A Simple Geometric Representative for μ\mu of a Point

For $SU(2)$ (or $SO(3)$) Donaldson theory on a 4-manifold $X$, we construct a simple geometric representative for $\mu$ of a point. Let $p$ be a generic point in $X$. Then the set $\{ [A] | F_A^-(p)$ is reducible $\}$, with coefficient -1/4 and appropriate orientation, is our desired geometric representative.Comment: Updated 2018 to published version. 8 pages, AmS-TeX, no figure

Global W2,pW^{2,p} estimates for solutions to the linearized Monge--Amp\`ere equations

In this paper, we establish global $W^{2,p}$ estimates for solutions to the linearized Monge-Amp\`ere equations under natural assumptions on the domain, Monge-Amp\`ere measures and boundary data. Our estimates are affine invariant analogues of the global $W^{2,p}$ estimates of Winter for fully nonlinear, uniformly elliptic equations, and also linearized counterparts of Savin's global $W^{2,p}$ estimates for the Monge-Amp\`ere equations.Comment: v2: presentation improve

Moduli spaces of vector bundles over a Klein surface

A compact topological surface S, possibly non-orientable and with non-empty boundary, always admits a Klein surface structure (an atlas whose transition maps are dianalytic). Its complex cover is, by definition, a compact Riemann surface M endowed with an anti-holomorphic involution which determines topologically the original surface S. In this paper, we compare dianalytic vector bundles over S and holomorphic vector bundles over M, devoting special attention to the implications that this has for moduli varieties of semistable vector bundles over M. We construct, starting from S, totally real, totally geodesic, Lagrangian submanifolds of moduli varieties of semistable vector bundles of fixed rank and degree over M. This relates the present work to the constructions of Ho and Liu over non-orientable compact surfaces with empty boundary (arXiv:math/0605587) .Comment: 19 pages, 1 figur

N=2 Topological Yang-Mills Theory on Compact K\"{a}hler Surfaces

We study a topological Yang-Mills theory with $N=2$ fermionic symmetry. Our formalism is a field theoretical interpretation of the Donaldson polynomial invariants on compact K\"{a}hler surfaces. We also study an analogous theory on compact oriented Riemann surfaces and briefly discuss a possible application of the Witten's non-Abelian localization formula to the problems in the case of compact K\"{a}hler surfaces.Comment: ESENAT-93-01 & YUMS-93-10, 34pages: [Final Version] to appear in Comm. Math. Phy

On complex surfaces diffeomorphic to rational surfaces

In this paper we prove that no complex surface of general type is diffeomorphic to a rational surface, thereby completing the smooth classification of rational surfaces and the proof of the Van de Ven conjecture on the smooth invariance of Kodaira dimension.Comment: 34 pages, AMS-Te

Numerical solution to the hermitian Yang-Mills equation on the Fermat quintic

We develop an iterative method for finding solutions to the hermitian Yang-Mills equation on stable holomorphic vector bundles, following ideas recently developed by Donaldson. As illustrations, we construct numerically the hermitian Einstein metrics on the tangent bundle and a rank three vector bundle on P^2. In addition, we find a hermitian Yang-Mills connection on a stable rank three vector bundle on the Fermat quintic.Comment: 25 pages, 2 figure

Degenerations of LeBrun twistor spaces

We investigate various limits of the twistor spaces associated to the self-dual metrics on n CP ^2, the connected sum of the complex projective planes, constructed by C. LeBrun. In particular, we explicitly present the following 3 kinds of degenerations whose limits of the metrics are: (a) LeBrun metrics on (n-1) CP ^2$, (b) (Another) LeBrun metrics on the total space of the line bundle O(-n) over CP ^1 (c) The hyper-Kaehler metrics on the small resolution of rational double points of type A_{n-1}, constructed by Gibbons and Hawking.Comment: 21 pages, 7 figures. V2: A new section added at the end of the article. V3: Reference slightly update

Three Applications of Instanton Numbers

We use instanton numbers to: (i) stratify moduli of vector bundles, (ii) calculate relative homology of moduli spaces and (iii) distinguish curve singularities.Comment: To appear in Communications in Mathematical Physic

The Nekrasov Conjecture for Toric Surfaces

The Nekrasov conjecture predicts a relation between the partition function for N=2 supersymmetric Yang-Mills theory and the Seiberg-Witten prepotential. For instantons on R^4, the conjecture was proved, independently and using different methods, by Nekrasov-Okounkov, Nakajima-Yoshioka, and Braverman-Etingof. We prove a generalized version of the conjecture for instantons on noncompact toric surfaces.Comment: 38 pages; typos corrected, references added, minor changes (e.g. minor change of convention in Definition 5.13, 5.19, 6.5

Calorons, Nahm's equations on S^1 and bundles over P^1xP^1

The moduli space of solutions to Nahm's equations of rank (k,k+j) on the circle, and hence, of SU(2) calorons of charge (k,j), is shown to be equivalent to the moduli of holomorphic rank 2 bundles on P^1xP^1 trivialized at infinity with c_2=k and equipped with a flag of degree j along P^1x{0}. An explicit matrix description of these spaces is given by a monad constructio