal K_{\mathcal X/S}^{\otimes m})$, $m > 0$, carry natural hermitian metrics. We prove an explicit formula for the curvature tensor of these direct images. We apply it to the morphisms $S^p \mathcal T_S \to R^pf_*\Lambda^p\mathcal T_{\mathcal X/S}$ that are induced by the Kodaira-Spencer map and obtain a differential geometric proof for hyperbolicity properties of $\mathcal M_{\text{can}}$.Comment: Supercedes arXiv:0808.3259v4 and arXiv:1002.4858v2. To appear in Invent. mat

Geometric Aspects of the Moduli Space of Riemann Surfaces

This is a survey of our recent results on the geometry of moduli spaces and Teichmuller spaces of Riemann surfaces appeared in math.DG/0403068 and math.DG/0409220. We introduce new metrics on the moduli and the Teichmuller spaces of Riemann surfaces with very good properties, study their curvatures and boundary behaviors in great detail. Based on the careful analysis of these new metrics, we have a good understanding of the Kahler-Einstein metric from which we prove that the logarithmic cotangent bundle of the moduli space is stable. Another corolary is a proof of the equivalences of all of the known classical complete metrics to the new metrics, in particular Yau's conjectures in the early 80s on the equivalences of the Kahler-Einstein metric to the Teichmuller and the Bergman metric.Comment: Survey article of our recent results on the subject. Typoes corrrecte

Lagrangian Floer superpotentials and crepant resolutions for toric orbifolds

We investigate the relationship between the Lagrangian Floer superpotentials for a toric orbifold and its toric crepant resolutions. More specifically, we study an open string version of the crepant resolution conjecture (CRC) which states that the Lagrangian Floer superpotential of a Gorenstein toric orbifold $\mathcal{X}$ and that of its toric crepant resolution $Y$ coincide after analytic continuation of quantum parameters and a change of variables. Relating this conjecture with the closed CRC, we find that the change of variable formula which appears in closed CRC can be explained by relations between open (orbifold) Gromov-Witten invariants. We also discover a geometric explanation (in terms of virtual counting of stable orbi-discs) for the specialization of quantum parameters to roots of unity which appears in Y. Ruan's original CRC ["The cohomology ring of crepant resolutions of orbifolds", Gromov-Witten theory of spin curves and orbifolds, 117-126, Contemp. Math., 403, Amer. Math. Soc., Providence, RI, 2006]. We prove the open CRC for the weighted projective spaces $\mathcal{X}=\mathbb{P}(1,\ldots,1,n)$ using an equality between open and closed orbifold Gromov-Witten invariants. Along the way, we also prove an open mirror theorem for these toric orbifolds.Comment: 48 pages, 1 figure; v2: references added and updated, final version, to appear in CM

Extension of holomorphic functions and cohomology classes from non reduced analytic subvarieties

The goal of this survey is to describe some recent results concerning the L 2 extension of holomorphic sections or cohomology classes with values in vector bundles satisfying weak semi-positivity properties. The results presented here are generalized versions of the Ohsawa-Takegoshi extension theorem, and borrow many techniques from the long series of papers by T. Ohsawa. The recent achievement that we want to point out is that the surjectivity property holds true for restriction morphisms to non necessarily reduced subvarieties, provided these are defined as zero varieties of multiplier ideal sheaves. The new idea involved to approach the existence problem is to make use of L 2 approximation in the Bochner-Kodaira technique. The extension results hold under curvature conditions that look pretty optimal. However, a major unsolved problem is to obtain natural (and hopefully best possible) L 2 estimates for the extension in the case of non reduced subvarieties -- the case when Y has singularities or several irreducible components is also a substantial issue.Comment: arXiv admin note: text overlap with arXiv:1703.00292, arXiv:1510.0523

Cohomological aspects on complex and symplectic manifolds

We discuss how quantitative cohomological informations could provide qualitative properties on complex and symplectic manifolds. In particular we focus on the Bott-Chern and the Aeppli cohomology groups in both cases, since they represent useful tools in studying non K\"ahler geometry. We give an overview on the comparisons among the dimensions of the cohomology groups that can be defined and we show how we reach the $\partial\overline\partial$-lemma in complex geometry and the Hard-Lefschetz condition in symplectic geometry. For more details we refer to [6] and [29].Comment: The present paper is a proceeding written on the occasion of the "INdAM Meeting Complex and Symplectic Geometry" held in Cortona. It is going to be published on the "Springer INdAM Series