Weighted interpolation from certain singular affine hypersurfaces

We prove that square integrable holomorphic functions (with respect to a plurisubharmonic weight) can be extended in a square integrable manner from certain singular hypersurfaces (which include uniformly flat, normal crossing divisors) to entire functions in affine space. This provides evidence for a conjecture regarding the positivity of the curvature of the weight under consideration.Comment: 7 page

A note on the deformed Hermitian Yang-Mills PDE

We prove a priori estimates for a generalised Monge-Amp\`ere PDE with "non-constant coefficients" thus improving a result of Sun in the K\"ahler case. We apply this result to the deformed Hermitian Yang-Mills (dHYM) equation of Jacob-Yau to obtain an existence result and a priori estimates for some ranges of the phase angle assuming the existence of a subsolution. We then generalise a theorem of Collins-Sz\`ekelyhidi on toric varieties and use it to address a conjecture of Collins-Jacob-Yau.Comment: Final version. 14 pages. To appear in Complex Variables and Elliptic equation

Representability of Chern-Weil forms

In this paper we look at two naturally occurring situations where the following question arises. When one can find a metric so that a Chern-Weil form can be represented by a given form ? The first setting is semi-stable Hartshorne-ample vector bundles on complex surfaces where we provide evidence for a conjecture of Griffiths by producing metrics whose Chern forms are positive. The second scenario deals with a particular rank-2 bundle (related to the vortex equations) over a product of a Riemann surface and the sphere.Comment: Final version. To appear in Mathematische Zeitschrif

Agricultural Growth and Economic Development: A View through the Globalization Lens

This paper re-visits the age old proposition that agriculture growth contributes to overall economic development, and asks whether the relationship still holds in an increasingly globalized world. There is overwhelming empirical support for the above proposition, indeed, it is hard to find exceptions, barring a few city states, where sustained economic development has not been preceded by robust agricultural growth. However, there are a large number of countries that have witnessed neither agricultural growth nor economic development. Even in countries where agricultural growth has been significant, dramatic inter-regional differences persist. This paper examines the factors that contribute to or constrain the process of agricultural transformation. Does the process of globalization, and the resultant changes in agrifood systems, offer new opportunities for agriculture led growth, or will it further marginalize excluded countries, regions and groups? The factors that cause exclusion are examined both in terms of globalization forces and in terms of domestic shortcomings in policies and governance. Policy interventions that attempt to reduce the costs of transition to a globalized agricultural system are explored, including safety nets for those left behind.International Development,

Bargmann-Fock extension from Singular Hypersurfaces

We establish sufficient conditions for extension of weighted square integrable holomorphic functions from a possibly singular hypersurface to the ambient affine space. The norms we use are the so-called Bargmann-Fock norms, and thus there are restrictions on the singularities and the density of the hypersurface. Our sufficient conditions are that it has density less than 1, and is uniformly flat in a sense that extends to singular varieties the notion of uniform flatness introduced earlier. We present an example of Ohsawa showing that uniform flatness is not necessary for extension in the singular case, and find an example showing that, for rather different reasons, it is also not necessary for the smooth case. The latter answers in the negative a question posed in an earlier paper of the second author.Comment: Corrected the email address of the second author. To appear in Crelle's Journa