We have mapped the central region of the Seyfert 1 galaxy NGC 1097 in 12CO(J=2-1) with the Submillieter Array (SMA). The 12CO(J=2-1) map shows a central concentration and a surrounding ring, which coincide respectively with the Seyfert nucleus and a starburst ring. The line intensity peaks at the nucleus, whereas in a previously published 12CO(J=1-0) map the intensity peaks at the starburst ring. The molecular ring has an azimuthally averaged 12CO(J=2-1)/(J=1-0) intensity ratio (R21) of about unity, which is similar to those in nearby active star forming galaxies, suggesting that most of the molecular mass in the ring is involved in fueling the starburst. The molecular gas can last for only about 1.2times10^8 years without further replenishment assuming a constant star formation rate and a perfect conversion of gas to stars. The velocity map shows that the central molecular gas is rotating with the molecular ring in the same direction, while its velocity gradient is much steeper than that of the ring. This velocity gradient of the central gas is similar to what is usually observed in some Seyfert 2 galaxies. To view the active nucleus directly in the optical, the central molecular gas structure can either be a low-inclined disk or torus but not too low to be less massive than the mass of the host galaxy itself, be a highly-inclined thin disk or clumpy and thick torus, or be an inner part of the galactic disk. The R21 value of ~1.9 of the central molecular gas component, which is significantly higher than the value found at the molecular gas ring, indicates that the activity of the Seyfert nucleus may have a significant influence on the conditions of the molecular gas in the central component.
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