• 충청수학회지
• /
• 제36권4호
• /
• pp.231-244
• /
• 2023

We study the modularity and holomorphic anomaly equation for Hodge-Gromov-Witten invariants of elliptic curves.

• 대한수학회보
• /
• 제37권4호
• /
• pp.843-854
• /
• 2000

S. Turner has shown that a Neron symbol can be calculated from the values of K-meromorphic theta functions corresponding to divisors on K-holomorphic torus of strongly diagonal type. Using an isogeny to a K-holomorphic torus of strongly diagonal type, he constructed a Neron symbol on K-holomorphic torus of diagonal type. In this work, we provide a simple formula of the Neron symbol on the Tate curve. And then we construct the Neron symbol on K-holomorphic torus of diagonal or st rongly diagonal type without using isogenies.

• 대한수학회보
• /
• 제57권5호
• /
• pp.1195-1204
• /
• 2020

In this paper we consider the holomorphic curves (or derived holomorphic curves introduced by Toda in [15]) with maximal defect sum in the complex plane. Some well-known theorems on meromorphic functions of finite order with maximal sum of defects are extended to holomorphic curves in projective space.

• 대한수학회보
• /
• 제59권5호
• /
• pp.1105-1117
• /
• 2022

In this paper, we obtain a second main theorem for holomorphic curves and moving hyperplanes of Pⁿ(C) where the counting functions are truncated multiplicity and have different weights. As its application, we prove a uniqueness theorem for holomorphic curves of finite growth index sharing moving hyperplanes with different multiple values.

• 대한수학회보
• /
• 제51권4호
• /
• pp.1101-1113
• /
• 2014

We give analogous criterion to admit a real parabolic connection on real parabolic bundles over a real curve. As an application of this criterion, if real curve has a real point, then we proved that a real vector bundle E of rank r and degree d with gcd(r, d) = 1 is real indecomposable if and only if it admits a real logarithmic connection singular exactly over one point with residue given as multiplication by $-\frac{d}{r}$. We also give an equivalent condition for real indecomposable vector bundle in the case when real curve has no real points.

• 대한수학회보
• /
• 제38권2호
• /
• pp.347-355
• /
• 2001

Let ($X, \omega$) be a closed symplectic 4-manifold. Let a finite cyclic group G act semifreely, holomorphically on X as isometries with fixed point set $\Sigma$(may be empty) which is a 2-dimension submanifold. Then there is a smooth structure on the quotient X'=X/G such that the projection $\pi$:X$\rightarrow$X' is a Lipschitz map. Let L$\rightarrow$X be the Spin$^c$ -structure on X pulled back from a Spin$^c$-structure L'$\rightarrow$X' and b_2^$+(X')>1. If the Seiberg-Witten invariant SW(L')$\neq$0 of L' is non-zero and $L=E\bigotimesK^-1\bigotimesE$ then there is a G-invariant pseudo-holomorphic curve u:$C\rightarrowX$,/TEX> such that the image u(C) represents the fundamental class of the Poincare dual $c_1$(E). This is an equivariant version of the Taubes' Theorem.

• 대한수학회지
• /
• 제31권4호
• /
• pp.569-608
• /
• 1994

The subject matter of this survey has to do with holomorphic maps from a compact Riemann surface to projective space, which are also called algebrac curves; the theory we survey lies at the crossroads of function theory, projective geometry, and commutative algebra (although we should mention that the present survey de-emphasizes the algebraic aspect). Algebraic curves have been vigorously and continuously investigated since the time of Riemann. The reasons for the preoccupation with algebraic curves amongst mathematicians perhaps have to do with-other than the usual usual reason, namely, the herd mentality prompting us to follow the leads of a few great pioneering methematicians in the field-the fact that algebraic curves possess a certain simple unity together with a rich and complex structure. From a differential-topological standpoint algebraic curves are quite simple as they are neatly parameterized by a single discrete invariant, the genus. Even the possible complex structures of a fixed genus curve afford a fairly complete description. Yet there are a multitude of diverse perspectives (algebraic, function theoretic, and geometric) often coalescing to yield a spectacular result.